Research ArticleA New Approach and Solution Technique to Solve TimeFractional Nonlinear Reaction-Diffusion Equations
Inci Cilingir Sungu1 and Huseyin Demir2
1Department of Elementary School Mathematics Education Education Faculty Ondokuz Mayıs University 55139 Samsun Turkey2Department of Mathematics Arts and Science Faculty Ondokuz Mayıs University 55139 Samsun Turkey
Correspondence should be addressed to Huseyin Demir hdemiromuedutr
Received 19 August 2014 Accepted 20 November 2014
Academic Editor Samir B Belhaouari
Copyright copy 2015 I Cilingir Sungu and H Demir This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
A new application of the hybrid generalized differential transform and finite difference method is proposed by solving timefractional nonlinear reaction-diffusion equations This method is a combination of the multi-time-stepping temporal generalizeddifferential transform and the spatial finite difference methods The procedure first converts the time-evolutionary equations intoPoisson equations which are then solved using the central difference method The temporal differential transform method as usedin the paper takes care of stability and the finite difference method on the resulting equation results in a system of diagonallydominant linear algebraic equations The Gauss-Seidel iterative procedure then used to solve the linear system thus has assuredconvergence To have optimized convergence rate numerical experiments were done by using a combination of factors involvingmulti-time-stepping spatial step size and degree of the polynomial fit in time It is shown that the hybrid technique is reliableaccurate and easy to apply
1 Introduction
The nonlinear reaction-diffusion equations have foundnumerous applications in pattern formation in manybranches of biology chemistry and physics [1ndash4] Reaction-diffusion (RD) equations have also been applied to otherareas of science and can be successfully modelled by theuse of fractional order derivatives [5ndash18] for example theRD equations are employed to describe the CO oxidationon Pt (110) [5] the study of Ca
2
+ waves on Xenopus oocytes[11] and the study of reentry in heart tissue [7 13] A greatdeal of effort has been expended over the last 10 years inattempting to find robust and stable numerical and analyticalmethods for solving fractional partial differential equationsof physical interest There has also been a wide variety ofnumerical methods for example finite difference techniquesfinite element methods spectral techniques adaptive andnonadaptive algorithms and so forth which have beendeveloped for RDrsquos numerical solution [19 20]
The differential transformmethod was used first by Zhou[21] who solved linear and nonlinear initial value problems in
electric circuit analysis This method constructs an analyticalsolution in the form of a polynomial It is different from thetraditional higher order Taylor series method which requiressymbolic computation of the necessary derivatives of thedata functions The Taylor series method computationallytakes long time for large orders The differential transformis an iterative procedure for obtaining analytic Taylor seriessolution of ordinary or partial differential equations Themethod is well addressed in [22ndash26] Recently the applicationof differential transform method is successfully extended toobtain analytical approximate solutions to ordinary differ-ential equations of fractional order Fractional differentialtransform method (FDTM) is a method that Arikoglu andOzkol [23] developed for solving linear and nonlinear inte-grodifferential equations of fractional order This methodsolves problems with high accuracy while constructing semi-analytic solutions in the polynomial forms FDTM is basedon classical differential transform method fractional powerseries and Caputo fractional derivative [27] Arikoglu andOzkol [23] tested their approach on several examples andthe results obtained are in good agreement with the existing
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 457013 13 pageshttpdxdoiorg1011552015457013
2 Mathematical Problems in Engineering
ones in the open literature Momani et al [28] presenteda new generalization of the differential transform methodthat extended the application of the method to differentialequations of fractional order The new technique is namedgeneralized differential transform method (GDTM) and isbased on one-dimensional differential transform generalizedTaylorrsquos formula [29] and Caputo fractional derivative [27]
Fractional partial differential equations (FPDEs) are alsoan interesting and an important topic The fractional deriva-tives and integrals have been occurring in many physicaland engineering problems with noninteger orders Fractionalcalculus is based on the definition of the fractional deriva-tives and integrals They play a major role in engineeringphysics and applied mathematics FPDEs are used to modelcomplex phenomena since the fractional order differentialequations are naturally related to the systems with memoryand nonlocal relations in space and time which exist in mostphysical phenomena Fractional order differential equationsare as stable as their integer order counterpart One of thefundamental equations of physics is the Schrodinger equationwhich describes how the quantum state of physical systemchanges with time The fractional Schrodinger equationprovides us with a general point of view on the relationshipbetween statistical properties of quantum mechanical pathand structure of fundamental equations of quantummechan-ics [30] In other words the fractional quantum mechanicsincludes the standard quantum mechanics as a particularcase So FPDEs are obtained by generalizing differential equa-tions to an arbitrary order There are three popular methodsfor seeking approximate solutions for FPDEs which are thefinite difference finite element and spectral methods In theliterature there are many papers on these three methods Inthese papers the authors proposed the use of least squaresfinite element solution and fully discrete Galerkin method tosolve nonlinear space fractional partial differential equations[20 31]
The differential transform is well suited to combine withother numerical techniques as shown by Yu and Chen [32]who applied the hybridmethod to solve the transient thermalstress distribution in a perfectly elastic isotropic annularfin Kuo and Chen [33] employed the hybrid method tosolve Burgerrsquos equation for flow systems with high Reynoldsnumbers This method was also employed to analyze thedynamic response of an electrostatically actuatedmicro fixed-fixed beam [34]
In the current study the hybrid generalized differen-tial transformfinite difference method is used for solvingtime fractional nonlinear RD equations The validity of theproposed approach has been confirmed by comparing theresults derived in the literature using theGDTMmethod [19]homotopy perturbation method (HPM) [35] and fractionalvariational method (FVIM) [36]
There are several approaches of definitions for the frac-tional derivative Among them one is called Riemann-Liouville fractional derivatives and defined by
119863120572119910 (119909) =
119889119898
119889119909119898119869119898minus120572
119910 (119909) 119898 minus 1 lt 120572 le 119898
119898 isin N 119909 gt 0
(1)
Here 119869120573 is the 120573-order Riemann-Liouville integral operatorwhich is expressed as follows
119869120573119891 (119909) =
1
Γ (120573)[int
119909
0
(119909 minus 119905)120573minus1
119891 (119905) 119889 (119905)] 120573 gt 0 (2)
If we use this definition we must know the initial value ofsome fractional order derivative of the unknown functionor we must have homogenous initial conditions Unlikethe Riemann-Liouville approach in the case of the Caputoderivative there are no restrictions on the initial conditionsThus the following definition is used in this study
119863120572119910 (119909) = 119869
119898minus120572119910(119898)
(119909) 119898 minus 1 lt 120572 le 119898
119898 isin N 119909 gt 0
(3)
This operator is generally called ldquo120572-order Caputo differentialoperatorrdquo [37 38]
For Caputo derivative we have
119863120572119905119896=
0 119896 le 120572 minus 1
Γ (119896 + 1)
Γ (119896 minus 120572 + 1)119905119896minus120572
We firstly introduce the main features of GDTM [19 23 2839ndash41] according to the generalized differential transform ofthe 119896th derivative of a function of one variable defined asfollows
where 0 lt 120572 le 1 (119863120572)119896 = 119863120572119863120572sdot sdot sdot 119863120572119896 times In (5) 119910(119909)
is the original function 119884120572(119896) is the transformed function
and the differential inverse transform of 119884120572(119896) is defined as
follows
119910 (119909) =
infin
sum
119896=0
119884120572(119896) (119909 minus 119909
0)120572119896
(6)
In case of 120572 = 1 GDTM reduces to the classical DTM Fromdefinitions (2) and (3) all fundamental properties of GDTMcan be obtained easily [19 39] Since lim
120572rarr1119863120572119906 = 119863119906 has
been proved from the definitions of fractional calculus thefractional solutions 119906
120572(119909 119905) reduce to the standard solution
119906(119909 119905)In this study we use a hybrid method that is a combi-
nation of generalized temporal differential transform andspatial finite difference methods to solve nonlinear fractionalreaction diffusion equations
We present a solution of a more general model of RDequation
where 1198631is the diffusion coefficient and 119891(119906) is a nonlinear
function We consider two different forms of 119891(119906) which arecalled time fractional Fisher equation and time fractionalFitzHugh-Nagumo equation
We apply GDTM to discretize fractional order timederivative and central difference method to discretize deriva-tives in 119909 direction respectively After transforming (7) usingthe GDTM we get the following equation
(119896) minus 2119880120572119894(119896) + 119880
120572(119894minus1)(119896)
ℎ2+ 119865 (119880
120572119894)
(8)
where 119880120572119894(119896) is the transformed function of 119906(119909
119894 119905) and
119865(119880120572119894) is the transformed function of 119891(119906) The region 0 lt
119909 lt 119886 is divided into several equal intervals and each intervalhas a width ℎ The time interval of interest is discretizedusing a time step Δ119905 After discretization of the equationwe get solution at time Δ119905 and these results are adoptedas the initial values for the next time interval This time-stepping procedure assists in obtaining a converged solutionto a desired accuracy [39] An appendix has been added to thepaper to show that the error if any in themethod is boundedThis implies stability of the scheme and by Lax equivalencetheorem it thereby implies convergence
The new algorithm has been developed to solve thenonlinear reaction diffusion equation and our aim of thisapproach is to combine the flexibility of differential transformand the efficiency of finite differences This algorithm alsoprovides an iterative procedure to calculate the numericalsolutions therefore it is not necessary to carry out com-plicated symbolic computation On applying the differentialtransform method with respect to time on the equation weare basically transforming the time-evolutionary equationto an elliptic type In essence this means that the centralfinite difference approximation that is subsequently used onthe transformed equation is a Poisson solver The resultingsystem of linear algebraic equations is then diagonally dom-inant and hence the Gauss-Seidel iterative method used forsolving the same has assured convergence as the coefficientmatrix remains nonsingular throughout the computationThe algorithm used thus succeeds in segregating the timediscretization from explicitly influencing the computation inthe spatial domain and this presents a situation wherein thetwo can be handled independent of each other in the courseof computation without having to bother about the stabilityof the solution if the differential transform part is properlyhandled The latter is achieved deftly in the differentialtransform part of the algorithm by using the multisteppingprocedure as first enunciated by Yu and Chen [32] in theirphenomenal work and used subsequently by Odibat et al[29 39] In summary this means that convergence is neverin doubt in the algorithm but slow convergence can be ifthe time and spatial discretizations are badly handled Theconvergence is optimized in the paper computationally byproper selection of time step in the differential transformpart of the algorithm and then the spatial step size in the
finite difference part of the algorithm Next important stepin the algorithm is the decision on the number of terms tobe adopted in the inverse differential transform that givesus the solution of the problem as a power series in timeThe aforementioned three vital components of the algorithmhave been meticulously handled and a brief summary of thenumerical experiment undertaken concerning the same ispresented in a table The numerical study recommends thatthe combination of 5 10 and 50 spatial step size with a timestep of 00005 or 0001 assures the best rate of convergence ifwe take minimum ten terms in the time series
3 Illustration of Generalized DifferentialTransform and Finite Difference Method
To show effectiveness of the proposed numerical solutionusing the temporal generalized differential transform and thespatial finite differencemethod and to give an understandableoverview of the methodology two examples of the reactiondiffusion equations will be discussed in the following sectionThen our results will be compared with published work ofRida et al [19] in which GDTM was used to solve the sameequations
Example 1 The time fractional Fisher equation is
119863120572
119905119906 = 119863
119909119909119906 + 6119906 (1 minus 119906) 119909 isin R
119905 gt 0 (0 lt 120572 le 1)
(9)
In this example we have the nonlinear function 119891(119906) =
6119906(1 minus 119906) The initial condition used is
119906 (119909 0) =1
(1 + 119890119909)2 (10)
Operating the generalized differential transform on (9)gives us the following equation
(119896) minus 2119880120572119894(119896) + 119880
120572(119894minus1)(119896)
ℎ2
+ 6119880120572119894(119896) minus 6
119896
sum
119897=0
119880120572119894(119896 minus 119897) 119880
120572119894(119897)
(12)
4 Mathematical Problems in Engineering
10
075
05
025
00minus10
minus5
5
10
x
00
0005
01
t
(a)
10
075
05
025
00minus10
minus5
5
10
x
00
0005
01
t
(b)
Figure 1 Numerical solution for the time fractional Fisher equation with 120572 rarr 1 (a) comparison with the analytical solution (b)
Table 1 Some values of 119880120572119894(119896) of Example 1
119894119896
0 1 2
0 0250000 1249791
Γ (120572 + 1)
3126294
Γ (2120572 + 1)
1 0225644 1184363
Γ (120572 + 1)
3406094
Γ (2120572 + 1)
2 0202649 1114004
Γ (120572 + 1)
3618706
Γ (2120572 + 1)
The initial condition on discretization yields
119880120572119894(0) =
1
(1 + 119890119894ℎ)2 (13)
Equation (12) is a recurrence relation The time seriessolution of the given equation is then obtained by using (12)and (13) with ℎ = 01 to obtain 119880
120572119894(119896) Some of 119880
120572119894(119896) are
recorded in Table 1The time series solutions of (12) with the initial condition
(13) are obtained as follows
119906 (0 119905) = 0250000 +1249791
Γ (120572 + 1)119905120572+
3126294
Γ (2120572 + 1)1199052120572
+ sdot sdot sdot
119906 (01 119905) = 0225644 +1184363
Γ (120572 + 1)119905120572+
3406094
Γ (2120572 + 1)1199052120572
+ sdot sdot sdot
119906 (02 119905) = 0202649 +1114004
Γ (120572 + 1)119905120572+
3618706
Γ (2120572 + 1)1199052120572
+ sdot sdot sdot
(14)
The numerical calculation results are shown in Figures1 and 2 respectively Our results are in agreement with the
published work of Rida et al [19] who considered the sameequation An exact solution of the standard form of Fisherequation for 120572 rarr 1 is
119906 (119909 119905) =1
(1 + 119890119909minus5119905)2 (15)
The comparison of our results with the exact solution isshown in Figure 1 for ℎ = 01 and quite clearly goodagreement is found
Approximate solutions are shown in Figure 2 for 120572 = 099
and 120572 = 095The influence of 120572 on the function 119906(119909 119905) is shown in
Figure 3 This figure indicates a decrease in the fractionalorder 120572 by choosing the fixed 119909 = 5 that corresponds to anincrease in the function and also indicates a slow diffusion forthe values of 120572 = 1 and 120572 = 09 and a fast diffusion for thevalues of 120572 = 08 07 06 respectively It is clearly seen that119906(119909 119905) increase for 120572 = 1 09 08 07 06 with the increasesin 119905
Numerical comparison between GDTM [19] HPM [35]FVIM [36] and hybrid method are found in Table 2 whichshows hybrid method is more promising
It is also found that the result is in complete agreementwith the result of HPM [42 43] and ADM [44] for 120572 = 1
We investigate convergence criteria of our solutions fordifferent values of ℎ and 119899 To illustrate this we comparedour results with the analytical solution in case of 120572 = 1 Here119899 is order of differential transformation method and denotesthe number of terms to be calculated
In Figures 4 5 and 6 the difference between the resultsobtained in this study and the results of the analytical solutionis of the order of 10minus5 This is a pointer to the fact that there isconvergence and is a restatement in numerical terms of whatwas shown in the Appendix
Mathematical Problems in Engineering 5
10
075
05
025
00minus10
minus5
5
10
x
00
0005
01
t
(a)
10
075
05
025
00minus10
minus5
5
10
x
00
0005
01
t
(b)
Figure 2 Numerical solution for the time fractional Fisher equation with (a) 120572 rarr 099 and (b) 120572 rarr 095
0007
0006
0005
0004
0003
0002
0001
u(xt)
0 01 02 03 04
t
120572 = 06120572 = 07
120572 = 08
120572 = 09
120572 = 1
(a)
0007
0006
0005
0004
0003
0002
0001
000000 01 02 03 04
u
t
120572 = 710120572 = 810
120572 = 910
120572 = 1
(b)
Figure 3 Approximate solution for the time fractional Fisher equation with different 120572 values at 119909 = 5 (a) present and (b) [35]
Table 2 Comparison of numerical results between different methods for the time fractional Fisher equation GDTM generalized differentialtransformmethod Rida et al [19] HPM homotpy perturbation method Khan et al [35] and FVIM fractional variational iteration methodMerdan [36]
Table 3 Some values of 119880120572119894(119896) of Example 2
119894119896
0 1 2
0 05(0125 minus 025120583)
Γ (120572 + 1)
(0000011 minus 0000052120583)
Γ (2120572 + 1)
1 0517670(0124847 minus 0249687120583)
Γ (120572 + 1)
(minus0002179 + 0008772120583 minus 00088241205832)
Γ (2120572 + 1)
2 0535296(0124384 minus 0248754120583)
Γ (120572 + 1)
(minus0004379 + 0017511120583 minus 00175601205832)
Γ (2120572 + 1)
One important observation made from the computationis that when the number of mesh points was increased lessnumber of terms was required in the time series solution tohave convergence for a predetermined accuracy The hybridmethod of the present study gives faster convergence thanother traditional methods for example if we take ℎ = 002
(mesh point is 50) then the solution converges for 119899 = 3 Wenow consider another example
Example 2 The time fractional FitzHugh-Nagumo equationis
119863120572
119905119906 = 119863
2
119909119906 + 119906 (1 minus 119906) (119906 minus 120583) 120583 gt 0
0 lt 120572 le 1 119909 isin R 119905 gt 0
(16)
In this type of equation the nonlinear function depends on 120583and it is 119891(119906) = 119906(1 minus 119906)(119906 minus 120583) The initial condition is
119906 (119909 0) =1
(1 + 119890minus119909radic2)
(17)
Using the hybrid method on the above initial boundary valueproblem (IBVP) as done in the previous example we get
(119896) minus 2119880120572119894(119896) + 119880
120572(119894minus1)(119896)
ℎ2
minus 120583119880120572119894(119896) + (1 + 120583)
119896
sum
119897=0
119880120572119894(119896 minus 119897) 119880
120572119894(119897)
minus
119896
sum
119904=0
119904
sum
119897=0
119880120572119894(119896 minus 119904)119880
120572119894(119904 minus 119897) 119880
120572119894(119897)
119880120572119894(0) =
1
1 + 119890minus119894ℎradic2
(18)
Using second order finite difference method the boundaryvalues were obtained as follows
1198801205720(119896) = 3119880
1205721(119896) minus 3119880
1205722(119896) + 119880
1205723(119896)
119880120572119873
(119896) = 3119880120572(119873minus1)
(119896) minus 3119880120572(119873minus2)
(119896) + 119880120572(119873minus3)
(119896)
(19)
Table 3 presents some of the 119880120572119894(119896)rsquos
The time series solution for the above IBVP at differenttimes is
119906 (0 119905)
= 05 +(0125 minus 025120583)
Γ (120572 + 1)119905120572
+(0000011 minus 0000052120583)
Γ (2120572 + 1)1199052120572
+ sdot sdot sdot
119906 (01 119905)
= 0517670 +(0124847 minus 0249687120583)
Γ (120572 + 1)119905120572
+
(minus0002179 + 0008772120583 minus 00088241205832)
Γ (2120572 + 1)1199052120572
+ sdot sdot sdot
119906 (02 119905)
= 0535296 +(0124384 minus 0248754120583)
Γ (120572 + 1)119905120572
+
(minus0004379 + 001751120583 minus 00175601205832)
Γ (2120572 + 1)1199052120572
+ sdot sdot sdot
(20)
Numerical solutions for the time fractional FitzHugh-Nagumo equation with various 120572 values are shown in Figures7 and 8 A comparison of the results in a limiting case whereinan analytical solution exists is shown in Figure 7 The resultsare in close agreement with those of Rida et al [19] for thesame equation
For 120572 rarr 1 it can easily be seen that the exact solution ofFitzHugh-Nagumo equation is
Figure 9 is prepared to show the influence of 120572 on thefunction 119906(119909 119905) It is clearly seen that 119906(119909 119905) decrease for120572 = 1 095 085 075 065 with the decreases in 119905
As shown in the Table 4 our results show close agreementwith the exact solution and agree with those of Rida et al [19]
Mathematical Problems in Engineering 7
Table 4 Coefficients of 1 119905 1199052 for some 119894 values and comparisonwith exact and Ridarsquos solution 120583 = 07 [19]
Rida et al [19] Exact Present119894 = 0
Coef of 1199050 05 05 05Coef of 1199051 minus005 minus005 minus0049999Coef of 1199052 002 0 minus0000011
119894 = 1
Coef of 1199050 0517670 0517670 0517670Coef of 1199051 minus0049937 minus00499937 minus0049933Coef of 1199052 0020327 minus0000176 minus0000181
119894 = 2
Coef of 1199050 0535296 0535296 0535296Coef of 1199051 minus0049937 minus0049937 minus0049743Coef of 1199052 0020602 minus0000351 minus0000352
119894 = 3
Coef of 1199050 0552835 0552835 0552835Coef of 1199051 minus0049441 minus0049441 minus0049430Coef of 1199052 0020821 minus0000522 minus0000542
026
024
022
PresentAnalytic
020
018
0 002 004 006 008 010
t
Figure 4 Comparison of present results for ℎ = 02 and 119899 = 10withthe analytical solution in case of 120572 rarr 1 at 119909 = 04
From this table it is clear that the present work gives betterapproximation than GDTM as we increase 119899
Numerical comparison between GDTM FVIM andhybrid method is shown in Table 5 which indicates hybridmethod is more promising
4 Conclusion
Many real physical problems can be best modelled withfractional differential equations but the fact is when the
026
024
022
PresentAnalytic
020
018
0 002 004 006 008 010
t
Figure 5 Comparison of present results for ℎ = 01 and 119899 = 5 withthe analytical solution in case of 120572 rarr 1 at 119909 = 04
026
024
022
PresentAnalytic
020
018
0 002 004 006 008 010
t
Figure 6 Comparison of present results for ℎ = 002 and 119899 = 3withthe analytical solution in case of 120572 rarr 1 at 119909 = 04
equation is nonlinear there are very few reliable methodsThe numerical methods that can be used to solve frac-tional differential equations are known to have problems ofconvergence and stability These aspects are well addressedin the paper by suggesting a new procedure that uses acombination of the generalized differential transform andcentral difference methods The Appendix clearly spells out
8 Mathematical Problems in Engineering
10
075
05
025
0010
5
x
01
0
005
00
t
minus10
minus5
(a)
10
075
05
025
0010
minus10
5
minus5x 010
005
00
t
(b)
Figure 7 Numerical solution for the time fractional FitzHugh-Nagumo equation with 120572 rarr 1 (a) comparison with the analytical solution(b) with 120583 = 05
10
075
05
025
0010
minus10
5
minus5x01
0
005
00
t
(a)
10
075
05
025
0010
minus10
5
minus5x
01
0005
00
t
(b)
Figure 8 Numerical solution for the time fractional FitzHugh-Nagumo equation with 120572 = 095 (a) and 120572 = 099 (b)
the fact that the error as a result of discretization and compu-tation is bounded and hence implies stability of the methodLax equivalence theorem further implies convergence of thescheme Two time fractional nonlinear reaction-diffusionequations considered for illustration of the hybrid methodhighlight the usefulness of the method in obtaining thesolution of IBVPs involving time fractional derivatives Thecontrol of convergence through a judicious choice of time andspatial step sizes and also the number of terms in the timeseries solution spells assured convergence The segregationof the time domain from the spatial domain in the solution
method ensures the fact that problem of stability does notarise Diagonal dominance of the coefficient matrix in thesystem of linear algebraic equations resulting from the use ofthe central difference approximation in the Poisson equationensures the fact that the matrix remains nonsingular duringiterations and hence has assured convergence An appropriatecomputational decision on the number of terms to be takenin the time series solution results in a convergent solutionwith fast convergence Excellent comparison of the presentresults with the previous works on generalized differentialtransform method [19] and homotopy perturbation method
Mathematical Problems in Engineering 9
09716
09714
09712
09710
u(xt)
09708
09706
120572 = 095
120572 = 1
120572 = 085
120572 = 075
120572 = 06509704
0 01 02 03
t
(a)
120572 = 095
120572 = 085
120572 = 075
120572 = 065
120572 = 1
09716
09714
09712
0971
u(xt)
09708
09706
09704
0 01005 02015 03025
t
(b)
Figure 9 Approximate solution for the time fractional FitzHugh-Nagumo equation with different 120572 values at 119909 = 5 (a) present and (b) [36]
[35] and fractional variational iterationmethod [36] providesconfidence in the methodology adopted for the solution oftime fractional differential equations
Appendix
Estimation of Bounds on Truncation Error
Consider the fractional differential equations (9) and (16) ina general form as
119863120572
119905119906 = 119863
119909119909119906 + 119891 (119906) 119905 ge 0
119909 isin R (0 lt 120572 le 1)
(A1)
The differential transform of (A1) at the spatially discretizedpoints 119909
We now follow Jang et al [45] and move on to arrive at anestimate on the bounds for the truncation error in a generalway by considering the Taylor series expansion of119880
Table 5 Comparison of numerical results between differentmethods for the time fractional FitzHugh-Nagumo equationGDTM generalizeddifferential transform method Rida et al [19] and FVIM fractional variational iteration method Merdan [36]
We so far addressed the local error due to two differenttruncations in the time series In what follows we estimatethe bounds on the cumulative error that includes the errordiscussed above
Let 119910119894(119905119896) denote the solution of (A2) The local error in
119908119894(119905119896) relative to 119910
119894(119905119896) is
1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908119894(119905119896+ Δ119905)
1003816100381610038161003816
le1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
+1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
(A13)
Since 119908119894(119905119896+ Δ119905) is a better approximation than 119908
119894(119905119896+ Δ119905)
we may assume that1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816 ≪ 1 (A14)
Mathematical Problems in Engineering 11
In view of (A14) we now have1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816 ≐1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
(A15)
Using (A8) and noting that (Δ119905)119899+1 is quite small in (A15)we may take 120573 to be
120573 ≐
1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908119894(119905119896+ Δ119905)
1003816100381610038161003816
(Δ119905)119899+1
(A16)
Thus 120573(Δ119905)119899+1 = 120576 is the bound on the tolerance in the to-be-obtained solution When using different number of terms inthe Taylor series expansion earlier we denoted the solutionsusing a time stepΔ119905 by119908
119894(119905119896+Δ119905) and119908
119894(119905119896+Δ119905) respectively
The paper uses an adaptive step size in computing theresults This is because such a procedure succeeds in keepingthe error bounded and ensures convergence as a consequenceof Lax equivalence theorem To see what the adaptive stepsize produces and to show how such a procedure keepsthe error bounded we start with the premise that Δ119905 isthe most appropriate step size for the problem This stepsize is determined using the definition of inverse differentialtransform
119906119894(Δ119905) =
infin
sum
119896=0
119880120572119894(119896) (Δ119905)
119896 (A17)
In our actual calculation we will not be able to considerinfinite number of terms We consider ldquo119899rdquo terms in respectof 119908119894and ldquo119899 + 119898rdquo terms in respect of 119908
119894
Thus1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
=
119899+119898
sum
119895=0
119880120572119894(119895) (Δ119905)
119895minus
119899
sum
119895=0
119880120572119894(119895) (Δ119905)
119895
=
119899+119898
sum
119895=119899+1
119880120572119894(119895) (Δ119905)
119895
(A18)
To write down a simpler expression we change the summa-tion index from 119895 to 119901 = 119899 + 119895 So we have from (A18) thefollowing
1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908119894(119905119896+ Δ119905)
Thus the above proceedings tell us that if criterion (A23) issatisfied then the error is bounded In effect this means thatthe scheme is convergent in lieu of Lax equivalence theoremIn our computations Δ119905 has been always chosen to satisfyinequality (A23)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors are thankful to Ondokuz Mayıs UniversitySamsun Turkey for providing financial support to carryout this work under a major research project (Grant nopyofen190113003)
References
[1] W Hundsdorfer and J G Verwer Numerical Solution of TimeDependent Advection-Diffusion-Reaction Equations SpringerBerlin Germany 2003
[2] Y Kuramoto Chemical Oscillations Waves and TurbulenceDover Mineola NY USA 2003
[3] J DMurrayMathematical Biology II vol 18 of InterdisciplinaryApplied Mathematics Springer New York NY USA 3rd edi-tion 2003
[4] H Wilhelmsson and E Lazzaro Reaction-Diffusion Problemsin the Physics of Hot Plasmas Institute of Physics PublishingPhiladelphia Pa USA 2001
[5] M Bar N Gottschalk M Eiswirth and G Ertl ldquoSpiral waves ina surface reaction model calculationsrdquoThe Journal of ChemicalPhysics vol 100 no 2 pp 1202ndash1214 1994
[6] D Barkley ldquoA model for fast computer simulation of waves inexcitable mediardquo Physica D Nonlinear Phenomena vol 49 no1-2 pp 61ndash70 1991
[7] F H Fenton E M Cherry H M Hastings and S J EvansldquoMultiple mechanisms of spiral wave breakup in a model ofcardiac electrical activityrdquo Chaos vol 12 no 3 pp 852ndash8922002
[8] M GosakMMarhl andM Perc ldquoSpatial coherence resonancein excitable biochemical media induced by internal noiserdquoBiophysical Chemistry vol 128 no 2-3 pp 210ndash214 2007
[9] A Karma ldquoMeandering transition in two-dimensionalexcitable mediardquo Physical Review Letters vol 65 no 22 pp2824ndash2827 1990
12 Mathematical Problems in Engineering
[10] J P Keener ldquoA geometrical theory for spiral waves in excitablemediardquo SIAM Journal onAppliedMathematics vol 46 no 6 pp1039ndash1056 1986
[11] J P Keener Mathematical Physiology Interdisciplinary AppliedMathematics Springer New York NY USA 1998
[12] V M Kenkre and M N Kuperman ldquoApplicability of the Fisherequation to bacterial population dynamicsrdquo Physical Review Evol 67 no 5 Article ID 051921 5 pages 2003
[13] V Krinsky and A Pumir ldquoModels of defibrillation of cardiactissuerdquo Chaos vol 8 no 1 pp 188ndash203 1998
[14] N F Otani ldquoA primary mechanism for spiral wave meander-ingrdquo Chaos vol 12 no 3 pp 829ndash842 2002
[15] M Perc ldquoSpatial coherence resonance in excitable mediardquoPhysical Review E vol 72 no 3 Article ID 016207 2005
[16] M Perc ldquoStochastic resonance on excitable small-world net-works via a pacemakerrdquo Physical Review E vol 76 no 6 ArticleID 066203 2007
[17] M Perc ldquoEffects of small-world connectivity on noise-inducedtemporal and spatial order in neural mediardquo Chaos Solitons ampFractals vol 31 no 2 pp 280ndash291 2007
[18] J J Tyson ldquoWhat everyone should know about the Belousov-Zhabotinsky reactionrdquo in Frontiers inMathematical Biology vol100 of Lecture Notes in Biomathematics pp 569ndash587 SpringerBerlin Germany 1994
[19] S Z Rida A M El-Sayed and A A Arafa ldquoOn the solutions oftime-fractional reaction-diffusion equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 15 no 12pp 3847ndash3854 2010
[20] Y Zheng and Z Zhao ldquoA fully discrete Galerkin method fora nonlinear space-fractional diffusion equationrdquo MathematicalProblems in Engineering vol 2011 Article ID 171620 20 pages2011
[21] J K Zhou Differential Transformation and Its Applications forElectrical Circuits Huazhang University Press Wuhan China1986 (Chinese)
[22] I H Abdel-Halim Hassan ldquoComparison differential transfor-mation technique with Adomian decomposition method forlinear and nonlinear initial value problemsrdquoChaos Solitons andFractals vol 36 no 1 pp 53ndash65 2008
[23] A Arikoglu and I Ozkol ldquoSolution of boundary value problemsfor integro-differential equations by using differential transformmethodrdquoAppliedMathematics and Computation vol 168 no 2pp 1145ndash1158 2005
[24] F Ayaz ldquoSolutions of the system of differential equationsby differential transform methodrdquo Applied Mathematics andComputation vol 147 no 2 pp 547ndash567 2004
[25] N Bildik A Konuralp F Orak Bek and S KucukarslanldquoSolution of different type of the partial differential equationby differential transformmethod andAdomianrsquos decompositionmethodrdquo Applied Mathematics and Computation vol 172 no 1pp 551ndash567 2006
[26] H Liu and Y Song ldquoDifferential transform method applied tohigh index differential-algebraic equationsrdquoAppliedMathemat-ics and Computation vol 184 no 2 pp 748ndash753 2007
[27] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent Part IIrdquo Geophysical Journal of theRoyal Astronomical Society vol 13 no 5 pp 529ndash539 1967
[28] SMomani ZOdibat andV S Erturk ldquoGeneralized differentialtransform method for solving a space- and time-fractionaldiffusion-wave equationrdquo Physics Letters A vol 370 no 5-6 pp379ndash387 2007
[29] Z M Odibat C Bertelle M A Aziz-Alaoui and G HDuchamp ldquoA multi-step differential transform method andapplication to non-chaotic or chaotic systemsrdquo Computers ampMathematics with Applications vol 59 no 4 pp 1462ndash14722010
[30] N Laskin ldquoFractional Schrodinger Equationrdquo Physical ReviewE vol 66 no 5 Article ID 056108 2002
[31] G J Fix and J P Roop ldquoLeast squares finite-element solutionof a fractional order two-point boundary value problemrdquoComputers ampMathematics with Applications vol 48 no 7-8 pp1017ndash1033 2004
[32] L-T Yu and C-K Chen ldquoApplication of the hybrid methodto the transient thermal stresses response in isotropic annularfinsrdquo Journal of Applied Mechanics vol 66 no 2 pp 340ndash3471999
[33] B-L Kuo and C-K Chen ldquoApplication of a hybrid methodto the solution of the nonlinear burgersrsquo equationrdquo Journal ofApplied Mechanics Transactions ASME vol 70 no 6 pp 926ndash929 2003
[34] C K Chen H Y Lai and C C Liu ldquoApplication of hybrid dif-ferential transformationfinite difference method to nonlinearanalysis of micro fixed-fixed beamrdquo Microsystem Technologiesvol 15 no 6 pp 813ndash820 2009
[35] N A Khan M Ayaz L Jin and A Yildirim ldquoOn approximatesolutions for the time-fractional reaction-diffusion equation ofFisher typerdquo International Journal of Physical Sciences vol 6 no10 pp 2483ndash2496 2011
[36] M Merdan ldquoSolutions of time-fractional reaction-diffusionequation withmodified Riemann-Liouville derivativerdquo Interna-tional Journal of Physical Sciences vol 7 no 15 pp 2317ndash23262012
[37] C Li and G Peng ldquoChaos in Chenrsquos system with a fractionalorderrdquo Chaos Solitons amp Fractals vol 22 no 2 pp 443ndash4502004
[38] Y Luchko and R Gorenflo ldquoAn operational method for solvingfractional differential equations with the Caputo derivativesrdquoActa Mathematica Vietnamica vol 24 no 2 pp 207ndash233 1999
[39] Z Odibat and S Momani ldquoA generalized differential transformmethod for linear partial differential equations of fractionalorderrdquo Applied Mathematics Letters vol 21 no 2 pp 194ndash1992008
[40] Z M Odibat and N T Shawagfeh ldquoGeneralized Taylorrsquosformulardquo Applied Mathematics and Computation vol 186 no1 pp 286ndash293 2007
[41] D Nazari and S Shahmorad ldquoApplication of the fractionaldifferential transform method to fractional-order integro-differential equations with nonlocal boundary conditionsrdquoJournal of Computational andAppliedMathematics vol 234 no3 pp 883ndash891 2010
[42] D Agırseven and T Ozis ldquoAn analytical study for Fisher typeequations by using homotopy perturbationmethodrdquoComputersamp Mathematics with Applications vol 60 no 3 pp 602ndash6092010
[43] M Dehghan J M Heris and A Saadatmandi ldquoApplicationof semi-analytic methods for the Fitzhugh-Nagumo equation
Mathematical Problems in Engineering 13
which models the transmission of nerve impulsesrdquoMathemati-calMethods in the Applied Sciences vol 33 no 11 pp 1384ndash13982010
[44] A-M Wazwaz and A Gorguis ldquoAn analytic study of Fisherrsquosequation by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 154 no 3 pp 609ndash6202004
[45] M-J Jang C-L Chen and Y-C Liy ldquoOn solving the initial-value problems using the differential transformation methodrdquoAppliedMathematics andComputation vol 115 no 2-3 pp 145ndash160 2000
ones in the open literature Momani et al [28] presenteda new generalization of the differential transform methodthat extended the application of the method to differentialequations of fractional order The new technique is namedgeneralized differential transform method (GDTM) and isbased on one-dimensional differential transform generalizedTaylorrsquos formula [29] and Caputo fractional derivative [27]
Fractional partial differential equations (FPDEs) are alsoan interesting and an important topic The fractional deriva-tives and integrals have been occurring in many physicaland engineering problems with noninteger orders Fractionalcalculus is based on the definition of the fractional deriva-tives and integrals They play a major role in engineeringphysics and applied mathematics FPDEs are used to modelcomplex phenomena since the fractional order differentialequations are naturally related to the systems with memoryand nonlocal relations in space and time which exist in mostphysical phenomena Fractional order differential equationsare as stable as their integer order counterpart One of thefundamental equations of physics is the Schrodinger equationwhich describes how the quantum state of physical systemchanges with time The fractional Schrodinger equationprovides us with a general point of view on the relationshipbetween statistical properties of quantum mechanical pathand structure of fundamental equations of quantummechan-ics [30] In other words the fractional quantum mechanicsincludes the standard quantum mechanics as a particularcase So FPDEs are obtained by generalizing differential equa-tions to an arbitrary order There are three popular methodsfor seeking approximate solutions for FPDEs which are thefinite difference finite element and spectral methods In theliterature there are many papers on these three methods Inthese papers the authors proposed the use of least squaresfinite element solution and fully discrete Galerkin method tosolve nonlinear space fractional partial differential equations[20 31]
The differential transform is well suited to combine withother numerical techniques as shown by Yu and Chen [32]who applied the hybridmethod to solve the transient thermalstress distribution in a perfectly elastic isotropic annularfin Kuo and Chen [33] employed the hybrid method tosolve Burgerrsquos equation for flow systems with high Reynoldsnumbers This method was also employed to analyze thedynamic response of an electrostatically actuatedmicro fixed-fixed beam [34]
In the current study the hybrid generalized differen-tial transformfinite difference method is used for solvingtime fractional nonlinear RD equations The validity of theproposed approach has been confirmed by comparing theresults derived in the literature using theGDTMmethod [19]homotopy perturbation method (HPM) [35] and fractionalvariational method (FVIM) [36]
There are several approaches of definitions for the frac-tional derivative Among them one is called Riemann-Liouville fractional derivatives and defined by
119863120572119910 (119909) =
119889119898
119889119909119898119869119898minus120572
119910 (119909) 119898 minus 1 lt 120572 le 119898
119898 isin N 119909 gt 0
(1)
Here 119869120573 is the 120573-order Riemann-Liouville integral operatorwhich is expressed as follows
119869120573119891 (119909) =
1
Γ (120573)[int
119909
0
(119909 minus 119905)120573minus1
119891 (119905) 119889 (119905)] 120573 gt 0 (2)
If we use this definition we must know the initial value ofsome fractional order derivative of the unknown functionor we must have homogenous initial conditions Unlikethe Riemann-Liouville approach in the case of the Caputoderivative there are no restrictions on the initial conditionsThus the following definition is used in this study
119863120572119910 (119909) = 119869
119898minus120572119910(119898)
(119909) 119898 minus 1 lt 120572 le 119898
119898 isin N 119909 gt 0
(3)
This operator is generally called ldquo120572-order Caputo differentialoperatorrdquo [37 38]
For Caputo derivative we have
119863120572119905119896=
0 119896 le 120572 minus 1
Γ (119896 + 1)
Γ (119896 minus 120572 + 1)119905119896minus120572
We firstly introduce the main features of GDTM [19 23 2839ndash41] according to the generalized differential transform ofthe 119896th derivative of a function of one variable defined asfollows
where 0 lt 120572 le 1 (119863120572)119896 = 119863120572119863120572sdot sdot sdot 119863120572119896 times In (5) 119910(119909)
is the original function 119884120572(119896) is the transformed function
and the differential inverse transform of 119884120572(119896) is defined as
follows
119910 (119909) =
infin
sum
119896=0
119884120572(119896) (119909 minus 119909
0)120572119896
(6)
In case of 120572 = 1 GDTM reduces to the classical DTM Fromdefinitions (2) and (3) all fundamental properties of GDTMcan be obtained easily [19 39] Since lim
120572rarr1119863120572119906 = 119863119906 has
been proved from the definitions of fractional calculus thefractional solutions 119906
120572(119909 119905) reduce to the standard solution
119906(119909 119905)In this study we use a hybrid method that is a combi-
nation of generalized temporal differential transform andspatial finite difference methods to solve nonlinear fractionalreaction diffusion equations
We present a solution of a more general model of RDequation
where 1198631is the diffusion coefficient and 119891(119906) is a nonlinear
function We consider two different forms of 119891(119906) which arecalled time fractional Fisher equation and time fractionalFitzHugh-Nagumo equation
We apply GDTM to discretize fractional order timederivative and central difference method to discretize deriva-tives in 119909 direction respectively After transforming (7) usingthe GDTM we get the following equation
(119896) minus 2119880120572119894(119896) + 119880
120572(119894minus1)(119896)
ℎ2+ 119865 (119880
120572119894)
(8)
where 119880120572119894(119896) is the transformed function of 119906(119909
119894 119905) and
119865(119880120572119894) is the transformed function of 119891(119906) The region 0 lt
119909 lt 119886 is divided into several equal intervals and each intervalhas a width ℎ The time interval of interest is discretizedusing a time step Δ119905 After discretization of the equationwe get solution at time Δ119905 and these results are adoptedas the initial values for the next time interval This time-stepping procedure assists in obtaining a converged solutionto a desired accuracy [39] An appendix has been added to thepaper to show that the error if any in themethod is boundedThis implies stability of the scheme and by Lax equivalencetheorem it thereby implies convergence
The new algorithm has been developed to solve thenonlinear reaction diffusion equation and our aim of thisapproach is to combine the flexibility of differential transformand the efficiency of finite differences This algorithm alsoprovides an iterative procedure to calculate the numericalsolutions therefore it is not necessary to carry out com-plicated symbolic computation On applying the differentialtransform method with respect to time on the equation weare basically transforming the time-evolutionary equationto an elliptic type In essence this means that the centralfinite difference approximation that is subsequently used onthe transformed equation is a Poisson solver The resultingsystem of linear algebraic equations is then diagonally dom-inant and hence the Gauss-Seidel iterative method used forsolving the same has assured convergence as the coefficientmatrix remains nonsingular throughout the computationThe algorithm used thus succeeds in segregating the timediscretization from explicitly influencing the computation inthe spatial domain and this presents a situation wherein thetwo can be handled independent of each other in the courseof computation without having to bother about the stabilityof the solution if the differential transform part is properlyhandled The latter is achieved deftly in the differentialtransform part of the algorithm by using the multisteppingprocedure as first enunciated by Yu and Chen [32] in theirphenomenal work and used subsequently by Odibat et al[29 39] In summary this means that convergence is neverin doubt in the algorithm but slow convergence can be ifthe time and spatial discretizations are badly handled Theconvergence is optimized in the paper computationally byproper selection of time step in the differential transformpart of the algorithm and then the spatial step size in the
finite difference part of the algorithm Next important stepin the algorithm is the decision on the number of terms tobe adopted in the inverse differential transform that givesus the solution of the problem as a power series in timeThe aforementioned three vital components of the algorithmhave been meticulously handled and a brief summary of thenumerical experiment undertaken concerning the same ispresented in a table The numerical study recommends thatthe combination of 5 10 and 50 spatial step size with a timestep of 00005 or 0001 assures the best rate of convergence ifwe take minimum ten terms in the time series
3 Illustration of Generalized DifferentialTransform and Finite Difference Method
To show effectiveness of the proposed numerical solutionusing the temporal generalized differential transform and thespatial finite differencemethod and to give an understandableoverview of the methodology two examples of the reactiondiffusion equations will be discussed in the following sectionThen our results will be compared with published work ofRida et al [19] in which GDTM was used to solve the sameequations
Example 1 The time fractional Fisher equation is
119863120572
119905119906 = 119863
119909119909119906 + 6119906 (1 minus 119906) 119909 isin R
119905 gt 0 (0 lt 120572 le 1)
(9)
In this example we have the nonlinear function 119891(119906) =
6119906(1 minus 119906) The initial condition used is
119906 (119909 0) =1
(1 + 119890119909)2 (10)
Operating the generalized differential transform on (9)gives us the following equation
(119896) minus 2119880120572119894(119896) + 119880
120572(119894minus1)(119896)
ℎ2
+ 6119880120572119894(119896) minus 6
119896
sum
119897=0
119880120572119894(119896 minus 119897) 119880
120572119894(119897)
(12)
4 Mathematical Problems in Engineering
10
075
05
025
00minus10
minus5
5
10
x
00
0005
01
t
(a)
10
075
05
025
00minus10
minus5
5
10
x
00
0005
01
t
(b)
Figure 1 Numerical solution for the time fractional Fisher equation with 120572 rarr 1 (a) comparison with the analytical solution (b)
Table 1 Some values of 119880120572119894(119896) of Example 1
119894119896
0 1 2
0 0250000 1249791
Γ (120572 + 1)
3126294
Γ (2120572 + 1)
1 0225644 1184363
Γ (120572 + 1)
3406094
Γ (2120572 + 1)
2 0202649 1114004
Γ (120572 + 1)
3618706
Γ (2120572 + 1)
The initial condition on discretization yields
119880120572119894(0) =
1
(1 + 119890119894ℎ)2 (13)
Equation (12) is a recurrence relation The time seriessolution of the given equation is then obtained by using (12)and (13) with ℎ = 01 to obtain 119880
120572119894(119896) Some of 119880
120572119894(119896) are
recorded in Table 1The time series solutions of (12) with the initial condition
(13) are obtained as follows
119906 (0 119905) = 0250000 +1249791
Γ (120572 + 1)119905120572+
3126294
Γ (2120572 + 1)1199052120572
+ sdot sdot sdot
119906 (01 119905) = 0225644 +1184363
Γ (120572 + 1)119905120572+
3406094
Γ (2120572 + 1)1199052120572
+ sdot sdot sdot
119906 (02 119905) = 0202649 +1114004
Γ (120572 + 1)119905120572+
3618706
Γ (2120572 + 1)1199052120572
+ sdot sdot sdot
(14)
The numerical calculation results are shown in Figures1 and 2 respectively Our results are in agreement with the
published work of Rida et al [19] who considered the sameequation An exact solution of the standard form of Fisherequation for 120572 rarr 1 is
119906 (119909 119905) =1
(1 + 119890119909minus5119905)2 (15)
The comparison of our results with the exact solution isshown in Figure 1 for ℎ = 01 and quite clearly goodagreement is found
Approximate solutions are shown in Figure 2 for 120572 = 099
and 120572 = 095The influence of 120572 on the function 119906(119909 119905) is shown in
Figure 3 This figure indicates a decrease in the fractionalorder 120572 by choosing the fixed 119909 = 5 that corresponds to anincrease in the function and also indicates a slow diffusion forthe values of 120572 = 1 and 120572 = 09 and a fast diffusion for thevalues of 120572 = 08 07 06 respectively It is clearly seen that119906(119909 119905) increase for 120572 = 1 09 08 07 06 with the increasesin 119905
Numerical comparison between GDTM [19] HPM [35]FVIM [36] and hybrid method are found in Table 2 whichshows hybrid method is more promising
It is also found that the result is in complete agreementwith the result of HPM [42 43] and ADM [44] for 120572 = 1
We investigate convergence criteria of our solutions fordifferent values of ℎ and 119899 To illustrate this we comparedour results with the analytical solution in case of 120572 = 1 Here119899 is order of differential transformation method and denotesthe number of terms to be calculated
In Figures 4 5 and 6 the difference between the resultsobtained in this study and the results of the analytical solutionis of the order of 10minus5 This is a pointer to the fact that there isconvergence and is a restatement in numerical terms of whatwas shown in the Appendix
Mathematical Problems in Engineering 5
10
075
05
025
00minus10
minus5
5
10
x
00
0005
01
t
(a)
10
075
05
025
00minus10
minus5
5
10
x
00
0005
01
t
(b)
Figure 2 Numerical solution for the time fractional Fisher equation with (a) 120572 rarr 099 and (b) 120572 rarr 095
0007
0006
0005
0004
0003
0002
0001
u(xt)
0 01 02 03 04
t
120572 = 06120572 = 07
120572 = 08
120572 = 09
120572 = 1
(a)
0007
0006
0005
0004
0003
0002
0001
000000 01 02 03 04
u
t
120572 = 710120572 = 810
120572 = 910
120572 = 1
(b)
Figure 3 Approximate solution for the time fractional Fisher equation with different 120572 values at 119909 = 5 (a) present and (b) [35]
Table 2 Comparison of numerical results between different methods for the time fractional Fisher equation GDTM generalized differentialtransformmethod Rida et al [19] HPM homotpy perturbation method Khan et al [35] and FVIM fractional variational iteration methodMerdan [36]
Table 3 Some values of 119880120572119894(119896) of Example 2
119894119896
0 1 2
0 05(0125 minus 025120583)
Γ (120572 + 1)
(0000011 minus 0000052120583)
Γ (2120572 + 1)
1 0517670(0124847 minus 0249687120583)
Γ (120572 + 1)
(minus0002179 + 0008772120583 minus 00088241205832)
Γ (2120572 + 1)
2 0535296(0124384 minus 0248754120583)
Γ (120572 + 1)
(minus0004379 + 0017511120583 minus 00175601205832)
Γ (2120572 + 1)
One important observation made from the computationis that when the number of mesh points was increased lessnumber of terms was required in the time series solution tohave convergence for a predetermined accuracy The hybridmethod of the present study gives faster convergence thanother traditional methods for example if we take ℎ = 002
(mesh point is 50) then the solution converges for 119899 = 3 Wenow consider another example
Example 2 The time fractional FitzHugh-Nagumo equationis
119863120572
119905119906 = 119863
2
119909119906 + 119906 (1 minus 119906) (119906 minus 120583) 120583 gt 0
0 lt 120572 le 1 119909 isin R 119905 gt 0
(16)
In this type of equation the nonlinear function depends on 120583and it is 119891(119906) = 119906(1 minus 119906)(119906 minus 120583) The initial condition is
119906 (119909 0) =1
(1 + 119890minus119909radic2)
(17)
Using the hybrid method on the above initial boundary valueproblem (IBVP) as done in the previous example we get
(119896) minus 2119880120572119894(119896) + 119880
120572(119894minus1)(119896)
ℎ2
minus 120583119880120572119894(119896) + (1 + 120583)
119896
sum
119897=0
119880120572119894(119896 minus 119897) 119880
120572119894(119897)
minus
119896
sum
119904=0
119904
sum
119897=0
119880120572119894(119896 minus 119904)119880
120572119894(119904 minus 119897) 119880
120572119894(119897)
119880120572119894(0) =
1
1 + 119890minus119894ℎradic2
(18)
Using second order finite difference method the boundaryvalues were obtained as follows
1198801205720(119896) = 3119880
1205721(119896) minus 3119880
1205722(119896) + 119880
1205723(119896)
119880120572119873
(119896) = 3119880120572(119873minus1)
(119896) minus 3119880120572(119873minus2)
(119896) + 119880120572(119873minus3)
(119896)
(19)
Table 3 presents some of the 119880120572119894(119896)rsquos
The time series solution for the above IBVP at differenttimes is
119906 (0 119905)
= 05 +(0125 minus 025120583)
Γ (120572 + 1)119905120572
+(0000011 minus 0000052120583)
Γ (2120572 + 1)1199052120572
+ sdot sdot sdot
119906 (01 119905)
= 0517670 +(0124847 minus 0249687120583)
Γ (120572 + 1)119905120572
+
(minus0002179 + 0008772120583 minus 00088241205832)
Γ (2120572 + 1)1199052120572
+ sdot sdot sdot
119906 (02 119905)
= 0535296 +(0124384 minus 0248754120583)
Γ (120572 + 1)119905120572
+
(minus0004379 + 001751120583 minus 00175601205832)
Γ (2120572 + 1)1199052120572
+ sdot sdot sdot
(20)
Numerical solutions for the time fractional FitzHugh-Nagumo equation with various 120572 values are shown in Figures7 and 8 A comparison of the results in a limiting case whereinan analytical solution exists is shown in Figure 7 The resultsare in close agreement with those of Rida et al [19] for thesame equation
For 120572 rarr 1 it can easily be seen that the exact solution ofFitzHugh-Nagumo equation is
Figure 9 is prepared to show the influence of 120572 on thefunction 119906(119909 119905) It is clearly seen that 119906(119909 119905) decrease for120572 = 1 095 085 075 065 with the decreases in 119905
As shown in the Table 4 our results show close agreementwith the exact solution and agree with those of Rida et al [19]
Mathematical Problems in Engineering 7
Table 4 Coefficients of 1 119905 1199052 for some 119894 values and comparisonwith exact and Ridarsquos solution 120583 = 07 [19]
Rida et al [19] Exact Present119894 = 0
Coef of 1199050 05 05 05Coef of 1199051 minus005 minus005 minus0049999Coef of 1199052 002 0 minus0000011
119894 = 1
Coef of 1199050 0517670 0517670 0517670Coef of 1199051 minus0049937 minus00499937 minus0049933Coef of 1199052 0020327 minus0000176 minus0000181
119894 = 2
Coef of 1199050 0535296 0535296 0535296Coef of 1199051 minus0049937 minus0049937 minus0049743Coef of 1199052 0020602 minus0000351 minus0000352
119894 = 3
Coef of 1199050 0552835 0552835 0552835Coef of 1199051 minus0049441 minus0049441 minus0049430Coef of 1199052 0020821 minus0000522 minus0000542
026
024
022
PresentAnalytic
020
018
0 002 004 006 008 010
t
Figure 4 Comparison of present results for ℎ = 02 and 119899 = 10withthe analytical solution in case of 120572 rarr 1 at 119909 = 04
From this table it is clear that the present work gives betterapproximation than GDTM as we increase 119899
Numerical comparison between GDTM FVIM andhybrid method is shown in Table 5 which indicates hybridmethod is more promising
4 Conclusion
Many real physical problems can be best modelled withfractional differential equations but the fact is when the
026
024
022
PresentAnalytic
020
018
0 002 004 006 008 010
t
Figure 5 Comparison of present results for ℎ = 01 and 119899 = 5 withthe analytical solution in case of 120572 rarr 1 at 119909 = 04
026
024
022
PresentAnalytic
020
018
0 002 004 006 008 010
t
Figure 6 Comparison of present results for ℎ = 002 and 119899 = 3withthe analytical solution in case of 120572 rarr 1 at 119909 = 04
equation is nonlinear there are very few reliable methodsThe numerical methods that can be used to solve frac-tional differential equations are known to have problems ofconvergence and stability These aspects are well addressedin the paper by suggesting a new procedure that uses acombination of the generalized differential transform andcentral difference methods The Appendix clearly spells out
8 Mathematical Problems in Engineering
10
075
05
025
0010
5
x
01
0
005
00
t
minus10
minus5
(a)
10
075
05
025
0010
minus10
5
minus5x 010
005
00
t
(b)
Figure 7 Numerical solution for the time fractional FitzHugh-Nagumo equation with 120572 rarr 1 (a) comparison with the analytical solution(b) with 120583 = 05
10
075
05
025
0010
minus10
5
minus5x01
0
005
00
t
(a)
10
075
05
025
0010
minus10
5
minus5x
01
0005
00
t
(b)
Figure 8 Numerical solution for the time fractional FitzHugh-Nagumo equation with 120572 = 095 (a) and 120572 = 099 (b)
the fact that the error as a result of discretization and compu-tation is bounded and hence implies stability of the methodLax equivalence theorem further implies convergence of thescheme Two time fractional nonlinear reaction-diffusionequations considered for illustration of the hybrid methodhighlight the usefulness of the method in obtaining thesolution of IBVPs involving time fractional derivatives Thecontrol of convergence through a judicious choice of time andspatial step sizes and also the number of terms in the timeseries solution spells assured convergence The segregationof the time domain from the spatial domain in the solution
method ensures the fact that problem of stability does notarise Diagonal dominance of the coefficient matrix in thesystem of linear algebraic equations resulting from the use ofthe central difference approximation in the Poisson equationensures the fact that the matrix remains nonsingular duringiterations and hence has assured convergence An appropriatecomputational decision on the number of terms to be takenin the time series solution results in a convergent solutionwith fast convergence Excellent comparison of the presentresults with the previous works on generalized differentialtransform method [19] and homotopy perturbation method
Mathematical Problems in Engineering 9
09716
09714
09712
09710
u(xt)
09708
09706
120572 = 095
120572 = 1
120572 = 085
120572 = 075
120572 = 06509704
0 01 02 03
t
(a)
120572 = 095
120572 = 085
120572 = 075
120572 = 065
120572 = 1
09716
09714
09712
0971
u(xt)
09708
09706
09704
0 01005 02015 03025
t
(b)
Figure 9 Approximate solution for the time fractional FitzHugh-Nagumo equation with different 120572 values at 119909 = 5 (a) present and (b) [36]
[35] and fractional variational iterationmethod [36] providesconfidence in the methodology adopted for the solution oftime fractional differential equations
Appendix
Estimation of Bounds on Truncation Error
Consider the fractional differential equations (9) and (16) ina general form as
119863120572
119905119906 = 119863
119909119909119906 + 119891 (119906) 119905 ge 0
119909 isin R (0 lt 120572 le 1)
(A1)
The differential transform of (A1) at the spatially discretizedpoints 119909
We now follow Jang et al [45] and move on to arrive at anestimate on the bounds for the truncation error in a generalway by considering the Taylor series expansion of119880
Table 5 Comparison of numerical results between differentmethods for the time fractional FitzHugh-Nagumo equationGDTM generalizeddifferential transform method Rida et al [19] and FVIM fractional variational iteration method Merdan [36]
We so far addressed the local error due to two differenttruncations in the time series In what follows we estimatethe bounds on the cumulative error that includes the errordiscussed above
Let 119910119894(119905119896) denote the solution of (A2) The local error in
119908119894(119905119896) relative to 119910
119894(119905119896) is
1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908119894(119905119896+ Δ119905)
1003816100381610038161003816
le1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
+1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
(A13)
Since 119908119894(119905119896+ Δ119905) is a better approximation than 119908
119894(119905119896+ Δ119905)
we may assume that1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816 ≪ 1 (A14)
Mathematical Problems in Engineering 11
In view of (A14) we now have1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816 ≐1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
(A15)
Using (A8) and noting that (Δ119905)119899+1 is quite small in (A15)we may take 120573 to be
120573 ≐
1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908119894(119905119896+ Δ119905)
1003816100381610038161003816
(Δ119905)119899+1
(A16)
Thus 120573(Δ119905)119899+1 = 120576 is the bound on the tolerance in the to-be-obtained solution When using different number of terms inthe Taylor series expansion earlier we denoted the solutionsusing a time stepΔ119905 by119908
119894(119905119896+Δ119905) and119908
119894(119905119896+Δ119905) respectively
The paper uses an adaptive step size in computing theresults This is because such a procedure succeeds in keepingthe error bounded and ensures convergence as a consequenceof Lax equivalence theorem To see what the adaptive stepsize produces and to show how such a procedure keepsthe error bounded we start with the premise that Δ119905 isthe most appropriate step size for the problem This stepsize is determined using the definition of inverse differentialtransform
119906119894(Δ119905) =
infin
sum
119896=0
119880120572119894(119896) (Δ119905)
119896 (A17)
In our actual calculation we will not be able to considerinfinite number of terms We consider ldquo119899rdquo terms in respectof 119908119894and ldquo119899 + 119898rdquo terms in respect of 119908
119894
Thus1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
=
119899+119898
sum
119895=0
119880120572119894(119895) (Δ119905)
119895minus
119899
sum
119895=0
119880120572119894(119895) (Δ119905)
119895
=
119899+119898
sum
119895=119899+1
119880120572119894(119895) (Δ119905)
119895
(A18)
To write down a simpler expression we change the summa-tion index from 119895 to 119901 = 119899 + 119895 So we have from (A18) thefollowing
1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908119894(119905119896+ Δ119905)
Thus the above proceedings tell us that if criterion (A23) issatisfied then the error is bounded In effect this means thatthe scheme is convergent in lieu of Lax equivalence theoremIn our computations Δ119905 has been always chosen to satisfyinequality (A23)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors are thankful to Ondokuz Mayıs UniversitySamsun Turkey for providing financial support to carryout this work under a major research project (Grant nopyofen190113003)
References
[1] W Hundsdorfer and J G Verwer Numerical Solution of TimeDependent Advection-Diffusion-Reaction Equations SpringerBerlin Germany 2003
[2] Y Kuramoto Chemical Oscillations Waves and TurbulenceDover Mineola NY USA 2003
[3] J DMurrayMathematical Biology II vol 18 of InterdisciplinaryApplied Mathematics Springer New York NY USA 3rd edi-tion 2003
[4] H Wilhelmsson and E Lazzaro Reaction-Diffusion Problemsin the Physics of Hot Plasmas Institute of Physics PublishingPhiladelphia Pa USA 2001
[5] M Bar N Gottschalk M Eiswirth and G Ertl ldquoSpiral waves ina surface reaction model calculationsrdquoThe Journal of ChemicalPhysics vol 100 no 2 pp 1202ndash1214 1994
[6] D Barkley ldquoA model for fast computer simulation of waves inexcitable mediardquo Physica D Nonlinear Phenomena vol 49 no1-2 pp 61ndash70 1991
[7] F H Fenton E M Cherry H M Hastings and S J EvansldquoMultiple mechanisms of spiral wave breakup in a model ofcardiac electrical activityrdquo Chaos vol 12 no 3 pp 852ndash8922002
[8] M GosakMMarhl andM Perc ldquoSpatial coherence resonancein excitable biochemical media induced by internal noiserdquoBiophysical Chemistry vol 128 no 2-3 pp 210ndash214 2007
[9] A Karma ldquoMeandering transition in two-dimensionalexcitable mediardquo Physical Review Letters vol 65 no 22 pp2824ndash2827 1990
12 Mathematical Problems in Engineering
[10] J P Keener ldquoA geometrical theory for spiral waves in excitablemediardquo SIAM Journal onAppliedMathematics vol 46 no 6 pp1039ndash1056 1986
[11] J P Keener Mathematical Physiology Interdisciplinary AppliedMathematics Springer New York NY USA 1998
[12] V M Kenkre and M N Kuperman ldquoApplicability of the Fisherequation to bacterial population dynamicsrdquo Physical Review Evol 67 no 5 Article ID 051921 5 pages 2003
[13] V Krinsky and A Pumir ldquoModels of defibrillation of cardiactissuerdquo Chaos vol 8 no 1 pp 188ndash203 1998
[14] N F Otani ldquoA primary mechanism for spiral wave meander-ingrdquo Chaos vol 12 no 3 pp 829ndash842 2002
[15] M Perc ldquoSpatial coherence resonance in excitable mediardquoPhysical Review E vol 72 no 3 Article ID 016207 2005
[16] M Perc ldquoStochastic resonance on excitable small-world net-works via a pacemakerrdquo Physical Review E vol 76 no 6 ArticleID 066203 2007
[17] M Perc ldquoEffects of small-world connectivity on noise-inducedtemporal and spatial order in neural mediardquo Chaos Solitons ampFractals vol 31 no 2 pp 280ndash291 2007
[18] J J Tyson ldquoWhat everyone should know about the Belousov-Zhabotinsky reactionrdquo in Frontiers inMathematical Biology vol100 of Lecture Notes in Biomathematics pp 569ndash587 SpringerBerlin Germany 1994
[19] S Z Rida A M El-Sayed and A A Arafa ldquoOn the solutions oftime-fractional reaction-diffusion equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 15 no 12pp 3847ndash3854 2010
[20] Y Zheng and Z Zhao ldquoA fully discrete Galerkin method fora nonlinear space-fractional diffusion equationrdquo MathematicalProblems in Engineering vol 2011 Article ID 171620 20 pages2011
[21] J K Zhou Differential Transformation and Its Applications forElectrical Circuits Huazhang University Press Wuhan China1986 (Chinese)
[22] I H Abdel-Halim Hassan ldquoComparison differential transfor-mation technique with Adomian decomposition method forlinear and nonlinear initial value problemsrdquoChaos Solitons andFractals vol 36 no 1 pp 53ndash65 2008
[23] A Arikoglu and I Ozkol ldquoSolution of boundary value problemsfor integro-differential equations by using differential transformmethodrdquoAppliedMathematics and Computation vol 168 no 2pp 1145ndash1158 2005
[24] F Ayaz ldquoSolutions of the system of differential equationsby differential transform methodrdquo Applied Mathematics andComputation vol 147 no 2 pp 547ndash567 2004
[25] N Bildik A Konuralp F Orak Bek and S KucukarslanldquoSolution of different type of the partial differential equationby differential transformmethod andAdomianrsquos decompositionmethodrdquo Applied Mathematics and Computation vol 172 no 1pp 551ndash567 2006
[26] H Liu and Y Song ldquoDifferential transform method applied tohigh index differential-algebraic equationsrdquoAppliedMathemat-ics and Computation vol 184 no 2 pp 748ndash753 2007
[27] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent Part IIrdquo Geophysical Journal of theRoyal Astronomical Society vol 13 no 5 pp 529ndash539 1967
[28] SMomani ZOdibat andV S Erturk ldquoGeneralized differentialtransform method for solving a space- and time-fractionaldiffusion-wave equationrdquo Physics Letters A vol 370 no 5-6 pp379ndash387 2007
[29] Z M Odibat C Bertelle M A Aziz-Alaoui and G HDuchamp ldquoA multi-step differential transform method andapplication to non-chaotic or chaotic systemsrdquo Computers ampMathematics with Applications vol 59 no 4 pp 1462ndash14722010
[30] N Laskin ldquoFractional Schrodinger Equationrdquo Physical ReviewE vol 66 no 5 Article ID 056108 2002
[31] G J Fix and J P Roop ldquoLeast squares finite-element solutionof a fractional order two-point boundary value problemrdquoComputers ampMathematics with Applications vol 48 no 7-8 pp1017ndash1033 2004
[32] L-T Yu and C-K Chen ldquoApplication of the hybrid methodto the transient thermal stresses response in isotropic annularfinsrdquo Journal of Applied Mechanics vol 66 no 2 pp 340ndash3471999
[33] B-L Kuo and C-K Chen ldquoApplication of a hybrid methodto the solution of the nonlinear burgersrsquo equationrdquo Journal ofApplied Mechanics Transactions ASME vol 70 no 6 pp 926ndash929 2003
[34] C K Chen H Y Lai and C C Liu ldquoApplication of hybrid dif-ferential transformationfinite difference method to nonlinearanalysis of micro fixed-fixed beamrdquo Microsystem Technologiesvol 15 no 6 pp 813ndash820 2009
[35] N A Khan M Ayaz L Jin and A Yildirim ldquoOn approximatesolutions for the time-fractional reaction-diffusion equation ofFisher typerdquo International Journal of Physical Sciences vol 6 no10 pp 2483ndash2496 2011
[36] M Merdan ldquoSolutions of time-fractional reaction-diffusionequation withmodified Riemann-Liouville derivativerdquo Interna-tional Journal of Physical Sciences vol 7 no 15 pp 2317ndash23262012
[37] C Li and G Peng ldquoChaos in Chenrsquos system with a fractionalorderrdquo Chaos Solitons amp Fractals vol 22 no 2 pp 443ndash4502004
[38] Y Luchko and R Gorenflo ldquoAn operational method for solvingfractional differential equations with the Caputo derivativesrdquoActa Mathematica Vietnamica vol 24 no 2 pp 207ndash233 1999
[39] Z Odibat and S Momani ldquoA generalized differential transformmethod for linear partial differential equations of fractionalorderrdquo Applied Mathematics Letters vol 21 no 2 pp 194ndash1992008
[40] Z M Odibat and N T Shawagfeh ldquoGeneralized Taylorrsquosformulardquo Applied Mathematics and Computation vol 186 no1 pp 286ndash293 2007
[41] D Nazari and S Shahmorad ldquoApplication of the fractionaldifferential transform method to fractional-order integro-differential equations with nonlocal boundary conditionsrdquoJournal of Computational andAppliedMathematics vol 234 no3 pp 883ndash891 2010
[42] D Agırseven and T Ozis ldquoAn analytical study for Fisher typeequations by using homotopy perturbationmethodrdquoComputersamp Mathematics with Applications vol 60 no 3 pp 602ndash6092010
[43] M Dehghan J M Heris and A Saadatmandi ldquoApplicationof semi-analytic methods for the Fitzhugh-Nagumo equation
Mathematical Problems in Engineering 13
which models the transmission of nerve impulsesrdquoMathemati-calMethods in the Applied Sciences vol 33 no 11 pp 1384ndash13982010
[44] A-M Wazwaz and A Gorguis ldquoAn analytic study of Fisherrsquosequation by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 154 no 3 pp 609ndash6202004
[45] M-J Jang C-L Chen and Y-C Liy ldquoOn solving the initial-value problems using the differential transformation methodrdquoAppliedMathematics andComputation vol 115 no 2-3 pp 145ndash160 2000
where 1198631is the diffusion coefficient and 119891(119906) is a nonlinear
function We consider two different forms of 119891(119906) which arecalled time fractional Fisher equation and time fractionalFitzHugh-Nagumo equation
We apply GDTM to discretize fractional order timederivative and central difference method to discretize deriva-tives in 119909 direction respectively After transforming (7) usingthe GDTM we get the following equation
(119896) minus 2119880120572119894(119896) + 119880
120572(119894minus1)(119896)
ℎ2+ 119865 (119880
120572119894)
(8)
where 119880120572119894(119896) is the transformed function of 119906(119909
119894 119905) and
119865(119880120572119894) is the transformed function of 119891(119906) The region 0 lt
119909 lt 119886 is divided into several equal intervals and each intervalhas a width ℎ The time interval of interest is discretizedusing a time step Δ119905 After discretization of the equationwe get solution at time Δ119905 and these results are adoptedas the initial values for the next time interval This time-stepping procedure assists in obtaining a converged solutionto a desired accuracy [39] An appendix has been added to thepaper to show that the error if any in themethod is boundedThis implies stability of the scheme and by Lax equivalencetheorem it thereby implies convergence
The new algorithm has been developed to solve thenonlinear reaction diffusion equation and our aim of thisapproach is to combine the flexibility of differential transformand the efficiency of finite differences This algorithm alsoprovides an iterative procedure to calculate the numericalsolutions therefore it is not necessary to carry out com-plicated symbolic computation On applying the differentialtransform method with respect to time on the equation weare basically transforming the time-evolutionary equationto an elliptic type In essence this means that the centralfinite difference approximation that is subsequently used onthe transformed equation is a Poisson solver The resultingsystem of linear algebraic equations is then diagonally dom-inant and hence the Gauss-Seidel iterative method used forsolving the same has assured convergence as the coefficientmatrix remains nonsingular throughout the computationThe algorithm used thus succeeds in segregating the timediscretization from explicitly influencing the computation inthe spatial domain and this presents a situation wherein thetwo can be handled independent of each other in the courseof computation without having to bother about the stabilityof the solution if the differential transform part is properlyhandled The latter is achieved deftly in the differentialtransform part of the algorithm by using the multisteppingprocedure as first enunciated by Yu and Chen [32] in theirphenomenal work and used subsequently by Odibat et al[29 39] In summary this means that convergence is neverin doubt in the algorithm but slow convergence can be ifthe time and spatial discretizations are badly handled Theconvergence is optimized in the paper computationally byproper selection of time step in the differential transformpart of the algorithm and then the spatial step size in the
finite difference part of the algorithm Next important stepin the algorithm is the decision on the number of terms tobe adopted in the inverse differential transform that givesus the solution of the problem as a power series in timeThe aforementioned three vital components of the algorithmhave been meticulously handled and a brief summary of thenumerical experiment undertaken concerning the same ispresented in a table The numerical study recommends thatthe combination of 5 10 and 50 spatial step size with a timestep of 00005 or 0001 assures the best rate of convergence ifwe take minimum ten terms in the time series
3 Illustration of Generalized DifferentialTransform and Finite Difference Method
To show effectiveness of the proposed numerical solutionusing the temporal generalized differential transform and thespatial finite differencemethod and to give an understandableoverview of the methodology two examples of the reactiondiffusion equations will be discussed in the following sectionThen our results will be compared with published work ofRida et al [19] in which GDTM was used to solve the sameequations
Example 1 The time fractional Fisher equation is
119863120572
119905119906 = 119863
119909119909119906 + 6119906 (1 minus 119906) 119909 isin R
119905 gt 0 (0 lt 120572 le 1)
(9)
In this example we have the nonlinear function 119891(119906) =
6119906(1 minus 119906) The initial condition used is
119906 (119909 0) =1
(1 + 119890119909)2 (10)
Operating the generalized differential transform on (9)gives us the following equation
(119896) minus 2119880120572119894(119896) + 119880
120572(119894minus1)(119896)
ℎ2
+ 6119880120572119894(119896) minus 6
119896
sum
119897=0
119880120572119894(119896 minus 119897) 119880
120572119894(119897)
(12)
4 Mathematical Problems in Engineering
10
075
05
025
00minus10
minus5
5
10
x
00
0005
01
t
(a)
10
075
05
025
00minus10
minus5
5
10
x
00
0005
01
t
(b)
Figure 1 Numerical solution for the time fractional Fisher equation with 120572 rarr 1 (a) comparison with the analytical solution (b)
Table 1 Some values of 119880120572119894(119896) of Example 1
119894119896
0 1 2
0 0250000 1249791
Γ (120572 + 1)
3126294
Γ (2120572 + 1)
1 0225644 1184363
Γ (120572 + 1)
3406094
Γ (2120572 + 1)
2 0202649 1114004
Γ (120572 + 1)
3618706
Γ (2120572 + 1)
The initial condition on discretization yields
119880120572119894(0) =
1
(1 + 119890119894ℎ)2 (13)
Equation (12) is a recurrence relation The time seriessolution of the given equation is then obtained by using (12)and (13) with ℎ = 01 to obtain 119880
120572119894(119896) Some of 119880
120572119894(119896) are
recorded in Table 1The time series solutions of (12) with the initial condition
(13) are obtained as follows
119906 (0 119905) = 0250000 +1249791
Γ (120572 + 1)119905120572+
3126294
Γ (2120572 + 1)1199052120572
+ sdot sdot sdot
119906 (01 119905) = 0225644 +1184363
Γ (120572 + 1)119905120572+
3406094
Γ (2120572 + 1)1199052120572
+ sdot sdot sdot
119906 (02 119905) = 0202649 +1114004
Γ (120572 + 1)119905120572+
3618706
Γ (2120572 + 1)1199052120572
+ sdot sdot sdot
(14)
The numerical calculation results are shown in Figures1 and 2 respectively Our results are in agreement with the
published work of Rida et al [19] who considered the sameequation An exact solution of the standard form of Fisherequation for 120572 rarr 1 is
119906 (119909 119905) =1
(1 + 119890119909minus5119905)2 (15)
The comparison of our results with the exact solution isshown in Figure 1 for ℎ = 01 and quite clearly goodagreement is found
Approximate solutions are shown in Figure 2 for 120572 = 099
and 120572 = 095The influence of 120572 on the function 119906(119909 119905) is shown in
Figure 3 This figure indicates a decrease in the fractionalorder 120572 by choosing the fixed 119909 = 5 that corresponds to anincrease in the function and also indicates a slow diffusion forthe values of 120572 = 1 and 120572 = 09 and a fast diffusion for thevalues of 120572 = 08 07 06 respectively It is clearly seen that119906(119909 119905) increase for 120572 = 1 09 08 07 06 with the increasesin 119905
Numerical comparison between GDTM [19] HPM [35]FVIM [36] and hybrid method are found in Table 2 whichshows hybrid method is more promising
It is also found that the result is in complete agreementwith the result of HPM [42 43] and ADM [44] for 120572 = 1
We investigate convergence criteria of our solutions fordifferent values of ℎ and 119899 To illustrate this we comparedour results with the analytical solution in case of 120572 = 1 Here119899 is order of differential transformation method and denotesthe number of terms to be calculated
In Figures 4 5 and 6 the difference between the resultsobtained in this study and the results of the analytical solutionis of the order of 10minus5 This is a pointer to the fact that there isconvergence and is a restatement in numerical terms of whatwas shown in the Appendix
Mathematical Problems in Engineering 5
10
075
05
025
00minus10
minus5
5
10
x
00
0005
01
t
(a)
10
075
05
025
00minus10
minus5
5
10
x
00
0005
01
t
(b)
Figure 2 Numerical solution for the time fractional Fisher equation with (a) 120572 rarr 099 and (b) 120572 rarr 095
0007
0006
0005
0004
0003
0002
0001
u(xt)
0 01 02 03 04
t
120572 = 06120572 = 07
120572 = 08
120572 = 09
120572 = 1
(a)
0007
0006
0005
0004
0003
0002
0001
000000 01 02 03 04
u
t
120572 = 710120572 = 810
120572 = 910
120572 = 1
(b)
Figure 3 Approximate solution for the time fractional Fisher equation with different 120572 values at 119909 = 5 (a) present and (b) [35]
Table 2 Comparison of numerical results between different methods for the time fractional Fisher equation GDTM generalized differentialtransformmethod Rida et al [19] HPM homotpy perturbation method Khan et al [35] and FVIM fractional variational iteration methodMerdan [36]
Table 3 Some values of 119880120572119894(119896) of Example 2
119894119896
0 1 2
0 05(0125 minus 025120583)
Γ (120572 + 1)
(0000011 minus 0000052120583)
Γ (2120572 + 1)
1 0517670(0124847 minus 0249687120583)
Γ (120572 + 1)
(minus0002179 + 0008772120583 minus 00088241205832)
Γ (2120572 + 1)
2 0535296(0124384 minus 0248754120583)
Γ (120572 + 1)
(minus0004379 + 0017511120583 minus 00175601205832)
Γ (2120572 + 1)
One important observation made from the computationis that when the number of mesh points was increased lessnumber of terms was required in the time series solution tohave convergence for a predetermined accuracy The hybridmethod of the present study gives faster convergence thanother traditional methods for example if we take ℎ = 002
(mesh point is 50) then the solution converges for 119899 = 3 Wenow consider another example
Example 2 The time fractional FitzHugh-Nagumo equationis
119863120572
119905119906 = 119863
2
119909119906 + 119906 (1 minus 119906) (119906 minus 120583) 120583 gt 0
0 lt 120572 le 1 119909 isin R 119905 gt 0
(16)
In this type of equation the nonlinear function depends on 120583and it is 119891(119906) = 119906(1 minus 119906)(119906 minus 120583) The initial condition is
119906 (119909 0) =1
(1 + 119890minus119909radic2)
(17)
Using the hybrid method on the above initial boundary valueproblem (IBVP) as done in the previous example we get
(119896) minus 2119880120572119894(119896) + 119880
120572(119894minus1)(119896)
ℎ2
minus 120583119880120572119894(119896) + (1 + 120583)
119896
sum
119897=0
119880120572119894(119896 minus 119897) 119880
120572119894(119897)
minus
119896
sum
119904=0
119904
sum
119897=0
119880120572119894(119896 minus 119904)119880
120572119894(119904 minus 119897) 119880
120572119894(119897)
119880120572119894(0) =
1
1 + 119890minus119894ℎradic2
(18)
Using second order finite difference method the boundaryvalues were obtained as follows
1198801205720(119896) = 3119880
1205721(119896) minus 3119880
1205722(119896) + 119880
1205723(119896)
119880120572119873
(119896) = 3119880120572(119873minus1)
(119896) minus 3119880120572(119873minus2)
(119896) + 119880120572(119873minus3)
(119896)
(19)
Table 3 presents some of the 119880120572119894(119896)rsquos
The time series solution for the above IBVP at differenttimes is
119906 (0 119905)
= 05 +(0125 minus 025120583)
Γ (120572 + 1)119905120572
+(0000011 minus 0000052120583)
Γ (2120572 + 1)1199052120572
+ sdot sdot sdot
119906 (01 119905)
= 0517670 +(0124847 minus 0249687120583)
Γ (120572 + 1)119905120572
+
(minus0002179 + 0008772120583 minus 00088241205832)
Γ (2120572 + 1)1199052120572
+ sdot sdot sdot
119906 (02 119905)
= 0535296 +(0124384 minus 0248754120583)
Γ (120572 + 1)119905120572
+
(minus0004379 + 001751120583 minus 00175601205832)
Γ (2120572 + 1)1199052120572
+ sdot sdot sdot
(20)
Numerical solutions for the time fractional FitzHugh-Nagumo equation with various 120572 values are shown in Figures7 and 8 A comparison of the results in a limiting case whereinan analytical solution exists is shown in Figure 7 The resultsare in close agreement with those of Rida et al [19] for thesame equation
For 120572 rarr 1 it can easily be seen that the exact solution ofFitzHugh-Nagumo equation is
Figure 9 is prepared to show the influence of 120572 on thefunction 119906(119909 119905) It is clearly seen that 119906(119909 119905) decrease for120572 = 1 095 085 075 065 with the decreases in 119905
As shown in the Table 4 our results show close agreementwith the exact solution and agree with those of Rida et al [19]
Mathematical Problems in Engineering 7
Table 4 Coefficients of 1 119905 1199052 for some 119894 values and comparisonwith exact and Ridarsquos solution 120583 = 07 [19]
Rida et al [19] Exact Present119894 = 0
Coef of 1199050 05 05 05Coef of 1199051 minus005 minus005 minus0049999Coef of 1199052 002 0 minus0000011
119894 = 1
Coef of 1199050 0517670 0517670 0517670Coef of 1199051 minus0049937 minus00499937 minus0049933Coef of 1199052 0020327 minus0000176 minus0000181
119894 = 2
Coef of 1199050 0535296 0535296 0535296Coef of 1199051 minus0049937 minus0049937 minus0049743Coef of 1199052 0020602 minus0000351 minus0000352
119894 = 3
Coef of 1199050 0552835 0552835 0552835Coef of 1199051 minus0049441 minus0049441 minus0049430Coef of 1199052 0020821 minus0000522 minus0000542
026
024
022
PresentAnalytic
020
018
0 002 004 006 008 010
t
Figure 4 Comparison of present results for ℎ = 02 and 119899 = 10withthe analytical solution in case of 120572 rarr 1 at 119909 = 04
From this table it is clear that the present work gives betterapproximation than GDTM as we increase 119899
Numerical comparison between GDTM FVIM andhybrid method is shown in Table 5 which indicates hybridmethod is more promising
4 Conclusion
Many real physical problems can be best modelled withfractional differential equations but the fact is when the
026
024
022
PresentAnalytic
020
018
0 002 004 006 008 010
t
Figure 5 Comparison of present results for ℎ = 01 and 119899 = 5 withthe analytical solution in case of 120572 rarr 1 at 119909 = 04
026
024
022
PresentAnalytic
020
018
0 002 004 006 008 010
t
Figure 6 Comparison of present results for ℎ = 002 and 119899 = 3withthe analytical solution in case of 120572 rarr 1 at 119909 = 04
equation is nonlinear there are very few reliable methodsThe numerical methods that can be used to solve frac-tional differential equations are known to have problems ofconvergence and stability These aspects are well addressedin the paper by suggesting a new procedure that uses acombination of the generalized differential transform andcentral difference methods The Appendix clearly spells out
8 Mathematical Problems in Engineering
10
075
05
025
0010
5
x
01
0
005
00
t
minus10
minus5
(a)
10
075
05
025
0010
minus10
5
minus5x 010
005
00
t
(b)
Figure 7 Numerical solution for the time fractional FitzHugh-Nagumo equation with 120572 rarr 1 (a) comparison with the analytical solution(b) with 120583 = 05
10
075
05
025
0010
minus10
5
minus5x01
0
005
00
t
(a)
10
075
05
025
0010
minus10
5
minus5x
01
0005
00
t
(b)
Figure 8 Numerical solution for the time fractional FitzHugh-Nagumo equation with 120572 = 095 (a) and 120572 = 099 (b)
the fact that the error as a result of discretization and compu-tation is bounded and hence implies stability of the methodLax equivalence theorem further implies convergence of thescheme Two time fractional nonlinear reaction-diffusionequations considered for illustration of the hybrid methodhighlight the usefulness of the method in obtaining thesolution of IBVPs involving time fractional derivatives Thecontrol of convergence through a judicious choice of time andspatial step sizes and also the number of terms in the timeseries solution spells assured convergence The segregationof the time domain from the spatial domain in the solution
method ensures the fact that problem of stability does notarise Diagonal dominance of the coefficient matrix in thesystem of linear algebraic equations resulting from the use ofthe central difference approximation in the Poisson equationensures the fact that the matrix remains nonsingular duringiterations and hence has assured convergence An appropriatecomputational decision on the number of terms to be takenin the time series solution results in a convergent solutionwith fast convergence Excellent comparison of the presentresults with the previous works on generalized differentialtransform method [19] and homotopy perturbation method
Mathematical Problems in Engineering 9
09716
09714
09712
09710
u(xt)
09708
09706
120572 = 095
120572 = 1
120572 = 085
120572 = 075
120572 = 06509704
0 01 02 03
t
(a)
120572 = 095
120572 = 085
120572 = 075
120572 = 065
120572 = 1
09716
09714
09712
0971
u(xt)
09708
09706
09704
0 01005 02015 03025
t
(b)
Figure 9 Approximate solution for the time fractional FitzHugh-Nagumo equation with different 120572 values at 119909 = 5 (a) present and (b) [36]
[35] and fractional variational iterationmethod [36] providesconfidence in the methodology adopted for the solution oftime fractional differential equations
Appendix
Estimation of Bounds on Truncation Error
Consider the fractional differential equations (9) and (16) ina general form as
119863120572
119905119906 = 119863
119909119909119906 + 119891 (119906) 119905 ge 0
119909 isin R (0 lt 120572 le 1)
(A1)
The differential transform of (A1) at the spatially discretizedpoints 119909
We now follow Jang et al [45] and move on to arrive at anestimate on the bounds for the truncation error in a generalway by considering the Taylor series expansion of119880
Table 5 Comparison of numerical results between differentmethods for the time fractional FitzHugh-Nagumo equationGDTM generalizeddifferential transform method Rida et al [19] and FVIM fractional variational iteration method Merdan [36]
We so far addressed the local error due to two differenttruncations in the time series In what follows we estimatethe bounds on the cumulative error that includes the errordiscussed above
Let 119910119894(119905119896) denote the solution of (A2) The local error in
119908119894(119905119896) relative to 119910
119894(119905119896) is
1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908119894(119905119896+ Δ119905)
1003816100381610038161003816
le1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
+1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
(A13)
Since 119908119894(119905119896+ Δ119905) is a better approximation than 119908
119894(119905119896+ Δ119905)
we may assume that1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816 ≪ 1 (A14)
Mathematical Problems in Engineering 11
In view of (A14) we now have1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816 ≐1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
(A15)
Using (A8) and noting that (Δ119905)119899+1 is quite small in (A15)we may take 120573 to be
120573 ≐
1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908119894(119905119896+ Δ119905)
1003816100381610038161003816
(Δ119905)119899+1
(A16)
Thus 120573(Δ119905)119899+1 = 120576 is the bound on the tolerance in the to-be-obtained solution When using different number of terms inthe Taylor series expansion earlier we denoted the solutionsusing a time stepΔ119905 by119908
119894(119905119896+Δ119905) and119908
119894(119905119896+Δ119905) respectively
The paper uses an adaptive step size in computing theresults This is because such a procedure succeeds in keepingthe error bounded and ensures convergence as a consequenceof Lax equivalence theorem To see what the adaptive stepsize produces and to show how such a procedure keepsthe error bounded we start with the premise that Δ119905 isthe most appropriate step size for the problem This stepsize is determined using the definition of inverse differentialtransform
119906119894(Δ119905) =
infin
sum
119896=0
119880120572119894(119896) (Δ119905)
119896 (A17)
In our actual calculation we will not be able to considerinfinite number of terms We consider ldquo119899rdquo terms in respectof 119908119894and ldquo119899 + 119898rdquo terms in respect of 119908
119894
Thus1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
=
119899+119898
sum
119895=0
119880120572119894(119895) (Δ119905)
119895minus
119899
sum
119895=0
119880120572119894(119895) (Δ119905)
119895
=
119899+119898
sum
119895=119899+1
119880120572119894(119895) (Δ119905)
119895
(A18)
To write down a simpler expression we change the summa-tion index from 119895 to 119901 = 119899 + 119895 So we have from (A18) thefollowing
1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908119894(119905119896+ Δ119905)
Thus the above proceedings tell us that if criterion (A23) issatisfied then the error is bounded In effect this means thatthe scheme is convergent in lieu of Lax equivalence theoremIn our computations Δ119905 has been always chosen to satisfyinequality (A23)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors are thankful to Ondokuz Mayıs UniversitySamsun Turkey for providing financial support to carryout this work under a major research project (Grant nopyofen190113003)
References
[1] W Hundsdorfer and J G Verwer Numerical Solution of TimeDependent Advection-Diffusion-Reaction Equations SpringerBerlin Germany 2003
[2] Y Kuramoto Chemical Oscillations Waves and TurbulenceDover Mineola NY USA 2003
[3] J DMurrayMathematical Biology II vol 18 of InterdisciplinaryApplied Mathematics Springer New York NY USA 3rd edi-tion 2003
[4] H Wilhelmsson and E Lazzaro Reaction-Diffusion Problemsin the Physics of Hot Plasmas Institute of Physics PublishingPhiladelphia Pa USA 2001
[5] M Bar N Gottschalk M Eiswirth and G Ertl ldquoSpiral waves ina surface reaction model calculationsrdquoThe Journal of ChemicalPhysics vol 100 no 2 pp 1202ndash1214 1994
[6] D Barkley ldquoA model for fast computer simulation of waves inexcitable mediardquo Physica D Nonlinear Phenomena vol 49 no1-2 pp 61ndash70 1991
[7] F H Fenton E M Cherry H M Hastings and S J EvansldquoMultiple mechanisms of spiral wave breakup in a model ofcardiac electrical activityrdquo Chaos vol 12 no 3 pp 852ndash8922002
[8] M GosakMMarhl andM Perc ldquoSpatial coherence resonancein excitable biochemical media induced by internal noiserdquoBiophysical Chemistry vol 128 no 2-3 pp 210ndash214 2007
[9] A Karma ldquoMeandering transition in two-dimensionalexcitable mediardquo Physical Review Letters vol 65 no 22 pp2824ndash2827 1990
12 Mathematical Problems in Engineering
[10] J P Keener ldquoA geometrical theory for spiral waves in excitablemediardquo SIAM Journal onAppliedMathematics vol 46 no 6 pp1039ndash1056 1986
[11] J P Keener Mathematical Physiology Interdisciplinary AppliedMathematics Springer New York NY USA 1998
[12] V M Kenkre and M N Kuperman ldquoApplicability of the Fisherequation to bacterial population dynamicsrdquo Physical Review Evol 67 no 5 Article ID 051921 5 pages 2003
[13] V Krinsky and A Pumir ldquoModels of defibrillation of cardiactissuerdquo Chaos vol 8 no 1 pp 188ndash203 1998
[14] N F Otani ldquoA primary mechanism for spiral wave meander-ingrdquo Chaos vol 12 no 3 pp 829ndash842 2002
[15] M Perc ldquoSpatial coherence resonance in excitable mediardquoPhysical Review E vol 72 no 3 Article ID 016207 2005
[16] M Perc ldquoStochastic resonance on excitable small-world net-works via a pacemakerrdquo Physical Review E vol 76 no 6 ArticleID 066203 2007
[17] M Perc ldquoEffects of small-world connectivity on noise-inducedtemporal and spatial order in neural mediardquo Chaos Solitons ampFractals vol 31 no 2 pp 280ndash291 2007
[18] J J Tyson ldquoWhat everyone should know about the Belousov-Zhabotinsky reactionrdquo in Frontiers inMathematical Biology vol100 of Lecture Notes in Biomathematics pp 569ndash587 SpringerBerlin Germany 1994
[19] S Z Rida A M El-Sayed and A A Arafa ldquoOn the solutions oftime-fractional reaction-diffusion equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 15 no 12pp 3847ndash3854 2010
[20] Y Zheng and Z Zhao ldquoA fully discrete Galerkin method fora nonlinear space-fractional diffusion equationrdquo MathematicalProblems in Engineering vol 2011 Article ID 171620 20 pages2011
[21] J K Zhou Differential Transformation and Its Applications forElectrical Circuits Huazhang University Press Wuhan China1986 (Chinese)
[22] I H Abdel-Halim Hassan ldquoComparison differential transfor-mation technique with Adomian decomposition method forlinear and nonlinear initial value problemsrdquoChaos Solitons andFractals vol 36 no 1 pp 53ndash65 2008
[23] A Arikoglu and I Ozkol ldquoSolution of boundary value problemsfor integro-differential equations by using differential transformmethodrdquoAppliedMathematics and Computation vol 168 no 2pp 1145ndash1158 2005
[24] F Ayaz ldquoSolutions of the system of differential equationsby differential transform methodrdquo Applied Mathematics andComputation vol 147 no 2 pp 547ndash567 2004
[25] N Bildik A Konuralp F Orak Bek and S KucukarslanldquoSolution of different type of the partial differential equationby differential transformmethod andAdomianrsquos decompositionmethodrdquo Applied Mathematics and Computation vol 172 no 1pp 551ndash567 2006
[26] H Liu and Y Song ldquoDifferential transform method applied tohigh index differential-algebraic equationsrdquoAppliedMathemat-ics and Computation vol 184 no 2 pp 748ndash753 2007
[27] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent Part IIrdquo Geophysical Journal of theRoyal Astronomical Society vol 13 no 5 pp 529ndash539 1967
[28] SMomani ZOdibat andV S Erturk ldquoGeneralized differentialtransform method for solving a space- and time-fractionaldiffusion-wave equationrdquo Physics Letters A vol 370 no 5-6 pp379ndash387 2007
[29] Z M Odibat C Bertelle M A Aziz-Alaoui and G HDuchamp ldquoA multi-step differential transform method andapplication to non-chaotic or chaotic systemsrdquo Computers ampMathematics with Applications vol 59 no 4 pp 1462ndash14722010
[30] N Laskin ldquoFractional Schrodinger Equationrdquo Physical ReviewE vol 66 no 5 Article ID 056108 2002
[31] G J Fix and J P Roop ldquoLeast squares finite-element solutionof a fractional order two-point boundary value problemrdquoComputers ampMathematics with Applications vol 48 no 7-8 pp1017ndash1033 2004
[32] L-T Yu and C-K Chen ldquoApplication of the hybrid methodto the transient thermal stresses response in isotropic annularfinsrdquo Journal of Applied Mechanics vol 66 no 2 pp 340ndash3471999
[33] B-L Kuo and C-K Chen ldquoApplication of a hybrid methodto the solution of the nonlinear burgersrsquo equationrdquo Journal ofApplied Mechanics Transactions ASME vol 70 no 6 pp 926ndash929 2003
[34] C K Chen H Y Lai and C C Liu ldquoApplication of hybrid dif-ferential transformationfinite difference method to nonlinearanalysis of micro fixed-fixed beamrdquo Microsystem Technologiesvol 15 no 6 pp 813ndash820 2009
[35] N A Khan M Ayaz L Jin and A Yildirim ldquoOn approximatesolutions for the time-fractional reaction-diffusion equation ofFisher typerdquo International Journal of Physical Sciences vol 6 no10 pp 2483ndash2496 2011
[36] M Merdan ldquoSolutions of time-fractional reaction-diffusionequation withmodified Riemann-Liouville derivativerdquo Interna-tional Journal of Physical Sciences vol 7 no 15 pp 2317ndash23262012
[37] C Li and G Peng ldquoChaos in Chenrsquos system with a fractionalorderrdquo Chaos Solitons amp Fractals vol 22 no 2 pp 443ndash4502004
[38] Y Luchko and R Gorenflo ldquoAn operational method for solvingfractional differential equations with the Caputo derivativesrdquoActa Mathematica Vietnamica vol 24 no 2 pp 207ndash233 1999
[39] Z Odibat and S Momani ldquoA generalized differential transformmethod for linear partial differential equations of fractionalorderrdquo Applied Mathematics Letters vol 21 no 2 pp 194ndash1992008
[40] Z M Odibat and N T Shawagfeh ldquoGeneralized Taylorrsquosformulardquo Applied Mathematics and Computation vol 186 no1 pp 286ndash293 2007
[41] D Nazari and S Shahmorad ldquoApplication of the fractionaldifferential transform method to fractional-order integro-differential equations with nonlocal boundary conditionsrdquoJournal of Computational andAppliedMathematics vol 234 no3 pp 883ndash891 2010
[42] D Agırseven and T Ozis ldquoAn analytical study for Fisher typeequations by using homotopy perturbationmethodrdquoComputersamp Mathematics with Applications vol 60 no 3 pp 602ndash6092010
[43] M Dehghan J M Heris and A Saadatmandi ldquoApplicationof semi-analytic methods for the Fitzhugh-Nagumo equation
Mathematical Problems in Engineering 13
which models the transmission of nerve impulsesrdquoMathemati-calMethods in the Applied Sciences vol 33 no 11 pp 1384ndash13982010
[44] A-M Wazwaz and A Gorguis ldquoAn analytic study of Fisherrsquosequation by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 154 no 3 pp 609ndash6202004
[45] M-J Jang C-L Chen and Y-C Liy ldquoOn solving the initial-value problems using the differential transformation methodrdquoAppliedMathematics andComputation vol 115 no 2-3 pp 145ndash160 2000
Figure 1 Numerical solution for the time fractional Fisher equation with 120572 rarr 1 (a) comparison with the analytical solution (b)
Table 1 Some values of 119880120572119894(119896) of Example 1
119894119896
0 1 2
0 0250000 1249791
Γ (120572 + 1)
3126294
Γ (2120572 + 1)
1 0225644 1184363
Γ (120572 + 1)
3406094
Γ (2120572 + 1)
2 0202649 1114004
Γ (120572 + 1)
3618706
Γ (2120572 + 1)
The initial condition on discretization yields
119880120572119894(0) =
1
(1 + 119890119894ℎ)2 (13)
Equation (12) is a recurrence relation The time seriessolution of the given equation is then obtained by using (12)and (13) with ℎ = 01 to obtain 119880
120572119894(119896) Some of 119880
120572119894(119896) are
recorded in Table 1The time series solutions of (12) with the initial condition
(13) are obtained as follows
119906 (0 119905) = 0250000 +1249791
Γ (120572 + 1)119905120572+
3126294
Γ (2120572 + 1)1199052120572
+ sdot sdot sdot
119906 (01 119905) = 0225644 +1184363
Γ (120572 + 1)119905120572+
3406094
Γ (2120572 + 1)1199052120572
+ sdot sdot sdot
119906 (02 119905) = 0202649 +1114004
Γ (120572 + 1)119905120572+
3618706
Γ (2120572 + 1)1199052120572
+ sdot sdot sdot
(14)
The numerical calculation results are shown in Figures1 and 2 respectively Our results are in agreement with the
published work of Rida et al [19] who considered the sameequation An exact solution of the standard form of Fisherequation for 120572 rarr 1 is
119906 (119909 119905) =1
(1 + 119890119909minus5119905)2 (15)
The comparison of our results with the exact solution isshown in Figure 1 for ℎ = 01 and quite clearly goodagreement is found
Approximate solutions are shown in Figure 2 for 120572 = 099
and 120572 = 095The influence of 120572 on the function 119906(119909 119905) is shown in
Figure 3 This figure indicates a decrease in the fractionalorder 120572 by choosing the fixed 119909 = 5 that corresponds to anincrease in the function and also indicates a slow diffusion forthe values of 120572 = 1 and 120572 = 09 and a fast diffusion for thevalues of 120572 = 08 07 06 respectively It is clearly seen that119906(119909 119905) increase for 120572 = 1 09 08 07 06 with the increasesin 119905
Numerical comparison between GDTM [19] HPM [35]FVIM [36] and hybrid method are found in Table 2 whichshows hybrid method is more promising
It is also found that the result is in complete agreementwith the result of HPM [42 43] and ADM [44] for 120572 = 1
We investigate convergence criteria of our solutions fordifferent values of ℎ and 119899 To illustrate this we comparedour results with the analytical solution in case of 120572 = 1 Here119899 is order of differential transformation method and denotesthe number of terms to be calculated
In Figures 4 5 and 6 the difference between the resultsobtained in this study and the results of the analytical solutionis of the order of 10minus5 This is a pointer to the fact that there isconvergence and is a restatement in numerical terms of whatwas shown in the Appendix
Mathematical Problems in Engineering 5
10
075
05
025
00minus10
minus5
5
10
x
00
0005
01
t
(a)
10
075
05
025
00minus10
minus5
5
10
x
00
0005
01
t
(b)
Figure 2 Numerical solution for the time fractional Fisher equation with (a) 120572 rarr 099 and (b) 120572 rarr 095
0007
0006
0005
0004
0003
0002
0001
u(xt)
0 01 02 03 04
t
120572 = 06120572 = 07
120572 = 08
120572 = 09
120572 = 1
(a)
0007
0006
0005
0004
0003
0002
0001
000000 01 02 03 04
u
t
120572 = 710120572 = 810
120572 = 910
120572 = 1
(b)
Figure 3 Approximate solution for the time fractional Fisher equation with different 120572 values at 119909 = 5 (a) present and (b) [35]
Table 2 Comparison of numerical results between different methods for the time fractional Fisher equation GDTM generalized differentialtransformmethod Rida et al [19] HPM homotpy perturbation method Khan et al [35] and FVIM fractional variational iteration methodMerdan [36]
Table 3 Some values of 119880120572119894(119896) of Example 2
119894119896
0 1 2
0 05(0125 minus 025120583)
Γ (120572 + 1)
(0000011 minus 0000052120583)
Γ (2120572 + 1)
1 0517670(0124847 minus 0249687120583)
Γ (120572 + 1)
(minus0002179 + 0008772120583 minus 00088241205832)
Γ (2120572 + 1)
2 0535296(0124384 minus 0248754120583)
Γ (120572 + 1)
(minus0004379 + 0017511120583 minus 00175601205832)
Γ (2120572 + 1)
One important observation made from the computationis that when the number of mesh points was increased lessnumber of terms was required in the time series solution tohave convergence for a predetermined accuracy The hybridmethod of the present study gives faster convergence thanother traditional methods for example if we take ℎ = 002
(mesh point is 50) then the solution converges for 119899 = 3 Wenow consider another example
Example 2 The time fractional FitzHugh-Nagumo equationis
119863120572
119905119906 = 119863
2
119909119906 + 119906 (1 minus 119906) (119906 minus 120583) 120583 gt 0
0 lt 120572 le 1 119909 isin R 119905 gt 0
(16)
In this type of equation the nonlinear function depends on 120583and it is 119891(119906) = 119906(1 minus 119906)(119906 minus 120583) The initial condition is
119906 (119909 0) =1
(1 + 119890minus119909radic2)
(17)
Using the hybrid method on the above initial boundary valueproblem (IBVP) as done in the previous example we get
(119896) minus 2119880120572119894(119896) + 119880
120572(119894minus1)(119896)
ℎ2
minus 120583119880120572119894(119896) + (1 + 120583)
119896
sum
119897=0
119880120572119894(119896 minus 119897) 119880
120572119894(119897)
minus
119896
sum
119904=0
119904
sum
119897=0
119880120572119894(119896 minus 119904)119880
120572119894(119904 minus 119897) 119880
120572119894(119897)
119880120572119894(0) =
1
1 + 119890minus119894ℎradic2
(18)
Using second order finite difference method the boundaryvalues were obtained as follows
1198801205720(119896) = 3119880
1205721(119896) minus 3119880
1205722(119896) + 119880
1205723(119896)
119880120572119873
(119896) = 3119880120572(119873minus1)
(119896) minus 3119880120572(119873minus2)
(119896) + 119880120572(119873minus3)
(119896)
(19)
Table 3 presents some of the 119880120572119894(119896)rsquos
The time series solution for the above IBVP at differenttimes is
119906 (0 119905)
= 05 +(0125 minus 025120583)
Γ (120572 + 1)119905120572
+(0000011 minus 0000052120583)
Γ (2120572 + 1)1199052120572
+ sdot sdot sdot
119906 (01 119905)
= 0517670 +(0124847 minus 0249687120583)
Γ (120572 + 1)119905120572
+
(minus0002179 + 0008772120583 minus 00088241205832)
Γ (2120572 + 1)1199052120572
+ sdot sdot sdot
119906 (02 119905)
= 0535296 +(0124384 minus 0248754120583)
Γ (120572 + 1)119905120572
+
(minus0004379 + 001751120583 minus 00175601205832)
Γ (2120572 + 1)1199052120572
+ sdot sdot sdot
(20)
Numerical solutions for the time fractional FitzHugh-Nagumo equation with various 120572 values are shown in Figures7 and 8 A comparison of the results in a limiting case whereinan analytical solution exists is shown in Figure 7 The resultsare in close agreement with those of Rida et al [19] for thesame equation
For 120572 rarr 1 it can easily be seen that the exact solution ofFitzHugh-Nagumo equation is
Figure 9 is prepared to show the influence of 120572 on thefunction 119906(119909 119905) It is clearly seen that 119906(119909 119905) decrease for120572 = 1 095 085 075 065 with the decreases in 119905
As shown in the Table 4 our results show close agreementwith the exact solution and agree with those of Rida et al [19]
Mathematical Problems in Engineering 7
Table 4 Coefficients of 1 119905 1199052 for some 119894 values and comparisonwith exact and Ridarsquos solution 120583 = 07 [19]
Rida et al [19] Exact Present119894 = 0
Coef of 1199050 05 05 05Coef of 1199051 minus005 minus005 minus0049999Coef of 1199052 002 0 minus0000011
119894 = 1
Coef of 1199050 0517670 0517670 0517670Coef of 1199051 minus0049937 minus00499937 minus0049933Coef of 1199052 0020327 minus0000176 minus0000181
119894 = 2
Coef of 1199050 0535296 0535296 0535296Coef of 1199051 minus0049937 minus0049937 minus0049743Coef of 1199052 0020602 minus0000351 minus0000352
119894 = 3
Coef of 1199050 0552835 0552835 0552835Coef of 1199051 minus0049441 minus0049441 minus0049430Coef of 1199052 0020821 minus0000522 minus0000542
026
024
022
PresentAnalytic
020
018
0 002 004 006 008 010
t
Figure 4 Comparison of present results for ℎ = 02 and 119899 = 10withthe analytical solution in case of 120572 rarr 1 at 119909 = 04
From this table it is clear that the present work gives betterapproximation than GDTM as we increase 119899
Numerical comparison between GDTM FVIM andhybrid method is shown in Table 5 which indicates hybridmethod is more promising
4 Conclusion
Many real physical problems can be best modelled withfractional differential equations but the fact is when the
026
024
022
PresentAnalytic
020
018
0 002 004 006 008 010
t
Figure 5 Comparison of present results for ℎ = 01 and 119899 = 5 withthe analytical solution in case of 120572 rarr 1 at 119909 = 04
026
024
022
PresentAnalytic
020
018
0 002 004 006 008 010
t
Figure 6 Comparison of present results for ℎ = 002 and 119899 = 3withthe analytical solution in case of 120572 rarr 1 at 119909 = 04
equation is nonlinear there are very few reliable methodsThe numerical methods that can be used to solve frac-tional differential equations are known to have problems ofconvergence and stability These aspects are well addressedin the paper by suggesting a new procedure that uses acombination of the generalized differential transform andcentral difference methods The Appendix clearly spells out
8 Mathematical Problems in Engineering
10
075
05
025
0010
5
x
01
0
005
00
t
minus10
minus5
(a)
10
075
05
025
0010
minus10
5
minus5x 010
005
00
t
(b)
Figure 7 Numerical solution for the time fractional FitzHugh-Nagumo equation with 120572 rarr 1 (a) comparison with the analytical solution(b) with 120583 = 05
10
075
05
025
0010
minus10
5
minus5x01
0
005
00
t
(a)
10
075
05
025
0010
minus10
5
minus5x
01
0005
00
t
(b)
Figure 8 Numerical solution for the time fractional FitzHugh-Nagumo equation with 120572 = 095 (a) and 120572 = 099 (b)
the fact that the error as a result of discretization and compu-tation is bounded and hence implies stability of the methodLax equivalence theorem further implies convergence of thescheme Two time fractional nonlinear reaction-diffusionequations considered for illustration of the hybrid methodhighlight the usefulness of the method in obtaining thesolution of IBVPs involving time fractional derivatives Thecontrol of convergence through a judicious choice of time andspatial step sizes and also the number of terms in the timeseries solution spells assured convergence The segregationof the time domain from the spatial domain in the solution
method ensures the fact that problem of stability does notarise Diagonal dominance of the coefficient matrix in thesystem of linear algebraic equations resulting from the use ofthe central difference approximation in the Poisson equationensures the fact that the matrix remains nonsingular duringiterations and hence has assured convergence An appropriatecomputational decision on the number of terms to be takenin the time series solution results in a convergent solutionwith fast convergence Excellent comparison of the presentresults with the previous works on generalized differentialtransform method [19] and homotopy perturbation method
Mathematical Problems in Engineering 9
09716
09714
09712
09710
u(xt)
09708
09706
120572 = 095
120572 = 1
120572 = 085
120572 = 075
120572 = 06509704
0 01 02 03
t
(a)
120572 = 095
120572 = 085
120572 = 075
120572 = 065
120572 = 1
09716
09714
09712
0971
u(xt)
09708
09706
09704
0 01005 02015 03025
t
(b)
Figure 9 Approximate solution for the time fractional FitzHugh-Nagumo equation with different 120572 values at 119909 = 5 (a) present and (b) [36]
[35] and fractional variational iterationmethod [36] providesconfidence in the methodology adopted for the solution oftime fractional differential equations
Appendix
Estimation of Bounds on Truncation Error
Consider the fractional differential equations (9) and (16) ina general form as
119863120572
119905119906 = 119863
119909119909119906 + 119891 (119906) 119905 ge 0
119909 isin R (0 lt 120572 le 1)
(A1)
The differential transform of (A1) at the spatially discretizedpoints 119909
We now follow Jang et al [45] and move on to arrive at anestimate on the bounds for the truncation error in a generalway by considering the Taylor series expansion of119880
Table 5 Comparison of numerical results between differentmethods for the time fractional FitzHugh-Nagumo equationGDTM generalizeddifferential transform method Rida et al [19] and FVIM fractional variational iteration method Merdan [36]
We so far addressed the local error due to two differenttruncations in the time series In what follows we estimatethe bounds on the cumulative error that includes the errordiscussed above
Let 119910119894(119905119896) denote the solution of (A2) The local error in
119908119894(119905119896) relative to 119910
119894(119905119896) is
1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908119894(119905119896+ Δ119905)
1003816100381610038161003816
le1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
+1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
(A13)
Since 119908119894(119905119896+ Δ119905) is a better approximation than 119908
119894(119905119896+ Δ119905)
we may assume that1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816 ≪ 1 (A14)
Mathematical Problems in Engineering 11
In view of (A14) we now have1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816 ≐1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
(A15)
Using (A8) and noting that (Δ119905)119899+1 is quite small in (A15)we may take 120573 to be
120573 ≐
1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908119894(119905119896+ Δ119905)
1003816100381610038161003816
(Δ119905)119899+1
(A16)
Thus 120573(Δ119905)119899+1 = 120576 is the bound on the tolerance in the to-be-obtained solution When using different number of terms inthe Taylor series expansion earlier we denoted the solutionsusing a time stepΔ119905 by119908
119894(119905119896+Δ119905) and119908
119894(119905119896+Δ119905) respectively
The paper uses an adaptive step size in computing theresults This is because such a procedure succeeds in keepingthe error bounded and ensures convergence as a consequenceof Lax equivalence theorem To see what the adaptive stepsize produces and to show how such a procedure keepsthe error bounded we start with the premise that Δ119905 isthe most appropriate step size for the problem This stepsize is determined using the definition of inverse differentialtransform
119906119894(Δ119905) =
infin
sum
119896=0
119880120572119894(119896) (Δ119905)
119896 (A17)
In our actual calculation we will not be able to considerinfinite number of terms We consider ldquo119899rdquo terms in respectof 119908119894and ldquo119899 + 119898rdquo terms in respect of 119908
119894
Thus1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
=
119899+119898
sum
119895=0
119880120572119894(119895) (Δ119905)
119895minus
119899
sum
119895=0
119880120572119894(119895) (Δ119905)
119895
=
119899+119898
sum
119895=119899+1
119880120572119894(119895) (Δ119905)
119895
(A18)
To write down a simpler expression we change the summa-tion index from 119895 to 119901 = 119899 + 119895 So we have from (A18) thefollowing
1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908119894(119905119896+ Δ119905)
Thus the above proceedings tell us that if criterion (A23) issatisfied then the error is bounded In effect this means thatthe scheme is convergent in lieu of Lax equivalence theoremIn our computations Δ119905 has been always chosen to satisfyinequality (A23)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors are thankful to Ondokuz Mayıs UniversitySamsun Turkey for providing financial support to carryout this work under a major research project (Grant nopyofen190113003)
References
[1] W Hundsdorfer and J G Verwer Numerical Solution of TimeDependent Advection-Diffusion-Reaction Equations SpringerBerlin Germany 2003
[2] Y Kuramoto Chemical Oscillations Waves and TurbulenceDover Mineola NY USA 2003
[3] J DMurrayMathematical Biology II vol 18 of InterdisciplinaryApplied Mathematics Springer New York NY USA 3rd edi-tion 2003
[4] H Wilhelmsson and E Lazzaro Reaction-Diffusion Problemsin the Physics of Hot Plasmas Institute of Physics PublishingPhiladelphia Pa USA 2001
[5] M Bar N Gottschalk M Eiswirth and G Ertl ldquoSpiral waves ina surface reaction model calculationsrdquoThe Journal of ChemicalPhysics vol 100 no 2 pp 1202ndash1214 1994
[6] D Barkley ldquoA model for fast computer simulation of waves inexcitable mediardquo Physica D Nonlinear Phenomena vol 49 no1-2 pp 61ndash70 1991
[7] F H Fenton E M Cherry H M Hastings and S J EvansldquoMultiple mechanisms of spiral wave breakup in a model ofcardiac electrical activityrdquo Chaos vol 12 no 3 pp 852ndash8922002
[8] M GosakMMarhl andM Perc ldquoSpatial coherence resonancein excitable biochemical media induced by internal noiserdquoBiophysical Chemistry vol 128 no 2-3 pp 210ndash214 2007
[9] A Karma ldquoMeandering transition in two-dimensionalexcitable mediardquo Physical Review Letters vol 65 no 22 pp2824ndash2827 1990
12 Mathematical Problems in Engineering
[10] J P Keener ldquoA geometrical theory for spiral waves in excitablemediardquo SIAM Journal onAppliedMathematics vol 46 no 6 pp1039ndash1056 1986
[11] J P Keener Mathematical Physiology Interdisciplinary AppliedMathematics Springer New York NY USA 1998
[12] V M Kenkre and M N Kuperman ldquoApplicability of the Fisherequation to bacterial population dynamicsrdquo Physical Review Evol 67 no 5 Article ID 051921 5 pages 2003
[13] V Krinsky and A Pumir ldquoModels of defibrillation of cardiactissuerdquo Chaos vol 8 no 1 pp 188ndash203 1998
[14] N F Otani ldquoA primary mechanism for spiral wave meander-ingrdquo Chaos vol 12 no 3 pp 829ndash842 2002
[15] M Perc ldquoSpatial coherence resonance in excitable mediardquoPhysical Review E vol 72 no 3 Article ID 016207 2005
[16] M Perc ldquoStochastic resonance on excitable small-world net-works via a pacemakerrdquo Physical Review E vol 76 no 6 ArticleID 066203 2007
[17] M Perc ldquoEffects of small-world connectivity on noise-inducedtemporal and spatial order in neural mediardquo Chaos Solitons ampFractals vol 31 no 2 pp 280ndash291 2007
[18] J J Tyson ldquoWhat everyone should know about the Belousov-Zhabotinsky reactionrdquo in Frontiers inMathematical Biology vol100 of Lecture Notes in Biomathematics pp 569ndash587 SpringerBerlin Germany 1994
[19] S Z Rida A M El-Sayed and A A Arafa ldquoOn the solutions oftime-fractional reaction-diffusion equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 15 no 12pp 3847ndash3854 2010
[20] Y Zheng and Z Zhao ldquoA fully discrete Galerkin method fora nonlinear space-fractional diffusion equationrdquo MathematicalProblems in Engineering vol 2011 Article ID 171620 20 pages2011
[21] J K Zhou Differential Transformation and Its Applications forElectrical Circuits Huazhang University Press Wuhan China1986 (Chinese)
[22] I H Abdel-Halim Hassan ldquoComparison differential transfor-mation technique with Adomian decomposition method forlinear and nonlinear initial value problemsrdquoChaos Solitons andFractals vol 36 no 1 pp 53ndash65 2008
[23] A Arikoglu and I Ozkol ldquoSolution of boundary value problemsfor integro-differential equations by using differential transformmethodrdquoAppliedMathematics and Computation vol 168 no 2pp 1145ndash1158 2005
[24] F Ayaz ldquoSolutions of the system of differential equationsby differential transform methodrdquo Applied Mathematics andComputation vol 147 no 2 pp 547ndash567 2004
[25] N Bildik A Konuralp F Orak Bek and S KucukarslanldquoSolution of different type of the partial differential equationby differential transformmethod andAdomianrsquos decompositionmethodrdquo Applied Mathematics and Computation vol 172 no 1pp 551ndash567 2006
[26] H Liu and Y Song ldquoDifferential transform method applied tohigh index differential-algebraic equationsrdquoAppliedMathemat-ics and Computation vol 184 no 2 pp 748ndash753 2007
[27] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent Part IIrdquo Geophysical Journal of theRoyal Astronomical Society vol 13 no 5 pp 529ndash539 1967
[28] SMomani ZOdibat andV S Erturk ldquoGeneralized differentialtransform method for solving a space- and time-fractionaldiffusion-wave equationrdquo Physics Letters A vol 370 no 5-6 pp379ndash387 2007
[29] Z M Odibat C Bertelle M A Aziz-Alaoui and G HDuchamp ldquoA multi-step differential transform method andapplication to non-chaotic or chaotic systemsrdquo Computers ampMathematics with Applications vol 59 no 4 pp 1462ndash14722010
[30] N Laskin ldquoFractional Schrodinger Equationrdquo Physical ReviewE vol 66 no 5 Article ID 056108 2002
[31] G J Fix and J P Roop ldquoLeast squares finite-element solutionof a fractional order two-point boundary value problemrdquoComputers ampMathematics with Applications vol 48 no 7-8 pp1017ndash1033 2004
[32] L-T Yu and C-K Chen ldquoApplication of the hybrid methodto the transient thermal stresses response in isotropic annularfinsrdquo Journal of Applied Mechanics vol 66 no 2 pp 340ndash3471999
[33] B-L Kuo and C-K Chen ldquoApplication of a hybrid methodto the solution of the nonlinear burgersrsquo equationrdquo Journal ofApplied Mechanics Transactions ASME vol 70 no 6 pp 926ndash929 2003
[34] C K Chen H Y Lai and C C Liu ldquoApplication of hybrid dif-ferential transformationfinite difference method to nonlinearanalysis of micro fixed-fixed beamrdquo Microsystem Technologiesvol 15 no 6 pp 813ndash820 2009
[35] N A Khan M Ayaz L Jin and A Yildirim ldquoOn approximatesolutions for the time-fractional reaction-diffusion equation ofFisher typerdquo International Journal of Physical Sciences vol 6 no10 pp 2483ndash2496 2011
[36] M Merdan ldquoSolutions of time-fractional reaction-diffusionequation withmodified Riemann-Liouville derivativerdquo Interna-tional Journal of Physical Sciences vol 7 no 15 pp 2317ndash23262012
[37] C Li and G Peng ldquoChaos in Chenrsquos system with a fractionalorderrdquo Chaos Solitons amp Fractals vol 22 no 2 pp 443ndash4502004
[38] Y Luchko and R Gorenflo ldquoAn operational method for solvingfractional differential equations with the Caputo derivativesrdquoActa Mathematica Vietnamica vol 24 no 2 pp 207ndash233 1999
[39] Z Odibat and S Momani ldquoA generalized differential transformmethod for linear partial differential equations of fractionalorderrdquo Applied Mathematics Letters vol 21 no 2 pp 194ndash1992008
[40] Z M Odibat and N T Shawagfeh ldquoGeneralized Taylorrsquosformulardquo Applied Mathematics and Computation vol 186 no1 pp 286ndash293 2007
[41] D Nazari and S Shahmorad ldquoApplication of the fractionaldifferential transform method to fractional-order integro-differential equations with nonlocal boundary conditionsrdquoJournal of Computational andAppliedMathematics vol 234 no3 pp 883ndash891 2010
[42] D Agırseven and T Ozis ldquoAn analytical study for Fisher typeequations by using homotopy perturbationmethodrdquoComputersamp Mathematics with Applications vol 60 no 3 pp 602ndash6092010
[43] M Dehghan J M Heris and A Saadatmandi ldquoApplicationof semi-analytic methods for the Fitzhugh-Nagumo equation
Mathematical Problems in Engineering 13
which models the transmission of nerve impulsesrdquoMathemati-calMethods in the Applied Sciences vol 33 no 11 pp 1384ndash13982010
[44] A-M Wazwaz and A Gorguis ldquoAn analytic study of Fisherrsquosequation by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 154 no 3 pp 609ndash6202004
[45] M-J Jang C-L Chen and Y-C Liy ldquoOn solving the initial-value problems using the differential transformation methodrdquoAppliedMathematics andComputation vol 115 no 2-3 pp 145ndash160 2000
Figure 2 Numerical solution for the time fractional Fisher equation with (a) 120572 rarr 099 and (b) 120572 rarr 095
0007
0006
0005
0004
0003
0002
0001
u(xt)
0 01 02 03 04
t
120572 = 06120572 = 07
120572 = 08
120572 = 09
120572 = 1
(a)
0007
0006
0005
0004
0003
0002
0001
000000 01 02 03 04
u
t
120572 = 710120572 = 810
120572 = 910
120572 = 1
(b)
Figure 3 Approximate solution for the time fractional Fisher equation with different 120572 values at 119909 = 5 (a) present and (b) [35]
Table 2 Comparison of numerical results between different methods for the time fractional Fisher equation GDTM generalized differentialtransformmethod Rida et al [19] HPM homotpy perturbation method Khan et al [35] and FVIM fractional variational iteration methodMerdan [36]
Table 3 Some values of 119880120572119894(119896) of Example 2
119894119896
0 1 2
0 05(0125 minus 025120583)
Γ (120572 + 1)
(0000011 minus 0000052120583)
Γ (2120572 + 1)
1 0517670(0124847 minus 0249687120583)
Γ (120572 + 1)
(minus0002179 + 0008772120583 minus 00088241205832)
Γ (2120572 + 1)
2 0535296(0124384 minus 0248754120583)
Γ (120572 + 1)
(minus0004379 + 0017511120583 minus 00175601205832)
Γ (2120572 + 1)
One important observation made from the computationis that when the number of mesh points was increased lessnumber of terms was required in the time series solution tohave convergence for a predetermined accuracy The hybridmethod of the present study gives faster convergence thanother traditional methods for example if we take ℎ = 002
(mesh point is 50) then the solution converges for 119899 = 3 Wenow consider another example
Example 2 The time fractional FitzHugh-Nagumo equationis
119863120572
119905119906 = 119863
2
119909119906 + 119906 (1 minus 119906) (119906 minus 120583) 120583 gt 0
0 lt 120572 le 1 119909 isin R 119905 gt 0
(16)
In this type of equation the nonlinear function depends on 120583and it is 119891(119906) = 119906(1 minus 119906)(119906 minus 120583) The initial condition is
119906 (119909 0) =1
(1 + 119890minus119909radic2)
(17)
Using the hybrid method on the above initial boundary valueproblem (IBVP) as done in the previous example we get
(119896) minus 2119880120572119894(119896) + 119880
120572(119894minus1)(119896)
ℎ2
minus 120583119880120572119894(119896) + (1 + 120583)
119896
sum
119897=0
119880120572119894(119896 minus 119897) 119880
120572119894(119897)
minus
119896
sum
119904=0
119904
sum
119897=0
119880120572119894(119896 minus 119904)119880
120572119894(119904 minus 119897) 119880
120572119894(119897)
119880120572119894(0) =
1
1 + 119890minus119894ℎradic2
(18)
Using second order finite difference method the boundaryvalues were obtained as follows
1198801205720(119896) = 3119880
1205721(119896) minus 3119880
1205722(119896) + 119880
1205723(119896)
119880120572119873
(119896) = 3119880120572(119873minus1)
(119896) minus 3119880120572(119873minus2)
(119896) + 119880120572(119873minus3)
(119896)
(19)
Table 3 presents some of the 119880120572119894(119896)rsquos
The time series solution for the above IBVP at differenttimes is
119906 (0 119905)
= 05 +(0125 minus 025120583)
Γ (120572 + 1)119905120572
+(0000011 minus 0000052120583)
Γ (2120572 + 1)1199052120572
+ sdot sdot sdot
119906 (01 119905)
= 0517670 +(0124847 minus 0249687120583)
Γ (120572 + 1)119905120572
+
(minus0002179 + 0008772120583 minus 00088241205832)
Γ (2120572 + 1)1199052120572
+ sdot sdot sdot
119906 (02 119905)
= 0535296 +(0124384 minus 0248754120583)
Γ (120572 + 1)119905120572
+
(minus0004379 + 001751120583 minus 00175601205832)
Γ (2120572 + 1)1199052120572
+ sdot sdot sdot
(20)
Numerical solutions for the time fractional FitzHugh-Nagumo equation with various 120572 values are shown in Figures7 and 8 A comparison of the results in a limiting case whereinan analytical solution exists is shown in Figure 7 The resultsare in close agreement with those of Rida et al [19] for thesame equation
For 120572 rarr 1 it can easily be seen that the exact solution ofFitzHugh-Nagumo equation is
Figure 9 is prepared to show the influence of 120572 on thefunction 119906(119909 119905) It is clearly seen that 119906(119909 119905) decrease for120572 = 1 095 085 075 065 with the decreases in 119905
As shown in the Table 4 our results show close agreementwith the exact solution and agree with those of Rida et al [19]
Mathematical Problems in Engineering 7
Table 4 Coefficients of 1 119905 1199052 for some 119894 values and comparisonwith exact and Ridarsquos solution 120583 = 07 [19]
Rida et al [19] Exact Present119894 = 0
Coef of 1199050 05 05 05Coef of 1199051 minus005 minus005 minus0049999Coef of 1199052 002 0 minus0000011
119894 = 1
Coef of 1199050 0517670 0517670 0517670Coef of 1199051 minus0049937 minus00499937 minus0049933Coef of 1199052 0020327 minus0000176 minus0000181
119894 = 2
Coef of 1199050 0535296 0535296 0535296Coef of 1199051 minus0049937 minus0049937 minus0049743Coef of 1199052 0020602 minus0000351 minus0000352
119894 = 3
Coef of 1199050 0552835 0552835 0552835Coef of 1199051 minus0049441 minus0049441 minus0049430Coef of 1199052 0020821 minus0000522 minus0000542
026
024
022
PresentAnalytic
020
018
0 002 004 006 008 010
t
Figure 4 Comparison of present results for ℎ = 02 and 119899 = 10withthe analytical solution in case of 120572 rarr 1 at 119909 = 04
From this table it is clear that the present work gives betterapproximation than GDTM as we increase 119899
Numerical comparison between GDTM FVIM andhybrid method is shown in Table 5 which indicates hybridmethod is more promising
4 Conclusion
Many real physical problems can be best modelled withfractional differential equations but the fact is when the
026
024
022
PresentAnalytic
020
018
0 002 004 006 008 010
t
Figure 5 Comparison of present results for ℎ = 01 and 119899 = 5 withthe analytical solution in case of 120572 rarr 1 at 119909 = 04
026
024
022
PresentAnalytic
020
018
0 002 004 006 008 010
t
Figure 6 Comparison of present results for ℎ = 002 and 119899 = 3withthe analytical solution in case of 120572 rarr 1 at 119909 = 04
equation is nonlinear there are very few reliable methodsThe numerical methods that can be used to solve frac-tional differential equations are known to have problems ofconvergence and stability These aspects are well addressedin the paper by suggesting a new procedure that uses acombination of the generalized differential transform andcentral difference methods The Appendix clearly spells out
8 Mathematical Problems in Engineering
10
075
05
025
0010
5
x
01
0
005
00
t
minus10
minus5
(a)
10
075
05
025
0010
minus10
5
minus5x 010
005
00
t
(b)
Figure 7 Numerical solution for the time fractional FitzHugh-Nagumo equation with 120572 rarr 1 (a) comparison with the analytical solution(b) with 120583 = 05
10
075
05
025
0010
minus10
5
minus5x01
0
005
00
t
(a)
10
075
05
025
0010
minus10
5
minus5x
01
0005
00
t
(b)
Figure 8 Numerical solution for the time fractional FitzHugh-Nagumo equation with 120572 = 095 (a) and 120572 = 099 (b)
the fact that the error as a result of discretization and compu-tation is bounded and hence implies stability of the methodLax equivalence theorem further implies convergence of thescheme Two time fractional nonlinear reaction-diffusionequations considered for illustration of the hybrid methodhighlight the usefulness of the method in obtaining thesolution of IBVPs involving time fractional derivatives Thecontrol of convergence through a judicious choice of time andspatial step sizes and also the number of terms in the timeseries solution spells assured convergence The segregationof the time domain from the spatial domain in the solution
method ensures the fact that problem of stability does notarise Diagonal dominance of the coefficient matrix in thesystem of linear algebraic equations resulting from the use ofthe central difference approximation in the Poisson equationensures the fact that the matrix remains nonsingular duringiterations and hence has assured convergence An appropriatecomputational decision on the number of terms to be takenin the time series solution results in a convergent solutionwith fast convergence Excellent comparison of the presentresults with the previous works on generalized differentialtransform method [19] and homotopy perturbation method
Mathematical Problems in Engineering 9
09716
09714
09712
09710
u(xt)
09708
09706
120572 = 095
120572 = 1
120572 = 085
120572 = 075
120572 = 06509704
0 01 02 03
t
(a)
120572 = 095
120572 = 085
120572 = 075
120572 = 065
120572 = 1
09716
09714
09712
0971
u(xt)
09708
09706
09704
0 01005 02015 03025
t
(b)
Figure 9 Approximate solution for the time fractional FitzHugh-Nagumo equation with different 120572 values at 119909 = 5 (a) present and (b) [36]
[35] and fractional variational iterationmethod [36] providesconfidence in the methodology adopted for the solution oftime fractional differential equations
Appendix
Estimation of Bounds on Truncation Error
Consider the fractional differential equations (9) and (16) ina general form as
119863120572
119905119906 = 119863
119909119909119906 + 119891 (119906) 119905 ge 0
119909 isin R (0 lt 120572 le 1)
(A1)
The differential transform of (A1) at the spatially discretizedpoints 119909
We now follow Jang et al [45] and move on to arrive at anestimate on the bounds for the truncation error in a generalway by considering the Taylor series expansion of119880
Table 5 Comparison of numerical results between differentmethods for the time fractional FitzHugh-Nagumo equationGDTM generalizeddifferential transform method Rida et al [19] and FVIM fractional variational iteration method Merdan [36]
We so far addressed the local error due to two differenttruncations in the time series In what follows we estimatethe bounds on the cumulative error that includes the errordiscussed above
Let 119910119894(119905119896) denote the solution of (A2) The local error in
119908119894(119905119896) relative to 119910
119894(119905119896) is
1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908119894(119905119896+ Δ119905)
1003816100381610038161003816
le1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
+1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
(A13)
Since 119908119894(119905119896+ Δ119905) is a better approximation than 119908
119894(119905119896+ Δ119905)
we may assume that1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816 ≪ 1 (A14)
Mathematical Problems in Engineering 11
In view of (A14) we now have1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816 ≐1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
(A15)
Using (A8) and noting that (Δ119905)119899+1 is quite small in (A15)we may take 120573 to be
120573 ≐
1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908119894(119905119896+ Δ119905)
1003816100381610038161003816
(Δ119905)119899+1
(A16)
Thus 120573(Δ119905)119899+1 = 120576 is the bound on the tolerance in the to-be-obtained solution When using different number of terms inthe Taylor series expansion earlier we denoted the solutionsusing a time stepΔ119905 by119908
119894(119905119896+Δ119905) and119908
119894(119905119896+Δ119905) respectively
The paper uses an adaptive step size in computing theresults This is because such a procedure succeeds in keepingthe error bounded and ensures convergence as a consequenceof Lax equivalence theorem To see what the adaptive stepsize produces and to show how such a procedure keepsthe error bounded we start with the premise that Δ119905 isthe most appropriate step size for the problem This stepsize is determined using the definition of inverse differentialtransform
119906119894(Δ119905) =
infin
sum
119896=0
119880120572119894(119896) (Δ119905)
119896 (A17)
In our actual calculation we will not be able to considerinfinite number of terms We consider ldquo119899rdquo terms in respectof 119908119894and ldquo119899 + 119898rdquo terms in respect of 119908
119894
Thus1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
=
119899+119898
sum
119895=0
119880120572119894(119895) (Δ119905)
119895minus
119899
sum
119895=0
119880120572119894(119895) (Δ119905)
119895
=
119899+119898
sum
119895=119899+1
119880120572119894(119895) (Δ119905)
119895
(A18)
To write down a simpler expression we change the summa-tion index from 119895 to 119901 = 119899 + 119895 So we have from (A18) thefollowing
1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908119894(119905119896+ Δ119905)
Thus the above proceedings tell us that if criterion (A23) issatisfied then the error is bounded In effect this means thatthe scheme is convergent in lieu of Lax equivalence theoremIn our computations Δ119905 has been always chosen to satisfyinequality (A23)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors are thankful to Ondokuz Mayıs UniversitySamsun Turkey for providing financial support to carryout this work under a major research project (Grant nopyofen190113003)
References
[1] W Hundsdorfer and J G Verwer Numerical Solution of TimeDependent Advection-Diffusion-Reaction Equations SpringerBerlin Germany 2003
[2] Y Kuramoto Chemical Oscillations Waves and TurbulenceDover Mineola NY USA 2003
[3] J DMurrayMathematical Biology II vol 18 of InterdisciplinaryApplied Mathematics Springer New York NY USA 3rd edi-tion 2003
[4] H Wilhelmsson and E Lazzaro Reaction-Diffusion Problemsin the Physics of Hot Plasmas Institute of Physics PublishingPhiladelphia Pa USA 2001
[5] M Bar N Gottschalk M Eiswirth and G Ertl ldquoSpiral waves ina surface reaction model calculationsrdquoThe Journal of ChemicalPhysics vol 100 no 2 pp 1202ndash1214 1994
[6] D Barkley ldquoA model for fast computer simulation of waves inexcitable mediardquo Physica D Nonlinear Phenomena vol 49 no1-2 pp 61ndash70 1991
[7] F H Fenton E M Cherry H M Hastings and S J EvansldquoMultiple mechanisms of spiral wave breakup in a model ofcardiac electrical activityrdquo Chaos vol 12 no 3 pp 852ndash8922002
[8] M GosakMMarhl andM Perc ldquoSpatial coherence resonancein excitable biochemical media induced by internal noiserdquoBiophysical Chemistry vol 128 no 2-3 pp 210ndash214 2007
[9] A Karma ldquoMeandering transition in two-dimensionalexcitable mediardquo Physical Review Letters vol 65 no 22 pp2824ndash2827 1990
12 Mathematical Problems in Engineering
[10] J P Keener ldquoA geometrical theory for spiral waves in excitablemediardquo SIAM Journal onAppliedMathematics vol 46 no 6 pp1039ndash1056 1986
[11] J P Keener Mathematical Physiology Interdisciplinary AppliedMathematics Springer New York NY USA 1998
[12] V M Kenkre and M N Kuperman ldquoApplicability of the Fisherequation to bacterial population dynamicsrdquo Physical Review Evol 67 no 5 Article ID 051921 5 pages 2003
[13] V Krinsky and A Pumir ldquoModels of defibrillation of cardiactissuerdquo Chaos vol 8 no 1 pp 188ndash203 1998
[14] N F Otani ldquoA primary mechanism for spiral wave meander-ingrdquo Chaos vol 12 no 3 pp 829ndash842 2002
[15] M Perc ldquoSpatial coherence resonance in excitable mediardquoPhysical Review E vol 72 no 3 Article ID 016207 2005
[16] M Perc ldquoStochastic resonance on excitable small-world net-works via a pacemakerrdquo Physical Review E vol 76 no 6 ArticleID 066203 2007
[17] M Perc ldquoEffects of small-world connectivity on noise-inducedtemporal and spatial order in neural mediardquo Chaos Solitons ampFractals vol 31 no 2 pp 280ndash291 2007
[18] J J Tyson ldquoWhat everyone should know about the Belousov-Zhabotinsky reactionrdquo in Frontiers inMathematical Biology vol100 of Lecture Notes in Biomathematics pp 569ndash587 SpringerBerlin Germany 1994
[19] S Z Rida A M El-Sayed and A A Arafa ldquoOn the solutions oftime-fractional reaction-diffusion equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 15 no 12pp 3847ndash3854 2010
[20] Y Zheng and Z Zhao ldquoA fully discrete Galerkin method fora nonlinear space-fractional diffusion equationrdquo MathematicalProblems in Engineering vol 2011 Article ID 171620 20 pages2011
[21] J K Zhou Differential Transformation and Its Applications forElectrical Circuits Huazhang University Press Wuhan China1986 (Chinese)
[22] I H Abdel-Halim Hassan ldquoComparison differential transfor-mation technique with Adomian decomposition method forlinear and nonlinear initial value problemsrdquoChaos Solitons andFractals vol 36 no 1 pp 53ndash65 2008
[23] A Arikoglu and I Ozkol ldquoSolution of boundary value problemsfor integro-differential equations by using differential transformmethodrdquoAppliedMathematics and Computation vol 168 no 2pp 1145ndash1158 2005
[24] F Ayaz ldquoSolutions of the system of differential equationsby differential transform methodrdquo Applied Mathematics andComputation vol 147 no 2 pp 547ndash567 2004
[25] N Bildik A Konuralp F Orak Bek and S KucukarslanldquoSolution of different type of the partial differential equationby differential transformmethod andAdomianrsquos decompositionmethodrdquo Applied Mathematics and Computation vol 172 no 1pp 551ndash567 2006
[26] H Liu and Y Song ldquoDifferential transform method applied tohigh index differential-algebraic equationsrdquoAppliedMathemat-ics and Computation vol 184 no 2 pp 748ndash753 2007
[27] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent Part IIrdquo Geophysical Journal of theRoyal Astronomical Society vol 13 no 5 pp 529ndash539 1967
[28] SMomani ZOdibat andV S Erturk ldquoGeneralized differentialtransform method for solving a space- and time-fractionaldiffusion-wave equationrdquo Physics Letters A vol 370 no 5-6 pp379ndash387 2007
[29] Z M Odibat C Bertelle M A Aziz-Alaoui and G HDuchamp ldquoA multi-step differential transform method andapplication to non-chaotic or chaotic systemsrdquo Computers ampMathematics with Applications vol 59 no 4 pp 1462ndash14722010
[30] N Laskin ldquoFractional Schrodinger Equationrdquo Physical ReviewE vol 66 no 5 Article ID 056108 2002
[31] G J Fix and J P Roop ldquoLeast squares finite-element solutionof a fractional order two-point boundary value problemrdquoComputers ampMathematics with Applications vol 48 no 7-8 pp1017ndash1033 2004
[32] L-T Yu and C-K Chen ldquoApplication of the hybrid methodto the transient thermal stresses response in isotropic annularfinsrdquo Journal of Applied Mechanics vol 66 no 2 pp 340ndash3471999
[33] B-L Kuo and C-K Chen ldquoApplication of a hybrid methodto the solution of the nonlinear burgersrsquo equationrdquo Journal ofApplied Mechanics Transactions ASME vol 70 no 6 pp 926ndash929 2003
[34] C K Chen H Y Lai and C C Liu ldquoApplication of hybrid dif-ferential transformationfinite difference method to nonlinearanalysis of micro fixed-fixed beamrdquo Microsystem Technologiesvol 15 no 6 pp 813ndash820 2009
[35] N A Khan M Ayaz L Jin and A Yildirim ldquoOn approximatesolutions for the time-fractional reaction-diffusion equation ofFisher typerdquo International Journal of Physical Sciences vol 6 no10 pp 2483ndash2496 2011
[36] M Merdan ldquoSolutions of time-fractional reaction-diffusionequation withmodified Riemann-Liouville derivativerdquo Interna-tional Journal of Physical Sciences vol 7 no 15 pp 2317ndash23262012
[37] C Li and G Peng ldquoChaos in Chenrsquos system with a fractionalorderrdquo Chaos Solitons amp Fractals vol 22 no 2 pp 443ndash4502004
[38] Y Luchko and R Gorenflo ldquoAn operational method for solvingfractional differential equations with the Caputo derivativesrdquoActa Mathematica Vietnamica vol 24 no 2 pp 207ndash233 1999
[39] Z Odibat and S Momani ldquoA generalized differential transformmethod for linear partial differential equations of fractionalorderrdquo Applied Mathematics Letters vol 21 no 2 pp 194ndash1992008
[40] Z M Odibat and N T Shawagfeh ldquoGeneralized Taylorrsquosformulardquo Applied Mathematics and Computation vol 186 no1 pp 286ndash293 2007
[41] D Nazari and S Shahmorad ldquoApplication of the fractionaldifferential transform method to fractional-order integro-differential equations with nonlocal boundary conditionsrdquoJournal of Computational andAppliedMathematics vol 234 no3 pp 883ndash891 2010
[42] D Agırseven and T Ozis ldquoAn analytical study for Fisher typeequations by using homotopy perturbationmethodrdquoComputersamp Mathematics with Applications vol 60 no 3 pp 602ndash6092010
[43] M Dehghan J M Heris and A Saadatmandi ldquoApplicationof semi-analytic methods for the Fitzhugh-Nagumo equation
Mathematical Problems in Engineering 13
which models the transmission of nerve impulsesrdquoMathemati-calMethods in the Applied Sciences vol 33 no 11 pp 1384ndash13982010
[44] A-M Wazwaz and A Gorguis ldquoAn analytic study of Fisherrsquosequation by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 154 no 3 pp 609ndash6202004
[45] M-J Jang C-L Chen and Y-C Liy ldquoOn solving the initial-value problems using the differential transformation methodrdquoAppliedMathematics andComputation vol 115 no 2-3 pp 145ndash160 2000
Table 3 Some values of 119880120572119894(119896) of Example 2
119894119896
0 1 2
0 05(0125 minus 025120583)
Γ (120572 + 1)
(0000011 minus 0000052120583)
Γ (2120572 + 1)
1 0517670(0124847 minus 0249687120583)
Γ (120572 + 1)
(minus0002179 + 0008772120583 minus 00088241205832)
Γ (2120572 + 1)
2 0535296(0124384 minus 0248754120583)
Γ (120572 + 1)
(minus0004379 + 0017511120583 minus 00175601205832)
Γ (2120572 + 1)
One important observation made from the computationis that when the number of mesh points was increased lessnumber of terms was required in the time series solution tohave convergence for a predetermined accuracy The hybridmethod of the present study gives faster convergence thanother traditional methods for example if we take ℎ = 002
(mesh point is 50) then the solution converges for 119899 = 3 Wenow consider another example
Example 2 The time fractional FitzHugh-Nagumo equationis
119863120572
119905119906 = 119863
2
119909119906 + 119906 (1 minus 119906) (119906 minus 120583) 120583 gt 0
0 lt 120572 le 1 119909 isin R 119905 gt 0
(16)
In this type of equation the nonlinear function depends on 120583and it is 119891(119906) = 119906(1 minus 119906)(119906 minus 120583) The initial condition is
119906 (119909 0) =1
(1 + 119890minus119909radic2)
(17)
Using the hybrid method on the above initial boundary valueproblem (IBVP) as done in the previous example we get
(119896) minus 2119880120572119894(119896) + 119880
120572(119894minus1)(119896)
ℎ2
minus 120583119880120572119894(119896) + (1 + 120583)
119896
sum
119897=0
119880120572119894(119896 minus 119897) 119880
120572119894(119897)
minus
119896
sum
119904=0
119904
sum
119897=0
119880120572119894(119896 minus 119904)119880
120572119894(119904 minus 119897) 119880
120572119894(119897)
119880120572119894(0) =
1
1 + 119890minus119894ℎradic2
(18)
Using second order finite difference method the boundaryvalues were obtained as follows
1198801205720(119896) = 3119880
1205721(119896) minus 3119880
1205722(119896) + 119880
1205723(119896)
119880120572119873
(119896) = 3119880120572(119873minus1)
(119896) minus 3119880120572(119873minus2)
(119896) + 119880120572(119873minus3)
(119896)
(19)
Table 3 presents some of the 119880120572119894(119896)rsquos
The time series solution for the above IBVP at differenttimes is
119906 (0 119905)
= 05 +(0125 minus 025120583)
Γ (120572 + 1)119905120572
+(0000011 minus 0000052120583)
Γ (2120572 + 1)1199052120572
+ sdot sdot sdot
119906 (01 119905)
= 0517670 +(0124847 minus 0249687120583)
Γ (120572 + 1)119905120572
+
(minus0002179 + 0008772120583 minus 00088241205832)
Γ (2120572 + 1)1199052120572
+ sdot sdot sdot
119906 (02 119905)
= 0535296 +(0124384 minus 0248754120583)
Γ (120572 + 1)119905120572
+
(minus0004379 + 001751120583 minus 00175601205832)
Γ (2120572 + 1)1199052120572
+ sdot sdot sdot
(20)
Numerical solutions for the time fractional FitzHugh-Nagumo equation with various 120572 values are shown in Figures7 and 8 A comparison of the results in a limiting case whereinan analytical solution exists is shown in Figure 7 The resultsare in close agreement with those of Rida et al [19] for thesame equation
For 120572 rarr 1 it can easily be seen that the exact solution ofFitzHugh-Nagumo equation is
Figure 9 is prepared to show the influence of 120572 on thefunction 119906(119909 119905) It is clearly seen that 119906(119909 119905) decrease for120572 = 1 095 085 075 065 with the decreases in 119905
As shown in the Table 4 our results show close agreementwith the exact solution and agree with those of Rida et al [19]
Mathematical Problems in Engineering 7
Table 4 Coefficients of 1 119905 1199052 for some 119894 values and comparisonwith exact and Ridarsquos solution 120583 = 07 [19]
Rida et al [19] Exact Present119894 = 0
Coef of 1199050 05 05 05Coef of 1199051 minus005 minus005 minus0049999Coef of 1199052 002 0 minus0000011
119894 = 1
Coef of 1199050 0517670 0517670 0517670Coef of 1199051 minus0049937 minus00499937 minus0049933Coef of 1199052 0020327 minus0000176 minus0000181
119894 = 2
Coef of 1199050 0535296 0535296 0535296Coef of 1199051 minus0049937 minus0049937 minus0049743Coef of 1199052 0020602 minus0000351 minus0000352
119894 = 3
Coef of 1199050 0552835 0552835 0552835Coef of 1199051 minus0049441 minus0049441 minus0049430Coef of 1199052 0020821 minus0000522 minus0000542
026
024
022
PresentAnalytic
020
018
0 002 004 006 008 010
t
Figure 4 Comparison of present results for ℎ = 02 and 119899 = 10withthe analytical solution in case of 120572 rarr 1 at 119909 = 04
From this table it is clear that the present work gives betterapproximation than GDTM as we increase 119899
Numerical comparison between GDTM FVIM andhybrid method is shown in Table 5 which indicates hybridmethod is more promising
4 Conclusion
Many real physical problems can be best modelled withfractional differential equations but the fact is when the
026
024
022
PresentAnalytic
020
018
0 002 004 006 008 010
t
Figure 5 Comparison of present results for ℎ = 01 and 119899 = 5 withthe analytical solution in case of 120572 rarr 1 at 119909 = 04
026
024
022
PresentAnalytic
020
018
0 002 004 006 008 010
t
Figure 6 Comparison of present results for ℎ = 002 and 119899 = 3withthe analytical solution in case of 120572 rarr 1 at 119909 = 04
equation is nonlinear there are very few reliable methodsThe numerical methods that can be used to solve frac-tional differential equations are known to have problems ofconvergence and stability These aspects are well addressedin the paper by suggesting a new procedure that uses acombination of the generalized differential transform andcentral difference methods The Appendix clearly spells out
8 Mathematical Problems in Engineering
10
075
05
025
0010
5
x
01
0
005
00
t
minus10
minus5
(a)
10
075
05
025
0010
minus10
5
minus5x 010
005
00
t
(b)
Figure 7 Numerical solution for the time fractional FitzHugh-Nagumo equation with 120572 rarr 1 (a) comparison with the analytical solution(b) with 120583 = 05
10
075
05
025
0010
minus10
5
minus5x01
0
005
00
t
(a)
10
075
05
025
0010
minus10
5
minus5x
01
0005
00
t
(b)
Figure 8 Numerical solution for the time fractional FitzHugh-Nagumo equation with 120572 = 095 (a) and 120572 = 099 (b)
the fact that the error as a result of discretization and compu-tation is bounded and hence implies stability of the methodLax equivalence theorem further implies convergence of thescheme Two time fractional nonlinear reaction-diffusionequations considered for illustration of the hybrid methodhighlight the usefulness of the method in obtaining thesolution of IBVPs involving time fractional derivatives Thecontrol of convergence through a judicious choice of time andspatial step sizes and also the number of terms in the timeseries solution spells assured convergence The segregationof the time domain from the spatial domain in the solution
method ensures the fact that problem of stability does notarise Diagonal dominance of the coefficient matrix in thesystem of linear algebraic equations resulting from the use ofthe central difference approximation in the Poisson equationensures the fact that the matrix remains nonsingular duringiterations and hence has assured convergence An appropriatecomputational decision on the number of terms to be takenin the time series solution results in a convergent solutionwith fast convergence Excellent comparison of the presentresults with the previous works on generalized differentialtransform method [19] and homotopy perturbation method
Mathematical Problems in Engineering 9
09716
09714
09712
09710
u(xt)
09708
09706
120572 = 095
120572 = 1
120572 = 085
120572 = 075
120572 = 06509704
0 01 02 03
t
(a)
120572 = 095
120572 = 085
120572 = 075
120572 = 065
120572 = 1
09716
09714
09712
0971
u(xt)
09708
09706
09704
0 01005 02015 03025
t
(b)
Figure 9 Approximate solution for the time fractional FitzHugh-Nagumo equation with different 120572 values at 119909 = 5 (a) present and (b) [36]
[35] and fractional variational iterationmethod [36] providesconfidence in the methodology adopted for the solution oftime fractional differential equations
Appendix
Estimation of Bounds on Truncation Error
Consider the fractional differential equations (9) and (16) ina general form as
119863120572
119905119906 = 119863
119909119909119906 + 119891 (119906) 119905 ge 0
119909 isin R (0 lt 120572 le 1)
(A1)
The differential transform of (A1) at the spatially discretizedpoints 119909
We now follow Jang et al [45] and move on to arrive at anestimate on the bounds for the truncation error in a generalway by considering the Taylor series expansion of119880
Table 5 Comparison of numerical results between differentmethods for the time fractional FitzHugh-Nagumo equationGDTM generalizeddifferential transform method Rida et al [19] and FVIM fractional variational iteration method Merdan [36]
We so far addressed the local error due to two differenttruncations in the time series In what follows we estimatethe bounds on the cumulative error that includes the errordiscussed above
Let 119910119894(119905119896) denote the solution of (A2) The local error in
119908119894(119905119896) relative to 119910
119894(119905119896) is
1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908119894(119905119896+ Δ119905)
1003816100381610038161003816
le1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
+1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
(A13)
Since 119908119894(119905119896+ Δ119905) is a better approximation than 119908
119894(119905119896+ Δ119905)
we may assume that1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816 ≪ 1 (A14)
Mathematical Problems in Engineering 11
In view of (A14) we now have1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816 ≐1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
(A15)
Using (A8) and noting that (Δ119905)119899+1 is quite small in (A15)we may take 120573 to be
120573 ≐
1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908119894(119905119896+ Δ119905)
1003816100381610038161003816
(Δ119905)119899+1
(A16)
Thus 120573(Δ119905)119899+1 = 120576 is the bound on the tolerance in the to-be-obtained solution When using different number of terms inthe Taylor series expansion earlier we denoted the solutionsusing a time stepΔ119905 by119908
119894(119905119896+Δ119905) and119908
119894(119905119896+Δ119905) respectively
The paper uses an adaptive step size in computing theresults This is because such a procedure succeeds in keepingthe error bounded and ensures convergence as a consequenceof Lax equivalence theorem To see what the adaptive stepsize produces and to show how such a procedure keepsthe error bounded we start with the premise that Δ119905 isthe most appropriate step size for the problem This stepsize is determined using the definition of inverse differentialtransform
119906119894(Δ119905) =
infin
sum
119896=0
119880120572119894(119896) (Δ119905)
119896 (A17)
In our actual calculation we will not be able to considerinfinite number of terms We consider ldquo119899rdquo terms in respectof 119908119894and ldquo119899 + 119898rdquo terms in respect of 119908
119894
Thus1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
=
119899+119898
sum
119895=0
119880120572119894(119895) (Δ119905)
119895minus
119899
sum
119895=0
119880120572119894(119895) (Δ119905)
119895
=
119899+119898
sum
119895=119899+1
119880120572119894(119895) (Δ119905)
119895
(A18)
To write down a simpler expression we change the summa-tion index from 119895 to 119901 = 119899 + 119895 So we have from (A18) thefollowing
1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908119894(119905119896+ Δ119905)
Thus the above proceedings tell us that if criterion (A23) issatisfied then the error is bounded In effect this means thatthe scheme is convergent in lieu of Lax equivalence theoremIn our computations Δ119905 has been always chosen to satisfyinequality (A23)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors are thankful to Ondokuz Mayıs UniversitySamsun Turkey for providing financial support to carryout this work under a major research project (Grant nopyofen190113003)
References
[1] W Hundsdorfer and J G Verwer Numerical Solution of TimeDependent Advection-Diffusion-Reaction Equations SpringerBerlin Germany 2003
[2] Y Kuramoto Chemical Oscillations Waves and TurbulenceDover Mineola NY USA 2003
[3] J DMurrayMathematical Biology II vol 18 of InterdisciplinaryApplied Mathematics Springer New York NY USA 3rd edi-tion 2003
[4] H Wilhelmsson and E Lazzaro Reaction-Diffusion Problemsin the Physics of Hot Plasmas Institute of Physics PublishingPhiladelphia Pa USA 2001
[5] M Bar N Gottschalk M Eiswirth and G Ertl ldquoSpiral waves ina surface reaction model calculationsrdquoThe Journal of ChemicalPhysics vol 100 no 2 pp 1202ndash1214 1994
[6] D Barkley ldquoA model for fast computer simulation of waves inexcitable mediardquo Physica D Nonlinear Phenomena vol 49 no1-2 pp 61ndash70 1991
[7] F H Fenton E M Cherry H M Hastings and S J EvansldquoMultiple mechanisms of spiral wave breakup in a model ofcardiac electrical activityrdquo Chaos vol 12 no 3 pp 852ndash8922002
[8] M GosakMMarhl andM Perc ldquoSpatial coherence resonancein excitable biochemical media induced by internal noiserdquoBiophysical Chemistry vol 128 no 2-3 pp 210ndash214 2007
[9] A Karma ldquoMeandering transition in two-dimensionalexcitable mediardquo Physical Review Letters vol 65 no 22 pp2824ndash2827 1990
12 Mathematical Problems in Engineering
[10] J P Keener ldquoA geometrical theory for spiral waves in excitablemediardquo SIAM Journal onAppliedMathematics vol 46 no 6 pp1039ndash1056 1986
[11] J P Keener Mathematical Physiology Interdisciplinary AppliedMathematics Springer New York NY USA 1998
[12] V M Kenkre and M N Kuperman ldquoApplicability of the Fisherequation to bacterial population dynamicsrdquo Physical Review Evol 67 no 5 Article ID 051921 5 pages 2003
[13] V Krinsky and A Pumir ldquoModels of defibrillation of cardiactissuerdquo Chaos vol 8 no 1 pp 188ndash203 1998
[14] N F Otani ldquoA primary mechanism for spiral wave meander-ingrdquo Chaos vol 12 no 3 pp 829ndash842 2002
[15] M Perc ldquoSpatial coherence resonance in excitable mediardquoPhysical Review E vol 72 no 3 Article ID 016207 2005
[16] M Perc ldquoStochastic resonance on excitable small-world net-works via a pacemakerrdquo Physical Review E vol 76 no 6 ArticleID 066203 2007
[17] M Perc ldquoEffects of small-world connectivity on noise-inducedtemporal and spatial order in neural mediardquo Chaos Solitons ampFractals vol 31 no 2 pp 280ndash291 2007
[18] J J Tyson ldquoWhat everyone should know about the Belousov-Zhabotinsky reactionrdquo in Frontiers inMathematical Biology vol100 of Lecture Notes in Biomathematics pp 569ndash587 SpringerBerlin Germany 1994
[19] S Z Rida A M El-Sayed and A A Arafa ldquoOn the solutions oftime-fractional reaction-diffusion equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 15 no 12pp 3847ndash3854 2010
[20] Y Zheng and Z Zhao ldquoA fully discrete Galerkin method fora nonlinear space-fractional diffusion equationrdquo MathematicalProblems in Engineering vol 2011 Article ID 171620 20 pages2011
[21] J K Zhou Differential Transformation and Its Applications forElectrical Circuits Huazhang University Press Wuhan China1986 (Chinese)
[22] I H Abdel-Halim Hassan ldquoComparison differential transfor-mation technique with Adomian decomposition method forlinear and nonlinear initial value problemsrdquoChaos Solitons andFractals vol 36 no 1 pp 53ndash65 2008
[23] A Arikoglu and I Ozkol ldquoSolution of boundary value problemsfor integro-differential equations by using differential transformmethodrdquoAppliedMathematics and Computation vol 168 no 2pp 1145ndash1158 2005
[24] F Ayaz ldquoSolutions of the system of differential equationsby differential transform methodrdquo Applied Mathematics andComputation vol 147 no 2 pp 547ndash567 2004
[25] N Bildik A Konuralp F Orak Bek and S KucukarslanldquoSolution of different type of the partial differential equationby differential transformmethod andAdomianrsquos decompositionmethodrdquo Applied Mathematics and Computation vol 172 no 1pp 551ndash567 2006
[26] H Liu and Y Song ldquoDifferential transform method applied tohigh index differential-algebraic equationsrdquoAppliedMathemat-ics and Computation vol 184 no 2 pp 748ndash753 2007
[27] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent Part IIrdquo Geophysical Journal of theRoyal Astronomical Society vol 13 no 5 pp 529ndash539 1967
[28] SMomani ZOdibat andV S Erturk ldquoGeneralized differentialtransform method for solving a space- and time-fractionaldiffusion-wave equationrdquo Physics Letters A vol 370 no 5-6 pp379ndash387 2007
[29] Z M Odibat C Bertelle M A Aziz-Alaoui and G HDuchamp ldquoA multi-step differential transform method andapplication to non-chaotic or chaotic systemsrdquo Computers ampMathematics with Applications vol 59 no 4 pp 1462ndash14722010
[30] N Laskin ldquoFractional Schrodinger Equationrdquo Physical ReviewE vol 66 no 5 Article ID 056108 2002
[31] G J Fix and J P Roop ldquoLeast squares finite-element solutionof a fractional order two-point boundary value problemrdquoComputers ampMathematics with Applications vol 48 no 7-8 pp1017ndash1033 2004
[32] L-T Yu and C-K Chen ldquoApplication of the hybrid methodto the transient thermal stresses response in isotropic annularfinsrdquo Journal of Applied Mechanics vol 66 no 2 pp 340ndash3471999
[33] B-L Kuo and C-K Chen ldquoApplication of a hybrid methodto the solution of the nonlinear burgersrsquo equationrdquo Journal ofApplied Mechanics Transactions ASME vol 70 no 6 pp 926ndash929 2003
[34] C K Chen H Y Lai and C C Liu ldquoApplication of hybrid dif-ferential transformationfinite difference method to nonlinearanalysis of micro fixed-fixed beamrdquo Microsystem Technologiesvol 15 no 6 pp 813ndash820 2009
[35] N A Khan M Ayaz L Jin and A Yildirim ldquoOn approximatesolutions for the time-fractional reaction-diffusion equation ofFisher typerdquo International Journal of Physical Sciences vol 6 no10 pp 2483ndash2496 2011
[36] M Merdan ldquoSolutions of time-fractional reaction-diffusionequation withmodified Riemann-Liouville derivativerdquo Interna-tional Journal of Physical Sciences vol 7 no 15 pp 2317ndash23262012
[37] C Li and G Peng ldquoChaos in Chenrsquos system with a fractionalorderrdquo Chaos Solitons amp Fractals vol 22 no 2 pp 443ndash4502004
[38] Y Luchko and R Gorenflo ldquoAn operational method for solvingfractional differential equations with the Caputo derivativesrdquoActa Mathematica Vietnamica vol 24 no 2 pp 207ndash233 1999
[39] Z Odibat and S Momani ldquoA generalized differential transformmethod for linear partial differential equations of fractionalorderrdquo Applied Mathematics Letters vol 21 no 2 pp 194ndash1992008
[40] Z M Odibat and N T Shawagfeh ldquoGeneralized Taylorrsquosformulardquo Applied Mathematics and Computation vol 186 no1 pp 286ndash293 2007
[41] D Nazari and S Shahmorad ldquoApplication of the fractionaldifferential transform method to fractional-order integro-differential equations with nonlocal boundary conditionsrdquoJournal of Computational andAppliedMathematics vol 234 no3 pp 883ndash891 2010
[42] D Agırseven and T Ozis ldquoAn analytical study for Fisher typeequations by using homotopy perturbationmethodrdquoComputersamp Mathematics with Applications vol 60 no 3 pp 602ndash6092010
[43] M Dehghan J M Heris and A Saadatmandi ldquoApplicationof semi-analytic methods for the Fitzhugh-Nagumo equation
Mathematical Problems in Engineering 13
which models the transmission of nerve impulsesrdquoMathemati-calMethods in the Applied Sciences vol 33 no 11 pp 1384ndash13982010
[44] A-M Wazwaz and A Gorguis ldquoAn analytic study of Fisherrsquosequation by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 154 no 3 pp 609ndash6202004
[45] M-J Jang C-L Chen and Y-C Liy ldquoOn solving the initial-value problems using the differential transformation methodrdquoAppliedMathematics andComputation vol 115 no 2-3 pp 145ndash160 2000
Table 4 Coefficients of 1 119905 1199052 for some 119894 values and comparisonwith exact and Ridarsquos solution 120583 = 07 [19]
Rida et al [19] Exact Present119894 = 0
Coef of 1199050 05 05 05Coef of 1199051 minus005 minus005 minus0049999Coef of 1199052 002 0 minus0000011
119894 = 1
Coef of 1199050 0517670 0517670 0517670Coef of 1199051 minus0049937 minus00499937 minus0049933Coef of 1199052 0020327 minus0000176 minus0000181
119894 = 2
Coef of 1199050 0535296 0535296 0535296Coef of 1199051 minus0049937 minus0049937 minus0049743Coef of 1199052 0020602 minus0000351 minus0000352
119894 = 3
Coef of 1199050 0552835 0552835 0552835Coef of 1199051 minus0049441 minus0049441 minus0049430Coef of 1199052 0020821 minus0000522 minus0000542
026
024
022
PresentAnalytic
020
018
0 002 004 006 008 010
t
Figure 4 Comparison of present results for ℎ = 02 and 119899 = 10withthe analytical solution in case of 120572 rarr 1 at 119909 = 04
From this table it is clear that the present work gives betterapproximation than GDTM as we increase 119899
Numerical comparison between GDTM FVIM andhybrid method is shown in Table 5 which indicates hybridmethod is more promising
4 Conclusion
Many real physical problems can be best modelled withfractional differential equations but the fact is when the
026
024
022
PresentAnalytic
020
018
0 002 004 006 008 010
t
Figure 5 Comparison of present results for ℎ = 01 and 119899 = 5 withthe analytical solution in case of 120572 rarr 1 at 119909 = 04
026
024
022
PresentAnalytic
020
018
0 002 004 006 008 010
t
Figure 6 Comparison of present results for ℎ = 002 and 119899 = 3withthe analytical solution in case of 120572 rarr 1 at 119909 = 04
equation is nonlinear there are very few reliable methodsThe numerical methods that can be used to solve frac-tional differential equations are known to have problems ofconvergence and stability These aspects are well addressedin the paper by suggesting a new procedure that uses acombination of the generalized differential transform andcentral difference methods The Appendix clearly spells out
8 Mathematical Problems in Engineering
10
075
05
025
0010
5
x
01
0
005
00
t
minus10
minus5
(a)
10
075
05
025
0010
minus10
5
minus5x 010
005
00
t
(b)
Figure 7 Numerical solution for the time fractional FitzHugh-Nagumo equation with 120572 rarr 1 (a) comparison with the analytical solution(b) with 120583 = 05
10
075
05
025
0010
minus10
5
minus5x01
0
005
00
t
(a)
10
075
05
025
0010
minus10
5
minus5x
01
0005
00
t
(b)
Figure 8 Numerical solution for the time fractional FitzHugh-Nagumo equation with 120572 = 095 (a) and 120572 = 099 (b)
the fact that the error as a result of discretization and compu-tation is bounded and hence implies stability of the methodLax equivalence theorem further implies convergence of thescheme Two time fractional nonlinear reaction-diffusionequations considered for illustration of the hybrid methodhighlight the usefulness of the method in obtaining thesolution of IBVPs involving time fractional derivatives Thecontrol of convergence through a judicious choice of time andspatial step sizes and also the number of terms in the timeseries solution spells assured convergence The segregationof the time domain from the spatial domain in the solution
method ensures the fact that problem of stability does notarise Diagonal dominance of the coefficient matrix in thesystem of linear algebraic equations resulting from the use ofthe central difference approximation in the Poisson equationensures the fact that the matrix remains nonsingular duringiterations and hence has assured convergence An appropriatecomputational decision on the number of terms to be takenin the time series solution results in a convergent solutionwith fast convergence Excellent comparison of the presentresults with the previous works on generalized differentialtransform method [19] and homotopy perturbation method
Mathematical Problems in Engineering 9
09716
09714
09712
09710
u(xt)
09708
09706
120572 = 095
120572 = 1
120572 = 085
120572 = 075
120572 = 06509704
0 01 02 03
t
(a)
120572 = 095
120572 = 085
120572 = 075
120572 = 065
120572 = 1
09716
09714
09712
0971
u(xt)
09708
09706
09704
0 01005 02015 03025
t
(b)
Figure 9 Approximate solution for the time fractional FitzHugh-Nagumo equation with different 120572 values at 119909 = 5 (a) present and (b) [36]
[35] and fractional variational iterationmethod [36] providesconfidence in the methodology adopted for the solution oftime fractional differential equations
Appendix
Estimation of Bounds on Truncation Error
Consider the fractional differential equations (9) and (16) ina general form as
119863120572
119905119906 = 119863
119909119909119906 + 119891 (119906) 119905 ge 0
119909 isin R (0 lt 120572 le 1)
(A1)
The differential transform of (A1) at the spatially discretizedpoints 119909
We now follow Jang et al [45] and move on to arrive at anestimate on the bounds for the truncation error in a generalway by considering the Taylor series expansion of119880
Table 5 Comparison of numerical results between differentmethods for the time fractional FitzHugh-Nagumo equationGDTM generalizeddifferential transform method Rida et al [19] and FVIM fractional variational iteration method Merdan [36]
We so far addressed the local error due to two differenttruncations in the time series In what follows we estimatethe bounds on the cumulative error that includes the errordiscussed above
Let 119910119894(119905119896) denote the solution of (A2) The local error in
119908119894(119905119896) relative to 119910
119894(119905119896) is
1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908119894(119905119896+ Δ119905)
1003816100381610038161003816
le1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
+1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
(A13)
Since 119908119894(119905119896+ Δ119905) is a better approximation than 119908
119894(119905119896+ Δ119905)
we may assume that1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816 ≪ 1 (A14)
Mathematical Problems in Engineering 11
In view of (A14) we now have1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816 ≐1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
(A15)
Using (A8) and noting that (Δ119905)119899+1 is quite small in (A15)we may take 120573 to be
120573 ≐
1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908119894(119905119896+ Δ119905)
1003816100381610038161003816
(Δ119905)119899+1
(A16)
Thus 120573(Δ119905)119899+1 = 120576 is the bound on the tolerance in the to-be-obtained solution When using different number of terms inthe Taylor series expansion earlier we denoted the solutionsusing a time stepΔ119905 by119908
119894(119905119896+Δ119905) and119908
119894(119905119896+Δ119905) respectively
The paper uses an adaptive step size in computing theresults This is because such a procedure succeeds in keepingthe error bounded and ensures convergence as a consequenceof Lax equivalence theorem To see what the adaptive stepsize produces and to show how such a procedure keepsthe error bounded we start with the premise that Δ119905 isthe most appropriate step size for the problem This stepsize is determined using the definition of inverse differentialtransform
119906119894(Δ119905) =
infin
sum
119896=0
119880120572119894(119896) (Δ119905)
119896 (A17)
In our actual calculation we will not be able to considerinfinite number of terms We consider ldquo119899rdquo terms in respectof 119908119894and ldquo119899 + 119898rdquo terms in respect of 119908
119894
Thus1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
=
119899+119898
sum
119895=0
119880120572119894(119895) (Δ119905)
119895minus
119899
sum
119895=0
119880120572119894(119895) (Δ119905)
119895
=
119899+119898
sum
119895=119899+1
119880120572119894(119895) (Δ119905)
119895
(A18)
To write down a simpler expression we change the summa-tion index from 119895 to 119901 = 119899 + 119895 So we have from (A18) thefollowing
1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908119894(119905119896+ Δ119905)
Thus the above proceedings tell us that if criterion (A23) issatisfied then the error is bounded In effect this means thatthe scheme is convergent in lieu of Lax equivalence theoremIn our computations Δ119905 has been always chosen to satisfyinequality (A23)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors are thankful to Ondokuz Mayıs UniversitySamsun Turkey for providing financial support to carryout this work under a major research project (Grant nopyofen190113003)
References
[1] W Hundsdorfer and J G Verwer Numerical Solution of TimeDependent Advection-Diffusion-Reaction Equations SpringerBerlin Germany 2003
[2] Y Kuramoto Chemical Oscillations Waves and TurbulenceDover Mineola NY USA 2003
[3] J DMurrayMathematical Biology II vol 18 of InterdisciplinaryApplied Mathematics Springer New York NY USA 3rd edi-tion 2003
[4] H Wilhelmsson and E Lazzaro Reaction-Diffusion Problemsin the Physics of Hot Plasmas Institute of Physics PublishingPhiladelphia Pa USA 2001
[5] M Bar N Gottschalk M Eiswirth and G Ertl ldquoSpiral waves ina surface reaction model calculationsrdquoThe Journal of ChemicalPhysics vol 100 no 2 pp 1202ndash1214 1994
[6] D Barkley ldquoA model for fast computer simulation of waves inexcitable mediardquo Physica D Nonlinear Phenomena vol 49 no1-2 pp 61ndash70 1991
[7] F H Fenton E M Cherry H M Hastings and S J EvansldquoMultiple mechanisms of spiral wave breakup in a model ofcardiac electrical activityrdquo Chaos vol 12 no 3 pp 852ndash8922002
[8] M GosakMMarhl andM Perc ldquoSpatial coherence resonancein excitable biochemical media induced by internal noiserdquoBiophysical Chemistry vol 128 no 2-3 pp 210ndash214 2007
[9] A Karma ldquoMeandering transition in two-dimensionalexcitable mediardquo Physical Review Letters vol 65 no 22 pp2824ndash2827 1990
12 Mathematical Problems in Engineering
[10] J P Keener ldquoA geometrical theory for spiral waves in excitablemediardquo SIAM Journal onAppliedMathematics vol 46 no 6 pp1039ndash1056 1986
[11] J P Keener Mathematical Physiology Interdisciplinary AppliedMathematics Springer New York NY USA 1998
[12] V M Kenkre and M N Kuperman ldquoApplicability of the Fisherequation to bacterial population dynamicsrdquo Physical Review Evol 67 no 5 Article ID 051921 5 pages 2003
[13] V Krinsky and A Pumir ldquoModels of defibrillation of cardiactissuerdquo Chaos vol 8 no 1 pp 188ndash203 1998
[14] N F Otani ldquoA primary mechanism for spiral wave meander-ingrdquo Chaos vol 12 no 3 pp 829ndash842 2002
[15] M Perc ldquoSpatial coherence resonance in excitable mediardquoPhysical Review E vol 72 no 3 Article ID 016207 2005
[16] M Perc ldquoStochastic resonance on excitable small-world net-works via a pacemakerrdquo Physical Review E vol 76 no 6 ArticleID 066203 2007
[17] M Perc ldquoEffects of small-world connectivity on noise-inducedtemporal and spatial order in neural mediardquo Chaos Solitons ampFractals vol 31 no 2 pp 280ndash291 2007
[18] J J Tyson ldquoWhat everyone should know about the Belousov-Zhabotinsky reactionrdquo in Frontiers inMathematical Biology vol100 of Lecture Notes in Biomathematics pp 569ndash587 SpringerBerlin Germany 1994
[19] S Z Rida A M El-Sayed and A A Arafa ldquoOn the solutions oftime-fractional reaction-diffusion equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 15 no 12pp 3847ndash3854 2010
[20] Y Zheng and Z Zhao ldquoA fully discrete Galerkin method fora nonlinear space-fractional diffusion equationrdquo MathematicalProblems in Engineering vol 2011 Article ID 171620 20 pages2011
[21] J K Zhou Differential Transformation and Its Applications forElectrical Circuits Huazhang University Press Wuhan China1986 (Chinese)
[22] I H Abdel-Halim Hassan ldquoComparison differential transfor-mation technique with Adomian decomposition method forlinear and nonlinear initial value problemsrdquoChaos Solitons andFractals vol 36 no 1 pp 53ndash65 2008
[23] A Arikoglu and I Ozkol ldquoSolution of boundary value problemsfor integro-differential equations by using differential transformmethodrdquoAppliedMathematics and Computation vol 168 no 2pp 1145ndash1158 2005
[24] F Ayaz ldquoSolutions of the system of differential equationsby differential transform methodrdquo Applied Mathematics andComputation vol 147 no 2 pp 547ndash567 2004
[25] N Bildik A Konuralp F Orak Bek and S KucukarslanldquoSolution of different type of the partial differential equationby differential transformmethod andAdomianrsquos decompositionmethodrdquo Applied Mathematics and Computation vol 172 no 1pp 551ndash567 2006
[26] H Liu and Y Song ldquoDifferential transform method applied tohigh index differential-algebraic equationsrdquoAppliedMathemat-ics and Computation vol 184 no 2 pp 748ndash753 2007
[27] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent Part IIrdquo Geophysical Journal of theRoyal Astronomical Society vol 13 no 5 pp 529ndash539 1967
[28] SMomani ZOdibat andV S Erturk ldquoGeneralized differentialtransform method for solving a space- and time-fractionaldiffusion-wave equationrdquo Physics Letters A vol 370 no 5-6 pp379ndash387 2007
[29] Z M Odibat C Bertelle M A Aziz-Alaoui and G HDuchamp ldquoA multi-step differential transform method andapplication to non-chaotic or chaotic systemsrdquo Computers ampMathematics with Applications vol 59 no 4 pp 1462ndash14722010
[30] N Laskin ldquoFractional Schrodinger Equationrdquo Physical ReviewE vol 66 no 5 Article ID 056108 2002
[31] G J Fix and J P Roop ldquoLeast squares finite-element solutionof a fractional order two-point boundary value problemrdquoComputers ampMathematics with Applications vol 48 no 7-8 pp1017ndash1033 2004
[32] L-T Yu and C-K Chen ldquoApplication of the hybrid methodto the transient thermal stresses response in isotropic annularfinsrdquo Journal of Applied Mechanics vol 66 no 2 pp 340ndash3471999
[33] B-L Kuo and C-K Chen ldquoApplication of a hybrid methodto the solution of the nonlinear burgersrsquo equationrdquo Journal ofApplied Mechanics Transactions ASME vol 70 no 6 pp 926ndash929 2003
[34] C K Chen H Y Lai and C C Liu ldquoApplication of hybrid dif-ferential transformationfinite difference method to nonlinearanalysis of micro fixed-fixed beamrdquo Microsystem Technologiesvol 15 no 6 pp 813ndash820 2009
[35] N A Khan M Ayaz L Jin and A Yildirim ldquoOn approximatesolutions for the time-fractional reaction-diffusion equation ofFisher typerdquo International Journal of Physical Sciences vol 6 no10 pp 2483ndash2496 2011
[36] M Merdan ldquoSolutions of time-fractional reaction-diffusionequation withmodified Riemann-Liouville derivativerdquo Interna-tional Journal of Physical Sciences vol 7 no 15 pp 2317ndash23262012
[37] C Li and G Peng ldquoChaos in Chenrsquos system with a fractionalorderrdquo Chaos Solitons amp Fractals vol 22 no 2 pp 443ndash4502004
[38] Y Luchko and R Gorenflo ldquoAn operational method for solvingfractional differential equations with the Caputo derivativesrdquoActa Mathematica Vietnamica vol 24 no 2 pp 207ndash233 1999
[39] Z Odibat and S Momani ldquoA generalized differential transformmethod for linear partial differential equations of fractionalorderrdquo Applied Mathematics Letters vol 21 no 2 pp 194ndash1992008
[40] Z M Odibat and N T Shawagfeh ldquoGeneralized Taylorrsquosformulardquo Applied Mathematics and Computation vol 186 no1 pp 286ndash293 2007
[41] D Nazari and S Shahmorad ldquoApplication of the fractionaldifferential transform method to fractional-order integro-differential equations with nonlocal boundary conditionsrdquoJournal of Computational andAppliedMathematics vol 234 no3 pp 883ndash891 2010
[42] D Agırseven and T Ozis ldquoAn analytical study for Fisher typeequations by using homotopy perturbationmethodrdquoComputersamp Mathematics with Applications vol 60 no 3 pp 602ndash6092010
[43] M Dehghan J M Heris and A Saadatmandi ldquoApplicationof semi-analytic methods for the Fitzhugh-Nagumo equation
Mathematical Problems in Engineering 13
which models the transmission of nerve impulsesrdquoMathemati-calMethods in the Applied Sciences vol 33 no 11 pp 1384ndash13982010
[44] A-M Wazwaz and A Gorguis ldquoAn analytic study of Fisherrsquosequation by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 154 no 3 pp 609ndash6202004
[45] M-J Jang C-L Chen and Y-C Liy ldquoOn solving the initial-value problems using the differential transformation methodrdquoAppliedMathematics andComputation vol 115 no 2-3 pp 145ndash160 2000
Figure 7 Numerical solution for the time fractional FitzHugh-Nagumo equation with 120572 rarr 1 (a) comparison with the analytical solution(b) with 120583 = 05
10
075
05
025
0010
minus10
5
minus5x01
0
005
00
t
(a)
10
075
05
025
0010
minus10
5
minus5x
01
0005
00
t
(b)
Figure 8 Numerical solution for the time fractional FitzHugh-Nagumo equation with 120572 = 095 (a) and 120572 = 099 (b)
the fact that the error as a result of discretization and compu-tation is bounded and hence implies stability of the methodLax equivalence theorem further implies convergence of thescheme Two time fractional nonlinear reaction-diffusionequations considered for illustration of the hybrid methodhighlight the usefulness of the method in obtaining thesolution of IBVPs involving time fractional derivatives Thecontrol of convergence through a judicious choice of time andspatial step sizes and also the number of terms in the timeseries solution spells assured convergence The segregationof the time domain from the spatial domain in the solution
method ensures the fact that problem of stability does notarise Diagonal dominance of the coefficient matrix in thesystem of linear algebraic equations resulting from the use ofthe central difference approximation in the Poisson equationensures the fact that the matrix remains nonsingular duringiterations and hence has assured convergence An appropriatecomputational decision on the number of terms to be takenin the time series solution results in a convergent solutionwith fast convergence Excellent comparison of the presentresults with the previous works on generalized differentialtransform method [19] and homotopy perturbation method
Mathematical Problems in Engineering 9
09716
09714
09712
09710
u(xt)
09708
09706
120572 = 095
120572 = 1
120572 = 085
120572 = 075
120572 = 06509704
0 01 02 03
t
(a)
120572 = 095
120572 = 085
120572 = 075
120572 = 065
120572 = 1
09716
09714
09712
0971
u(xt)
09708
09706
09704
0 01005 02015 03025
t
(b)
Figure 9 Approximate solution for the time fractional FitzHugh-Nagumo equation with different 120572 values at 119909 = 5 (a) present and (b) [36]
[35] and fractional variational iterationmethod [36] providesconfidence in the methodology adopted for the solution oftime fractional differential equations
Appendix
Estimation of Bounds on Truncation Error
Consider the fractional differential equations (9) and (16) ina general form as
119863120572
119905119906 = 119863
119909119909119906 + 119891 (119906) 119905 ge 0
119909 isin R (0 lt 120572 le 1)
(A1)
The differential transform of (A1) at the spatially discretizedpoints 119909
We now follow Jang et al [45] and move on to arrive at anestimate on the bounds for the truncation error in a generalway by considering the Taylor series expansion of119880
Table 5 Comparison of numerical results between differentmethods for the time fractional FitzHugh-Nagumo equationGDTM generalizeddifferential transform method Rida et al [19] and FVIM fractional variational iteration method Merdan [36]
We so far addressed the local error due to two differenttruncations in the time series In what follows we estimatethe bounds on the cumulative error that includes the errordiscussed above
Let 119910119894(119905119896) denote the solution of (A2) The local error in
119908119894(119905119896) relative to 119910
119894(119905119896) is
1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908119894(119905119896+ Δ119905)
1003816100381610038161003816
le1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
+1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
(A13)
Since 119908119894(119905119896+ Δ119905) is a better approximation than 119908
119894(119905119896+ Δ119905)
we may assume that1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816 ≪ 1 (A14)
Mathematical Problems in Engineering 11
In view of (A14) we now have1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816 ≐1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
(A15)
Using (A8) and noting that (Δ119905)119899+1 is quite small in (A15)we may take 120573 to be
120573 ≐
1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908119894(119905119896+ Δ119905)
1003816100381610038161003816
(Δ119905)119899+1
(A16)
Thus 120573(Δ119905)119899+1 = 120576 is the bound on the tolerance in the to-be-obtained solution When using different number of terms inthe Taylor series expansion earlier we denoted the solutionsusing a time stepΔ119905 by119908
119894(119905119896+Δ119905) and119908
119894(119905119896+Δ119905) respectively
The paper uses an adaptive step size in computing theresults This is because such a procedure succeeds in keepingthe error bounded and ensures convergence as a consequenceof Lax equivalence theorem To see what the adaptive stepsize produces and to show how such a procedure keepsthe error bounded we start with the premise that Δ119905 isthe most appropriate step size for the problem This stepsize is determined using the definition of inverse differentialtransform
119906119894(Δ119905) =
infin
sum
119896=0
119880120572119894(119896) (Δ119905)
119896 (A17)
In our actual calculation we will not be able to considerinfinite number of terms We consider ldquo119899rdquo terms in respectof 119908119894and ldquo119899 + 119898rdquo terms in respect of 119908
119894
Thus1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
=
119899+119898
sum
119895=0
119880120572119894(119895) (Δ119905)
119895minus
119899
sum
119895=0
119880120572119894(119895) (Δ119905)
119895
=
119899+119898
sum
119895=119899+1
119880120572119894(119895) (Δ119905)
119895
(A18)
To write down a simpler expression we change the summa-tion index from 119895 to 119901 = 119899 + 119895 So we have from (A18) thefollowing
1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908119894(119905119896+ Δ119905)
Thus the above proceedings tell us that if criterion (A23) issatisfied then the error is bounded In effect this means thatthe scheme is convergent in lieu of Lax equivalence theoremIn our computations Δ119905 has been always chosen to satisfyinequality (A23)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors are thankful to Ondokuz Mayıs UniversitySamsun Turkey for providing financial support to carryout this work under a major research project (Grant nopyofen190113003)
References
[1] W Hundsdorfer and J G Verwer Numerical Solution of TimeDependent Advection-Diffusion-Reaction Equations SpringerBerlin Germany 2003
[2] Y Kuramoto Chemical Oscillations Waves and TurbulenceDover Mineola NY USA 2003
[3] J DMurrayMathematical Biology II vol 18 of InterdisciplinaryApplied Mathematics Springer New York NY USA 3rd edi-tion 2003
[4] H Wilhelmsson and E Lazzaro Reaction-Diffusion Problemsin the Physics of Hot Plasmas Institute of Physics PublishingPhiladelphia Pa USA 2001
[5] M Bar N Gottschalk M Eiswirth and G Ertl ldquoSpiral waves ina surface reaction model calculationsrdquoThe Journal of ChemicalPhysics vol 100 no 2 pp 1202ndash1214 1994
[6] D Barkley ldquoA model for fast computer simulation of waves inexcitable mediardquo Physica D Nonlinear Phenomena vol 49 no1-2 pp 61ndash70 1991
[7] F H Fenton E M Cherry H M Hastings and S J EvansldquoMultiple mechanisms of spiral wave breakup in a model ofcardiac electrical activityrdquo Chaos vol 12 no 3 pp 852ndash8922002
[8] M GosakMMarhl andM Perc ldquoSpatial coherence resonancein excitable biochemical media induced by internal noiserdquoBiophysical Chemistry vol 128 no 2-3 pp 210ndash214 2007
[9] A Karma ldquoMeandering transition in two-dimensionalexcitable mediardquo Physical Review Letters vol 65 no 22 pp2824ndash2827 1990
12 Mathematical Problems in Engineering
[10] J P Keener ldquoA geometrical theory for spiral waves in excitablemediardquo SIAM Journal onAppliedMathematics vol 46 no 6 pp1039ndash1056 1986
[11] J P Keener Mathematical Physiology Interdisciplinary AppliedMathematics Springer New York NY USA 1998
[12] V M Kenkre and M N Kuperman ldquoApplicability of the Fisherequation to bacterial population dynamicsrdquo Physical Review Evol 67 no 5 Article ID 051921 5 pages 2003
[13] V Krinsky and A Pumir ldquoModels of defibrillation of cardiactissuerdquo Chaos vol 8 no 1 pp 188ndash203 1998
[14] N F Otani ldquoA primary mechanism for spiral wave meander-ingrdquo Chaos vol 12 no 3 pp 829ndash842 2002
[15] M Perc ldquoSpatial coherence resonance in excitable mediardquoPhysical Review E vol 72 no 3 Article ID 016207 2005
[16] M Perc ldquoStochastic resonance on excitable small-world net-works via a pacemakerrdquo Physical Review E vol 76 no 6 ArticleID 066203 2007
[17] M Perc ldquoEffects of small-world connectivity on noise-inducedtemporal and spatial order in neural mediardquo Chaos Solitons ampFractals vol 31 no 2 pp 280ndash291 2007
[18] J J Tyson ldquoWhat everyone should know about the Belousov-Zhabotinsky reactionrdquo in Frontiers inMathematical Biology vol100 of Lecture Notes in Biomathematics pp 569ndash587 SpringerBerlin Germany 1994
[19] S Z Rida A M El-Sayed and A A Arafa ldquoOn the solutions oftime-fractional reaction-diffusion equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 15 no 12pp 3847ndash3854 2010
[20] Y Zheng and Z Zhao ldquoA fully discrete Galerkin method fora nonlinear space-fractional diffusion equationrdquo MathematicalProblems in Engineering vol 2011 Article ID 171620 20 pages2011
[21] J K Zhou Differential Transformation and Its Applications forElectrical Circuits Huazhang University Press Wuhan China1986 (Chinese)
[22] I H Abdel-Halim Hassan ldquoComparison differential transfor-mation technique with Adomian decomposition method forlinear and nonlinear initial value problemsrdquoChaos Solitons andFractals vol 36 no 1 pp 53ndash65 2008
[23] A Arikoglu and I Ozkol ldquoSolution of boundary value problemsfor integro-differential equations by using differential transformmethodrdquoAppliedMathematics and Computation vol 168 no 2pp 1145ndash1158 2005
[24] F Ayaz ldquoSolutions of the system of differential equationsby differential transform methodrdquo Applied Mathematics andComputation vol 147 no 2 pp 547ndash567 2004
[25] N Bildik A Konuralp F Orak Bek and S KucukarslanldquoSolution of different type of the partial differential equationby differential transformmethod andAdomianrsquos decompositionmethodrdquo Applied Mathematics and Computation vol 172 no 1pp 551ndash567 2006
[26] H Liu and Y Song ldquoDifferential transform method applied tohigh index differential-algebraic equationsrdquoAppliedMathemat-ics and Computation vol 184 no 2 pp 748ndash753 2007
[27] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent Part IIrdquo Geophysical Journal of theRoyal Astronomical Society vol 13 no 5 pp 529ndash539 1967
[28] SMomani ZOdibat andV S Erturk ldquoGeneralized differentialtransform method for solving a space- and time-fractionaldiffusion-wave equationrdquo Physics Letters A vol 370 no 5-6 pp379ndash387 2007
[29] Z M Odibat C Bertelle M A Aziz-Alaoui and G HDuchamp ldquoA multi-step differential transform method andapplication to non-chaotic or chaotic systemsrdquo Computers ampMathematics with Applications vol 59 no 4 pp 1462ndash14722010
[30] N Laskin ldquoFractional Schrodinger Equationrdquo Physical ReviewE vol 66 no 5 Article ID 056108 2002
[31] G J Fix and J P Roop ldquoLeast squares finite-element solutionof a fractional order two-point boundary value problemrdquoComputers ampMathematics with Applications vol 48 no 7-8 pp1017ndash1033 2004
[32] L-T Yu and C-K Chen ldquoApplication of the hybrid methodto the transient thermal stresses response in isotropic annularfinsrdquo Journal of Applied Mechanics vol 66 no 2 pp 340ndash3471999
[33] B-L Kuo and C-K Chen ldquoApplication of a hybrid methodto the solution of the nonlinear burgersrsquo equationrdquo Journal ofApplied Mechanics Transactions ASME vol 70 no 6 pp 926ndash929 2003
[34] C K Chen H Y Lai and C C Liu ldquoApplication of hybrid dif-ferential transformationfinite difference method to nonlinearanalysis of micro fixed-fixed beamrdquo Microsystem Technologiesvol 15 no 6 pp 813ndash820 2009
[35] N A Khan M Ayaz L Jin and A Yildirim ldquoOn approximatesolutions for the time-fractional reaction-diffusion equation ofFisher typerdquo International Journal of Physical Sciences vol 6 no10 pp 2483ndash2496 2011
[36] M Merdan ldquoSolutions of time-fractional reaction-diffusionequation withmodified Riemann-Liouville derivativerdquo Interna-tional Journal of Physical Sciences vol 7 no 15 pp 2317ndash23262012
[37] C Li and G Peng ldquoChaos in Chenrsquos system with a fractionalorderrdquo Chaos Solitons amp Fractals vol 22 no 2 pp 443ndash4502004
[38] Y Luchko and R Gorenflo ldquoAn operational method for solvingfractional differential equations with the Caputo derivativesrdquoActa Mathematica Vietnamica vol 24 no 2 pp 207ndash233 1999
[39] Z Odibat and S Momani ldquoA generalized differential transformmethod for linear partial differential equations of fractionalorderrdquo Applied Mathematics Letters vol 21 no 2 pp 194ndash1992008
[40] Z M Odibat and N T Shawagfeh ldquoGeneralized Taylorrsquosformulardquo Applied Mathematics and Computation vol 186 no1 pp 286ndash293 2007
[41] D Nazari and S Shahmorad ldquoApplication of the fractionaldifferential transform method to fractional-order integro-differential equations with nonlocal boundary conditionsrdquoJournal of Computational andAppliedMathematics vol 234 no3 pp 883ndash891 2010
[42] D Agırseven and T Ozis ldquoAn analytical study for Fisher typeequations by using homotopy perturbationmethodrdquoComputersamp Mathematics with Applications vol 60 no 3 pp 602ndash6092010
[43] M Dehghan J M Heris and A Saadatmandi ldquoApplicationof semi-analytic methods for the Fitzhugh-Nagumo equation
Mathematical Problems in Engineering 13
which models the transmission of nerve impulsesrdquoMathemati-calMethods in the Applied Sciences vol 33 no 11 pp 1384ndash13982010
[44] A-M Wazwaz and A Gorguis ldquoAn analytic study of Fisherrsquosequation by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 154 no 3 pp 609ndash6202004
[45] M-J Jang C-L Chen and Y-C Liy ldquoOn solving the initial-value problems using the differential transformation methodrdquoAppliedMathematics andComputation vol 115 no 2-3 pp 145ndash160 2000
Figure 9 Approximate solution for the time fractional FitzHugh-Nagumo equation with different 120572 values at 119909 = 5 (a) present and (b) [36]
[35] and fractional variational iterationmethod [36] providesconfidence in the methodology adopted for the solution oftime fractional differential equations
Appendix
Estimation of Bounds on Truncation Error
Consider the fractional differential equations (9) and (16) ina general form as
119863120572
119905119906 = 119863
119909119909119906 + 119891 (119906) 119905 ge 0
119909 isin R (0 lt 120572 le 1)
(A1)
The differential transform of (A1) at the spatially discretizedpoints 119909
We now follow Jang et al [45] and move on to arrive at anestimate on the bounds for the truncation error in a generalway by considering the Taylor series expansion of119880
Table 5 Comparison of numerical results between differentmethods for the time fractional FitzHugh-Nagumo equationGDTM generalizeddifferential transform method Rida et al [19] and FVIM fractional variational iteration method Merdan [36]
We so far addressed the local error due to two differenttruncations in the time series In what follows we estimatethe bounds on the cumulative error that includes the errordiscussed above
Let 119910119894(119905119896) denote the solution of (A2) The local error in
119908119894(119905119896) relative to 119910
119894(119905119896) is
1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908119894(119905119896+ Δ119905)
1003816100381610038161003816
le1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
+1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
(A13)
Since 119908119894(119905119896+ Δ119905) is a better approximation than 119908
119894(119905119896+ Δ119905)
we may assume that1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816 ≪ 1 (A14)
Mathematical Problems in Engineering 11
In view of (A14) we now have1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816 ≐1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
(A15)
Using (A8) and noting that (Δ119905)119899+1 is quite small in (A15)we may take 120573 to be
120573 ≐
1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908119894(119905119896+ Δ119905)
1003816100381610038161003816
(Δ119905)119899+1
(A16)
Thus 120573(Δ119905)119899+1 = 120576 is the bound on the tolerance in the to-be-obtained solution When using different number of terms inthe Taylor series expansion earlier we denoted the solutionsusing a time stepΔ119905 by119908
119894(119905119896+Δ119905) and119908
119894(119905119896+Δ119905) respectively
The paper uses an adaptive step size in computing theresults This is because such a procedure succeeds in keepingthe error bounded and ensures convergence as a consequenceof Lax equivalence theorem To see what the adaptive stepsize produces and to show how such a procedure keepsthe error bounded we start with the premise that Δ119905 isthe most appropriate step size for the problem This stepsize is determined using the definition of inverse differentialtransform
119906119894(Δ119905) =
infin
sum
119896=0
119880120572119894(119896) (Δ119905)
119896 (A17)
In our actual calculation we will not be able to considerinfinite number of terms We consider ldquo119899rdquo terms in respectof 119908119894and ldquo119899 + 119898rdquo terms in respect of 119908
119894
Thus1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
=
119899+119898
sum
119895=0
119880120572119894(119895) (Δ119905)
119895minus
119899
sum
119895=0
119880120572119894(119895) (Δ119905)
119895
=
119899+119898
sum
119895=119899+1
119880120572119894(119895) (Δ119905)
119895
(A18)
To write down a simpler expression we change the summa-tion index from 119895 to 119901 = 119899 + 119895 So we have from (A18) thefollowing
1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908119894(119905119896+ Δ119905)
Thus the above proceedings tell us that if criterion (A23) issatisfied then the error is bounded In effect this means thatthe scheme is convergent in lieu of Lax equivalence theoremIn our computations Δ119905 has been always chosen to satisfyinequality (A23)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors are thankful to Ondokuz Mayıs UniversitySamsun Turkey for providing financial support to carryout this work under a major research project (Grant nopyofen190113003)
References
[1] W Hundsdorfer and J G Verwer Numerical Solution of TimeDependent Advection-Diffusion-Reaction Equations SpringerBerlin Germany 2003
[2] Y Kuramoto Chemical Oscillations Waves and TurbulenceDover Mineola NY USA 2003
[3] J DMurrayMathematical Biology II vol 18 of InterdisciplinaryApplied Mathematics Springer New York NY USA 3rd edi-tion 2003
[4] H Wilhelmsson and E Lazzaro Reaction-Diffusion Problemsin the Physics of Hot Plasmas Institute of Physics PublishingPhiladelphia Pa USA 2001
[5] M Bar N Gottschalk M Eiswirth and G Ertl ldquoSpiral waves ina surface reaction model calculationsrdquoThe Journal of ChemicalPhysics vol 100 no 2 pp 1202ndash1214 1994
[6] D Barkley ldquoA model for fast computer simulation of waves inexcitable mediardquo Physica D Nonlinear Phenomena vol 49 no1-2 pp 61ndash70 1991
[7] F H Fenton E M Cherry H M Hastings and S J EvansldquoMultiple mechanisms of spiral wave breakup in a model ofcardiac electrical activityrdquo Chaos vol 12 no 3 pp 852ndash8922002
[8] M GosakMMarhl andM Perc ldquoSpatial coherence resonancein excitable biochemical media induced by internal noiserdquoBiophysical Chemistry vol 128 no 2-3 pp 210ndash214 2007
[9] A Karma ldquoMeandering transition in two-dimensionalexcitable mediardquo Physical Review Letters vol 65 no 22 pp2824ndash2827 1990
12 Mathematical Problems in Engineering
[10] J P Keener ldquoA geometrical theory for spiral waves in excitablemediardquo SIAM Journal onAppliedMathematics vol 46 no 6 pp1039ndash1056 1986
[11] J P Keener Mathematical Physiology Interdisciplinary AppliedMathematics Springer New York NY USA 1998
[12] V M Kenkre and M N Kuperman ldquoApplicability of the Fisherequation to bacterial population dynamicsrdquo Physical Review Evol 67 no 5 Article ID 051921 5 pages 2003
[13] V Krinsky and A Pumir ldquoModels of defibrillation of cardiactissuerdquo Chaos vol 8 no 1 pp 188ndash203 1998
[14] N F Otani ldquoA primary mechanism for spiral wave meander-ingrdquo Chaos vol 12 no 3 pp 829ndash842 2002
[15] M Perc ldquoSpatial coherence resonance in excitable mediardquoPhysical Review E vol 72 no 3 Article ID 016207 2005
[16] M Perc ldquoStochastic resonance on excitable small-world net-works via a pacemakerrdquo Physical Review E vol 76 no 6 ArticleID 066203 2007
[17] M Perc ldquoEffects of small-world connectivity on noise-inducedtemporal and spatial order in neural mediardquo Chaos Solitons ampFractals vol 31 no 2 pp 280ndash291 2007
[18] J J Tyson ldquoWhat everyone should know about the Belousov-Zhabotinsky reactionrdquo in Frontiers inMathematical Biology vol100 of Lecture Notes in Biomathematics pp 569ndash587 SpringerBerlin Germany 1994
[19] S Z Rida A M El-Sayed and A A Arafa ldquoOn the solutions oftime-fractional reaction-diffusion equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 15 no 12pp 3847ndash3854 2010
[20] Y Zheng and Z Zhao ldquoA fully discrete Galerkin method fora nonlinear space-fractional diffusion equationrdquo MathematicalProblems in Engineering vol 2011 Article ID 171620 20 pages2011
[21] J K Zhou Differential Transformation and Its Applications forElectrical Circuits Huazhang University Press Wuhan China1986 (Chinese)
[22] I H Abdel-Halim Hassan ldquoComparison differential transfor-mation technique with Adomian decomposition method forlinear and nonlinear initial value problemsrdquoChaos Solitons andFractals vol 36 no 1 pp 53ndash65 2008
[23] A Arikoglu and I Ozkol ldquoSolution of boundary value problemsfor integro-differential equations by using differential transformmethodrdquoAppliedMathematics and Computation vol 168 no 2pp 1145ndash1158 2005
[24] F Ayaz ldquoSolutions of the system of differential equationsby differential transform methodrdquo Applied Mathematics andComputation vol 147 no 2 pp 547ndash567 2004
[25] N Bildik A Konuralp F Orak Bek and S KucukarslanldquoSolution of different type of the partial differential equationby differential transformmethod andAdomianrsquos decompositionmethodrdquo Applied Mathematics and Computation vol 172 no 1pp 551ndash567 2006
[26] H Liu and Y Song ldquoDifferential transform method applied tohigh index differential-algebraic equationsrdquoAppliedMathemat-ics and Computation vol 184 no 2 pp 748ndash753 2007
[27] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent Part IIrdquo Geophysical Journal of theRoyal Astronomical Society vol 13 no 5 pp 529ndash539 1967
[28] SMomani ZOdibat andV S Erturk ldquoGeneralized differentialtransform method for solving a space- and time-fractionaldiffusion-wave equationrdquo Physics Letters A vol 370 no 5-6 pp379ndash387 2007
[29] Z M Odibat C Bertelle M A Aziz-Alaoui and G HDuchamp ldquoA multi-step differential transform method andapplication to non-chaotic or chaotic systemsrdquo Computers ampMathematics with Applications vol 59 no 4 pp 1462ndash14722010
[30] N Laskin ldquoFractional Schrodinger Equationrdquo Physical ReviewE vol 66 no 5 Article ID 056108 2002
[31] G J Fix and J P Roop ldquoLeast squares finite-element solutionof a fractional order two-point boundary value problemrdquoComputers ampMathematics with Applications vol 48 no 7-8 pp1017ndash1033 2004
[32] L-T Yu and C-K Chen ldquoApplication of the hybrid methodto the transient thermal stresses response in isotropic annularfinsrdquo Journal of Applied Mechanics vol 66 no 2 pp 340ndash3471999
[33] B-L Kuo and C-K Chen ldquoApplication of a hybrid methodto the solution of the nonlinear burgersrsquo equationrdquo Journal ofApplied Mechanics Transactions ASME vol 70 no 6 pp 926ndash929 2003
[34] C K Chen H Y Lai and C C Liu ldquoApplication of hybrid dif-ferential transformationfinite difference method to nonlinearanalysis of micro fixed-fixed beamrdquo Microsystem Technologiesvol 15 no 6 pp 813ndash820 2009
[35] N A Khan M Ayaz L Jin and A Yildirim ldquoOn approximatesolutions for the time-fractional reaction-diffusion equation ofFisher typerdquo International Journal of Physical Sciences vol 6 no10 pp 2483ndash2496 2011
[36] M Merdan ldquoSolutions of time-fractional reaction-diffusionequation withmodified Riemann-Liouville derivativerdquo Interna-tional Journal of Physical Sciences vol 7 no 15 pp 2317ndash23262012
[37] C Li and G Peng ldquoChaos in Chenrsquos system with a fractionalorderrdquo Chaos Solitons amp Fractals vol 22 no 2 pp 443ndash4502004
[38] Y Luchko and R Gorenflo ldquoAn operational method for solvingfractional differential equations with the Caputo derivativesrdquoActa Mathematica Vietnamica vol 24 no 2 pp 207ndash233 1999
[39] Z Odibat and S Momani ldquoA generalized differential transformmethod for linear partial differential equations of fractionalorderrdquo Applied Mathematics Letters vol 21 no 2 pp 194ndash1992008
[40] Z M Odibat and N T Shawagfeh ldquoGeneralized Taylorrsquosformulardquo Applied Mathematics and Computation vol 186 no1 pp 286ndash293 2007
[41] D Nazari and S Shahmorad ldquoApplication of the fractionaldifferential transform method to fractional-order integro-differential equations with nonlocal boundary conditionsrdquoJournal of Computational andAppliedMathematics vol 234 no3 pp 883ndash891 2010
[42] D Agırseven and T Ozis ldquoAn analytical study for Fisher typeequations by using homotopy perturbationmethodrdquoComputersamp Mathematics with Applications vol 60 no 3 pp 602ndash6092010
[43] M Dehghan J M Heris and A Saadatmandi ldquoApplicationof semi-analytic methods for the Fitzhugh-Nagumo equation
Mathematical Problems in Engineering 13
which models the transmission of nerve impulsesrdquoMathemati-calMethods in the Applied Sciences vol 33 no 11 pp 1384ndash13982010
[44] A-M Wazwaz and A Gorguis ldquoAn analytic study of Fisherrsquosequation by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 154 no 3 pp 609ndash6202004
[45] M-J Jang C-L Chen and Y-C Liy ldquoOn solving the initial-value problems using the differential transformation methodrdquoAppliedMathematics andComputation vol 115 no 2-3 pp 145ndash160 2000
Table 5 Comparison of numerical results between differentmethods for the time fractional FitzHugh-Nagumo equationGDTM generalizeddifferential transform method Rida et al [19] and FVIM fractional variational iteration method Merdan [36]
We so far addressed the local error due to two differenttruncations in the time series In what follows we estimatethe bounds on the cumulative error that includes the errordiscussed above
Let 119910119894(119905119896) denote the solution of (A2) The local error in
119908119894(119905119896) relative to 119910
119894(119905119896) is
1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908119894(119905119896+ Δ119905)
1003816100381610038161003816
le1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
+1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
(A13)
Since 119908119894(119905119896+ Δ119905) is a better approximation than 119908
119894(119905119896+ Δ119905)
we may assume that1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816 ≪ 1 (A14)
Mathematical Problems in Engineering 11
In view of (A14) we now have1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816 ≐1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
(A15)
Using (A8) and noting that (Δ119905)119899+1 is quite small in (A15)we may take 120573 to be
120573 ≐
1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908119894(119905119896+ Δ119905)
1003816100381610038161003816
(Δ119905)119899+1
(A16)
Thus 120573(Δ119905)119899+1 = 120576 is the bound on the tolerance in the to-be-obtained solution When using different number of terms inthe Taylor series expansion earlier we denoted the solutionsusing a time stepΔ119905 by119908
119894(119905119896+Δ119905) and119908
119894(119905119896+Δ119905) respectively
The paper uses an adaptive step size in computing theresults This is because such a procedure succeeds in keepingthe error bounded and ensures convergence as a consequenceof Lax equivalence theorem To see what the adaptive stepsize produces and to show how such a procedure keepsthe error bounded we start with the premise that Δ119905 isthe most appropriate step size for the problem This stepsize is determined using the definition of inverse differentialtransform
119906119894(Δ119905) =
infin
sum
119896=0
119880120572119894(119896) (Δ119905)
119896 (A17)
In our actual calculation we will not be able to considerinfinite number of terms We consider ldquo119899rdquo terms in respectof 119908119894and ldquo119899 + 119898rdquo terms in respect of 119908
119894
Thus1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
=
119899+119898
sum
119895=0
119880120572119894(119895) (Δ119905)
119895minus
119899
sum
119895=0
119880120572119894(119895) (Δ119905)
119895
=
119899+119898
sum
119895=119899+1
119880120572119894(119895) (Δ119905)
119895
(A18)
To write down a simpler expression we change the summa-tion index from 119895 to 119901 = 119899 + 119895 So we have from (A18) thefollowing
1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908119894(119905119896+ Δ119905)
Thus the above proceedings tell us that if criterion (A23) issatisfied then the error is bounded In effect this means thatthe scheme is convergent in lieu of Lax equivalence theoremIn our computations Δ119905 has been always chosen to satisfyinequality (A23)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors are thankful to Ondokuz Mayıs UniversitySamsun Turkey for providing financial support to carryout this work under a major research project (Grant nopyofen190113003)
References
[1] W Hundsdorfer and J G Verwer Numerical Solution of TimeDependent Advection-Diffusion-Reaction Equations SpringerBerlin Germany 2003
[2] Y Kuramoto Chemical Oscillations Waves and TurbulenceDover Mineola NY USA 2003
[3] J DMurrayMathematical Biology II vol 18 of InterdisciplinaryApplied Mathematics Springer New York NY USA 3rd edi-tion 2003
[4] H Wilhelmsson and E Lazzaro Reaction-Diffusion Problemsin the Physics of Hot Plasmas Institute of Physics PublishingPhiladelphia Pa USA 2001
[5] M Bar N Gottschalk M Eiswirth and G Ertl ldquoSpiral waves ina surface reaction model calculationsrdquoThe Journal of ChemicalPhysics vol 100 no 2 pp 1202ndash1214 1994
[6] D Barkley ldquoA model for fast computer simulation of waves inexcitable mediardquo Physica D Nonlinear Phenomena vol 49 no1-2 pp 61ndash70 1991
[7] F H Fenton E M Cherry H M Hastings and S J EvansldquoMultiple mechanisms of spiral wave breakup in a model ofcardiac electrical activityrdquo Chaos vol 12 no 3 pp 852ndash8922002
[8] M GosakMMarhl andM Perc ldquoSpatial coherence resonancein excitable biochemical media induced by internal noiserdquoBiophysical Chemistry vol 128 no 2-3 pp 210ndash214 2007
[9] A Karma ldquoMeandering transition in two-dimensionalexcitable mediardquo Physical Review Letters vol 65 no 22 pp2824ndash2827 1990
12 Mathematical Problems in Engineering
[10] J P Keener ldquoA geometrical theory for spiral waves in excitablemediardquo SIAM Journal onAppliedMathematics vol 46 no 6 pp1039ndash1056 1986
[11] J P Keener Mathematical Physiology Interdisciplinary AppliedMathematics Springer New York NY USA 1998
[12] V M Kenkre and M N Kuperman ldquoApplicability of the Fisherequation to bacterial population dynamicsrdquo Physical Review Evol 67 no 5 Article ID 051921 5 pages 2003
[13] V Krinsky and A Pumir ldquoModels of defibrillation of cardiactissuerdquo Chaos vol 8 no 1 pp 188ndash203 1998
[14] N F Otani ldquoA primary mechanism for spiral wave meander-ingrdquo Chaos vol 12 no 3 pp 829ndash842 2002
[15] M Perc ldquoSpatial coherence resonance in excitable mediardquoPhysical Review E vol 72 no 3 Article ID 016207 2005
[16] M Perc ldquoStochastic resonance on excitable small-world net-works via a pacemakerrdquo Physical Review E vol 76 no 6 ArticleID 066203 2007
[17] M Perc ldquoEffects of small-world connectivity on noise-inducedtemporal and spatial order in neural mediardquo Chaos Solitons ampFractals vol 31 no 2 pp 280ndash291 2007
[18] J J Tyson ldquoWhat everyone should know about the Belousov-Zhabotinsky reactionrdquo in Frontiers inMathematical Biology vol100 of Lecture Notes in Biomathematics pp 569ndash587 SpringerBerlin Germany 1994
[19] S Z Rida A M El-Sayed and A A Arafa ldquoOn the solutions oftime-fractional reaction-diffusion equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 15 no 12pp 3847ndash3854 2010
[20] Y Zheng and Z Zhao ldquoA fully discrete Galerkin method fora nonlinear space-fractional diffusion equationrdquo MathematicalProblems in Engineering vol 2011 Article ID 171620 20 pages2011
[21] J K Zhou Differential Transformation and Its Applications forElectrical Circuits Huazhang University Press Wuhan China1986 (Chinese)
[22] I H Abdel-Halim Hassan ldquoComparison differential transfor-mation technique with Adomian decomposition method forlinear and nonlinear initial value problemsrdquoChaos Solitons andFractals vol 36 no 1 pp 53ndash65 2008
[23] A Arikoglu and I Ozkol ldquoSolution of boundary value problemsfor integro-differential equations by using differential transformmethodrdquoAppliedMathematics and Computation vol 168 no 2pp 1145ndash1158 2005
[24] F Ayaz ldquoSolutions of the system of differential equationsby differential transform methodrdquo Applied Mathematics andComputation vol 147 no 2 pp 547ndash567 2004
[25] N Bildik A Konuralp F Orak Bek and S KucukarslanldquoSolution of different type of the partial differential equationby differential transformmethod andAdomianrsquos decompositionmethodrdquo Applied Mathematics and Computation vol 172 no 1pp 551ndash567 2006
[26] H Liu and Y Song ldquoDifferential transform method applied tohigh index differential-algebraic equationsrdquoAppliedMathemat-ics and Computation vol 184 no 2 pp 748ndash753 2007
[27] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent Part IIrdquo Geophysical Journal of theRoyal Astronomical Society vol 13 no 5 pp 529ndash539 1967
[28] SMomani ZOdibat andV S Erturk ldquoGeneralized differentialtransform method for solving a space- and time-fractionaldiffusion-wave equationrdquo Physics Letters A vol 370 no 5-6 pp379ndash387 2007
[29] Z M Odibat C Bertelle M A Aziz-Alaoui and G HDuchamp ldquoA multi-step differential transform method andapplication to non-chaotic or chaotic systemsrdquo Computers ampMathematics with Applications vol 59 no 4 pp 1462ndash14722010
[30] N Laskin ldquoFractional Schrodinger Equationrdquo Physical ReviewE vol 66 no 5 Article ID 056108 2002
[31] G J Fix and J P Roop ldquoLeast squares finite-element solutionof a fractional order two-point boundary value problemrdquoComputers ampMathematics with Applications vol 48 no 7-8 pp1017ndash1033 2004
[32] L-T Yu and C-K Chen ldquoApplication of the hybrid methodto the transient thermal stresses response in isotropic annularfinsrdquo Journal of Applied Mechanics vol 66 no 2 pp 340ndash3471999
[33] B-L Kuo and C-K Chen ldquoApplication of a hybrid methodto the solution of the nonlinear burgersrsquo equationrdquo Journal ofApplied Mechanics Transactions ASME vol 70 no 6 pp 926ndash929 2003
[34] C K Chen H Y Lai and C C Liu ldquoApplication of hybrid dif-ferential transformationfinite difference method to nonlinearanalysis of micro fixed-fixed beamrdquo Microsystem Technologiesvol 15 no 6 pp 813ndash820 2009
[35] N A Khan M Ayaz L Jin and A Yildirim ldquoOn approximatesolutions for the time-fractional reaction-diffusion equation ofFisher typerdquo International Journal of Physical Sciences vol 6 no10 pp 2483ndash2496 2011
[36] M Merdan ldquoSolutions of time-fractional reaction-diffusionequation withmodified Riemann-Liouville derivativerdquo Interna-tional Journal of Physical Sciences vol 7 no 15 pp 2317ndash23262012
[37] C Li and G Peng ldquoChaos in Chenrsquos system with a fractionalorderrdquo Chaos Solitons amp Fractals vol 22 no 2 pp 443ndash4502004
[38] Y Luchko and R Gorenflo ldquoAn operational method for solvingfractional differential equations with the Caputo derivativesrdquoActa Mathematica Vietnamica vol 24 no 2 pp 207ndash233 1999
[39] Z Odibat and S Momani ldquoA generalized differential transformmethod for linear partial differential equations of fractionalorderrdquo Applied Mathematics Letters vol 21 no 2 pp 194ndash1992008
[40] Z M Odibat and N T Shawagfeh ldquoGeneralized Taylorrsquosformulardquo Applied Mathematics and Computation vol 186 no1 pp 286ndash293 2007
[41] D Nazari and S Shahmorad ldquoApplication of the fractionaldifferential transform method to fractional-order integro-differential equations with nonlocal boundary conditionsrdquoJournal of Computational andAppliedMathematics vol 234 no3 pp 883ndash891 2010
[42] D Agırseven and T Ozis ldquoAn analytical study for Fisher typeequations by using homotopy perturbationmethodrdquoComputersamp Mathematics with Applications vol 60 no 3 pp 602ndash6092010
[43] M Dehghan J M Heris and A Saadatmandi ldquoApplicationof semi-analytic methods for the Fitzhugh-Nagumo equation
Mathematical Problems in Engineering 13
which models the transmission of nerve impulsesrdquoMathemati-calMethods in the Applied Sciences vol 33 no 11 pp 1384ndash13982010
[44] A-M Wazwaz and A Gorguis ldquoAn analytic study of Fisherrsquosequation by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 154 no 3 pp 609ndash6202004
[45] M-J Jang C-L Chen and Y-C Liy ldquoOn solving the initial-value problems using the differential transformation methodrdquoAppliedMathematics andComputation vol 115 no 2-3 pp 145ndash160 2000
In view of (A14) we now have1003816100381610038161003816119910119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816 ≐1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
(A15)
Using (A8) and noting that (Δ119905)119899+1 is quite small in (A15)we may take 120573 to be
120573 ≐
1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908119894(119905119896+ Δ119905)
1003816100381610038161003816
(Δ119905)119899+1
(A16)
Thus 120573(Δ119905)119899+1 = 120576 is the bound on the tolerance in the to-be-obtained solution When using different number of terms inthe Taylor series expansion earlier we denoted the solutionsusing a time stepΔ119905 by119908
119894(119905119896+Δ119905) and119908
119894(119905119896+Δ119905) respectively
The paper uses an adaptive step size in computing theresults This is because such a procedure succeeds in keepingthe error bounded and ensures convergence as a consequenceof Lax equivalence theorem To see what the adaptive stepsize produces and to show how such a procedure keepsthe error bounded we start with the premise that Δ119905 isthe most appropriate step size for the problem This stepsize is determined using the definition of inverse differentialtransform
119906119894(Δ119905) =
infin
sum
119896=0
119880120572119894(119896) (Δ119905)
119896 (A17)
In our actual calculation we will not be able to considerinfinite number of terms We consider ldquo119899rdquo terms in respectof 119908119894and ldquo119899 + 119898rdquo terms in respect of 119908
119894
Thus1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908
119894(119905119896+ Δ119905)
1003816100381610038161003816
=
119899+119898
sum
119895=0
119880120572119894(119895) (Δ119905)
119895minus
119899
sum
119895=0
119880120572119894(119895) (Δ119905)
119895
=
119899+119898
sum
119895=119899+1
119880120572119894(119895) (Δ119905)
119895
(A18)
To write down a simpler expression we change the summa-tion index from 119895 to 119901 = 119899 + 119895 So we have from (A18) thefollowing
1003816100381610038161003816119908119894 (119905119896 + Δ119905) minus 119908119894(119905119896+ Δ119905)
Thus the above proceedings tell us that if criterion (A23) issatisfied then the error is bounded In effect this means thatthe scheme is convergent in lieu of Lax equivalence theoremIn our computations Δ119905 has been always chosen to satisfyinequality (A23)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors are thankful to Ondokuz Mayıs UniversitySamsun Turkey for providing financial support to carryout this work under a major research project (Grant nopyofen190113003)
References
[1] W Hundsdorfer and J G Verwer Numerical Solution of TimeDependent Advection-Diffusion-Reaction Equations SpringerBerlin Germany 2003
[2] Y Kuramoto Chemical Oscillations Waves and TurbulenceDover Mineola NY USA 2003
[3] J DMurrayMathematical Biology II vol 18 of InterdisciplinaryApplied Mathematics Springer New York NY USA 3rd edi-tion 2003
[4] H Wilhelmsson and E Lazzaro Reaction-Diffusion Problemsin the Physics of Hot Plasmas Institute of Physics PublishingPhiladelphia Pa USA 2001
[5] M Bar N Gottschalk M Eiswirth and G Ertl ldquoSpiral waves ina surface reaction model calculationsrdquoThe Journal of ChemicalPhysics vol 100 no 2 pp 1202ndash1214 1994
[6] D Barkley ldquoA model for fast computer simulation of waves inexcitable mediardquo Physica D Nonlinear Phenomena vol 49 no1-2 pp 61ndash70 1991
[7] F H Fenton E M Cherry H M Hastings and S J EvansldquoMultiple mechanisms of spiral wave breakup in a model ofcardiac electrical activityrdquo Chaos vol 12 no 3 pp 852ndash8922002
[8] M GosakMMarhl andM Perc ldquoSpatial coherence resonancein excitable biochemical media induced by internal noiserdquoBiophysical Chemistry vol 128 no 2-3 pp 210ndash214 2007
[9] A Karma ldquoMeandering transition in two-dimensionalexcitable mediardquo Physical Review Letters vol 65 no 22 pp2824ndash2827 1990
12 Mathematical Problems in Engineering
[10] J P Keener ldquoA geometrical theory for spiral waves in excitablemediardquo SIAM Journal onAppliedMathematics vol 46 no 6 pp1039ndash1056 1986
[11] J P Keener Mathematical Physiology Interdisciplinary AppliedMathematics Springer New York NY USA 1998
[12] V M Kenkre and M N Kuperman ldquoApplicability of the Fisherequation to bacterial population dynamicsrdquo Physical Review Evol 67 no 5 Article ID 051921 5 pages 2003
[13] V Krinsky and A Pumir ldquoModels of defibrillation of cardiactissuerdquo Chaos vol 8 no 1 pp 188ndash203 1998
[14] N F Otani ldquoA primary mechanism for spiral wave meander-ingrdquo Chaos vol 12 no 3 pp 829ndash842 2002
[15] M Perc ldquoSpatial coherence resonance in excitable mediardquoPhysical Review E vol 72 no 3 Article ID 016207 2005
[16] M Perc ldquoStochastic resonance on excitable small-world net-works via a pacemakerrdquo Physical Review E vol 76 no 6 ArticleID 066203 2007
[17] M Perc ldquoEffects of small-world connectivity on noise-inducedtemporal and spatial order in neural mediardquo Chaos Solitons ampFractals vol 31 no 2 pp 280ndash291 2007
[18] J J Tyson ldquoWhat everyone should know about the Belousov-Zhabotinsky reactionrdquo in Frontiers inMathematical Biology vol100 of Lecture Notes in Biomathematics pp 569ndash587 SpringerBerlin Germany 1994
[19] S Z Rida A M El-Sayed and A A Arafa ldquoOn the solutions oftime-fractional reaction-diffusion equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 15 no 12pp 3847ndash3854 2010
[20] Y Zheng and Z Zhao ldquoA fully discrete Galerkin method fora nonlinear space-fractional diffusion equationrdquo MathematicalProblems in Engineering vol 2011 Article ID 171620 20 pages2011
[21] J K Zhou Differential Transformation and Its Applications forElectrical Circuits Huazhang University Press Wuhan China1986 (Chinese)
[22] I H Abdel-Halim Hassan ldquoComparison differential transfor-mation technique with Adomian decomposition method forlinear and nonlinear initial value problemsrdquoChaos Solitons andFractals vol 36 no 1 pp 53ndash65 2008
[23] A Arikoglu and I Ozkol ldquoSolution of boundary value problemsfor integro-differential equations by using differential transformmethodrdquoAppliedMathematics and Computation vol 168 no 2pp 1145ndash1158 2005
[24] F Ayaz ldquoSolutions of the system of differential equationsby differential transform methodrdquo Applied Mathematics andComputation vol 147 no 2 pp 547ndash567 2004
[25] N Bildik A Konuralp F Orak Bek and S KucukarslanldquoSolution of different type of the partial differential equationby differential transformmethod andAdomianrsquos decompositionmethodrdquo Applied Mathematics and Computation vol 172 no 1pp 551ndash567 2006
[26] H Liu and Y Song ldquoDifferential transform method applied tohigh index differential-algebraic equationsrdquoAppliedMathemat-ics and Computation vol 184 no 2 pp 748ndash753 2007
[27] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent Part IIrdquo Geophysical Journal of theRoyal Astronomical Society vol 13 no 5 pp 529ndash539 1967
[28] SMomani ZOdibat andV S Erturk ldquoGeneralized differentialtransform method for solving a space- and time-fractionaldiffusion-wave equationrdquo Physics Letters A vol 370 no 5-6 pp379ndash387 2007
[29] Z M Odibat C Bertelle M A Aziz-Alaoui and G HDuchamp ldquoA multi-step differential transform method andapplication to non-chaotic or chaotic systemsrdquo Computers ampMathematics with Applications vol 59 no 4 pp 1462ndash14722010
[30] N Laskin ldquoFractional Schrodinger Equationrdquo Physical ReviewE vol 66 no 5 Article ID 056108 2002
[31] G J Fix and J P Roop ldquoLeast squares finite-element solutionof a fractional order two-point boundary value problemrdquoComputers ampMathematics with Applications vol 48 no 7-8 pp1017ndash1033 2004
[32] L-T Yu and C-K Chen ldquoApplication of the hybrid methodto the transient thermal stresses response in isotropic annularfinsrdquo Journal of Applied Mechanics vol 66 no 2 pp 340ndash3471999
[33] B-L Kuo and C-K Chen ldquoApplication of a hybrid methodto the solution of the nonlinear burgersrsquo equationrdquo Journal ofApplied Mechanics Transactions ASME vol 70 no 6 pp 926ndash929 2003
[34] C K Chen H Y Lai and C C Liu ldquoApplication of hybrid dif-ferential transformationfinite difference method to nonlinearanalysis of micro fixed-fixed beamrdquo Microsystem Technologiesvol 15 no 6 pp 813ndash820 2009
[35] N A Khan M Ayaz L Jin and A Yildirim ldquoOn approximatesolutions for the time-fractional reaction-diffusion equation ofFisher typerdquo International Journal of Physical Sciences vol 6 no10 pp 2483ndash2496 2011
[36] M Merdan ldquoSolutions of time-fractional reaction-diffusionequation withmodified Riemann-Liouville derivativerdquo Interna-tional Journal of Physical Sciences vol 7 no 15 pp 2317ndash23262012
[37] C Li and G Peng ldquoChaos in Chenrsquos system with a fractionalorderrdquo Chaos Solitons amp Fractals vol 22 no 2 pp 443ndash4502004
[38] Y Luchko and R Gorenflo ldquoAn operational method for solvingfractional differential equations with the Caputo derivativesrdquoActa Mathematica Vietnamica vol 24 no 2 pp 207ndash233 1999
[39] Z Odibat and S Momani ldquoA generalized differential transformmethod for linear partial differential equations of fractionalorderrdquo Applied Mathematics Letters vol 21 no 2 pp 194ndash1992008
[40] Z M Odibat and N T Shawagfeh ldquoGeneralized Taylorrsquosformulardquo Applied Mathematics and Computation vol 186 no1 pp 286ndash293 2007
[41] D Nazari and S Shahmorad ldquoApplication of the fractionaldifferential transform method to fractional-order integro-differential equations with nonlocal boundary conditionsrdquoJournal of Computational andAppliedMathematics vol 234 no3 pp 883ndash891 2010
[42] D Agırseven and T Ozis ldquoAn analytical study for Fisher typeequations by using homotopy perturbationmethodrdquoComputersamp Mathematics with Applications vol 60 no 3 pp 602ndash6092010
[43] M Dehghan J M Heris and A Saadatmandi ldquoApplicationof semi-analytic methods for the Fitzhugh-Nagumo equation
Mathematical Problems in Engineering 13
which models the transmission of nerve impulsesrdquoMathemati-calMethods in the Applied Sciences vol 33 no 11 pp 1384ndash13982010
[44] A-M Wazwaz and A Gorguis ldquoAn analytic study of Fisherrsquosequation by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 154 no 3 pp 609ndash6202004
[45] M-J Jang C-L Chen and Y-C Liy ldquoOn solving the initial-value problems using the differential transformation methodrdquoAppliedMathematics andComputation vol 115 no 2-3 pp 145ndash160 2000
[10] J P Keener ldquoA geometrical theory for spiral waves in excitablemediardquo SIAM Journal onAppliedMathematics vol 46 no 6 pp1039ndash1056 1986
[11] J P Keener Mathematical Physiology Interdisciplinary AppliedMathematics Springer New York NY USA 1998
[12] V M Kenkre and M N Kuperman ldquoApplicability of the Fisherequation to bacterial population dynamicsrdquo Physical Review Evol 67 no 5 Article ID 051921 5 pages 2003
[13] V Krinsky and A Pumir ldquoModels of defibrillation of cardiactissuerdquo Chaos vol 8 no 1 pp 188ndash203 1998
[14] N F Otani ldquoA primary mechanism for spiral wave meander-ingrdquo Chaos vol 12 no 3 pp 829ndash842 2002
[15] M Perc ldquoSpatial coherence resonance in excitable mediardquoPhysical Review E vol 72 no 3 Article ID 016207 2005
[16] M Perc ldquoStochastic resonance on excitable small-world net-works via a pacemakerrdquo Physical Review E vol 76 no 6 ArticleID 066203 2007
[17] M Perc ldquoEffects of small-world connectivity on noise-inducedtemporal and spatial order in neural mediardquo Chaos Solitons ampFractals vol 31 no 2 pp 280ndash291 2007
[18] J J Tyson ldquoWhat everyone should know about the Belousov-Zhabotinsky reactionrdquo in Frontiers inMathematical Biology vol100 of Lecture Notes in Biomathematics pp 569ndash587 SpringerBerlin Germany 1994
[19] S Z Rida A M El-Sayed and A A Arafa ldquoOn the solutions oftime-fractional reaction-diffusion equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 15 no 12pp 3847ndash3854 2010
[20] Y Zheng and Z Zhao ldquoA fully discrete Galerkin method fora nonlinear space-fractional diffusion equationrdquo MathematicalProblems in Engineering vol 2011 Article ID 171620 20 pages2011
[21] J K Zhou Differential Transformation and Its Applications forElectrical Circuits Huazhang University Press Wuhan China1986 (Chinese)
[22] I H Abdel-Halim Hassan ldquoComparison differential transfor-mation technique with Adomian decomposition method forlinear and nonlinear initial value problemsrdquoChaos Solitons andFractals vol 36 no 1 pp 53ndash65 2008
[23] A Arikoglu and I Ozkol ldquoSolution of boundary value problemsfor integro-differential equations by using differential transformmethodrdquoAppliedMathematics and Computation vol 168 no 2pp 1145ndash1158 2005
[24] F Ayaz ldquoSolutions of the system of differential equationsby differential transform methodrdquo Applied Mathematics andComputation vol 147 no 2 pp 547ndash567 2004
[25] N Bildik A Konuralp F Orak Bek and S KucukarslanldquoSolution of different type of the partial differential equationby differential transformmethod andAdomianrsquos decompositionmethodrdquo Applied Mathematics and Computation vol 172 no 1pp 551ndash567 2006
[26] H Liu and Y Song ldquoDifferential transform method applied tohigh index differential-algebraic equationsrdquoAppliedMathemat-ics and Computation vol 184 no 2 pp 748ndash753 2007
[27] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent Part IIrdquo Geophysical Journal of theRoyal Astronomical Society vol 13 no 5 pp 529ndash539 1967
[28] SMomani ZOdibat andV S Erturk ldquoGeneralized differentialtransform method for solving a space- and time-fractionaldiffusion-wave equationrdquo Physics Letters A vol 370 no 5-6 pp379ndash387 2007
[29] Z M Odibat C Bertelle M A Aziz-Alaoui and G HDuchamp ldquoA multi-step differential transform method andapplication to non-chaotic or chaotic systemsrdquo Computers ampMathematics with Applications vol 59 no 4 pp 1462ndash14722010
[30] N Laskin ldquoFractional Schrodinger Equationrdquo Physical ReviewE vol 66 no 5 Article ID 056108 2002
[31] G J Fix and J P Roop ldquoLeast squares finite-element solutionof a fractional order two-point boundary value problemrdquoComputers ampMathematics with Applications vol 48 no 7-8 pp1017ndash1033 2004
[32] L-T Yu and C-K Chen ldquoApplication of the hybrid methodto the transient thermal stresses response in isotropic annularfinsrdquo Journal of Applied Mechanics vol 66 no 2 pp 340ndash3471999
[33] B-L Kuo and C-K Chen ldquoApplication of a hybrid methodto the solution of the nonlinear burgersrsquo equationrdquo Journal ofApplied Mechanics Transactions ASME vol 70 no 6 pp 926ndash929 2003
[34] C K Chen H Y Lai and C C Liu ldquoApplication of hybrid dif-ferential transformationfinite difference method to nonlinearanalysis of micro fixed-fixed beamrdquo Microsystem Technologiesvol 15 no 6 pp 813ndash820 2009
[35] N A Khan M Ayaz L Jin and A Yildirim ldquoOn approximatesolutions for the time-fractional reaction-diffusion equation ofFisher typerdquo International Journal of Physical Sciences vol 6 no10 pp 2483ndash2496 2011
[36] M Merdan ldquoSolutions of time-fractional reaction-diffusionequation withmodified Riemann-Liouville derivativerdquo Interna-tional Journal of Physical Sciences vol 7 no 15 pp 2317ndash23262012
[37] C Li and G Peng ldquoChaos in Chenrsquos system with a fractionalorderrdquo Chaos Solitons amp Fractals vol 22 no 2 pp 443ndash4502004
[38] Y Luchko and R Gorenflo ldquoAn operational method for solvingfractional differential equations with the Caputo derivativesrdquoActa Mathematica Vietnamica vol 24 no 2 pp 207ndash233 1999
[39] Z Odibat and S Momani ldquoA generalized differential transformmethod for linear partial differential equations of fractionalorderrdquo Applied Mathematics Letters vol 21 no 2 pp 194ndash1992008
[40] Z M Odibat and N T Shawagfeh ldquoGeneralized Taylorrsquosformulardquo Applied Mathematics and Computation vol 186 no1 pp 286ndash293 2007
[41] D Nazari and S Shahmorad ldquoApplication of the fractionaldifferential transform method to fractional-order integro-differential equations with nonlocal boundary conditionsrdquoJournal of Computational andAppliedMathematics vol 234 no3 pp 883ndash891 2010
[42] D Agırseven and T Ozis ldquoAn analytical study for Fisher typeequations by using homotopy perturbationmethodrdquoComputersamp Mathematics with Applications vol 60 no 3 pp 602ndash6092010
[43] M Dehghan J M Heris and A Saadatmandi ldquoApplicationof semi-analytic methods for the Fitzhugh-Nagumo equation
Mathematical Problems in Engineering 13
which models the transmission of nerve impulsesrdquoMathemati-calMethods in the Applied Sciences vol 33 no 11 pp 1384ndash13982010
[44] A-M Wazwaz and A Gorguis ldquoAn analytic study of Fisherrsquosequation by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 154 no 3 pp 609ndash6202004
[45] M-J Jang C-L Chen and Y-C Liy ldquoOn solving the initial-value problems using the differential transformation methodrdquoAppliedMathematics andComputation vol 115 no 2-3 pp 145ndash160 2000
which models the transmission of nerve impulsesrdquoMathemati-calMethods in the Applied Sciences vol 33 no 11 pp 1384ndash13982010
[44] A-M Wazwaz and A Gorguis ldquoAn analytic study of Fisherrsquosequation by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 154 no 3 pp 609ndash6202004
[45] M-J Jang C-L Chen and Y-C Liy ldquoOn solving the initial-value problems using the differential transformation methodrdquoAppliedMathematics andComputation vol 115 no 2-3 pp 145ndash160 2000
