Research ArticleA Hybrid Natural Transform Homotopy Perturbation Methodfor Solving Fractional Partial Differential Equations

Shehu Maitama

Department of Mathematics Faculty of Science Northwest University Kano Nigeria

Correspondence should be addressed to Shehu Maitama smusman12scijustedujo

Received 1 July 2016 Revised 23 August 2016 Accepted 4 September 2016

Academic Editor Nasser-Eddine Tatar

Copyright copy 2016 Shehu Maitama This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A hybrid analytical method for solving linear and nonlinear fractional partial differential equations is presented The proposedanalytical approach is an elegant combination of the Natural Transform Method (NTM) and a well-known method HomotopyPerturbation Method (HPM) In this analytical method the fractional derivative is computed in Caputo sense and the nonlinearterm is calculated using Hersquos polynomial The proposed analytical method reduces the computational size and avoids round-offerrors Exact solution of linear and nonlinear fractional partial differential equations is successfully obtained using the analyticalmethod

1 Introduction

The concept of fractional calculus which deals with deriva-tives and integrals of arbitrary orders [1] plays a significantrole in many areas of physical science and engineeringRecently there is a rapid development in the concept offractional calculus and its applications [2ndash5] The linear andnonlinear fractional differential equations are used to modelsignificant problems in fluid mechanics acoustic electro-magnetism signal processing analytical chemistry biologyandmany other areas of physical science and engineering [6]In recent years many analytical and numerical methods havebeen used to solve linear and nonlinear fractional differentialequations such as Adomian Decomposition Method [7ndash10] Homotopy Analysis Method [11] Laplace Decomposi-tion Method [12] Homotopy Perturbation Method [13 14]and Yang Laplace transform [15] Moreover local fractionalvariational iteration method [16 17] Cylindrical-Coordinatemethod [18] modified Laplace Decomposition Method [1219] and fractional complex transform method [20] are alsoapplied to linear and nonlinear fractional partial differentialequationsThe asymptotic behavior of solutions of a weightedCauchy-type nonlinear fractional partial differential equa-tions is studied in [21ndash23]

It is evident that most of the existing methods haveso many deficiencies such as unnecessary linearization

discretization of variables transformation or taking somerestrictive assumptions

In this paper an analytical method called a Hybrid Nat-ural Transform Homotopy Perturbation Method for solvinglinear and nonlinear fractional partial differential equationsis introduced The proposed analytical method is applieddirectly without any linearization transformation discretiza-tion of variables and so on The analytical method givesa series solution which converges rapidly to an exact orapproximate solution with elegant computational terms Thenonlinear terms are elegantly computed usingHersquos polynomi-als [24ndash26] Exact solution of linear and nonlinear fractionalpartial differential equation is successfully obtained using thenew analytical method Thus the Hybrid Natural TransformHomotopy Perturbation Method is a powerful mathematicalmethod for solving linear and nonlinear fractional partialdifferential equations and is a refinement of the existingmethods

2 Natural Transform

In this section the basic definition and properties of theNatural Transform are presented

In the year 2008 Z H Khan and W A Khan [27]introduced an integral transform called 119873-transform and it

Hindawi Publishing CorporationInternational Journal of Differential EquationsVolume 2016 Article ID 9207869 7 pageshttpdxdoiorg10115520169207869

2 International Journal of Differential Equations

Table 1 List of some special natural transforms

Functional form Natural transform form

1 1

119904

119905119906

1199042

119890119886119905

1

119904 minus 119886119906

119905119899minus1

(119899 minus 1) 119899 = 1 2 119906

119899minus1

119904119899

sin(119905) 119906

1199042 + 1199062

was recently renamed as the Natural Transform by Belgacemand Silambarasan [28 29] In fact based on personal com-munications and Internet records CASM in LUMS and IOSSForman Christian College in Lahore lectures in 2004 and2005 and 3rd ICM4F COMSATs University conference sidelectures in Islamabad in 2006 indicate that Belgacem sharedand discussed various aspects of this transform with thevarious attending audiences The Natural Transform is anintegral transform which is similar to Laplace transform[30] and Sumudu integral transform [31 32] It converges toLaplace transform when 119906 = 1 and Sumudu transform when119904 = 1 Belgacem and Silambarasan [29 33] have proposeda detailed theory and applications of the Natural TransformRecently Natural Transform and Adomian DecompositionMethod are successfully combined and obtained an exactsolution of linear and nonlinear partial differential equations[34ndash39] More details about the Natural Transform and itsapplications can be seen inWikipedia note about the NaturalTransform [40]

Some useful Natural Transforms in this paper are pre-sented in Table 1

Definition 1 The Natural Transform of the function V(119905) gt 0and V(119905) = 0 for 119905 lt 0 is defined over the set of functions

119860 = V (119905) exist119872 1205911 1205912gt 0 |V (119905)| lt 119872119890|119905|120591119895 if 119905

isin (minus1)119895times [0infin) 119895 = 1 2

(1)

by the following integral

N+[V (119905)] = 119881 (119904 119906) =

1

119906int

infin

0

119890minus119904119905119906V (119905) 119889119905

119904 gt 0 119906 gt 0

(2)

And the inverse Natural Transform of the function V(119905) isdefined by

Nminus1[119881 (119904 119906)] = V (119905) =

1

2120587119894int

120574+119894infin

120574minus119894infin

119890119904119905119906119881 (119904 119906) 119889119904 (3)

where 119904 and 119906 are the Natural Transform variables [28 29]and 120574 is a real constant and the integral in (3) is taken along119904 = 120574 in the complex plane 119904 = 119909 + 119894119910

Some properties of the Natural Transform Method aregiven below

Property 1 Natural Transform of derivative if V119899(119905) is the 119899thderivative of the function V(119905) isin 119860 with respect to ldquo119905rdquo thenits Natural Transform is given by

N+[V119899 (119905)] =

119904119899

119906119899119881 (119904 119906) minus

119899minus1

sum

119896=0

119904119899minus(119896+1)

119906119899minus119896V119896 (0) (4)

When 119899 = 1 2 and 3 we have the following results

N+[V1015840 (119905)] =

119904

119906119881 (119904 119906) minus

1

119906V (0)

N+[V10158401015840 (119905)] =

1199042

1199062119881 (119904 119906) minus

119904

1199062V (0) minus

1

119906V1015840 (0)

N+[V101584010158401015840 (119905)] =

1199043

1199063119881 (119904 119906) minus

1199042

1199063V (0) minus

119904

1199062V1015840 (0)

minus1

119906V10158401015840 (0)

(5)

Property 2 If119881(119904 119906) is the Natural Transform and 119865(119904) is theLaplace transform of the function 119891(119905) isin 119860 then N+[119891(119905)] =

119881(119904 119906) = (1119906) intinfin

0119890minus119904119905119906

119891(119905)119889119905 = (1119906)119865(119904119906)

Property 3 If119881(119904 119906) is the Natural Transform and119866(119906) is theSumudu transform of the function V(119905) isin 119860 then N+[V(119905)] =119881(119904 119906) = (1119904) int

infin

0119890minus119905 V(119906119905119904)119889119905 = (1119904)119866(119906119904)

Property 4 IfN+[V(119905)] = 119881(119904 119906) thenN+[V(120573119905)] = (1120573)119881(119904120573 119906)

Property 5 The Natural Transform is a linear operator thatis if 120572 and 120573 are nonzero constants then

N+[120572119891 (119905) plusmn 120573119892 (119905)] = 120572N

+[119891 (119905)] plusmn 120573N

+[119892 (119905)]

= 120572119865+(119904 119906) plusmn 120573119866

+(119904 119906)

(6)

Moreover119865+(119904 119906) and119866+(119904 119906) are theNatural Transforms offunctions 119891(119905) and 119892(119905) respectively

3 Basic Definitions of Fractional Calculus

In this section the basic definitions of fractional calculus arepresented

Definition 2 Function 119891(119905) 119905 gt 0 is said to be in the space119862119898

120572119898 isin 119873 cup 0 if 119891(119898) isin 119862

120572

Definition 3 A real function 119891(119905) 119905 gt 0 is said to be in thespace 119862

120572120572 isin R if there exists a real number 119901 (gt120572) such that

119891(119905) = 1199051199011198911(119905) where 119891

1(119905) isin 119862[0infin) Clearly 119862

120572sub 119862120573if

120573 le 120572

International Journal of Differential Equations 3

Definition 4 The left sided Riemann-Liouville fractionalintegral operator of order 120583 gt 0 of a function 119891(119905) isin 119862

120572

and 120572 ge minus1 is defined as [19 41]

119868120583119891 (119905)

=

1

Γ (120583)int

119905

0

(119905 minus 120591)120583minus1

119891 (120591) 119889120591 120583 gt 0 119905 gt 0

119891 (119905) 120583 = 0

(7)

where Γ(sdot) is the well-known Gamma function

Definition 5 The left sided Caputo fractional derivative 119891119891 isin 119862

119898

1119898 isin N cup 0 is defined as [1 5]

119863120583

lowast119891 (119905)

=

119868119898minus120583

[120597119898119891 (119905)

120597119905119898] 119898 minus 1 lt 120583 lt 119898 119898 isin N

120597119898119891 (119905)

120597119905119898 120583 = 119898

(8)

Note that [1 5]

(i) 119868120583119905119891(119905) = (1Γ(120583)) int

119905

0(119891(119905)(119905 minus 119904)

1minus120583)119889119905 120583 gt 0 119905 gt 0

(ii) 119863120583lowast119891(119909 119905) = 119868

119898minus120583

119905[120597119898119891(119905)120597119905

119898]119898 minus 1 lt 120583 le 119898

Definition 6 TheNatural Transform of the Caputo fractionalderivative is defined as

N+[119863119899120572

119905V (119905)] =

119904119899120572

119906119899120572119881 (119904 119906) minus

119899minus1

sum

119896=0

119904119899120572minus(119896+1)

119906119899120572minus119896V(119896) (0+) (9)

((119899 minus 1)119899 lt 120572 le 1)

Definition 7 The series expansion defines a one-parameterMittag-Leffler function as [1]

119864120572(119911) =

infin

sum

119896=0

119911119896

Γ (120572119896 + 1) 120572 gt 0 119911 isin C (10)

4 Analysis of the Method

In this section the basic idea of theHybridNatural TransformHomotopy Perturbation Method is clearly illustrated by thestandard nonlinear fractional partial differential equation ofthe form

119863119899120572

119905V (119909 119905) + 119872 (V (119909 119905)) + 119865 (V (119909 119905)) = 119892 (119909 119905) (11)

subject to the initial condition

V (119909 0) = 119891 (119909) (12)

where 119865(V(119909 119905)) represents the nonlinear terms 119863119899120572119905

=

120597119899120572120597119905119899120572 is the Caputo fractional derivative of function V(119905)

119872(V(119909 119905)) is the linear differential operator and 119892(119909 119905) is asource term

Applying the Natural Transform to (11) subject to thegiven initial condition we get

119881 (119909 119904 119906) =1

119904119891 (119909) +

119906119899120572

119904119899120572N+[119892 (119909 119905)]

minus119906119899120572

119904119899120572N+[119872 (V (119909 119905)) + 119865 (V (119909 119905))]

(13)

Taking the inverse Natural Transform of (13) we get

V (119909 119905)

= 119866 (119909 119905)

minus Nminus1[119906119899120572

119904119899120572N+[119872 (V (119909 119905)) + 119865 (V (119909 119905))]]

(14)

where 119866(119909 119905) is a term arising from the source term and theprescribed initial condition

Now we apply the Homotopy Perturbation Method

V (119909 119905) =infin

sum

119899=0

119901119899V119899(119909 119905) (15)

The nonlinear term 119865(V(119909 119905)) is decomposed as

119865 (V (119909 119905)) =infin

sum

119899=0

119901119899119867119899(V) (16)

where119867119899(V) is Hersquos polynomial which is computed using the

following formula

119867119899(V1 V2 V

119899) =

1

119899

120597119899

120597119901119899[

[

119865(

119899

sum

119895=0

119901119895V119895)]

]119901=0

119899 = 0 1 2

(17)

Substituting (16) and (15) into (14) we get

infin

sum

119899=0

119901119899V119899(119909 119905) = 119866 (119909 119905) minus 119901(N

minus1[119906119899120572

119904119899120572

sdot N+[

infin

sum

119899=0

119901119899119872(V (119909 119905)) +

infin

sum

119899=0

119901119899119867119899(V)]])

(18)

Using the coefficient of like powers of 119901 in (18) we obtainedthe following approximations

1199010 V0(119909 119905) = 119866 (119909 119905)

1199011 V1(119909 119905)

= minusNminus1[119906119899120572

119904119899120572N+[119872 (V (119909 119905)) + 119867

0(V)]]

4 International Journal of Differential Equations

1199012 V2(119909 119905)

= minusNminus1[119906119899120572

119904119899120572N+[119872 (V (119909 119905)) + 119867

1(V)]]

1199013 V3(119909 119905)

= minusNminus1[119906119899120572

119904119899120572N+[119872 (V (119909 119905)) + 119867

2(V)]]

(19)

and so onHence the series solution of (11)-(12) is given by

V (119909 119905) = lim119873rarrinfin

119873

sum

119899=0

V119899(119909 119905) (20)

5 Applications

In this section the application of the Hybrid NaturalTransform Homotopy Perturbation Method to linear andnonlinear fractional partial differential equations is clearlydemonstrated to show its simplicity and high accuracy

Example 8 Consider the following fractional diffusion equa-tion of the form119863120572

119905V + V119909119909+ V119910119910+ V119911119911= 0

minusinfin lt 119909 119910 119911 lt infin 119905 gt 0

(21)

subject to the initial condition

V (119909 119910 119911 0) = 119890119909+119910+119911 120572 isin (0 1) (22)

Applying the Natural Transform to (21) subject to the giveninitial condition we get

119881 (119909 119910 119911 119904 119906) =119890119909+119910+119911

119904+119906120572

119904120572N+[V119909119909+ V119910119910+ V119911119911] (23)

Taking the inverse Natural Transform of (23) we get

V (119909 119910 119911 119905) = 119890119909+119910+119911

+ Nminus1[119906120572

119904120572N+[V119909119909+ V119910119910+ V119911119911]]

(24)

Now we apply Homotopy Perturbation Method

V (119909 119910 119911 119905) =infin

sum

119899=0

119901119899V119899(119909 119910 119911 119905) (25)

Then (24) will becomeinfin

sum

119899=0

119901119899V119899(119909 119910 119911 119905) = 119890

119909+119910+119911minus 119901(N

minus1[119906120572

119904120572

sdot N+[

infin

sum

119899=0

119901119899V119899119909119909

+

infin

sum

119899=0

119901119899V119899119910119910

+

infin

sum

119899=0

119901119899V119899119911119911]])

(26)

Using the coefficients of like powers of 119901 in (26) we obtainedthe following approximations

1199010 V0(119909 119910 119911 119905) = 119890

119909+119910+119911

1199011 V1(119909 119910 119911 119905)

= minusNminus1[119906120572

119904120572N+[V0119909119909

+ V0119910119910

+ V0119911119911]]

= minus119890119909+119910+119911

119905120572

Γ (120572 + 1)

1199012 V2(119909 119910 119911 119905)

= minusNminus1[119906120572

119904120572N+[V1119909119909

+ V1119910119910

+ V1119911119911]]

=119890119909+119910+119911

1199052120572

Γ (2120572 + 1)

1199013 V3(119909 119910 119911 119905)

= minusNminus1[119906120572

119904120572N+[V2119909119909

+ V2119910119910

+ V2119911119911]]

= minus119890119909+119910+119911

1199053120572

Γ (3120572 + 1)

(27)

and so onThen the series solution of (21)-(22) is given by

V (119909 119910 119911 119905) = lim119873rarrinfin

119873

sum

119899=0

V119899(119909 119910 119911 119905) = V

0(119909 119910 119911 119905)

+ V1(119909 119910 119911 119905) + V

2(119909 119910 119911 119905) + V

3(119909 119910 119911 119905) + sdot sdot sdot

= 119890119909+119910+119911

(1 minus119905120572

Γ (120572 + 1)+

1199052120572

Γ (2120572 + 1)

minus1199053120572

Γ (3120572 + 1)+ sdot sdot sdot) = 119890

119909+119910+119911(1

+

infin

sum

119898=1

(minus119905120572)119898

Γ (119898120572 + 1)) = 119890119909+119910+119911

119864120572(minus119905120572)

(28)

When 120572 = 1 the following result is obtained

V (119909 119910 119911 119905) = lim119873rarrinfin

119873

sum

119899=0

V119899(119909 119910 119911 119905)

= V0(119909 119910 119911 119905) + V

1(119909 119910 119911 119905)

+ V2(119909 119910 119911 119905) + V

3(119909 119910 119911 119905) + sdot sdot sdot

= 119890119909+119910+119911

(1 minus119905

1+1199052

2minus1199053

3+ sdot sdot sdot)

= 119890119909+119910+119911minus119905

(29)

which is the exact solution of (21)-(22) when 120572 = 1

International Journal of Differential Equations 5

Example 9 Consider the following coupled system of non-linear fractional partial differential equations of the form

119863120572

119905V minus V119909119909minus 2VV119909+ V119909119908119909= 0

119863120572

119905119908 minus 119908

119909119909minus 2119908119908

119909+ V119909119908119909= 0

(30)

subject to the initial conditions

V (119909 0) = sin (119909)

119908 (119909 0) = sin (119909) (31)

Applying the Natural Transform to (30) subject to giveninitial conditions we get

119881 (119909 119904 119906) =1

119904sin (119909) + 119906

120572

119904120572N+[V119909119909+ 2VV119909minus V119909119908119909]

119882 (119909 119904 119906) =1

119904sin (119909)

+119906120572

119904120572N+[119908119909119909+ 2119908119908

119909minus V119909119908119909]

(32)

Taking the inverse Natural Transform of (32) we have

V (119909 119905) = sin (119909)

+ Nminus1[119906120572

119904120572N+[V119909119909+ 2VV119909minus V119909119908119909]]

119908 (119909 119905) = sin (119909)

+ Nminus1[119906120572

119904120572N+[119908119908119909119909+ 2119908119908

119909minus V119909119908119909]]

(33)

Now we apply the Homotopy Perturbation Method

V (119909 119905) =infin

sum

119899=0

119901119899V119899(119909 119905)

119908 (119909 119905) =

infin

sum

119899=0

119901119899119908119899(119909 119905)

(34)

Then (33) will become

V (119909 119905) = sin (119909) + 119901(Nminus1 [119906120572

119904120572

sdot N+[

infin

sum

119899=0

119901119899V119899119909119909

+ 2

infin

sum

119899=0

119901119899119867119899minus

infin

sum

119899=0

11990111989911986710158401015840

119899]])

119908 (119909 119905) = sin (119909) + 119901(Nminus1 [119906120572

119904120572

sdot N+[

infin

sum

119899=0

119901119899119908119899119909119909

+ 2

infin

sum

119899=0

1199011198991198671015840

119899minus

infin

sum

119899=0

11990111989911986710158401015840

119899]])

(35)

where1198671198991198671015840119899 and11986710158401015840

119899are Hersquos polynomials which represent

the nonlinear terms VV119909 119908119908119909 and V

119909119908119909 respectively

Some few components of Hersquos polynomials of the nonlin-ear terms VV

119909 119908119908119909 and V

119909119908119909are given as follows

1198670= V0V0119909

1198671= V0V1119909+ V1V0119909

1198672= V0V2119909+ V1V1119909+ V2V0119909

1198671015840

0= 11990801199080119909

1198671015840

1= 11990801199081119909+ 11990811199080119909

1198671015840

2= 11990801199082119909+ 11990811199081119909+ 11990821199080119909

11986710158401015840

0= V01199091199080119909

11986710158401015840

1= V01199091199081119909+ V11199091199080119909

11986710158401015840

2= V21199091199080119909+ V11199091199081119909+ V01199091199082119909

(36)

and so onUsing the coefficients of the like powers of 119901 in (35) we

obtained the following approximations

1199010 V0(119909 119905) = sin (119909)

1199011 V1(119909 119905) = N

minus1[119906120572

119904120572N+[V0119909119909

+ 21198670minus 11986710158401015840

0]]

= minussin (119909) 119905120572

Γ (120572 + 1)

1199012 V2(119909 119905) = N

minus1[119906120572

119904120572N+[V1119909119909

+ 21198671minus 11986710158401015840

1]]

=sin (119909) 1199052120572

Γ (2120572 + 1)

1199013 V3(119909 119905) = N

minus1[119906120572

119904120572N+[V2119909119909

+ 21198672minus 11986710158401015840

2]]

= minussin (119909) 1199053120572

Γ (3120572 + 1)

(37)

and so onSimilarly

1199010 1199080(119909 119905) = sin (119909)

1199011 1199081(119909 119905) = N

minus1[119906120572

119904120572N+[1199080119909119909

+ 21198671015840

0minus 11986710158401015840

0]]

= minussin (119909) 119905120572

Γ (120572 + 1)

6 International Journal of Differential Equations

1199012 1199082(119909 119905) = N

minus1[119906120572

119904120572N+[1199081119909119909

+ 21198671015840

1minus 11986710158401015840

1]]

=sin (119909) 1199052120572

Γ (2120572 + 1)

1199013 1199083(119909 119905) = N

minus1[119906120572

119904120572N+[1199082119909119909

+ 21198671015840

2minus 11986710158401015840

2]]

= minussin (119909) 1199053120572

Γ (3120572 + 1)

(38)and so on

Thus the series solution of (30)-(31) is given by

V (119909 119905) = lim119873rarrinfin

119873

sum

119899=0

V119899(119909 119905) = V

0(119909 119905) + V

1(119909 119905)

+ V2(119909 119905) + sdot sdot sdot = sin (119909)

sdot (1 minus119905120572

Γ (120572 + 1)+

1199052120572

Γ (2120572 + 1)minus

1199053120572

Γ (3120572 + 1)+ sdot sdot sdot)

= sin (119909) (1 +infin

sum

119898=1

(minus119905120572)119898

Γ (119898120572 + 1)) = sin (119909) 119864

120572(minus119905120572)

119908 (119909 119905) = lim119873rarrinfin

119873

sum

119899=0

119908119899(119909 119905) = 119908

0(119909 119905) + 119908

1(119909 119905)

+ 1199082(119909 119905) + sdot sdot sdot = sin (119909)

sdot (1 minus119905120572

Γ (120572 + 1)+

1199052120572

Γ (2120572 + 1)minus

1199053120572

Γ (3120572 + 1)+ sdot sdot sdot)

= sin (119909) (1 +infin

sum

119898=1

(minus119905120572)119898

Γ (119898120572 + 1)) = sin (119909) 119864

120572(minus119905120572)

(39)

Hence the exact solutions of (30)-(31) are given by

V (119909 119905) = sin (119909) 119864120572(minus119905120572)

119908 (119909 119905) = sin (119909) 119864120572(minus119905120572)

(40)

When 120572 = 1 the following result is obtained

V (119909 119905) = lim119873rarrinfin

119873

sum

119899=0

V119899(119909 119905)

= V0(119909 119905) + V

1(119909 119905) + V

2(119909 119905) + sdot sdot sdot

= sin (119909) (1 minus 119905 + 1199052

2+ sdot sdot sdot) = 119890

minus119905 sin (119909)

119908 (119909 119905) = lim119873rarrinfin

119873

sum

119899=0

119908119899(119909 119905)

= 1199080(119909 119905) + 119908

1(119909 119905) + 119908

2(119909 119905) + sdot sdot sdot

= sin (119909) (1 minus 119905 + 1199052

2+ sdot sdot sdot) = 119890

minus119905 sin (119909)

(41)

which is the exact solution of (30)-(31) when 120572 = 1

6 Conclusion

In this paper Natural TransformMethod (NTM) andHomo-topy Perturbation Method (HPM) are successfully combinedto form a robust analytical method called a Hybrid NaturalTransformHomotopy PerturbationMethod for solving linearand nonlinear fractional partial differential equations Theanalytical method gives a series solution which convergesrapidly to an exact or approximate solution with elegantcomputational terms In this analyticalmethod the fractionalderivative is computed in Caputo sense while the nonlinearterm is calculated using Hersquos polynomials The analyticalprocedure is applied successfully and obtained an exactsolution of linear and nonlinear fractional partial differentialequations The simplicity and high accuracy of the analyticalmethod are clearly illustrated Thus the Hybrid NaturalTransform Homotopy Perturbation Method is a powerfulanalytical method for solving linear and nonlinear fractionalpartial differential equations

Competing Interests

The author declares that there is no conflict of interestsregarding the publication of this article

References

[1] I Podlubny Fractional differential equations vol 198 ofMathe-matics in Science and Engineering Academic Press San DiegoCalif USA 1999

[2] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 2003

[3] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[4] K Diethelm and N J Ford ldquoAnalysis of fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 265 no 2 pp 229ndash248 2002

[5] S G Samko A A Kilbas and O I Marichev FractionalIntegrals andDerivatives Gordon andBreach Yverdon Switzer-land 1993

[6] B J West M Bologna and P Grigolini Physics of Fractal Oper-ators Institute for Nonlinear Science Springer New York NYUSA 2003

[7] S S Ray ldquoAnalytical solution for the space fractional diffusionequation by two-step Adomian decomposition methodrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 4 pp 1295ndash1306 2009

[8] K Abbaoui and Y Cherruault ldquoNew ideas for proving conver-gence of Adomian Decomposition Methodsrdquo Computers andMathematics with Applications vol 32 pp 103ndash108 1995

[9] S Momani and Z Odibat ldquoAnalytical solution of a time-fractional Navier-Stokes equation by Adomian decompositionmethodrdquo Applied Mathematics and Computation vol 177 no 2pp 488ndash494 2006

[10] H Jafari and V Daftardar-Gejji ldquoSolving linear and nonlinearfractional diffusion and wave equations by Adomian decompo-sitionrdquo Applied Mathematics and Computation vol 180 no 2pp 488ndash497 2006

International Journal of Differential Equations 7

[11] M Ganjiani ldquoSolution of nonlinear fractional differential equa-tions using homotopy analysis methodrdquo Applied MathematicalModelling vol 34 no 6 pp 1634ndash1641 2010

[12] H Jafari C M Khalique and M Nazari ldquoApplication of theLaplace decomposition method for solving linear and nonlin-ear fractional diffusion-wave equationsrdquo Applied MathematicsLetters vol 24 no 11 pp 1799ndash1805 2011

[13] P K Gupta andM Singh ldquoHomotopy perturbationmethod forfractional Fornberg-Whitham equationrdquo Computers amp Mathe-matics with Applications vol 61 no 2 pp 250ndash254 2011

[14] S Momani and Z Odibat ldquoHomotopy perturbation methodfor nonlinear partial differential equations of fractional orderrdquoPhysics Letters A vol 365 no 5-6 pp 345ndash350 2007

[15] Y-Z Zhang A-M Yang and Y Long ldquoInitial boundary valueproblem for fractal heat equation in the semi-infinite region byyang-laplace transformrdquoThermal Science vol 18 no 2 pp 677ndash681 2014

[16] X-J Yang and D Baleanu ldquoFractal heat conduction problemsolved by local fractional variation iteration methodrdquo ThermalScience vol 17 no 2 pp 625ndash628 2013

[17] X-J Yang D Baleanu Y Khan and S T Mohyud-Din ldquoLocalfractional variational iteration method for diffusion and waveequations on Cantor setsrdquo Romanian Journal of Physics vol 59no 1-2 pp 36ndash48 2014

[18] X J Yang H M Srivastava J H He and D Baleanu ldquoCantor-type cylindrical-coordinate fractional derivativesrdquo Proceedingsof the Romanian Academy Series A vol 14 pp 127ndash133 2013

[19] S Kumar D Kumar S Abbasbandy and M M Rashidi ldquoAna-lytical solution of fractional Navier-Stokes equation by usingmodified Laplace decompositionmethodrdquoAin Shams Engineer-ing Journal vol 5 no 2 pp 569ndash574 2014

[20] B Ghazanfari and A G Ghazanfari ldquoSolving fractional non-linear Schrodinger equation by fractional complex transformmethodrdquo International Journal of Mathematical Modelling ampComputations vol 2 no 4 pp 277ndash281 2012

[21] K M Furati and N-E Tatar ldquoLongtime behavior for a non-linear fractional modelrdquo Journal of Mathematical Analysis andApplications vol 332 no 1 pp 441ndash454 2007

[22] K M Furati and N-E Tatar ldquoBehavior of solutions for aweighted Cauchy type fractional problemrdquo Journal of FractionalCalculus vol 28 pp 23ndash42 2005

[23] K M Furati and N-e Tatar ldquoAn existence result for a nonlocalfractional differential problemrdquo Journal of Fractional Calculusvol 26 pp 43ndash51 2004

[24] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[25] J H He ldquoRecent development of the homotopy perturbationmethodrdquo Topological Methods in Nonlinear Analysis vol 31 pp205ndash209 2008

[26] J-H He ldquoThe homotopy perturbation method for nonlinearoscillators with discontinuitiesrdquoAppliedMathematics and Com-putation vol 151 no 1 pp 287ndash292 2004

[27] Z H Khan andWA Khan ldquoN-transformproperties and appli-cationsrdquo NUST Journal of Engineering Sciences vol 1 pp 127ndash133 2008

[28] F B M Belgacem and R Silambarasan ldquoTheory of the naturaltransformrdquo Mathematics in Engineering Science and Aerospace(MESA) Journal vol 3 no 1 pp 99ndash124 2012

[29] F BM Belgacem andR Silambarasan ldquoAdvances in the naturaltransformrdquo in Proceedings of the 9th International Conference on

Mathematical Problems in Engineering Aerospace and Sciences(ICNPAA rsquo12) vol 1493 of AIP Conference Proceedings pp 106ndash110 Vienna Austria July 2012

[30] R Murray Spiegel Theory and Problems of Laplace TransformSchaumrsquos Outline Series McGraw-Hill New York NY USA1965

[31] F B Belgacem and A Karaballi ldquoSumudu transform funda-mental properties investigations and applicationsrdquo Journal ofApplied Mathematics and Stochastic Analysis vol 2006 ArticleID 91083 23 pages 2006

[32] G KWatugala ldquoSumudu transformmdasha new integral transformto solve differential equations and control engineering prob-lemsrdquo Mathematical Engineering in Industry vol 6 no 4 pp319ndash329 1998

[33] R Silambarasan and F B M Belgacem ldquoApplication of thenatural transform to Maxwellrsquos equationsrdquo in Proceedings of theProgress in Electromagnetics Research Symposium Proceedings(PIERS rsquo11) pp 899ndash902 Suzhou China September 2011

[34] M Rawashdeh and S Maitama ldquoFinding exact solutions ofnonlinear PDEs using the natural decomposition methodrdquoMathematical Methods in the Applied Sciences 2016

[35] S Maitama and S M Kurawa ldquoAn efficient technique forsolving gas dynamics equation using the natural decompositionmethodrdquo International Mathematical Forum vol 9 no 24 pp1177ndash1190 2014

[36] S Maitama ldquoExact solution of equation governing the unsteadyflow of a polytropic gas using the natural decompositionmethodrdquoAppliedMathematical Sciences vol 8 no 77 pp 3809ndash3823 2014

[37] M S Rawashdeh and S Maitama ldquoSolving PDEs using thenatural decomposition methodrdquo Nonlinear Studies vol 23 no1 pp 63ndash72 2016

[38] M S Rawashdeh and S Maitama ldquoSolving nonlinear ordinarydifferential equations using the NDMrdquo The Journal of AppliedAnalysis and Computation vol 5 no 1 pp 77ndash88 2015

[39] M S Rawashdeh and S Maitama ldquoSolving coupled systemof nonlinear PDErsquos using the natural decomposition methodrdquoInternational Journal of Pure and Applied Mathematics vol 92no 5 pp 757ndash776 2014

[40] Wikipedia note about the Natural transform August 2016httpsenwikipediaorgwikiN-transform

[41] 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

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 International Journal of Differential Equations

Table 1 List of some special natural transforms

Functional form Natural transform form

1 1

119904

119905119906

1199042

119890119886119905

1

119904 minus 119886119906

119905119899minus1

(119899 minus 1) 119899 = 1 2 119906

119899minus1

119904119899

sin(119905) 119906

1199042 + 1199062

was recently renamed as the Natural Transform by Belgacemand Silambarasan [28 29] In fact based on personal com-munications and Internet records CASM in LUMS and IOSSForman Christian College in Lahore lectures in 2004 and2005 and 3rd ICM4F COMSATs University conference sidelectures in Islamabad in 2006 indicate that Belgacem sharedand discussed various aspects of this transform with thevarious attending audiences The Natural Transform is anintegral transform which is similar to Laplace transform[30] and Sumudu integral transform [31 32] It converges toLaplace transform when 119906 = 1 and Sumudu transform when119904 = 1 Belgacem and Silambarasan [29 33] have proposeda detailed theory and applications of the Natural TransformRecently Natural Transform and Adomian DecompositionMethod are successfully combined and obtained an exactsolution of linear and nonlinear partial differential equations[34ndash39] More details about the Natural Transform and itsapplications can be seen inWikipedia note about the NaturalTransform [40]

Some useful Natural Transforms in this paper are pre-sented in Table 1

Definition 1 The Natural Transform of the function V(119905) gt 0and V(119905) = 0 for 119905 lt 0 is defined over the set of functions

119860 = V (119905) exist119872 1205911 1205912gt 0 |V (119905)| lt 119872119890|119905|120591119895 if 119905

isin (minus1)119895times [0infin) 119895 = 1 2

(1)

by the following integral

N+[V (119905)] = 119881 (119904 119906) =

1

119906int

infin

0

119890minus119904119905119906V (119905) 119889119905

119904 gt 0 119906 gt 0

(2)

And the inverse Natural Transform of the function V(119905) isdefined by

Nminus1[119881 (119904 119906)] = V (119905) =

1

2120587119894int

120574+119894infin

120574minus119894infin

119890119904119905119906119881 (119904 119906) 119889119904 (3)

where 119904 and 119906 are the Natural Transform variables [28 29]and 120574 is a real constant and the integral in (3) is taken along119904 = 120574 in the complex plane 119904 = 119909 + 119894119910

Some properties of the Natural Transform Method aregiven below

Property 1 Natural Transform of derivative if V119899(119905) is the 119899thderivative of the function V(119905) isin 119860 with respect to ldquo119905rdquo thenits Natural Transform is given by

N+[V119899 (119905)] =

119904119899

119906119899119881 (119904 119906) minus

119899minus1

sum

119896=0

119904119899minus(119896+1)

119906119899minus119896V119896 (0) (4)

When 119899 = 1 2 and 3 we have the following results

N+[V1015840 (119905)] =

119904

119906119881 (119904 119906) minus

1

119906V (0)

N+[V10158401015840 (119905)] =

1199042

1199062119881 (119904 119906) minus

119904

1199062V (0) minus

1

119906V1015840 (0)

N+[V101584010158401015840 (119905)] =

1199043

1199063119881 (119904 119906) minus

1199042

1199063V (0) minus

119904

1199062V1015840 (0)

minus1

119906V10158401015840 (0)

(5)

Property 2 If119881(119904 119906) is the Natural Transform and 119865(119904) is theLaplace transform of the function 119891(119905) isin 119860 then N+[119891(119905)] =

119881(119904 119906) = (1119906) intinfin

0119890minus119904119905119906

119891(119905)119889119905 = (1119906)119865(119904119906)

Property 3 If119881(119904 119906) is the Natural Transform and119866(119906) is theSumudu transform of the function V(119905) isin 119860 then N+[V(119905)] =119881(119904 119906) = (1119904) int

infin

0119890minus119905 V(119906119905119904)119889119905 = (1119904)119866(119906119904)

Property 4 IfN+[V(119905)] = 119881(119904 119906) thenN+[V(120573119905)] = (1120573)119881(119904120573 119906)

Property 5 The Natural Transform is a linear operator thatis if 120572 and 120573 are nonzero constants then

N+[120572119891 (119905) plusmn 120573119892 (119905)] = 120572N

+[119891 (119905)] plusmn 120573N

+[119892 (119905)]

= 120572119865+(119904 119906) plusmn 120573119866

+(119904 119906)

(6)

Moreover119865+(119904 119906) and119866+(119904 119906) are theNatural Transforms offunctions 119891(119905) and 119892(119905) respectively

3 Basic Definitions of Fractional Calculus

In this section the basic definitions of fractional calculus arepresented

Definition 2 Function 119891(119905) 119905 gt 0 is said to be in the space119862119898

120572119898 isin 119873 cup 0 if 119891(119898) isin 119862

120572

Definition 3 A real function 119891(119905) 119905 gt 0 is said to be in thespace 119862

120572120572 isin R if there exists a real number 119901 (gt120572) such that

119891(119905) = 1199051199011198911(119905) where 119891

1(119905) isin 119862[0infin) Clearly 119862

120572sub 119862120573if

120573 le 120572

International Journal of Differential Equations 3

Definition 4 The left sided Riemann-Liouville fractionalintegral operator of order 120583 gt 0 of a function 119891(119905) isin 119862

120572

and 120572 ge minus1 is defined as [19 41]

119868120583119891 (119905)

=

1

Γ (120583)int

119905

0

(119905 minus 120591)120583minus1

119891 (120591) 119889120591 120583 gt 0 119905 gt 0

119891 (119905) 120583 = 0

(7)

where Γ(sdot) is the well-known Gamma function

Definition 5 The left sided Caputo fractional derivative 119891119891 isin 119862

119898

1119898 isin N cup 0 is defined as [1 5]

119863120583

lowast119891 (119905)

=

119868119898minus120583

[120597119898119891 (119905)

120597119905119898] 119898 minus 1 lt 120583 lt 119898 119898 isin N

120597119898119891 (119905)

120597119905119898 120583 = 119898

(8)

Note that [1 5]

(i) 119868120583119905119891(119905) = (1Γ(120583)) int

119905

0(119891(119905)(119905 minus 119904)

1minus120583)119889119905 120583 gt 0 119905 gt 0

(ii) 119863120583lowast119891(119909 119905) = 119868

119898minus120583

119905[120597119898119891(119905)120597119905

119898]119898 minus 1 lt 120583 le 119898

Definition 6 TheNatural Transform of the Caputo fractionalderivative is defined as

N+[119863119899120572

119905V (119905)] =

119904119899120572

119906119899120572119881 (119904 119906) minus

119899minus1

sum

119896=0

119904119899120572minus(119896+1)

119906119899120572minus119896V(119896) (0+) (9)

((119899 minus 1)119899 lt 120572 le 1)

Definition 7 The series expansion defines a one-parameterMittag-Leffler function as [1]

119864120572(119911) =

infin

sum

119896=0

119911119896

Γ (120572119896 + 1) 120572 gt 0 119911 isin C (10)

4 Analysis of the Method

In this section the basic idea of theHybridNatural TransformHomotopy Perturbation Method is clearly illustrated by thestandard nonlinear fractional partial differential equation ofthe form

119863119899120572

119905V (119909 119905) + 119872 (V (119909 119905)) + 119865 (V (119909 119905)) = 119892 (119909 119905) (11)

subject to the initial condition

V (119909 0) = 119891 (119909) (12)

where 119865(V(119909 119905)) represents the nonlinear terms 119863119899120572119905

=

120597119899120572120597119905119899120572 is the Caputo fractional derivative of function V(119905)

119872(V(119909 119905)) is the linear differential operator and 119892(119909 119905) is asource term

Applying the Natural Transform to (11) subject to thegiven initial condition we get

119881 (119909 119904 119906) =1

119904119891 (119909) +

119906119899120572

119904119899120572N+[119892 (119909 119905)]

minus119906119899120572

119904119899120572N+[119872 (V (119909 119905)) + 119865 (V (119909 119905))]

(13)

Taking the inverse Natural Transform of (13) we get

V (119909 119905)

= 119866 (119909 119905)

minus Nminus1[119906119899120572

119904119899120572N+[119872 (V (119909 119905)) + 119865 (V (119909 119905))]]

(14)

where 119866(119909 119905) is a term arising from the source term and theprescribed initial condition

Now we apply the Homotopy Perturbation Method

V (119909 119905) =infin

sum

119899=0

119901119899V119899(119909 119905) (15)

The nonlinear term 119865(V(119909 119905)) is decomposed as

119865 (V (119909 119905)) =infin

sum

119899=0

119901119899119867119899(V) (16)

where119867119899(V) is Hersquos polynomial which is computed using the

following formula

119867119899(V1 V2 V

119899) =

1

119899

120597119899

120597119901119899[

[

119865(

119899

sum

119895=0

119901119895V119895)]

]119901=0

119899 = 0 1 2

(17)

Substituting (16) and (15) into (14) we get

infin

sum

119899=0

119901119899V119899(119909 119905) = 119866 (119909 119905) minus 119901(N

minus1[119906119899120572

119904119899120572

sdot N+[

infin

sum

119899=0

119901119899119872(V (119909 119905)) +

infin

sum

119899=0

119901119899119867119899(V)]])

(18)

Using the coefficient of like powers of 119901 in (18) we obtainedthe following approximations

1199010 V0(119909 119905) = 119866 (119909 119905)

1199011 V1(119909 119905)

= minusNminus1[119906119899120572

119904119899120572N+[119872 (V (119909 119905)) + 119867

0(V)]]

4 International Journal of Differential Equations

1199012 V2(119909 119905)

= minusNminus1[119906119899120572

119904119899120572N+[119872 (V (119909 119905)) + 119867

1(V)]]

1199013 V3(119909 119905)

= minusNminus1[119906119899120572

119904119899120572N+[119872 (V (119909 119905)) + 119867

2(V)]]

(19)

and so onHence the series solution of (11)-(12) is given by

V (119909 119905) = lim119873rarrinfin

119873

sum

119899=0

V119899(119909 119905) (20)

5 Applications

In this section the application of the Hybrid NaturalTransform Homotopy Perturbation Method to linear andnonlinear fractional partial differential equations is clearlydemonstrated to show its simplicity and high accuracy

Example 8 Consider the following fractional diffusion equa-tion of the form119863120572

119905V + V119909119909+ V119910119910+ V119911119911= 0

minusinfin lt 119909 119910 119911 lt infin 119905 gt 0

(21)

subject to the initial condition

V (119909 119910 119911 0) = 119890119909+119910+119911 120572 isin (0 1) (22)

Applying the Natural Transform to (21) subject to the giveninitial condition we get

119881 (119909 119910 119911 119904 119906) =119890119909+119910+119911

119904+119906120572

119904120572N+[V119909119909+ V119910119910+ V119911119911] (23)

Taking the inverse Natural Transform of (23) we get

V (119909 119910 119911 119905) = 119890119909+119910+119911

+ Nminus1[119906120572

119904120572N+[V119909119909+ V119910119910+ V119911119911]]

(24)

Now we apply Homotopy Perturbation Method

V (119909 119910 119911 119905) =infin

sum

119899=0

119901119899V119899(119909 119910 119911 119905) (25)

Then (24) will becomeinfin

sum

119899=0

119901119899V119899(119909 119910 119911 119905) = 119890

119909+119910+119911minus 119901(N

minus1[119906120572

119904120572

sdot N+[

infin

sum

119899=0

119901119899V119899119909119909

+

infin

sum

119899=0

119901119899V119899119910119910

+

infin

sum

119899=0

119901119899V119899119911119911]])

(26)

Using the coefficients of like powers of 119901 in (26) we obtainedthe following approximations

1199010 V0(119909 119910 119911 119905) = 119890

119909+119910+119911

1199011 V1(119909 119910 119911 119905)

= minusNminus1[119906120572

119904120572N+[V0119909119909

+ V0119910119910

+ V0119911119911]]

= minus119890119909+119910+119911

119905120572

Γ (120572 + 1)

1199012 V2(119909 119910 119911 119905)

= minusNminus1[119906120572

119904120572N+[V1119909119909

+ V1119910119910

+ V1119911119911]]

=119890119909+119910+119911

1199052120572

Γ (2120572 + 1)

1199013 V3(119909 119910 119911 119905)

= minusNminus1[119906120572

119904120572N+[V2119909119909

+ V2119910119910

+ V2119911119911]]

= minus119890119909+119910+119911

1199053120572

Γ (3120572 + 1)

(27)

and so onThen the series solution of (21)-(22) is given by

V (119909 119910 119911 119905) = lim119873rarrinfin

119873

sum

119899=0

V119899(119909 119910 119911 119905) = V

0(119909 119910 119911 119905)

+ V1(119909 119910 119911 119905) + V

2(119909 119910 119911 119905) + V

3(119909 119910 119911 119905) + sdot sdot sdot

= 119890119909+119910+119911

(1 minus119905120572

Γ (120572 + 1)+

1199052120572

Γ (2120572 + 1)

minus1199053120572

Γ (3120572 + 1)+ sdot sdot sdot) = 119890

119909+119910+119911(1

+

infin

sum

119898=1

(minus119905120572)119898

Γ (119898120572 + 1)) = 119890119909+119910+119911

119864120572(minus119905120572)

(28)

When 120572 = 1 the following result is obtained

V (119909 119910 119911 119905) = lim119873rarrinfin

119873

sum

119899=0

V119899(119909 119910 119911 119905)

= V0(119909 119910 119911 119905) + V

1(119909 119910 119911 119905)

+ V2(119909 119910 119911 119905) + V

3(119909 119910 119911 119905) + sdot sdot sdot

= 119890119909+119910+119911

(1 minus119905

1+1199052

2minus1199053

3+ sdot sdot sdot)

= 119890119909+119910+119911minus119905

(29)

which is the exact solution of (21)-(22) when 120572 = 1

International Journal of Differential Equations 5

Example 9 Consider the following coupled system of non-linear fractional partial differential equations of the form

119863120572

119905V minus V119909119909minus 2VV119909+ V119909119908119909= 0

119863120572

119905119908 minus 119908

119909119909minus 2119908119908

119909+ V119909119908119909= 0

(30)

subject to the initial conditions

V (119909 0) = sin (119909)

119908 (119909 0) = sin (119909) (31)

Applying the Natural Transform to (30) subject to giveninitial conditions we get

119881 (119909 119904 119906) =1

119904sin (119909) + 119906

120572

119904120572N+[V119909119909+ 2VV119909minus V119909119908119909]

119882 (119909 119904 119906) =1

119904sin (119909)

+119906120572

119904120572N+[119908119909119909+ 2119908119908

119909minus V119909119908119909]

(32)

Taking the inverse Natural Transform of (32) we have

V (119909 119905) = sin (119909)

+ Nminus1[119906120572

119904120572N+[V119909119909+ 2VV119909minus V119909119908119909]]

119908 (119909 119905) = sin (119909)

+ Nminus1[119906120572

119904120572N+[119908119908119909119909+ 2119908119908

119909minus V119909119908119909]]

(33)

Now we apply the Homotopy Perturbation Method

V (119909 119905) =infin

sum

119899=0

119901119899V119899(119909 119905)

119908 (119909 119905) =

infin

sum

119899=0

119901119899119908119899(119909 119905)

(34)

Then (33) will become

V (119909 119905) = sin (119909) + 119901(Nminus1 [119906120572

119904120572

sdot N+[

infin

sum

119899=0

119901119899V119899119909119909

+ 2

infin

sum

119899=0

119901119899119867119899minus

infin

sum

119899=0

11990111989911986710158401015840

119899]])

119908 (119909 119905) = sin (119909) + 119901(Nminus1 [119906120572

119904120572

sdot N+[

infin

sum

119899=0

119901119899119908119899119909119909

+ 2

infin

sum

119899=0

1199011198991198671015840

119899minus

infin

sum

119899=0

11990111989911986710158401015840

119899]])

(35)

where1198671198991198671015840119899 and11986710158401015840

119899are Hersquos polynomials which represent

the nonlinear terms VV119909 119908119908119909 and V

119909119908119909 respectively

Some few components of Hersquos polynomials of the nonlin-ear terms VV

119909 119908119908119909 and V

119909119908119909are given as follows

1198670= V0V0119909

1198671= V0V1119909+ V1V0119909

1198672= V0V2119909+ V1V1119909+ V2V0119909

1198671015840

0= 11990801199080119909

1198671015840

1= 11990801199081119909+ 11990811199080119909

1198671015840

2= 11990801199082119909+ 11990811199081119909+ 11990821199080119909

11986710158401015840

0= V01199091199080119909

11986710158401015840

1= V01199091199081119909+ V11199091199080119909

11986710158401015840

2= V21199091199080119909+ V11199091199081119909+ V01199091199082119909

(36)

and so onUsing the coefficients of the like powers of 119901 in (35) we

obtained the following approximations

1199010 V0(119909 119905) = sin (119909)

1199011 V1(119909 119905) = N

minus1[119906120572

119904120572N+[V0119909119909

+ 21198670minus 11986710158401015840

0]]

= minussin (119909) 119905120572

Γ (120572 + 1)

1199012 V2(119909 119905) = N

minus1[119906120572

119904120572N+[V1119909119909

+ 21198671minus 11986710158401015840

1]]

=sin (119909) 1199052120572

Γ (2120572 + 1)

1199013 V3(119909 119905) = N

minus1[119906120572

119904120572N+[V2119909119909

+ 21198672minus 11986710158401015840

2]]

= minussin (119909) 1199053120572

Γ (3120572 + 1)

(37)

and so onSimilarly

1199010 1199080(119909 119905) = sin (119909)

1199011 1199081(119909 119905) = N

minus1[119906120572

119904120572N+[1199080119909119909

+ 21198671015840

0minus 11986710158401015840

0]]

= minussin (119909) 119905120572

Γ (120572 + 1)

6 International Journal of Differential Equations

1199012 1199082(119909 119905) = N

minus1[119906120572

119904120572N+[1199081119909119909

+ 21198671015840

1minus 11986710158401015840

1]]

=sin (119909) 1199052120572

Γ (2120572 + 1)

1199013 1199083(119909 119905) = N

minus1[119906120572

119904120572N+[1199082119909119909

+ 21198671015840

2minus 11986710158401015840

2]]

= minussin (119909) 1199053120572

Γ (3120572 + 1)

(38)and so on

Thus the series solution of (30)-(31) is given by

V (119909 119905) = lim119873rarrinfin

119873

sum

119899=0

V119899(119909 119905) = V

0(119909 119905) + V

1(119909 119905)

+ V2(119909 119905) + sdot sdot sdot = sin (119909)

sdot (1 minus119905120572

Γ (120572 + 1)+

1199052120572

Γ (2120572 + 1)minus

1199053120572

Γ (3120572 + 1)+ sdot sdot sdot)

= sin (119909) (1 +infin

sum

119898=1

(minus119905120572)119898

Γ (119898120572 + 1)) = sin (119909) 119864

120572(minus119905120572)

119908 (119909 119905) = lim119873rarrinfin

119873

sum

119899=0

119908119899(119909 119905) = 119908

0(119909 119905) + 119908

1(119909 119905)

+ 1199082(119909 119905) + sdot sdot sdot = sin (119909)

sdot (1 minus119905120572

Γ (120572 + 1)+

1199052120572

Γ (2120572 + 1)minus

1199053120572

Γ (3120572 + 1)+ sdot sdot sdot)

= sin (119909) (1 +infin

sum

119898=1

(minus119905120572)119898

Γ (119898120572 + 1)) = sin (119909) 119864

120572(minus119905120572)

(39)

Hence the exact solutions of (30)-(31) are given by

V (119909 119905) = sin (119909) 119864120572(minus119905120572)

119908 (119909 119905) = sin (119909) 119864120572(minus119905120572)

(40)

When 120572 = 1 the following result is obtained

V (119909 119905) = lim119873rarrinfin

119873

sum

119899=0

V119899(119909 119905)

= V0(119909 119905) + V

1(119909 119905) + V

2(119909 119905) + sdot sdot sdot

= sin (119909) (1 minus 119905 + 1199052

2+ sdot sdot sdot) = 119890

minus119905 sin (119909)

119908 (119909 119905) = lim119873rarrinfin

119873

sum

119899=0

119908119899(119909 119905)

= 1199080(119909 119905) + 119908

1(119909 119905) + 119908

2(119909 119905) + sdot sdot sdot

= sin (119909) (1 minus 119905 + 1199052

2+ sdot sdot sdot) = 119890

minus119905 sin (119909)

(41)

which is the exact solution of (30)-(31) when 120572 = 1

6 Conclusion

In this paper Natural TransformMethod (NTM) andHomo-topy Perturbation Method (HPM) are successfully combinedto form a robust analytical method called a Hybrid NaturalTransformHomotopy PerturbationMethod for solving linearand nonlinear fractional partial differential equations Theanalytical method gives a series solution which convergesrapidly to an exact or approximate solution with elegantcomputational terms In this analyticalmethod the fractionalderivative is computed in Caputo sense while the nonlinearterm is calculated using Hersquos polynomials The analyticalprocedure is applied successfully and obtained an exactsolution of linear and nonlinear fractional partial differentialequations The simplicity and high accuracy of the analyticalmethod are clearly illustrated Thus the Hybrid NaturalTransform Homotopy Perturbation Method is a powerfulanalytical method for solving linear and nonlinear fractionalpartial differential equations

Competing Interests

The author declares that there is no conflict of interestsregarding the publication of this article

References

[1] I Podlubny Fractional differential equations vol 198 ofMathe-matics in Science and Engineering Academic Press San DiegoCalif USA 1999

[2] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 2003

[3] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[4] K Diethelm and N J Ford ldquoAnalysis of fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 265 no 2 pp 229ndash248 2002

[5] S G Samko A A Kilbas and O I Marichev FractionalIntegrals andDerivatives Gordon andBreach Yverdon Switzer-land 1993

[6] B J West M Bologna and P Grigolini Physics of Fractal Oper-ators Institute for Nonlinear Science Springer New York NYUSA 2003

[7] S S Ray ldquoAnalytical solution for the space fractional diffusionequation by two-step Adomian decomposition methodrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 4 pp 1295ndash1306 2009

[8] K Abbaoui and Y Cherruault ldquoNew ideas for proving conver-gence of Adomian Decomposition Methodsrdquo Computers andMathematics with Applications vol 32 pp 103ndash108 1995

[9] S Momani and Z Odibat ldquoAnalytical solution of a time-fractional Navier-Stokes equation by Adomian decompositionmethodrdquo Applied Mathematics and Computation vol 177 no 2pp 488ndash494 2006

[10] H Jafari and V Daftardar-Gejji ldquoSolving linear and nonlinearfractional diffusion and wave equations by Adomian decompo-sitionrdquo Applied Mathematics and Computation vol 180 no 2pp 488ndash497 2006

International Journal of Differential Equations 7

[11] M Ganjiani ldquoSolution of nonlinear fractional differential equa-tions using homotopy analysis methodrdquo Applied MathematicalModelling vol 34 no 6 pp 1634ndash1641 2010

[12] H Jafari C M Khalique and M Nazari ldquoApplication of theLaplace decomposition method for solving linear and nonlin-ear fractional diffusion-wave equationsrdquo Applied MathematicsLetters vol 24 no 11 pp 1799ndash1805 2011

[13] P K Gupta andM Singh ldquoHomotopy perturbationmethod forfractional Fornberg-Whitham equationrdquo Computers amp Mathe-matics with Applications vol 61 no 2 pp 250ndash254 2011

[14] S Momani and Z Odibat ldquoHomotopy perturbation methodfor nonlinear partial differential equations of fractional orderrdquoPhysics Letters A vol 365 no 5-6 pp 345ndash350 2007

[15] Y-Z Zhang A-M Yang and Y Long ldquoInitial boundary valueproblem for fractal heat equation in the semi-infinite region byyang-laplace transformrdquoThermal Science vol 18 no 2 pp 677ndash681 2014

[16] X-J Yang and D Baleanu ldquoFractal heat conduction problemsolved by local fractional variation iteration methodrdquo ThermalScience vol 17 no 2 pp 625ndash628 2013

[17] X-J Yang D Baleanu Y Khan and S T Mohyud-Din ldquoLocalfractional variational iteration method for diffusion and waveequations on Cantor setsrdquo Romanian Journal of Physics vol 59no 1-2 pp 36ndash48 2014

[18] X J Yang H M Srivastava J H He and D Baleanu ldquoCantor-type cylindrical-coordinate fractional derivativesrdquo Proceedingsof the Romanian Academy Series A vol 14 pp 127ndash133 2013

[19] S Kumar D Kumar S Abbasbandy and M M Rashidi ldquoAna-lytical solution of fractional Navier-Stokes equation by usingmodified Laplace decompositionmethodrdquoAin Shams Engineer-ing Journal vol 5 no 2 pp 569ndash574 2014

[20] B Ghazanfari and A G Ghazanfari ldquoSolving fractional non-linear Schrodinger equation by fractional complex transformmethodrdquo International Journal of Mathematical Modelling ampComputations vol 2 no 4 pp 277ndash281 2012

[21] K M Furati and N-E Tatar ldquoLongtime behavior for a non-linear fractional modelrdquo Journal of Mathematical Analysis andApplications vol 332 no 1 pp 441ndash454 2007

[22] K M Furati and N-E Tatar ldquoBehavior of solutions for aweighted Cauchy type fractional problemrdquo Journal of FractionalCalculus vol 28 pp 23ndash42 2005

[23] K M Furati and N-e Tatar ldquoAn existence result for a nonlocalfractional differential problemrdquo Journal of Fractional Calculusvol 26 pp 43ndash51 2004

[24] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[25] J H He ldquoRecent development of the homotopy perturbationmethodrdquo Topological Methods in Nonlinear Analysis vol 31 pp205ndash209 2008

[26] J-H He ldquoThe homotopy perturbation method for nonlinearoscillators with discontinuitiesrdquoAppliedMathematics and Com-putation vol 151 no 1 pp 287ndash292 2004

[27] Z H Khan andWA Khan ldquoN-transformproperties and appli-cationsrdquo NUST Journal of Engineering Sciences vol 1 pp 127ndash133 2008

[28] F B M Belgacem and R Silambarasan ldquoTheory of the naturaltransformrdquo Mathematics in Engineering Science and Aerospace(MESA) Journal vol 3 no 1 pp 99ndash124 2012

[29] F BM Belgacem andR Silambarasan ldquoAdvances in the naturaltransformrdquo in Proceedings of the 9th International Conference on

Mathematical Problems in Engineering Aerospace and Sciences(ICNPAA rsquo12) vol 1493 of AIP Conference Proceedings pp 106ndash110 Vienna Austria July 2012

[30] R Murray Spiegel Theory and Problems of Laplace TransformSchaumrsquos Outline Series McGraw-Hill New York NY USA1965

[31] F B Belgacem and A Karaballi ldquoSumudu transform funda-mental properties investigations and applicationsrdquo Journal ofApplied Mathematics and Stochastic Analysis vol 2006 ArticleID 91083 23 pages 2006

[32] G KWatugala ldquoSumudu transformmdasha new integral transformto solve differential equations and control engineering prob-lemsrdquo Mathematical Engineering in Industry vol 6 no 4 pp319ndash329 1998

[33] R Silambarasan and F B M Belgacem ldquoApplication of thenatural transform to Maxwellrsquos equationsrdquo in Proceedings of theProgress in Electromagnetics Research Symposium Proceedings(PIERS rsquo11) pp 899ndash902 Suzhou China September 2011

[34] M Rawashdeh and S Maitama ldquoFinding exact solutions ofnonlinear PDEs using the natural decomposition methodrdquoMathematical Methods in the Applied Sciences 2016

[35] S Maitama and S M Kurawa ldquoAn efficient technique forsolving gas dynamics equation using the natural decompositionmethodrdquo International Mathematical Forum vol 9 no 24 pp1177ndash1190 2014

[36] S Maitama ldquoExact solution of equation governing the unsteadyflow of a polytropic gas using the natural decompositionmethodrdquoAppliedMathematical Sciences vol 8 no 77 pp 3809ndash3823 2014

[37] M S Rawashdeh and S Maitama ldquoSolving PDEs using thenatural decomposition methodrdquo Nonlinear Studies vol 23 no1 pp 63ndash72 2016

[38] M S Rawashdeh and S Maitama ldquoSolving nonlinear ordinarydifferential equations using the NDMrdquo The Journal of AppliedAnalysis and Computation vol 5 no 1 pp 77ndash88 2015

[39] M S Rawashdeh and S Maitama ldquoSolving coupled systemof nonlinear PDErsquos using the natural decomposition methodrdquoInternational Journal of Pure and Applied Mathematics vol 92no 5 pp 757ndash776 2014

[40] Wikipedia note about the Natural transform August 2016httpsenwikipediaorgwikiN-transform

[41] 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

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Differential Equations 3

Definition 4 The left sided Riemann-Liouville fractionalintegral operator of order 120583 gt 0 of a function 119891(119905) isin 119862

120572

and 120572 ge minus1 is defined as [19 41]

119868120583119891 (119905)

=

1

Γ (120583)int

119905

0

(119905 minus 120591)120583minus1

119891 (120591) 119889120591 120583 gt 0 119905 gt 0

119891 (119905) 120583 = 0

(7)

where Γ(sdot) is the well-known Gamma function

Definition 5 The left sided Caputo fractional derivative 119891119891 isin 119862

119898

1119898 isin N cup 0 is defined as [1 5]

119863120583

lowast119891 (119905)

=

119868119898minus120583

[120597119898119891 (119905)

120597119905119898] 119898 minus 1 lt 120583 lt 119898 119898 isin N

120597119898119891 (119905)

120597119905119898 120583 = 119898

(8)

Note that [1 5]

(i) 119868120583119905119891(119905) = (1Γ(120583)) int

119905

0(119891(119905)(119905 minus 119904)

1minus120583)119889119905 120583 gt 0 119905 gt 0

(ii) 119863120583lowast119891(119909 119905) = 119868

119898minus120583

119905[120597119898119891(119905)120597119905

119898]119898 minus 1 lt 120583 le 119898

Definition 6 TheNatural Transform of the Caputo fractionalderivative is defined as

N+[119863119899120572

119905V (119905)] =

119904119899120572

119906119899120572119881 (119904 119906) minus

119899minus1

sum

119896=0

119904119899120572minus(119896+1)

119906119899120572minus119896V(119896) (0+) (9)

((119899 minus 1)119899 lt 120572 le 1)

Definition 7 The series expansion defines a one-parameterMittag-Leffler function as [1]

119864120572(119911) =

infin

sum

119896=0

119911119896

Γ (120572119896 + 1) 120572 gt 0 119911 isin C (10)

4 Analysis of the Method

In this section the basic idea of theHybridNatural TransformHomotopy Perturbation Method is clearly illustrated by thestandard nonlinear fractional partial differential equation ofthe form

119863119899120572

119905V (119909 119905) + 119872 (V (119909 119905)) + 119865 (V (119909 119905)) = 119892 (119909 119905) (11)

subject to the initial condition

V (119909 0) = 119891 (119909) (12)

where 119865(V(119909 119905)) represents the nonlinear terms 119863119899120572119905

=

120597119899120572120597119905119899120572 is the Caputo fractional derivative of function V(119905)

119872(V(119909 119905)) is the linear differential operator and 119892(119909 119905) is asource term

Applying the Natural Transform to (11) subject to thegiven initial condition we get

119881 (119909 119904 119906) =1

119904119891 (119909) +

119906119899120572

119904119899120572N+[119892 (119909 119905)]

minus119906119899120572

119904119899120572N+[119872 (V (119909 119905)) + 119865 (V (119909 119905))]

(13)

Taking the inverse Natural Transform of (13) we get

V (119909 119905)

= 119866 (119909 119905)

minus Nminus1[119906119899120572

119904119899120572N+[119872 (V (119909 119905)) + 119865 (V (119909 119905))]]

(14)

where 119866(119909 119905) is a term arising from the source term and theprescribed initial condition

Now we apply the Homotopy Perturbation Method

V (119909 119905) =infin

sum

119899=0

119901119899V119899(119909 119905) (15)

The nonlinear term 119865(V(119909 119905)) is decomposed as

119865 (V (119909 119905)) =infin

sum

119899=0

119901119899119867119899(V) (16)

where119867119899(V) is Hersquos polynomial which is computed using the

following formula

119867119899(V1 V2 V

119899) =

1

119899

120597119899

120597119901119899[

[

119865(

119899

sum

119895=0

119901119895V119895)]

]119901=0

119899 = 0 1 2

(17)

Substituting (16) and (15) into (14) we get

infin

sum

119899=0

119901119899V119899(119909 119905) = 119866 (119909 119905) minus 119901(N

minus1[119906119899120572

119904119899120572

sdot N+[

infin

sum

119899=0

119901119899119872(V (119909 119905)) +

infin

sum

119899=0

119901119899119867119899(V)]])

(18)

Using the coefficient of like powers of 119901 in (18) we obtainedthe following approximations

1199010 V0(119909 119905) = 119866 (119909 119905)

1199011 V1(119909 119905)

= minusNminus1[119906119899120572

119904119899120572N+[119872 (V (119909 119905)) + 119867

0(V)]]

4 International Journal of Differential Equations

1199012 V2(119909 119905)

= minusNminus1[119906119899120572

119904119899120572N+[119872 (V (119909 119905)) + 119867

1(V)]]

1199013 V3(119909 119905)

= minusNminus1[119906119899120572

119904119899120572N+[119872 (V (119909 119905)) + 119867

2(V)]]

(19)

and so onHence the series solution of (11)-(12) is given by

V (119909 119905) = lim119873rarrinfin

119873

sum

119899=0

V119899(119909 119905) (20)

5 Applications

In this section the application of the Hybrid NaturalTransform Homotopy Perturbation Method to linear andnonlinear fractional partial differential equations is clearlydemonstrated to show its simplicity and high accuracy

Example 8 Consider the following fractional diffusion equa-tion of the form119863120572

119905V + V119909119909+ V119910119910+ V119911119911= 0

minusinfin lt 119909 119910 119911 lt infin 119905 gt 0

(21)

subject to the initial condition

V (119909 119910 119911 0) = 119890119909+119910+119911 120572 isin (0 1) (22)

Applying the Natural Transform to (21) subject to the giveninitial condition we get

119881 (119909 119910 119911 119904 119906) =119890119909+119910+119911

119904+119906120572

119904120572N+[V119909119909+ V119910119910+ V119911119911] (23)

Taking the inverse Natural Transform of (23) we get

V (119909 119910 119911 119905) = 119890119909+119910+119911

+ Nminus1[119906120572

119904120572N+[V119909119909+ V119910119910+ V119911119911]]

(24)

Now we apply Homotopy Perturbation Method

V (119909 119910 119911 119905) =infin

sum

119899=0

119901119899V119899(119909 119910 119911 119905) (25)

Then (24) will becomeinfin

sum

119899=0

119901119899V119899(119909 119910 119911 119905) = 119890

119909+119910+119911minus 119901(N

minus1[119906120572

119904120572

sdot N+[

infin

sum

119899=0

119901119899V119899119909119909

+

infin

sum

119899=0

119901119899V119899119910119910

+

infin

sum

119899=0

119901119899V119899119911119911]])

(26)

Using the coefficients of like powers of 119901 in (26) we obtainedthe following approximations

1199010 V0(119909 119910 119911 119905) = 119890

119909+119910+119911

1199011 V1(119909 119910 119911 119905)

= minusNminus1[119906120572

119904120572N+[V0119909119909

+ V0119910119910

+ V0119911119911]]

= minus119890119909+119910+119911

119905120572

Γ (120572 + 1)

1199012 V2(119909 119910 119911 119905)

= minusNminus1[119906120572

119904120572N+[V1119909119909

+ V1119910119910

+ V1119911119911]]

=119890119909+119910+119911

1199052120572

Γ (2120572 + 1)

1199013 V3(119909 119910 119911 119905)

= minusNminus1[119906120572

119904120572N+[V2119909119909

+ V2119910119910

+ V2119911119911]]

= minus119890119909+119910+119911

1199053120572

Γ (3120572 + 1)

(27)

and so onThen the series solution of (21)-(22) is given by

V (119909 119910 119911 119905) = lim119873rarrinfin

119873

sum

119899=0

V119899(119909 119910 119911 119905) = V

0(119909 119910 119911 119905)

+ V1(119909 119910 119911 119905) + V

2(119909 119910 119911 119905) + V

3(119909 119910 119911 119905) + sdot sdot sdot

= 119890119909+119910+119911

(1 minus119905120572

Γ (120572 + 1)+

1199052120572

Γ (2120572 + 1)

minus1199053120572

Γ (3120572 + 1)+ sdot sdot sdot) = 119890

119909+119910+119911(1

+

infin

sum

119898=1

(minus119905120572)119898

Γ (119898120572 + 1)) = 119890119909+119910+119911

119864120572(minus119905120572)

(28)

When 120572 = 1 the following result is obtained

V (119909 119910 119911 119905) = lim119873rarrinfin

119873

sum

119899=0

V119899(119909 119910 119911 119905)

= V0(119909 119910 119911 119905) + V

1(119909 119910 119911 119905)

+ V2(119909 119910 119911 119905) + V

3(119909 119910 119911 119905) + sdot sdot sdot

= 119890119909+119910+119911

(1 minus119905

1+1199052

2minus1199053

3+ sdot sdot sdot)

= 119890119909+119910+119911minus119905

(29)

which is the exact solution of (21)-(22) when 120572 = 1

International Journal of Differential Equations 5

Example 9 Consider the following coupled system of non-linear fractional partial differential equations of the form

119863120572

119905V minus V119909119909minus 2VV119909+ V119909119908119909= 0

119863120572

119905119908 minus 119908

119909119909minus 2119908119908

119909+ V119909119908119909= 0

(30)

subject to the initial conditions

V (119909 0) = sin (119909)

119908 (119909 0) = sin (119909) (31)

Applying the Natural Transform to (30) subject to giveninitial conditions we get

119881 (119909 119904 119906) =1

119904sin (119909) + 119906

120572

119904120572N+[V119909119909+ 2VV119909minus V119909119908119909]

119882 (119909 119904 119906) =1

119904sin (119909)

+119906120572

119904120572N+[119908119909119909+ 2119908119908

119909minus V119909119908119909]

(32)

Taking the inverse Natural Transform of (32) we have

V (119909 119905) = sin (119909)

+ Nminus1[119906120572

119904120572N+[V119909119909+ 2VV119909minus V119909119908119909]]

119908 (119909 119905) = sin (119909)

+ Nminus1[119906120572

119904120572N+[119908119908119909119909+ 2119908119908

119909minus V119909119908119909]]

(33)

Now we apply the Homotopy Perturbation Method

V (119909 119905) =infin

sum

119899=0

119901119899V119899(119909 119905)

119908 (119909 119905) =

infin

sum

119899=0

119901119899119908119899(119909 119905)

(34)

Then (33) will become

V (119909 119905) = sin (119909) + 119901(Nminus1 [119906120572

119904120572

sdot N+[

infin

sum

119899=0

119901119899V119899119909119909

+ 2

infin

sum

119899=0

119901119899119867119899minus

infin

sum

119899=0

11990111989911986710158401015840

119899]])

119908 (119909 119905) = sin (119909) + 119901(Nminus1 [119906120572

119904120572

sdot N+[

infin

sum

119899=0

119901119899119908119899119909119909

+ 2

infin

sum

119899=0

1199011198991198671015840

119899minus

infin

sum

119899=0

11990111989911986710158401015840

119899]])

(35)

where1198671198991198671015840119899 and11986710158401015840

119899are Hersquos polynomials which represent

the nonlinear terms VV119909 119908119908119909 and V

119909119908119909 respectively

Some few components of Hersquos polynomials of the nonlin-ear terms VV

119909 119908119908119909 and V

119909119908119909are given as follows

1198670= V0V0119909

1198671= V0V1119909+ V1V0119909

1198672= V0V2119909+ V1V1119909+ V2V0119909

1198671015840

0= 11990801199080119909

1198671015840

1= 11990801199081119909+ 11990811199080119909

1198671015840

2= 11990801199082119909+ 11990811199081119909+ 11990821199080119909

11986710158401015840

0= V01199091199080119909

11986710158401015840

1= V01199091199081119909+ V11199091199080119909

11986710158401015840

2= V21199091199080119909+ V11199091199081119909+ V01199091199082119909

(36)

and so onUsing the coefficients of the like powers of 119901 in (35) we

obtained the following approximations

1199010 V0(119909 119905) = sin (119909)

1199011 V1(119909 119905) = N

minus1[119906120572

119904120572N+[V0119909119909

+ 21198670minus 11986710158401015840

0]]

= minussin (119909) 119905120572

Γ (120572 + 1)

1199012 V2(119909 119905) = N

minus1[119906120572

119904120572N+[V1119909119909

+ 21198671minus 11986710158401015840

1]]

=sin (119909) 1199052120572

Γ (2120572 + 1)

1199013 V3(119909 119905) = N

minus1[119906120572

119904120572N+[V2119909119909

+ 21198672minus 11986710158401015840

2]]

= minussin (119909) 1199053120572

Γ (3120572 + 1)

(37)

and so onSimilarly

1199010 1199080(119909 119905) = sin (119909)

1199011 1199081(119909 119905) = N

minus1[119906120572

119904120572N+[1199080119909119909

+ 21198671015840

0minus 11986710158401015840

0]]

= minussin (119909) 119905120572

Γ (120572 + 1)

6 International Journal of Differential Equations

1199012 1199082(119909 119905) = N

minus1[119906120572

119904120572N+[1199081119909119909

+ 21198671015840

1minus 11986710158401015840

1]]

=sin (119909) 1199052120572

Γ (2120572 + 1)

1199013 1199083(119909 119905) = N

minus1[119906120572

119904120572N+[1199082119909119909

+ 21198671015840

2minus 11986710158401015840

2]]

= minussin (119909) 1199053120572

Γ (3120572 + 1)

(38)and so on

Thus the series solution of (30)-(31) is given by

V (119909 119905) = lim119873rarrinfin

119873

sum

119899=0

V119899(119909 119905) = V

0(119909 119905) + V

1(119909 119905)

+ V2(119909 119905) + sdot sdot sdot = sin (119909)

sdot (1 minus119905120572

Γ (120572 + 1)+

1199052120572

Γ (2120572 + 1)minus

1199053120572

Γ (3120572 + 1)+ sdot sdot sdot)

= sin (119909) (1 +infin

sum

119898=1

(minus119905120572)119898

Γ (119898120572 + 1)) = sin (119909) 119864

120572(minus119905120572)

119908 (119909 119905) = lim119873rarrinfin

119873

sum

119899=0

119908119899(119909 119905) = 119908

0(119909 119905) + 119908

1(119909 119905)

+ 1199082(119909 119905) + sdot sdot sdot = sin (119909)

sdot (1 minus119905120572

Γ (120572 + 1)+

1199052120572

Γ (2120572 + 1)minus

1199053120572

Γ (3120572 + 1)+ sdot sdot sdot)

= sin (119909) (1 +infin

sum

119898=1

(minus119905120572)119898

Γ (119898120572 + 1)) = sin (119909) 119864

120572(minus119905120572)

(39)

Hence the exact solutions of (30)-(31) are given by

V (119909 119905) = sin (119909) 119864120572(minus119905120572)

119908 (119909 119905) = sin (119909) 119864120572(minus119905120572)

(40)

When 120572 = 1 the following result is obtained

V (119909 119905) = lim119873rarrinfin

119873

sum

119899=0

V119899(119909 119905)

= V0(119909 119905) + V

1(119909 119905) + V

2(119909 119905) + sdot sdot sdot

= sin (119909) (1 minus 119905 + 1199052

2+ sdot sdot sdot) = 119890

minus119905 sin (119909)

119908 (119909 119905) = lim119873rarrinfin

119873

sum

119899=0

119908119899(119909 119905)

= 1199080(119909 119905) + 119908

1(119909 119905) + 119908

2(119909 119905) + sdot sdot sdot

= sin (119909) (1 minus 119905 + 1199052

2+ sdot sdot sdot) = 119890

minus119905 sin (119909)

(41)

which is the exact solution of (30)-(31) when 120572 = 1

6 Conclusion

In this paper Natural TransformMethod (NTM) andHomo-topy Perturbation Method (HPM) are successfully combinedto form a robust analytical method called a Hybrid NaturalTransformHomotopy PerturbationMethod for solving linearand nonlinear fractional partial differential equations Theanalytical method gives a series solution which convergesrapidly to an exact or approximate solution with elegantcomputational terms In this analyticalmethod the fractionalderivative is computed in Caputo sense while the nonlinearterm is calculated using Hersquos polynomials The analyticalprocedure is applied successfully and obtained an exactsolution of linear and nonlinear fractional partial differentialequations The simplicity and high accuracy of the analyticalmethod are clearly illustrated Thus the Hybrid NaturalTransform Homotopy Perturbation Method is a powerfulanalytical method for solving linear and nonlinear fractionalpartial differential equations

Competing Interests

The author declares that there is no conflict of interestsregarding the publication of this article

References

[1] I Podlubny Fractional differential equations vol 198 ofMathe-matics in Science and Engineering Academic Press San DiegoCalif USA 1999

[2] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 2003

[3] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[4] K Diethelm and N J Ford ldquoAnalysis of fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 265 no 2 pp 229ndash248 2002

[5] S G Samko A A Kilbas and O I Marichev FractionalIntegrals andDerivatives Gordon andBreach Yverdon Switzer-land 1993

[6] B J West M Bologna and P Grigolini Physics of Fractal Oper-ators Institute for Nonlinear Science Springer New York NYUSA 2003

[7] S S Ray ldquoAnalytical solution for the space fractional diffusionequation by two-step Adomian decomposition methodrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 4 pp 1295ndash1306 2009

[8] K Abbaoui and Y Cherruault ldquoNew ideas for proving conver-gence of Adomian Decomposition Methodsrdquo Computers andMathematics with Applications vol 32 pp 103ndash108 1995

[9] S Momani and Z Odibat ldquoAnalytical solution of a time-fractional Navier-Stokes equation by Adomian decompositionmethodrdquo Applied Mathematics and Computation vol 177 no 2pp 488ndash494 2006

[10] H Jafari and V Daftardar-Gejji ldquoSolving linear and nonlinearfractional diffusion and wave equations by Adomian decompo-sitionrdquo Applied Mathematics and Computation vol 180 no 2pp 488ndash497 2006

International Journal of Differential Equations 7

[11] M Ganjiani ldquoSolution of nonlinear fractional differential equa-tions using homotopy analysis methodrdquo Applied MathematicalModelling vol 34 no 6 pp 1634ndash1641 2010

[12] H Jafari C M Khalique and M Nazari ldquoApplication of theLaplace decomposition method for solving linear and nonlin-ear fractional diffusion-wave equationsrdquo Applied MathematicsLetters vol 24 no 11 pp 1799ndash1805 2011

[13] P K Gupta andM Singh ldquoHomotopy perturbationmethod forfractional Fornberg-Whitham equationrdquo Computers amp Mathe-matics with Applications vol 61 no 2 pp 250ndash254 2011

[14] S Momani and Z Odibat ldquoHomotopy perturbation methodfor nonlinear partial differential equations of fractional orderrdquoPhysics Letters A vol 365 no 5-6 pp 345ndash350 2007

[15] Y-Z Zhang A-M Yang and Y Long ldquoInitial boundary valueproblem for fractal heat equation in the semi-infinite region byyang-laplace transformrdquoThermal Science vol 18 no 2 pp 677ndash681 2014

[16] X-J Yang and D Baleanu ldquoFractal heat conduction problemsolved by local fractional variation iteration methodrdquo ThermalScience vol 17 no 2 pp 625ndash628 2013

[17] X-J Yang D Baleanu Y Khan and S T Mohyud-Din ldquoLocalfractional variational iteration method for diffusion and waveequations on Cantor setsrdquo Romanian Journal of Physics vol 59no 1-2 pp 36ndash48 2014

[18] X J Yang H M Srivastava J H He and D Baleanu ldquoCantor-type cylindrical-coordinate fractional derivativesrdquo Proceedingsof the Romanian Academy Series A vol 14 pp 127ndash133 2013

[19] S Kumar D Kumar S Abbasbandy and M M Rashidi ldquoAna-lytical solution of fractional Navier-Stokes equation by usingmodified Laplace decompositionmethodrdquoAin Shams Engineer-ing Journal vol 5 no 2 pp 569ndash574 2014

[20] B Ghazanfari and A G Ghazanfari ldquoSolving fractional non-linear Schrodinger equation by fractional complex transformmethodrdquo International Journal of Mathematical Modelling ampComputations vol 2 no 4 pp 277ndash281 2012

[21] K M Furati and N-E Tatar ldquoLongtime behavior for a non-linear fractional modelrdquo Journal of Mathematical Analysis andApplications vol 332 no 1 pp 441ndash454 2007

[22] K M Furati and N-E Tatar ldquoBehavior of solutions for aweighted Cauchy type fractional problemrdquo Journal of FractionalCalculus vol 28 pp 23ndash42 2005

[23] K M Furati and N-e Tatar ldquoAn existence result for a nonlocalfractional differential problemrdquo Journal of Fractional Calculusvol 26 pp 43ndash51 2004

[24] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[25] J H He ldquoRecent development of the homotopy perturbationmethodrdquo Topological Methods in Nonlinear Analysis vol 31 pp205ndash209 2008

[26] J-H He ldquoThe homotopy perturbation method for nonlinearoscillators with discontinuitiesrdquoAppliedMathematics and Com-putation vol 151 no 1 pp 287ndash292 2004

[27] Z H Khan andWA Khan ldquoN-transformproperties and appli-cationsrdquo NUST Journal of Engineering Sciences vol 1 pp 127ndash133 2008

[28] F B M Belgacem and R Silambarasan ldquoTheory of the naturaltransformrdquo Mathematics in Engineering Science and Aerospace(MESA) Journal vol 3 no 1 pp 99ndash124 2012

[29] F BM Belgacem andR Silambarasan ldquoAdvances in the naturaltransformrdquo in Proceedings of the 9th International Conference on

Mathematical Problems in Engineering Aerospace and Sciences(ICNPAA rsquo12) vol 1493 of AIP Conference Proceedings pp 106ndash110 Vienna Austria July 2012

[30] R Murray Spiegel Theory and Problems of Laplace TransformSchaumrsquos Outline Series McGraw-Hill New York NY USA1965

[31] F B Belgacem and A Karaballi ldquoSumudu transform funda-mental properties investigations and applicationsrdquo Journal ofApplied Mathematics and Stochastic Analysis vol 2006 ArticleID 91083 23 pages 2006

[32] G KWatugala ldquoSumudu transformmdasha new integral transformto solve differential equations and control engineering prob-lemsrdquo Mathematical Engineering in Industry vol 6 no 4 pp319ndash329 1998

[33] R Silambarasan and F B M Belgacem ldquoApplication of thenatural transform to Maxwellrsquos equationsrdquo in Proceedings of theProgress in Electromagnetics Research Symposium Proceedings(PIERS rsquo11) pp 899ndash902 Suzhou China September 2011

[34] M Rawashdeh and S Maitama ldquoFinding exact solutions ofnonlinear PDEs using the natural decomposition methodrdquoMathematical Methods in the Applied Sciences 2016

[35] S Maitama and S M Kurawa ldquoAn efficient technique forsolving gas dynamics equation using the natural decompositionmethodrdquo International Mathematical Forum vol 9 no 24 pp1177ndash1190 2014

[36] S Maitama ldquoExact solution of equation governing the unsteadyflow of a polytropic gas using the natural decompositionmethodrdquoAppliedMathematical Sciences vol 8 no 77 pp 3809ndash3823 2014

[37] M S Rawashdeh and S Maitama ldquoSolving PDEs using thenatural decomposition methodrdquo Nonlinear Studies vol 23 no1 pp 63ndash72 2016

[38] M S Rawashdeh and S Maitama ldquoSolving nonlinear ordinarydifferential equations using the NDMrdquo The Journal of AppliedAnalysis and Computation vol 5 no 1 pp 77ndash88 2015

[39] M S Rawashdeh and S Maitama ldquoSolving coupled systemof nonlinear PDErsquos using the natural decomposition methodrdquoInternational Journal of Pure and Applied Mathematics vol 92no 5 pp 757ndash776 2014

[40] Wikipedia note about the Natural transform August 2016httpsenwikipediaorgwikiN-transform

[41] 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

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 International Journal of Differential Equations

1199012 V2(119909 119905)

= minusNminus1[119906119899120572

119904119899120572N+[119872 (V (119909 119905)) + 119867

1(V)]]

1199013 V3(119909 119905)

= minusNminus1[119906119899120572

119904119899120572N+[119872 (V (119909 119905)) + 119867

2(V)]]

(19)

and so onHence the series solution of (11)-(12) is given by

V (119909 119905) = lim119873rarrinfin

119873

sum

119899=0

V119899(119909 119905) (20)

5 Applications

In this section the application of the Hybrid NaturalTransform Homotopy Perturbation Method to linear andnonlinear fractional partial differential equations is clearlydemonstrated to show its simplicity and high accuracy

Example 8 Consider the following fractional diffusion equa-tion of the form119863120572

119905V + V119909119909+ V119910119910+ V119911119911= 0

minusinfin lt 119909 119910 119911 lt infin 119905 gt 0

(21)

subject to the initial condition

V (119909 119910 119911 0) = 119890119909+119910+119911 120572 isin (0 1) (22)

Applying the Natural Transform to (21) subject to the giveninitial condition we get

119881 (119909 119910 119911 119904 119906) =119890119909+119910+119911

119904+119906120572

119904120572N+[V119909119909+ V119910119910+ V119911119911] (23)

Taking the inverse Natural Transform of (23) we get

V (119909 119910 119911 119905) = 119890119909+119910+119911

+ Nminus1[119906120572

119904120572N+[V119909119909+ V119910119910+ V119911119911]]

(24)

Now we apply Homotopy Perturbation Method

V (119909 119910 119911 119905) =infin

sum

119899=0

119901119899V119899(119909 119910 119911 119905) (25)

Then (24) will becomeinfin

sum

119899=0

119901119899V119899(119909 119910 119911 119905) = 119890

119909+119910+119911minus 119901(N

minus1[119906120572

119904120572

sdot N+[

infin

sum

119899=0

119901119899V119899119909119909

+

infin

sum

119899=0

119901119899V119899119910119910

+

infin

sum

119899=0

119901119899V119899119911119911]])

(26)

Using the coefficients of like powers of 119901 in (26) we obtainedthe following approximations

1199010 V0(119909 119910 119911 119905) = 119890

119909+119910+119911

1199011 V1(119909 119910 119911 119905)

= minusNminus1[119906120572

119904120572N+[V0119909119909

+ V0119910119910

+ V0119911119911]]

= minus119890119909+119910+119911

119905120572

Γ (120572 + 1)

1199012 V2(119909 119910 119911 119905)

= minusNminus1[119906120572

119904120572N+[V1119909119909

+ V1119910119910

+ V1119911119911]]

=119890119909+119910+119911

1199052120572

Γ (2120572 + 1)

1199013 V3(119909 119910 119911 119905)

= minusNminus1[119906120572

119904120572N+[V2119909119909

+ V2119910119910

+ V2119911119911]]

= minus119890119909+119910+119911

1199053120572

Γ (3120572 + 1)

(27)

and so onThen the series solution of (21)-(22) is given by

V (119909 119910 119911 119905) = lim119873rarrinfin

119873

sum

119899=0

V119899(119909 119910 119911 119905) = V

0(119909 119910 119911 119905)

+ V1(119909 119910 119911 119905) + V

2(119909 119910 119911 119905) + V

3(119909 119910 119911 119905) + sdot sdot sdot

= 119890119909+119910+119911

(1 minus119905120572

Γ (120572 + 1)+

1199052120572

Γ (2120572 + 1)

minus1199053120572

Γ (3120572 + 1)+ sdot sdot sdot) = 119890

119909+119910+119911(1

+

infin

sum

119898=1

(minus119905120572)119898

Γ (119898120572 + 1)) = 119890119909+119910+119911

119864120572(minus119905120572)

(28)

When 120572 = 1 the following result is obtained

V (119909 119910 119911 119905) = lim119873rarrinfin

119873

sum

119899=0

V119899(119909 119910 119911 119905)

= V0(119909 119910 119911 119905) + V

1(119909 119910 119911 119905)

+ V2(119909 119910 119911 119905) + V

3(119909 119910 119911 119905) + sdot sdot sdot

= 119890119909+119910+119911

(1 minus119905

1+1199052

2minus1199053

3+ sdot sdot sdot)

= 119890119909+119910+119911minus119905

(29)

which is the exact solution of (21)-(22) when 120572 = 1

International Journal of Differential Equations 5

Example 9 Consider the following coupled system of non-linear fractional partial differential equations of the form

119863120572

119905V minus V119909119909minus 2VV119909+ V119909119908119909= 0

119863120572

119905119908 minus 119908

119909119909minus 2119908119908

119909+ V119909119908119909= 0

(30)

subject to the initial conditions

V (119909 0) = sin (119909)

119908 (119909 0) = sin (119909) (31)

Applying the Natural Transform to (30) subject to giveninitial conditions we get

119881 (119909 119904 119906) =1

119904sin (119909) + 119906

120572

119904120572N+[V119909119909+ 2VV119909minus V119909119908119909]

119882 (119909 119904 119906) =1

119904sin (119909)

+119906120572

119904120572N+[119908119909119909+ 2119908119908

119909minus V119909119908119909]

(32)

Taking the inverse Natural Transform of (32) we have

V (119909 119905) = sin (119909)

+ Nminus1[119906120572

119904120572N+[V119909119909+ 2VV119909minus V119909119908119909]]

119908 (119909 119905) = sin (119909)

+ Nminus1[119906120572

119904120572N+[119908119908119909119909+ 2119908119908

119909minus V119909119908119909]]

(33)

Now we apply the Homotopy Perturbation Method

V (119909 119905) =infin

sum

119899=0

119901119899V119899(119909 119905)

119908 (119909 119905) =

infin

sum

119899=0

119901119899119908119899(119909 119905)

(34)

Then (33) will become

V (119909 119905) = sin (119909) + 119901(Nminus1 [119906120572

119904120572

sdot N+[

infin

sum

119899=0

119901119899V119899119909119909

+ 2

infin

sum

119899=0

119901119899119867119899minus

infin

sum

119899=0

11990111989911986710158401015840

119899]])

119908 (119909 119905) = sin (119909) + 119901(Nminus1 [119906120572

119904120572

sdot N+[

infin

sum

119899=0

119901119899119908119899119909119909

+ 2

infin

sum

119899=0

1199011198991198671015840

119899minus

infin

sum

119899=0

11990111989911986710158401015840

119899]])

(35)

where1198671198991198671015840119899 and11986710158401015840

119899are Hersquos polynomials which represent

the nonlinear terms VV119909 119908119908119909 and V

119909119908119909 respectively

Some few components of Hersquos polynomials of the nonlin-ear terms VV

119909 119908119908119909 and V

119909119908119909are given as follows

1198670= V0V0119909

1198671= V0V1119909+ V1V0119909

1198672= V0V2119909+ V1V1119909+ V2V0119909

1198671015840

0= 11990801199080119909

1198671015840

1= 11990801199081119909+ 11990811199080119909

1198671015840

2= 11990801199082119909+ 11990811199081119909+ 11990821199080119909

11986710158401015840

0= V01199091199080119909

11986710158401015840

1= V01199091199081119909+ V11199091199080119909

11986710158401015840

2= V21199091199080119909+ V11199091199081119909+ V01199091199082119909

(36)

and so onUsing the coefficients of the like powers of 119901 in (35) we

obtained the following approximations

1199010 V0(119909 119905) = sin (119909)

1199011 V1(119909 119905) = N

minus1[119906120572

119904120572N+[V0119909119909

+ 21198670minus 11986710158401015840

0]]

= minussin (119909) 119905120572

Γ (120572 + 1)

1199012 V2(119909 119905) = N

minus1[119906120572

119904120572N+[V1119909119909

+ 21198671minus 11986710158401015840

1]]

=sin (119909) 1199052120572

Γ (2120572 + 1)

1199013 V3(119909 119905) = N

minus1[119906120572

119904120572N+[V2119909119909

+ 21198672minus 11986710158401015840

2]]

= minussin (119909) 1199053120572

Γ (3120572 + 1)

(37)

and so onSimilarly

1199010 1199080(119909 119905) = sin (119909)

1199011 1199081(119909 119905) = N

minus1[119906120572

119904120572N+[1199080119909119909

+ 21198671015840

0minus 11986710158401015840

0]]

= minussin (119909) 119905120572

Γ (120572 + 1)

6 International Journal of Differential Equations

1199012 1199082(119909 119905) = N

minus1[119906120572

119904120572N+[1199081119909119909

+ 21198671015840

1minus 11986710158401015840

1]]

=sin (119909) 1199052120572

Γ (2120572 + 1)

1199013 1199083(119909 119905) = N

minus1[119906120572

119904120572N+[1199082119909119909

+ 21198671015840

2minus 11986710158401015840

2]]

= minussin (119909) 1199053120572

Γ (3120572 + 1)

(38)and so on

Thus the series solution of (30)-(31) is given by

V (119909 119905) = lim119873rarrinfin

119873

sum

119899=0

V119899(119909 119905) = V

0(119909 119905) + V

1(119909 119905)

+ V2(119909 119905) + sdot sdot sdot = sin (119909)

sdot (1 minus119905120572

Γ (120572 + 1)+

1199052120572

Γ (2120572 + 1)minus

1199053120572

Γ (3120572 + 1)+ sdot sdot sdot)

= sin (119909) (1 +infin

sum

119898=1

(minus119905120572)119898

Γ (119898120572 + 1)) = sin (119909) 119864

120572(minus119905120572)

119908 (119909 119905) = lim119873rarrinfin

119873

sum

119899=0

119908119899(119909 119905) = 119908

0(119909 119905) + 119908

1(119909 119905)

+ 1199082(119909 119905) + sdot sdot sdot = sin (119909)

sdot (1 minus119905120572

Γ (120572 + 1)+

1199052120572

Γ (2120572 + 1)minus

1199053120572

Γ (3120572 + 1)+ sdot sdot sdot)

= sin (119909) (1 +infin

sum

119898=1

(minus119905120572)119898

Γ (119898120572 + 1)) = sin (119909) 119864

120572(minus119905120572)

(39)

Hence the exact solutions of (30)-(31) are given by

V (119909 119905) = sin (119909) 119864120572(minus119905120572)

119908 (119909 119905) = sin (119909) 119864120572(minus119905120572)

(40)

When 120572 = 1 the following result is obtained

V (119909 119905) = lim119873rarrinfin

119873

sum

119899=0

V119899(119909 119905)

= V0(119909 119905) + V

1(119909 119905) + V

2(119909 119905) + sdot sdot sdot

= sin (119909) (1 minus 119905 + 1199052

2+ sdot sdot sdot) = 119890

minus119905 sin (119909)

119908 (119909 119905) = lim119873rarrinfin

119873

sum

119899=0

119908119899(119909 119905)

= 1199080(119909 119905) + 119908

1(119909 119905) + 119908

2(119909 119905) + sdot sdot sdot

= sin (119909) (1 minus 119905 + 1199052

2+ sdot sdot sdot) = 119890

minus119905 sin (119909)

(41)

which is the exact solution of (30)-(31) when 120572 = 1

6 Conclusion

In this paper Natural TransformMethod (NTM) andHomo-topy Perturbation Method (HPM) are successfully combinedto form a robust analytical method called a Hybrid NaturalTransformHomotopy PerturbationMethod for solving linearand nonlinear fractional partial differential equations Theanalytical method gives a series solution which convergesrapidly to an exact or approximate solution with elegantcomputational terms In this analyticalmethod the fractionalderivative is computed in Caputo sense while the nonlinearterm is calculated using Hersquos polynomials The analyticalprocedure is applied successfully and obtained an exactsolution of linear and nonlinear fractional partial differentialequations The simplicity and high accuracy of the analyticalmethod are clearly illustrated Thus the Hybrid NaturalTransform Homotopy Perturbation Method is a powerfulanalytical method for solving linear and nonlinear fractionalpartial differential equations

Competing Interests

The author declares that there is no conflict of interestsregarding the publication of this article

References

[1] I Podlubny Fractional differential equations vol 198 ofMathe-matics in Science and Engineering Academic Press San DiegoCalif USA 1999

[2] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 2003

[3] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[4] K Diethelm and N J Ford ldquoAnalysis of fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 265 no 2 pp 229ndash248 2002

[5] S G Samko A A Kilbas and O I Marichev FractionalIntegrals andDerivatives Gordon andBreach Yverdon Switzer-land 1993

[6] B J West M Bologna and P Grigolini Physics of Fractal Oper-ators Institute for Nonlinear Science Springer New York NYUSA 2003

[7] S S Ray ldquoAnalytical solution for the space fractional diffusionequation by two-step Adomian decomposition methodrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 4 pp 1295ndash1306 2009

[8] K Abbaoui and Y Cherruault ldquoNew ideas for proving conver-gence of Adomian Decomposition Methodsrdquo Computers andMathematics with Applications vol 32 pp 103ndash108 1995

[9] S Momani and Z Odibat ldquoAnalytical solution of a time-fractional Navier-Stokes equation by Adomian decompositionmethodrdquo Applied Mathematics and Computation vol 177 no 2pp 488ndash494 2006

[10] H Jafari and V Daftardar-Gejji ldquoSolving linear and nonlinearfractional diffusion and wave equations by Adomian decompo-sitionrdquo Applied Mathematics and Computation vol 180 no 2pp 488ndash497 2006

International Journal of Differential Equations 7

[11] M Ganjiani ldquoSolution of nonlinear fractional differential equa-tions using homotopy analysis methodrdquo Applied MathematicalModelling vol 34 no 6 pp 1634ndash1641 2010

[12] H Jafari C M Khalique and M Nazari ldquoApplication of theLaplace decomposition method for solving linear and nonlin-ear fractional diffusion-wave equationsrdquo Applied MathematicsLetters vol 24 no 11 pp 1799ndash1805 2011

[13] P K Gupta andM Singh ldquoHomotopy perturbationmethod forfractional Fornberg-Whitham equationrdquo Computers amp Mathe-matics with Applications vol 61 no 2 pp 250ndash254 2011

[14] S Momani and Z Odibat ldquoHomotopy perturbation methodfor nonlinear partial differential equations of fractional orderrdquoPhysics Letters A vol 365 no 5-6 pp 345ndash350 2007

[15] Y-Z Zhang A-M Yang and Y Long ldquoInitial boundary valueproblem for fractal heat equation in the semi-infinite region byyang-laplace transformrdquoThermal Science vol 18 no 2 pp 677ndash681 2014

[16] X-J Yang and D Baleanu ldquoFractal heat conduction problemsolved by local fractional variation iteration methodrdquo ThermalScience vol 17 no 2 pp 625ndash628 2013

[17] X-J Yang D Baleanu Y Khan and S T Mohyud-Din ldquoLocalfractional variational iteration method for diffusion and waveequations on Cantor setsrdquo Romanian Journal of Physics vol 59no 1-2 pp 36ndash48 2014

[18] X J Yang H M Srivastava J H He and D Baleanu ldquoCantor-type cylindrical-coordinate fractional derivativesrdquo Proceedingsof the Romanian Academy Series A vol 14 pp 127ndash133 2013

[19] S Kumar D Kumar S Abbasbandy and M M Rashidi ldquoAna-lytical solution of fractional Navier-Stokes equation by usingmodified Laplace decompositionmethodrdquoAin Shams Engineer-ing Journal vol 5 no 2 pp 569ndash574 2014

[20] B Ghazanfari and A G Ghazanfari ldquoSolving fractional non-linear Schrodinger equation by fractional complex transformmethodrdquo International Journal of Mathematical Modelling ampComputations vol 2 no 4 pp 277ndash281 2012

[21] K M Furati and N-E Tatar ldquoLongtime behavior for a non-linear fractional modelrdquo Journal of Mathematical Analysis andApplications vol 332 no 1 pp 441ndash454 2007

[22] K M Furati and N-E Tatar ldquoBehavior of solutions for aweighted Cauchy type fractional problemrdquo Journal of FractionalCalculus vol 28 pp 23ndash42 2005

[23] K M Furati and N-e Tatar ldquoAn existence result for a nonlocalfractional differential problemrdquo Journal of Fractional Calculusvol 26 pp 43ndash51 2004

[24] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[25] J H He ldquoRecent development of the homotopy perturbationmethodrdquo Topological Methods in Nonlinear Analysis vol 31 pp205ndash209 2008

[26] J-H He ldquoThe homotopy perturbation method for nonlinearoscillators with discontinuitiesrdquoAppliedMathematics and Com-putation vol 151 no 1 pp 287ndash292 2004

[27] Z H Khan andWA Khan ldquoN-transformproperties and appli-cationsrdquo NUST Journal of Engineering Sciences vol 1 pp 127ndash133 2008

[28] F B M Belgacem and R Silambarasan ldquoTheory of the naturaltransformrdquo Mathematics in Engineering Science and Aerospace(MESA) Journal vol 3 no 1 pp 99ndash124 2012

[29] F BM Belgacem andR Silambarasan ldquoAdvances in the naturaltransformrdquo in Proceedings of the 9th International Conference on

Mathematical Problems in Engineering Aerospace and Sciences(ICNPAA rsquo12) vol 1493 of AIP Conference Proceedings pp 106ndash110 Vienna Austria July 2012

[30] R Murray Spiegel Theory and Problems of Laplace TransformSchaumrsquos Outline Series McGraw-Hill New York NY USA1965

[31] F B Belgacem and A Karaballi ldquoSumudu transform funda-mental properties investigations and applicationsrdquo Journal ofApplied Mathematics and Stochastic Analysis vol 2006 ArticleID 91083 23 pages 2006

[32] G KWatugala ldquoSumudu transformmdasha new integral transformto solve differential equations and control engineering prob-lemsrdquo Mathematical Engineering in Industry vol 6 no 4 pp319ndash329 1998

[33] R Silambarasan and F B M Belgacem ldquoApplication of thenatural transform to Maxwellrsquos equationsrdquo in Proceedings of theProgress in Electromagnetics Research Symposium Proceedings(PIERS rsquo11) pp 899ndash902 Suzhou China September 2011

[34] M Rawashdeh and S Maitama ldquoFinding exact solutions ofnonlinear PDEs using the natural decomposition methodrdquoMathematical Methods in the Applied Sciences 2016

[35] S Maitama and S M Kurawa ldquoAn efficient technique forsolving gas dynamics equation using the natural decompositionmethodrdquo International Mathematical Forum vol 9 no 24 pp1177ndash1190 2014

[36] S Maitama ldquoExact solution of equation governing the unsteadyflow of a polytropic gas using the natural decompositionmethodrdquoAppliedMathematical Sciences vol 8 no 77 pp 3809ndash3823 2014

[37] M S Rawashdeh and S Maitama ldquoSolving PDEs using thenatural decomposition methodrdquo Nonlinear Studies vol 23 no1 pp 63ndash72 2016

[38] M S Rawashdeh and S Maitama ldquoSolving nonlinear ordinarydifferential equations using the NDMrdquo The Journal of AppliedAnalysis and Computation vol 5 no 1 pp 77ndash88 2015

[39] M S Rawashdeh and S Maitama ldquoSolving coupled systemof nonlinear PDErsquos using the natural decomposition methodrdquoInternational Journal of Pure and Applied Mathematics vol 92no 5 pp 757ndash776 2014

[40] Wikipedia note about the Natural transform August 2016httpsenwikipediaorgwikiN-transform

[41] 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

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Differential Equations 5

Example 9 Consider the following coupled system of non-linear fractional partial differential equations of the form

119863120572

119905V minus V119909119909minus 2VV119909+ V119909119908119909= 0

119863120572

119905119908 minus 119908

119909119909minus 2119908119908

119909+ V119909119908119909= 0

(30)

subject to the initial conditions

V (119909 0) = sin (119909)

119908 (119909 0) = sin (119909) (31)

Applying the Natural Transform to (30) subject to giveninitial conditions we get

119881 (119909 119904 119906) =1

119904sin (119909) + 119906

120572

119904120572N+[V119909119909+ 2VV119909minus V119909119908119909]

119882 (119909 119904 119906) =1

119904sin (119909)

+119906120572

119904120572N+[119908119909119909+ 2119908119908

119909minus V119909119908119909]

(32)

Taking the inverse Natural Transform of (32) we have

V (119909 119905) = sin (119909)

+ Nminus1[119906120572

119904120572N+[V119909119909+ 2VV119909minus V119909119908119909]]

119908 (119909 119905) = sin (119909)

+ Nminus1[119906120572

119904120572N+[119908119908119909119909+ 2119908119908

119909minus V119909119908119909]]

(33)

Now we apply the Homotopy Perturbation Method

V (119909 119905) =infin

sum

119899=0

119901119899V119899(119909 119905)

119908 (119909 119905) =

infin

sum

119899=0

119901119899119908119899(119909 119905)

(34)

Then (33) will become

V (119909 119905) = sin (119909) + 119901(Nminus1 [119906120572

119904120572

sdot N+[

infin

sum

119899=0

119901119899V119899119909119909

+ 2

infin

sum

119899=0

119901119899119867119899minus

infin

sum

119899=0

11990111989911986710158401015840

119899]])

119908 (119909 119905) = sin (119909) + 119901(Nminus1 [119906120572

119904120572

sdot N+[

infin

sum

119899=0

119901119899119908119899119909119909

+ 2

infin

sum

119899=0

1199011198991198671015840

119899minus

infin

sum

119899=0

11990111989911986710158401015840

119899]])

(35)

where1198671198991198671015840119899 and11986710158401015840

119899are Hersquos polynomials which represent

the nonlinear terms VV119909 119908119908119909 and V

119909119908119909 respectively

Some few components of Hersquos polynomials of the nonlin-ear terms VV

119909 119908119908119909 and V

119909119908119909are given as follows

1198670= V0V0119909

1198671= V0V1119909+ V1V0119909

1198672= V0V2119909+ V1V1119909+ V2V0119909

1198671015840

0= 11990801199080119909

1198671015840

1= 11990801199081119909+ 11990811199080119909

1198671015840

2= 11990801199082119909+ 11990811199081119909+ 11990821199080119909

11986710158401015840

0= V01199091199080119909

11986710158401015840

1= V01199091199081119909+ V11199091199080119909

11986710158401015840

2= V21199091199080119909+ V11199091199081119909+ V01199091199082119909

(36)

and so onUsing the coefficients of the like powers of 119901 in (35) we

obtained the following approximations

1199010 V0(119909 119905) = sin (119909)

1199011 V1(119909 119905) = N

minus1[119906120572

119904120572N+[V0119909119909

+ 21198670minus 11986710158401015840

0]]

= minussin (119909) 119905120572

Γ (120572 + 1)

1199012 V2(119909 119905) = N

minus1[119906120572

119904120572N+[V1119909119909

+ 21198671minus 11986710158401015840

1]]

=sin (119909) 1199052120572

Γ (2120572 + 1)

1199013 V3(119909 119905) = N

minus1[119906120572

119904120572N+[V2119909119909

+ 21198672minus 11986710158401015840

2]]

= minussin (119909) 1199053120572

Γ (3120572 + 1)

(37)

and so onSimilarly

1199010 1199080(119909 119905) = sin (119909)

1199011 1199081(119909 119905) = N

minus1[119906120572

119904120572N+[1199080119909119909

+ 21198671015840

0minus 11986710158401015840

0]]

= minussin (119909) 119905120572

Γ (120572 + 1)

6 International Journal of Differential Equations

1199012 1199082(119909 119905) = N

minus1[119906120572

119904120572N+[1199081119909119909

+ 21198671015840

1minus 11986710158401015840

1]]

=sin (119909) 1199052120572

Γ (2120572 + 1)

1199013 1199083(119909 119905) = N

minus1[119906120572

119904120572N+[1199082119909119909

+ 21198671015840

2minus 11986710158401015840

2]]

= minussin (119909) 1199053120572

Γ (3120572 + 1)

(38)and so on

Thus the series solution of (30)-(31) is given by

V (119909 119905) = lim119873rarrinfin

119873

sum

119899=0

V119899(119909 119905) = V

0(119909 119905) + V

1(119909 119905)

+ V2(119909 119905) + sdot sdot sdot = sin (119909)

sdot (1 minus119905120572

Γ (120572 + 1)+

1199052120572

Γ (2120572 + 1)minus

1199053120572

Γ (3120572 + 1)+ sdot sdot sdot)

= sin (119909) (1 +infin

sum

119898=1

(minus119905120572)119898

Γ (119898120572 + 1)) = sin (119909) 119864

120572(minus119905120572)

119908 (119909 119905) = lim119873rarrinfin

119873

sum

119899=0

119908119899(119909 119905) = 119908

0(119909 119905) + 119908

1(119909 119905)

+ 1199082(119909 119905) + sdot sdot sdot = sin (119909)

sdot (1 minus119905120572

Γ (120572 + 1)+

1199052120572

Γ (2120572 + 1)minus

1199053120572

Γ (3120572 + 1)+ sdot sdot sdot)

= sin (119909) (1 +infin

sum

119898=1

(minus119905120572)119898

Γ (119898120572 + 1)) = sin (119909) 119864

120572(minus119905120572)

(39)

Hence the exact solutions of (30)-(31) are given by

V (119909 119905) = sin (119909) 119864120572(minus119905120572)

119908 (119909 119905) = sin (119909) 119864120572(minus119905120572)

(40)

When 120572 = 1 the following result is obtained

V (119909 119905) = lim119873rarrinfin

119873

sum

119899=0

V119899(119909 119905)

= V0(119909 119905) + V

1(119909 119905) + V

2(119909 119905) + sdot sdot sdot

= sin (119909) (1 minus 119905 + 1199052

2+ sdot sdot sdot) = 119890

minus119905 sin (119909)

119908 (119909 119905) = lim119873rarrinfin

119873

sum

119899=0

119908119899(119909 119905)

= 1199080(119909 119905) + 119908

1(119909 119905) + 119908

2(119909 119905) + sdot sdot sdot

= sin (119909) (1 minus 119905 + 1199052

2+ sdot sdot sdot) = 119890

minus119905 sin (119909)

(41)

which is the exact solution of (30)-(31) when 120572 = 1

6 Conclusion

In this paper Natural TransformMethod (NTM) andHomo-topy Perturbation Method (HPM) are successfully combinedto form a robust analytical method called a Hybrid NaturalTransformHomotopy PerturbationMethod for solving linearand nonlinear fractional partial differential equations Theanalytical method gives a series solution which convergesrapidly to an exact or approximate solution with elegantcomputational terms In this analyticalmethod the fractionalderivative is computed in Caputo sense while the nonlinearterm is calculated using Hersquos polynomials The analyticalprocedure is applied successfully and obtained an exactsolution of linear and nonlinear fractional partial differentialequations The simplicity and high accuracy of the analyticalmethod are clearly illustrated Thus the Hybrid NaturalTransform Homotopy Perturbation Method is a powerfulanalytical method for solving linear and nonlinear fractionalpartial differential equations

Competing Interests

The author declares that there is no conflict of interestsregarding the publication of this article

References

[1] I Podlubny Fractional differential equations vol 198 ofMathe-matics in Science and Engineering Academic Press San DiegoCalif USA 1999

[2] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 2003

[3] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[4] K Diethelm and N J Ford ldquoAnalysis of fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 265 no 2 pp 229ndash248 2002

[5] S G Samko A A Kilbas and O I Marichev FractionalIntegrals andDerivatives Gordon andBreach Yverdon Switzer-land 1993

[6] B J West M Bologna and P Grigolini Physics of Fractal Oper-ators Institute for Nonlinear Science Springer New York NYUSA 2003

[7] S S Ray ldquoAnalytical solution for the space fractional diffusionequation by two-step Adomian decomposition methodrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 4 pp 1295ndash1306 2009

[8] K Abbaoui and Y Cherruault ldquoNew ideas for proving conver-gence of Adomian Decomposition Methodsrdquo Computers andMathematics with Applications vol 32 pp 103ndash108 1995

[9] S Momani and Z Odibat ldquoAnalytical solution of a time-fractional Navier-Stokes equation by Adomian decompositionmethodrdquo Applied Mathematics and Computation vol 177 no 2pp 488ndash494 2006

[10] H Jafari and V Daftardar-Gejji ldquoSolving linear and nonlinearfractional diffusion and wave equations by Adomian decompo-sitionrdquo Applied Mathematics and Computation vol 180 no 2pp 488ndash497 2006

International Journal of Differential Equations 7

[11] M Ganjiani ldquoSolution of nonlinear fractional differential equa-tions using homotopy analysis methodrdquo Applied MathematicalModelling vol 34 no 6 pp 1634ndash1641 2010

[12] H Jafari C M Khalique and M Nazari ldquoApplication of theLaplace decomposition method for solving linear and nonlin-ear fractional diffusion-wave equationsrdquo Applied MathematicsLetters vol 24 no 11 pp 1799ndash1805 2011

[13] P K Gupta andM Singh ldquoHomotopy perturbationmethod forfractional Fornberg-Whitham equationrdquo Computers amp Mathe-matics with Applications vol 61 no 2 pp 250ndash254 2011

[14] S Momani and Z Odibat ldquoHomotopy perturbation methodfor nonlinear partial differential equations of fractional orderrdquoPhysics Letters A vol 365 no 5-6 pp 345ndash350 2007

[15] Y-Z Zhang A-M Yang and Y Long ldquoInitial boundary valueproblem for fractal heat equation in the semi-infinite region byyang-laplace transformrdquoThermal Science vol 18 no 2 pp 677ndash681 2014

[16] X-J Yang and D Baleanu ldquoFractal heat conduction problemsolved by local fractional variation iteration methodrdquo ThermalScience vol 17 no 2 pp 625ndash628 2013

[17] X-J Yang D Baleanu Y Khan and S T Mohyud-Din ldquoLocalfractional variational iteration method for diffusion and waveequations on Cantor setsrdquo Romanian Journal of Physics vol 59no 1-2 pp 36ndash48 2014

[18] X J Yang H M Srivastava J H He and D Baleanu ldquoCantor-type cylindrical-coordinate fractional derivativesrdquo Proceedingsof the Romanian Academy Series A vol 14 pp 127ndash133 2013

[19] S Kumar D Kumar S Abbasbandy and M M Rashidi ldquoAna-lytical solution of fractional Navier-Stokes equation by usingmodified Laplace decompositionmethodrdquoAin Shams Engineer-ing Journal vol 5 no 2 pp 569ndash574 2014

[20] B Ghazanfari and A G Ghazanfari ldquoSolving fractional non-linear Schrodinger equation by fractional complex transformmethodrdquo International Journal of Mathematical Modelling ampComputations vol 2 no 4 pp 277ndash281 2012

[21] K M Furati and N-E Tatar ldquoLongtime behavior for a non-linear fractional modelrdquo Journal of Mathematical Analysis andApplications vol 332 no 1 pp 441ndash454 2007

[22] K M Furati and N-E Tatar ldquoBehavior of solutions for aweighted Cauchy type fractional problemrdquo Journal of FractionalCalculus vol 28 pp 23ndash42 2005

[23] K M Furati and N-e Tatar ldquoAn existence result for a nonlocalfractional differential problemrdquo Journal of Fractional Calculusvol 26 pp 43ndash51 2004

[24] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[25] J H He ldquoRecent development of the homotopy perturbationmethodrdquo Topological Methods in Nonlinear Analysis vol 31 pp205ndash209 2008

[26] J-H He ldquoThe homotopy perturbation method for nonlinearoscillators with discontinuitiesrdquoAppliedMathematics and Com-putation vol 151 no 1 pp 287ndash292 2004

[27] Z H Khan andWA Khan ldquoN-transformproperties and appli-cationsrdquo NUST Journal of Engineering Sciences vol 1 pp 127ndash133 2008

[28] F B M Belgacem and R Silambarasan ldquoTheory of the naturaltransformrdquo Mathematics in Engineering Science and Aerospace(MESA) Journal vol 3 no 1 pp 99ndash124 2012

[29] F BM Belgacem andR Silambarasan ldquoAdvances in the naturaltransformrdquo in Proceedings of the 9th International Conference on

Mathematical Problems in Engineering Aerospace and Sciences(ICNPAA rsquo12) vol 1493 of AIP Conference Proceedings pp 106ndash110 Vienna Austria July 2012

[30] R Murray Spiegel Theory and Problems of Laplace TransformSchaumrsquos Outline Series McGraw-Hill New York NY USA1965

[31] F B Belgacem and A Karaballi ldquoSumudu transform funda-mental properties investigations and applicationsrdquo Journal ofApplied Mathematics and Stochastic Analysis vol 2006 ArticleID 91083 23 pages 2006

[32] G KWatugala ldquoSumudu transformmdasha new integral transformto solve differential equations and control engineering prob-lemsrdquo Mathematical Engineering in Industry vol 6 no 4 pp319ndash329 1998

[33] R Silambarasan and F B M Belgacem ldquoApplication of thenatural transform to Maxwellrsquos equationsrdquo in Proceedings of theProgress in Electromagnetics Research Symposium Proceedings(PIERS rsquo11) pp 899ndash902 Suzhou China September 2011

[34] M Rawashdeh and S Maitama ldquoFinding exact solutions ofnonlinear PDEs using the natural decomposition methodrdquoMathematical Methods in the Applied Sciences 2016

[35] S Maitama and S M Kurawa ldquoAn efficient technique forsolving gas dynamics equation using the natural decompositionmethodrdquo International Mathematical Forum vol 9 no 24 pp1177ndash1190 2014

[36] S Maitama ldquoExact solution of equation governing the unsteadyflow of a polytropic gas using the natural decompositionmethodrdquoAppliedMathematical Sciences vol 8 no 77 pp 3809ndash3823 2014

[37] M S Rawashdeh and S Maitama ldquoSolving PDEs using thenatural decomposition methodrdquo Nonlinear Studies vol 23 no1 pp 63ndash72 2016

[38] M S Rawashdeh and S Maitama ldquoSolving nonlinear ordinarydifferential equations using the NDMrdquo The Journal of AppliedAnalysis and Computation vol 5 no 1 pp 77ndash88 2015

[39] M S Rawashdeh and S Maitama ldquoSolving coupled systemof nonlinear PDErsquos using the natural decomposition methodrdquoInternational Journal of Pure and Applied Mathematics vol 92no 5 pp 757ndash776 2014

[40] Wikipedia note about the Natural transform August 2016httpsenwikipediaorgwikiN-transform

[41] 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

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 International Journal of Differential Equations

1199012 1199082(119909 119905) = N

minus1[119906120572

119904120572N+[1199081119909119909

+ 21198671015840

1minus 11986710158401015840

1]]

=sin (119909) 1199052120572

Γ (2120572 + 1)

1199013 1199083(119909 119905) = N

minus1[119906120572

119904120572N+[1199082119909119909

+ 21198671015840

2minus 11986710158401015840

2]]

= minussin (119909) 1199053120572

Γ (3120572 + 1)

(38)and so on

Thus the series solution of (30)-(31) is given by

V (119909 119905) = lim119873rarrinfin

119873

sum

119899=0

V119899(119909 119905) = V

0(119909 119905) + V

1(119909 119905)

+ V2(119909 119905) + sdot sdot sdot = sin (119909)

sdot (1 minus119905120572

Γ (120572 + 1)+

1199052120572

Γ (2120572 + 1)minus

1199053120572

Γ (3120572 + 1)+ sdot sdot sdot)

= sin (119909) (1 +infin

sum

119898=1

(minus119905120572)119898

Γ (119898120572 + 1)) = sin (119909) 119864

120572(minus119905120572)

119908 (119909 119905) = lim119873rarrinfin

119873

sum

119899=0

119908119899(119909 119905) = 119908

0(119909 119905) + 119908

1(119909 119905)

+ 1199082(119909 119905) + sdot sdot sdot = sin (119909)

sdot (1 minus119905120572

Γ (120572 + 1)+

1199052120572

Γ (2120572 + 1)minus

1199053120572

Γ (3120572 + 1)+ sdot sdot sdot)

= sin (119909) (1 +infin

sum

119898=1

(minus119905120572)119898

Γ (119898120572 + 1)) = sin (119909) 119864

120572(minus119905120572)

(39)

Hence the exact solutions of (30)-(31) are given by

V (119909 119905) = sin (119909) 119864120572(minus119905120572)

119908 (119909 119905) = sin (119909) 119864120572(minus119905120572)

(40)

When 120572 = 1 the following result is obtained

V (119909 119905) = lim119873rarrinfin

119873

sum

119899=0

V119899(119909 119905)

= V0(119909 119905) + V

1(119909 119905) + V

2(119909 119905) + sdot sdot sdot

= sin (119909) (1 minus 119905 + 1199052

2+ sdot sdot sdot) = 119890

minus119905 sin (119909)

119908 (119909 119905) = lim119873rarrinfin

119873

sum

119899=0

119908119899(119909 119905)

= 1199080(119909 119905) + 119908

1(119909 119905) + 119908

2(119909 119905) + sdot sdot sdot

= sin (119909) (1 minus 119905 + 1199052

2+ sdot sdot sdot) = 119890

minus119905 sin (119909)

(41)

which is the exact solution of (30)-(31) when 120572 = 1

6 Conclusion

In this paper Natural TransformMethod (NTM) andHomo-topy Perturbation Method (HPM) are successfully combinedto form a robust analytical method called a Hybrid NaturalTransformHomotopy PerturbationMethod for solving linearand nonlinear fractional partial differential equations Theanalytical method gives a series solution which convergesrapidly to an exact or approximate solution with elegantcomputational terms In this analyticalmethod the fractionalderivative is computed in Caputo sense while the nonlinearterm is calculated using Hersquos polynomials The analyticalprocedure is applied successfully and obtained an exactsolution of linear and nonlinear fractional partial differentialequations The simplicity and high accuracy of the analyticalmethod are clearly illustrated Thus the Hybrid NaturalTransform Homotopy Perturbation Method is a powerfulanalytical method for solving linear and nonlinear fractionalpartial differential equations

Competing Interests

The author declares that there is no conflict of interestsregarding the publication of this article

References

[1] I Podlubny Fractional differential equations vol 198 ofMathe-matics in Science and Engineering Academic Press San DiegoCalif USA 1999

[2] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 2003

[3] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[4] K Diethelm and N J Ford ldquoAnalysis of fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 265 no 2 pp 229ndash248 2002

[5] S G Samko A A Kilbas and O I Marichev FractionalIntegrals andDerivatives Gordon andBreach Yverdon Switzer-land 1993

[6] B J West M Bologna and P Grigolini Physics of Fractal Oper-ators Institute for Nonlinear Science Springer New York NYUSA 2003

[7] S S Ray ldquoAnalytical solution for the space fractional diffusionequation by two-step Adomian decomposition methodrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 4 pp 1295ndash1306 2009

[8] K Abbaoui and Y Cherruault ldquoNew ideas for proving conver-gence of Adomian Decomposition Methodsrdquo Computers andMathematics with Applications vol 32 pp 103ndash108 1995

[9] S Momani and Z Odibat ldquoAnalytical solution of a time-fractional Navier-Stokes equation by Adomian decompositionmethodrdquo Applied Mathematics and Computation vol 177 no 2pp 488ndash494 2006

[10] H Jafari and V Daftardar-Gejji ldquoSolving linear and nonlinearfractional diffusion and wave equations by Adomian decompo-sitionrdquo Applied Mathematics and Computation vol 180 no 2pp 488ndash497 2006

International Journal of Differential Equations 7

[11] M Ganjiani ldquoSolution of nonlinear fractional differential equa-tions using homotopy analysis methodrdquo Applied MathematicalModelling vol 34 no 6 pp 1634ndash1641 2010

[12] H Jafari C M Khalique and M Nazari ldquoApplication of theLaplace decomposition method for solving linear and nonlin-ear fractional diffusion-wave equationsrdquo Applied MathematicsLetters vol 24 no 11 pp 1799ndash1805 2011

[13] P K Gupta andM Singh ldquoHomotopy perturbationmethod forfractional Fornberg-Whitham equationrdquo Computers amp Mathe-matics with Applications vol 61 no 2 pp 250ndash254 2011

[14] S Momani and Z Odibat ldquoHomotopy perturbation methodfor nonlinear partial differential equations of fractional orderrdquoPhysics Letters A vol 365 no 5-6 pp 345ndash350 2007

[15] Y-Z Zhang A-M Yang and Y Long ldquoInitial boundary valueproblem for fractal heat equation in the semi-infinite region byyang-laplace transformrdquoThermal Science vol 18 no 2 pp 677ndash681 2014

[16] X-J Yang and D Baleanu ldquoFractal heat conduction problemsolved by local fractional variation iteration methodrdquo ThermalScience vol 17 no 2 pp 625ndash628 2013

[17] X-J Yang D Baleanu Y Khan and S T Mohyud-Din ldquoLocalfractional variational iteration method for diffusion and waveequations on Cantor setsrdquo Romanian Journal of Physics vol 59no 1-2 pp 36ndash48 2014

[18] X J Yang H M Srivastava J H He and D Baleanu ldquoCantor-type cylindrical-coordinate fractional derivativesrdquo Proceedingsof the Romanian Academy Series A vol 14 pp 127ndash133 2013

[19] S Kumar D Kumar S Abbasbandy and M M Rashidi ldquoAna-lytical solution of fractional Navier-Stokes equation by usingmodified Laplace decompositionmethodrdquoAin Shams Engineer-ing Journal vol 5 no 2 pp 569ndash574 2014

[20] B Ghazanfari and A G Ghazanfari ldquoSolving fractional non-linear Schrodinger equation by fractional complex transformmethodrdquo International Journal of Mathematical Modelling ampComputations vol 2 no 4 pp 277ndash281 2012

[21] K M Furati and N-E Tatar ldquoLongtime behavior for a non-linear fractional modelrdquo Journal of Mathematical Analysis andApplications vol 332 no 1 pp 441ndash454 2007

[22] K M Furati and N-E Tatar ldquoBehavior of solutions for aweighted Cauchy type fractional problemrdquo Journal of FractionalCalculus vol 28 pp 23ndash42 2005

[23] K M Furati and N-e Tatar ldquoAn existence result for a nonlocalfractional differential problemrdquo Journal of Fractional Calculusvol 26 pp 43ndash51 2004

[24] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[25] J H He ldquoRecent development of the homotopy perturbationmethodrdquo Topological Methods in Nonlinear Analysis vol 31 pp205ndash209 2008

[26] J-H He ldquoThe homotopy perturbation method for nonlinearoscillators with discontinuitiesrdquoAppliedMathematics and Com-putation vol 151 no 1 pp 287ndash292 2004

[27] Z H Khan andWA Khan ldquoN-transformproperties and appli-cationsrdquo NUST Journal of Engineering Sciences vol 1 pp 127ndash133 2008

[28] F B M Belgacem and R Silambarasan ldquoTheory of the naturaltransformrdquo Mathematics in Engineering Science and Aerospace(MESA) Journal vol 3 no 1 pp 99ndash124 2012

[29] F BM Belgacem andR Silambarasan ldquoAdvances in the naturaltransformrdquo in Proceedings of the 9th International Conference on

Mathematical Problems in Engineering Aerospace and Sciences(ICNPAA rsquo12) vol 1493 of AIP Conference Proceedings pp 106ndash110 Vienna Austria July 2012

[30] R Murray Spiegel Theory and Problems of Laplace TransformSchaumrsquos Outline Series McGraw-Hill New York NY USA1965

[31] F B Belgacem and A Karaballi ldquoSumudu transform funda-mental properties investigations and applicationsrdquo Journal ofApplied Mathematics and Stochastic Analysis vol 2006 ArticleID 91083 23 pages 2006

[32] G KWatugala ldquoSumudu transformmdasha new integral transformto solve differential equations and control engineering prob-lemsrdquo Mathematical Engineering in Industry vol 6 no 4 pp319ndash329 1998

[33] R Silambarasan and F B M Belgacem ldquoApplication of thenatural transform to Maxwellrsquos equationsrdquo in Proceedings of theProgress in Electromagnetics Research Symposium Proceedings(PIERS rsquo11) pp 899ndash902 Suzhou China September 2011

[34] M Rawashdeh and S Maitama ldquoFinding exact solutions ofnonlinear PDEs using the natural decomposition methodrdquoMathematical Methods in the Applied Sciences 2016

[35] S Maitama and S M Kurawa ldquoAn efficient technique forsolving gas dynamics equation using the natural decompositionmethodrdquo International Mathematical Forum vol 9 no 24 pp1177ndash1190 2014

[36] S Maitama ldquoExact solution of equation governing the unsteadyflow of a polytropic gas using the natural decompositionmethodrdquoAppliedMathematical Sciences vol 8 no 77 pp 3809ndash3823 2014

[37] M S Rawashdeh and S Maitama ldquoSolving PDEs using thenatural decomposition methodrdquo Nonlinear Studies vol 23 no1 pp 63ndash72 2016

[38] M S Rawashdeh and S Maitama ldquoSolving nonlinear ordinarydifferential equations using the NDMrdquo The Journal of AppliedAnalysis and Computation vol 5 no 1 pp 77ndash88 2015

[39] M S Rawashdeh and S Maitama ldquoSolving coupled systemof nonlinear PDErsquos using the natural decomposition methodrdquoInternational Journal of Pure and Applied Mathematics vol 92no 5 pp 757ndash776 2014

[40] Wikipedia note about the Natural transform August 2016httpsenwikipediaorgwikiN-transform

[41] 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

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Differential Equations 7

[11] M Ganjiani ldquoSolution of nonlinear fractional differential equa-tions using homotopy analysis methodrdquo Applied MathematicalModelling vol 34 no 6 pp 1634ndash1641 2010

[12] H Jafari C M Khalique and M Nazari ldquoApplication of theLaplace decomposition method for solving linear and nonlin-ear fractional diffusion-wave equationsrdquo Applied MathematicsLetters vol 24 no 11 pp 1799ndash1805 2011

[13] P K Gupta andM Singh ldquoHomotopy perturbationmethod forfractional Fornberg-Whitham equationrdquo Computers amp Mathe-matics with Applications vol 61 no 2 pp 250ndash254 2011

[14] S Momani and Z Odibat ldquoHomotopy perturbation methodfor nonlinear partial differential equations of fractional orderrdquoPhysics Letters A vol 365 no 5-6 pp 345ndash350 2007

[15] Y-Z Zhang A-M Yang and Y Long ldquoInitial boundary valueproblem for fractal heat equation in the semi-infinite region byyang-laplace transformrdquoThermal Science vol 18 no 2 pp 677ndash681 2014

[16] X-J Yang and D Baleanu ldquoFractal heat conduction problemsolved by local fractional variation iteration methodrdquo ThermalScience vol 17 no 2 pp 625ndash628 2013

[17] X-J Yang D Baleanu Y Khan and S T Mohyud-Din ldquoLocalfractional variational iteration method for diffusion and waveequations on Cantor setsrdquo Romanian Journal of Physics vol 59no 1-2 pp 36ndash48 2014

[18] X J Yang H M Srivastava J H He and D Baleanu ldquoCantor-type cylindrical-coordinate fractional derivativesrdquo Proceedingsof the Romanian Academy Series A vol 14 pp 127ndash133 2013

[19] S Kumar D Kumar S Abbasbandy and M M Rashidi ldquoAna-lytical solution of fractional Navier-Stokes equation by usingmodified Laplace decompositionmethodrdquoAin Shams Engineer-ing Journal vol 5 no 2 pp 569ndash574 2014

[20] B Ghazanfari and A G Ghazanfari ldquoSolving fractional non-linear Schrodinger equation by fractional complex transformmethodrdquo International Journal of Mathematical Modelling ampComputations vol 2 no 4 pp 277ndash281 2012

[21] K M Furati and N-E Tatar ldquoLongtime behavior for a non-linear fractional modelrdquo Journal of Mathematical Analysis andApplications vol 332 no 1 pp 441ndash454 2007

[22] K M Furati and N-E Tatar ldquoBehavior of solutions for aweighted Cauchy type fractional problemrdquo Journal of FractionalCalculus vol 28 pp 23ndash42 2005

[23] K M Furati and N-e Tatar ldquoAn existence result for a nonlocalfractional differential problemrdquo Journal of Fractional Calculusvol 26 pp 43ndash51 2004

[24] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[25] J H He ldquoRecent development of the homotopy perturbationmethodrdquo Topological Methods in Nonlinear Analysis vol 31 pp205ndash209 2008

[26] J-H He ldquoThe homotopy perturbation method for nonlinearoscillators with discontinuitiesrdquoAppliedMathematics and Com-putation vol 151 no 1 pp 287ndash292 2004

[27] Z H Khan andWA Khan ldquoN-transformproperties and appli-cationsrdquo NUST Journal of Engineering Sciences vol 1 pp 127ndash133 2008

[28] F B M Belgacem and R Silambarasan ldquoTheory of the naturaltransformrdquo Mathematics in Engineering Science and Aerospace(MESA) Journal vol 3 no 1 pp 99ndash124 2012

[29] F BM Belgacem andR Silambarasan ldquoAdvances in the naturaltransformrdquo in Proceedings of the 9th International Conference on

Mathematical Problems in Engineering Aerospace and Sciences(ICNPAA rsquo12) vol 1493 of AIP Conference Proceedings pp 106ndash110 Vienna Austria July 2012

[30] R Murray Spiegel Theory and Problems of Laplace TransformSchaumrsquos Outline Series McGraw-Hill New York NY USA1965

[31] F B Belgacem and A Karaballi ldquoSumudu transform funda-mental properties investigations and applicationsrdquo Journal ofApplied Mathematics and Stochastic Analysis vol 2006 ArticleID 91083 23 pages 2006

[32] G KWatugala ldquoSumudu transformmdasha new integral transformto solve differential equations and control engineering prob-lemsrdquo Mathematical Engineering in Industry vol 6 no 4 pp319ndash329 1998

[33] R Silambarasan and F B M Belgacem ldquoApplication of thenatural transform to Maxwellrsquos equationsrdquo in Proceedings of theProgress in Electromagnetics Research Symposium Proceedings(PIERS rsquo11) pp 899ndash902 Suzhou China September 2011

[34] M Rawashdeh and S Maitama ldquoFinding exact solutions ofnonlinear PDEs using the natural decomposition methodrdquoMathematical Methods in the Applied Sciences 2016

[35] S Maitama and S M Kurawa ldquoAn efficient technique forsolving gas dynamics equation using the natural decompositionmethodrdquo International Mathematical Forum vol 9 no 24 pp1177ndash1190 2014

[36] S Maitama ldquoExact solution of equation governing the unsteadyflow of a polytropic gas using the natural decompositionmethodrdquoAppliedMathematical Sciences vol 8 no 77 pp 3809ndash3823 2014

[37] M S Rawashdeh and S Maitama ldquoSolving PDEs using thenatural decomposition methodrdquo Nonlinear Studies vol 23 no1 pp 63ndash72 2016

[38] M S Rawashdeh and S Maitama ldquoSolving nonlinear ordinarydifferential equations using the NDMrdquo The Journal of AppliedAnalysis and Computation vol 5 no 1 pp 77ndash88 2015

[39] M S Rawashdeh and S Maitama ldquoSolving coupled systemof nonlinear PDErsquos using the natural decomposition methodrdquoInternational Journal of Pure and Applied Mathematics vol 92no 5 pp 757ndash776 2014

[40] Wikipedia note about the Natural transform August 2016httpsenwikipediaorgwikiN-transform

[41] 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

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of