Research Article Two Different Methods for Numerical...

Research ArticleTwo Different Methods for NumericalSolution of the Modified Burgersrsquo Equation

Seydi Battal Gazi Karakoccedil1 Ali BaGhan2 and Turabi Geyikli2

1 Department of Mathematics Faculty of Science and Art Nevsehir Haci Bektas Veli University 50300 Nevsehir Turkey2Department of Mathematics Faculty of Science and Art Inonu University 44280 Malatya Turkey

Correspondence should be addressed to Seydi Battal Gazi Karakoc sbgkarakocnevsehiredutr

Received 23 January 2014 Accepted 23 February 2014 Published 3 April 2014

Academic Editors D Baleanu and H Jafari

Copyright copy 2014 Seydi Battal Gazi Karakoc et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

A numerical solution of the modified Burgersrsquo equation (MBE) is obtained by using quartic B-spline subdomain finite elementmethod (SFEM) over which the nonlinear term is locally linearized and using quartic B-spline differential quadrature (QBDQM)method The accuracy and efficiency of the methods are discussed by computing 119871

2and 119871

infinerror norms Comparisons are made

with those of some earlier papers The obtained numerical results show that the methods are effective numerical schemes to solvethe MBE A linear stability analysis based on the von Neumann scheme shows the SFEM is unconditionally stable A rate ofconvergence analysis is also given for the DQM

1 Introduction

The one-dimensional Burgersrsquo equation first suggested byBateman [1] and later treated by Burgersrsquo [2] has the form

119880119905+ 119880119880119909minus V119880119909119909

= 0 (1)

where V is a positive parameter and the subscripts 119909 and119905 denote space and time derivatives respectively Burgersrsquomodel of turbulence is very important in fluid dynamicsmodel and study of this model and the theory of shock waveshas been considered by many authors for both conceptualunderstanding of a class of physical flows and for testingvarious numerical methods [3] Relationship between (1) andboth turbulence theory and shock wave theory was presentedby Cole [4] He also obtained an exact solution of theequation Analytical solutions of the equation were found forrestricted values of Vwhich represent the kinematics viscosityof the fluid motion So the numerical solution of Burgersrsquoequation has been subject of many papers Various numericalmethods have been studied based on finite difference [56] Runge-Kutta-Chebyshev method [7 8] group-theoreticmethods [9] and finite element methods including GalerkinPetrov-Galerkin least squares and collocation [10ndash13]

The modified Burgersrsquo equation (MBE) which we discuss inthis paper is based upon Burgersrsquo equation (BE) of the form

119880119905+ 1198802119880119909minus V119880119909119909

= 0 (2)

The equation has the strong nonlinear aspects and has beenused in many practical transport problems for instancenonlinear waves in a medium with low-frequency pumpingor absorption turbulence transport wave processes in ther-moelastic medium transport and dispersion of pollutants inrivers and sediment transport and ion reflection at quasi-perpendicular shocks Recently some numerical studies ofthe equation have been presented Ramadan and El-Danaf[14] obtained the numerical solutions of the MBE usingquintic B-spline collocation finite element method A speciallattice Boltzmannmodel is developed by Duan et al [15] Daget al [16] have developed a Galerkin finite element solution ofthe equation using quintic B-splines and time-split techniqueA solution based on sextic B-spline collocation method isproposed by Irk [17] Roshan and Bhamra [18] applied aPetrov-Galerkin method using a linear hat function as thetrial function and a cubic B-spline function as the test func-tionAdiscontinuousGalerkinmethod is presented byZhanget al [19] Bratsos [20] has used a finite difference scheme

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 780269 13 pageshttpdxdoiorg1011552014780269

2 The Scientific World Journal

based on fourth-order rational approximants to the matrix-exponential term in a two-time level recurrence relation forcalculating the numerical solution of the equation

Recently DQM has become a very efficient and effectivemethod to obtain the numerical solutions of various typesof partial differential equations In 1972 Bellman et al [21]first introduced differential quadrature method (DQM) forsolving partial differential equations The main idea behindthe method is to find out the weighting coefficients of thefunctional values at nodal points by using base functions ofwhich derivatives are already known at the same nodal pointsover the entire region Various researchers have developeddifferent types of DQMs by utilizing various test functionsBellman et al [22] have used Legendre polynomials andspline functions in order to get weighting coefficients Quanand Chang [23 24] have presented an explicit formulationfor determining the weighting coefficients using Lagrangeinterpolation polynomials Zhong [25] Guo and Zhong[26] and Zhong and Lan [27] have introduced anotherefficient DQM as spline based DQM and applied it todifferent problems Shu and Wu [28] have considered someof the implicit formulations of weighting coefficients with thehelp of radial basis functions Nonlinear Burgersrsquo equationis solved using polynomial based differential quadraturemethod by Korkmaz and Dag [29] The DQM has manyadvantages over the classical techniques mainly it preventslinearization and perturbation in order to find better solu-tions of given nonlinear equations Since QBDQM do notneed transforming for solving the equation the method hasbeen preferred

In the present work we have applied a subdomainfinite element method and a quartic B-spline differentialquadrature method to the MBE To show the performanceand accuracy of the methods and make comparisons ofnumerical solutions we have taken different values of V

2 Numerical Methods

To implement the numerical schemes the interval [119886 119887] issplitted up into uniformly sized intervals by the nodes 119909

119898

119898 = 1 2 119873 such that 119886 = 1199090lt 1199091sdot sdot sdot lt 119909

119873= 119887 where

ℎ = (119909119898+1

minus 119909119898)

21 Subdomain Finite Element Method (SFEM) We willconsider (2) with the boundary conditions chosen from

119880 (119886 119905) = 1205731 119880 (119887 119905) = 120573

2

119880119909(119886 119905) = 0 119880

119909(119887 119905) = 0

119880119909119909

(119886 119905) = 0 119880119909119909

(119887 119905) = 0 119905 gt 0

(3)

with the initial condition

119880 (119909 0) = 119891 (119909) 119886 le 119909 le 119887 (4)

where 1205731and 120573

2are constants The quartic B-splines 120601

119898(119909)

(119898 = minus2(1) 119873 + 1) at the knots 119909119898which form a basis over

the interval [119886 119887] are defined by the relationships [30]

120601119898(119909)

=1

ℎ4

(119909 minus 119909119898minus2

)4

119909 isin [119909119898minus2

119909119898minus1

]

(119909 minus 119909119898minus2

)4

minus 5(119909 minus 119909119898minus1

)4

119909 isin [119909119898minus1

119909119898]

(119909 minus 119909119898minus2

)4


)4

+10(119909 minus 119909119898)4

119909 isin [119909

119898 119909119898+1

]

(119909119898+3

minus 119909)4

minus 5(119909119898+2

minus 119909)4

119909 isin [119909119898+1

119909119898+2

]

(119909119898+3

minus 119909)4

119909 isin [119909119898+2

119909119898+3

]

0 otherwise(5)

Our numerical treatment for solving the MBE using thesubdomain finite element method with quartic B-splines isto find a global approximation 119880

119873(119909 119905) to the exact solution

119880(119909 119905) that can be expressed in the following form

119880119873(119909 119905) =

119873+1

sum

119895=minus2

120575119895(119905) 120601119895(119909) (6)

where 120575119895are time-dependent parameters to be determined

from both boundary and weighted residual conditions Thenodal values 119880

119898 1198801015840119898 11988010158401015840119898 and 119880

101584010158401015840

119898at the knots 119909

119898can be

obtained from (5) and (6) in the following form

119880119898

= 119880 (119909119898) = 120575119898minus2

+ 11120575119898minus1

+ 11120575119898+ 120575119898+1

1198801015840

119898= 1198801015840(119909119898) =

4

ℎ(minus120575119898minus2

minus 3120575119898minus1

+ 3120575119898+ 120575119898+1

)

11988010158401015840

119898= 11988010158401015840(119909119898) =

12

ℎ2(120575119898minus2


minus 120575119898+ 120575119898+1

)

119880101584010158401015840

119898= 119880101584010158401015840

(119909119898) =

24

ℎ3(minus120575119898minus2

+ 3120575119898minus1

minus 3120575119898+ 120575119898+1

)

(7)

For each element using the local coordinate transformation

ℎ120585 = 119909 minus 119909119898 0 le 120585 le 1 (8)

a typical finite interval [119909119898 119909119898+1

] is mapped into the interval[0 1]Therefore the quartic B-spline shape functions over theelement [0 1] can be defined as

120601119890=

120601119898minus2

= 1 minus 4120585 + 61205852minus 41205853+ 1205854

120601119898minus1

= 11 minus 12120585 minus 61205852+ 121205853minus 1205854

120601119898

= 11 + 12120585 minus 61205852minus 121205853+ 1205854

120601119898+1

= 1 + 4120585 + 61205852+ 41205853minus 1205854

120601119898+2

= 1205854

(9)

All other splines apart from 120601119898minus2

(119909) 120601119898minus1

(119909) 120601119898(119909)

120601119898+1

(119909) and 120601119898+2

(119909) are zero over the element [0 1] So the

The Scientific World Journal 3

approximation equation (6) over this element can be writtenin terms of basis functions given in (9) as

119880119873(120585 119905) =

119898+2

sum

119895=119898minus2

120575119895(119905) 120601119895(120585) (10)

where 120575119898minus2

120575119898minus1

120575119898 120575119898+1

and 120575119898+2

act as element param-eters and B-splines 120601

119898minus2(119909) 120601

119898minus1 120601119898 120601119898+1

and 120601119898+2

aselement shape functions Applying the subdomain approachto (33) with the weight function

119882119898(119909) =

1 119909 isin [119909119898 119909119898+1

]

0 otherwise(11)

we obtain the weak form of (2)

int

119909119898+1

119909119898

1 (119880119905+ 1198802119880119909minus V119880119909119909) 119889119909 = 0 (12)

Using the transformation (8) into the weak form (12) andthen integrating (12) term by term with some manipulationby parts result in

ℎ

5( 120575119898minus2

+ 26 120575119898minus1

+ 66 120575119898+ 26 120575

119898+1+ 120575119898+2

)

+ 119885119898(minus120575119898minus2

minus 10120575119898minus1

+ 10120575119898+1

+ 120575119898+2

)

minus4Vℎ

(120575119898minus2

+ 2120575119898minus1

minus 6120575119898+ 2120575119898+1

+ 120575119898+2

) = 0

(13)

where the dot denotes differentiation with respect to 119905 and

119885119898

= (120575119898minus2

+ 11120575119898minus1

+ 11120575119898+ 120575119898+1

)2

(14)

In (13) using the Crank-Nicolson formula and its time deriva-tive that is discretized by the forward difference approachrespectively

120575119898

=120575119899

119898+ 120575119899+1

119898

2 120575

119898=

120575119899+1

119898minus 120575119899

119898

Δ119905

(15)

we obtain a recurrence relationship between the two timelevels 119899 and 119899 + 1 relating two unknown parameters 120575119899+1

119894and

120575119899

119894 for 119894 = 119898 minus 2119898 minus 1 119898 + 2

1205721198981

120575119899+1

119898minus2+ 1205721198982

120575119899+1

119898minus1+ 1205721198983

120575119899+1

119898+ 1205721198984

120575119899+1

119898+1

+ 1205721198985

120575119899+1

119898+2

= 1205721198986

120575119899

119898minus2+ 1205721198987

120575119899

119898minus1+ 1205721198988

120575119899

119898+ 1205721198989

120575119899

119898+1

+ 12057211989810

120575119899

119898+2

119898 = 0 1 119873 minus 1

(16)

where

1205721198981

= 1 minus 119864119885119898minus 119872 120572

1198982= 26 minus 10119864119885

119898minus 2119872

1205721198983

= 66 + 6119872 1205721198984

= 26 + 10119864119885119898minus 2119872

1205721198985

= 1 + 119864119885119898minus 119872 120572

1198986= 1 + 119864119885

119898+ 119872

1205721198987

= 26 + 10119864119885119898+ 2119872 120572

1198988= 66 minus 6119872

1205721198989

= 26 minus 10119864119885119898+ 2119872 120572

11989810= 1 minus 119864119885

119898+ 119872

119864 =5Δ119905

2ℎ 119872 =

20VΔ1199052ℎ2

(17)

Obviously the system (16) consists of 119873 equations in the119873+4 unknowns (120575

minus2 120575minus1 120575

119873+1) To get a unique solution

of the system we need four additional constraints These areobtained from the boundary conditions (3) and can be used toeliminate 120575

minus2 120575minus1 120575119873 and 120575

119873+1from the system (16) which

then becomes a matrix equation for the 119873 unknowns 119889 =

(1205750 1205751 120575

119873minus1) of the form

119860119889119899+1

= 119861119889119899 (18)

A lumped value of 119885119898is obtained from (119880

119898+ 119880119898+1

)24 as

119885119898

=1

4(120575119898minus2

+ 12120575119898minus1

+ 22120575119898+ 12120575

119898+1+ 120575119898+2

)2

(19)

The resulting system can be solved with a variant of Thomasalgorithm and we need an inner iteration (120575

lowast)119899+1

= 120575119899+

(12)(120575119899+1

minus 120575119899) at each time step to cope with the nonlinear

term 119885119898 A typical member of the matrix system (16) is

rewritten in terms of the nodal parameters 120575119899119898as

1205741120575119899+1

119898minus2+ 1205742120575119899+1

119898minus1+ 1205743120575119899+1

119898+ 1205744120575119899+1

119898+1+ 1205745120575119899+1

119898+2

= 1205746120575119899

119898minus2+ 1205747120575119899

119898minus1+ 1205748120575119899

119898+ 1205749120575119899

119898+1+ 12057410120575119899

119898+2

(20)

where

1205741= 120572 minus 120573 minus 120582 120574

2= 26120572 minus 10120573 minus 2120582

1205743= 66120572 + 6120582 120574

4= 26120572 + 10120573 minus 2120582

1205745= 120572 + 120573 minus 120582 120574

6= 120572 + 120573 + 120582

1205747= 26120572 + 10120573 + 2120582 120574

8= 66120572 minus 6120582

1205749= 26120572 minus 10120573 + 2120582 120574

10= 120572 minus 120573 + 120582

120572 = 1 120573 = 119864119885119898 120582 = 119872

(21)


Before the solution process begins iteratively the initialvector 1205750 = (120575

0 1205751 120575

119873minus1) must be determined by means

of the following requirements

1198801015840(119886 0) =

4

ℎ(minus1205750

minus2minus 31205750

minus1+ 31205750

0+ 1205750

1) = 0

11988010158401015840(119886 0) =

12

ℎ2(1205750

minus2minus 1205750

minus1minus 1205750

0+ 1205750

1) = 0

119880 (119909119898 0) = 120575

0

119898minus2+ 11120575

0

119898minus1+ 11120575

0

119898+ 1205750

119898+1= 119891 (119909)

119898 = 0 1 119873 minus 1

1198801015840(119887 0) =

4

ℎ(minus1205750

119873minus2minus 31205750

119873minus1+ 31205750

119873+ 1205750

119873+1) = 0

11988010158401015840(119887 0) =

12

ℎ2(1205750

119873minus2minus 1205750


119873+ 1205750

119873+1) = 0

(22)

If we eliminate the parameters 1205750

minus2 1205750minus1 1205750119873 and 120575

0

119873+1

from the system (16) we obtain 119873 times 119873 matrix system of thefollowing form

1198601205750= 119861 (23)

where 119860 is

119860 =

[[[[[

[

18 6

115 115 1

1 11 11 1

1 11 11 1

2 14 8

]]]]]

]

(24)

1205750

= [1205750

0 1205750

1 120575

0

119873minus1]119879 and 119861 = [119880(119909

0 0) 119880(119909

1 0)

119880(119909119873minus1

0)]119879 This system is solved by using a variant of

Thomas algorithm

22 Linear Stability Analysis We have investigated stabilityof the scheme by using the vonNeumannmethod In order toapply the stability analysis theMBE needs to be linearized byassuming that the quantity 119880 in the nonlinear term 119880

2119880119909is

locally constant The growth factor of a typical Fourier modeis defined as

120575119899

119895= 120585119899119890119894119895119896ℎ

(25)

where 119896 is mode number and ℎ is the element size Substitut-ing (37) into the scheme (20) we have

119892 =1198601+ 119894119887

1198602minus 119894119887

(26)

where1198601= (120572 minus 120582) cos (2119896ℎ) + (26120572 minus 2120582) cos (119896ℎ) + 66 + 6120582

1198602= (120572 + 120582) cos (2119896ℎ) + (26120572 + 2120582) cos (119896ℎ) + 66 minus 6120582

119887 = sin (2119896ℎ) + 10 sin (119896ℎ)

(27)

We can see that11986021lt 1198602

2and taking themodulus of (38) gives

|119892| le 1 so we find that the scheme (20) is unconditionallystable

23 Quartic B-Spline Differential Quadrature Method(QBDQM) DQM can be defined as an approximation to aderivative of a given function by using the linear summationof its values at specific discrete nodal points over the solutiondomain of a problem Provided that any given function 119880(119909)

is enough smooth over the solution domain its derivativeswith respect to 119909 at a nodal point 119909

119894can be approximated

by a linear summation of all the functional values in thesolution domain namely

119880(119903)

119909(119909119894) =

119889119880(119903)

119889119909(119903)|119909119894

=

119873

sum

119895=1

119908(119903)

119894119895119880(119909119895)

119894 = 1 2 119873 119903 = 1 2 119873 minus 1

(28)

where 119903 denotes the order of the derivative 119908(119903)

119894119895repre-

sent the weighting coefficients of the 119903th order derivativeapproximation and 119873 denotes the number of nodal pointsin the solution domain Here the index 119895 represents thefact that 119908(119903)

119894119895is the corresponding weighting coefficient of

the functional value 119880(119909119895) We need first- and second-order

derivative of the function 119880(119909) So we will find value of (28)for the 119903 = 1 2 If we consider (28) then it is seen that thefundamental process for approximating the derivatives of anygiven function throughDQM is to find out the correspondingweighting coefficients 119908

(119903)

119894119895 The main idea of the DQM

approximation is to find out the corresponding weightingcoefficients 119908(119903)

119894119895by means of a set of base functions spanning

the problem domain While determining the correspondingweighting coefficients different basis may be used Using thequartic B-splines as test functions in the fundamental DQMequation (28) leads to the equation

119889(119903)119876119898(119909119894)

119889119909(119903)=

119898+2

sum

119895=119898minus1

119908(119903)

119894119895119876119898(119909119895)

119898 = minus1 0 119873 + 2 119894 = 1 2 119873

(29)

24 First-Order Derivative Approximation When DQMmethodology is applied the fundamental equality for deter-mining the corresponding weighting coefficients of the first-order derivative approximation is obtained as Korkmaz used[31]

119889119876119898(119909119894)

119889119909=

119898+2

sum

119895=119898minus1

119908(1)

119894119895119876119898(119909119895)

119898 = minus1 0 119873 + 1 119894 = 1 2 119873

(30)

In this process the initial step for finding out the correspond-ing weighting coefficients 119908(1)

119894119895 119895 = minus2 minus1 119873 + 3 of the

first nodal point 1199091is to apply the test functions 119876

119898 119898 =

minus1 0 119873 + 1 at the nodal point 1199091 After all the 119876

119898test


functions are applied we get the following system of algebraicequation system

[[[[[[[[[[[[[[[[[[[[[

[

1 11 11 1

1 11 11 1

d d d d

1 11 11 1

1 11 11 1

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(1)

1minus2

119908(1)

1minus1

119908(1)

10

119908(1)

11

119908(1)

12

119908(1)

1119873+2

119908(1)

1119873+3

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[[

[

minus4

ℎ

minus12

ℎ

12

ℎ

4

ℎ

0

0

0

]]]]]]]]]]]]]]]]]]]]]]

]

(31)

The weighting coefficients 119908(1)

1119895related to the first grid

point are determined by solving the system (31) This systemconsists of 119873 + 6 unknowns and 119873 + 3 equations To havea unique solution it is required to add three additionalequations to the system These are

119889(2)

119876minus1

(1199091)

119889119909(2)=

1

sum

119895=minus2

119908(1)

11198951198761015840

minus1(119909119895)

119889(2)

119876119873+1

(1199091)

119889119909(2)=

119873+3

sum

119895=119873

119908(1)

11198951198761015840

119873+1(119909119895)

119889(3)

119876119873+1

(1199091)

120597119909(3)=

119873+3

sum

119895=119873

119908(1)

111989511987610158401015840

119873+1(119909119895)

(32)

By using these equations which we obtained by derivationsthree unknown terms will be eliminated from the systemConsider

[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11 1

2 23 23

6 18

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(1)

1minus1

119908(1)

10

119908(1)

11

119908(1)

12

119908(1)

13

119908(1)

1119873

119908(1)

1119873+1

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[

[

minus7

ℎ

minus12

ℎ

12

ℎ

4

ℎ0

0

0

]]]]]]]]]]]]]]]]]]]]]

]

(33)

To determine the weighting coefficients 119908(1)

119896119895 119895 =

minus1 0 119873 + 1 at grid points 119909119896 2 le 119896 le 119873 minus 1 we got

the following algebraic equation system

[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11 1

2 23 23

6 18

]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

119908(1)

119896minus1

119908(1)

119896119896minus3

119908(1)

119896119896minus2

119908(1)

119896119896minus1

119908(1)

119896119896

119908(1)

119896119896+1

119908(1)

119896119896+2

119908(1)

119896119873+1

]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]


=

[[[[[[[[[[[[[[[[[[[[[[[[[[

[

0

0

minus4

ℎ

minus12

ℎ

12

ℎ

4

ℎ0

0

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

(34)

For the last grid point of the domain 119909119873 determine

weighting coefficients 119908(1)119873119895

119895 = minus1 0 119873 + 1 we got thefollowing algebraic equation system

[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11 1

2 23 23

6 18

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(1)

119873minus1

119908(1)

1198730

119908(1)

119873119873minus3

119908(1)

119873119873minus2

119908(1)

119873119873minus1

119908(1)

119873119873

119908(1)

119873119873+1

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[

[

0

0

0

minus4

ℎ

minus12

ℎ

53

2ℎ

17

ℎ

]]]]]]]]]]]]]]]]]]]]]

]

(35)

25 Second-Order Derivative Approximation The generalform of DQM approximation to the problem on the solutiondomain is

1198892119876119898

1198891199092(119909119894) =

119898+2

sum

119895=119898minus1

119908(2)

119894119895119876119898(119909119895)

119898 = minus1 0 119873 + 1 119894 = 1 2 119873

(36)

Table 1 1198712and 119871

infinerror norms for ℎ = 0005 Δ119905 = 001 and

V = 0001 (SFEM)

Time 1198712times 103

119871infin

times 103

2 00054945 002820493 00082404 004224214 00109858 005622805 00137296 007015666 00164729 008404277 00192154 009789758 00219573 011169349 00246985 0125446610 00274379 01391304

where 119908(2)

119894119895represents the corresponding weighting coeffi-

cients of the second-order derivative approximations Simi-larly for finding out the weighting coefficients of the first gridpoint 119909

1all test functions 119876

119898119898 = minus1 0 119873 + 1 are used

and the following algebraic equations system is obtained

[[[[[[[[[[[[[[[[[[[[[

[

1 11 11 1

1 11 11 1

d d d d

1 11 11 1

1 11 11 1

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(2)

1minus2

119908(2)

1minus1

119908(2)

10

119908(2)

11

119908(2)

12

119908(2)

1119873+2

119908(2)

1119873+3

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[

[

12

ℎ2

minus12

ℎ2

minus12

ℎ2

12

ℎ2

0

0

0

]]]]]]]]]]]]]]]]]]]]]

]

(37)

Here the system (37) consists of 119873 + 6 unknowns and119873 + 3 equations To have a unique solution it is required toadd new equations to the system These are

1198893119876minus1

(1199091)

1198891199093=

1

sum

119895=minus2

119908(1)

11198951198761015840

minus1(119909119895) (38)


1198893119876119873+1

(1199091)

1198891199093=

119873+3

sum

119895=119873

119908(1)

11198951198761015840

119873+1(119909119895) (39)

If we used (38) and (39) we obtain the following equationssystem

[[[[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11 1

2 14 8

]]]]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[[[

[

119908(2)

1minus1

119908(2)

10

119908(2)

11

119908(2)

12

119908(2)

13

119908(2)

1119873+1

119908(2)

1119873+2

]]]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[[

[

18

ℎ2

minus12

ℎ2

minus12

ℎ2

12

ℎ2

0

0

0

]]]]]]]]]]]]]]]]]]]]]]

]

(40)

Quartic B-splines have not got fourth-order derivations atthe grid points so we cannot eliminate the unknown term119908(2)

1119873+2by the one more derivation of (39) We will use

first-order weighting coefficients instead of second-orderweighting coefficients which are introduced by Shu [32]

119908(2)

119894119895= 2119908(1)

119894119895(119908(1)

119894119894minus

1

119909119894minus 119909119895

) 119894 = 119895 (41)

After we use (41)

1198601= 119908(2)

1119873+2= 2119908(1)

1119873+2(119908(1)

11minus

1

1199091minus 119909119873+2

)

[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11

2 14

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(2)

1minus1

119908(2)

10

119908(2)

11

119908(2)

12

119908(2)

13

119908(2)

1119873

119908(2)

1119873+1

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[

[

18

ℎ2

minus12

ℎ2

minus12

ℎ2

12

ℎ2

0

minus1198601

minus81198601

]]]]]]]]]]]]]]]]]]]]]

]

(42)

system (42) is obtained To determine the weighting coeffi-cients 119908(2)

119896119895 119895 = minus1 0 119873 + 1 at grid points 119909

119896 2 le 119896 le

119873 minus 1 we got the following algebraic system

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11

2 14

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

119908(2)

119896minus1

119908(2)

119896119896minus3

119908(2)

119896119896minus2

119908(2)

119896119896minus1

119908(2)

119896119896

119908(2)

119896119896+1

119908(2)

119896119896+2

119908(2)

119896119873minus1

119908(2)

119896119873

119908(2)

119896119873+1

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]


Table 2 1198712and 119871


V = 0005 (SFEM)


119871infin

times 103

2 00246966 008456893 00370384 012662224 00493707 016843625 00616997 021013196 00740253 025163927 00863444 029301788 00986573 033419229 01109636 0375245710 01232629 04160477

Table 3 1198712and 119871


V = 001 (SFEM)


119871infin

times 103

2 00978574 028062433 01467089 041859814 01955072 055502865 02442506 068987136 02929396 082386297 03415703 095666888 03901436 108812899 04386580 1218223110 04871136 13469237

=

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

0

12

ℎ2

minus12

ℎ2

minus12

ℎ2

12

ℎ2

0

0

minus119860119896

minus8119860119896

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

(43)

where119860119896equals119860

119896= 119908(2)

119896119873+2= 2119908(1)

119896119873+2(119908(1)

119896119896minus1(119909

119896minus119909119873+2

))For the last grid point of the domain 119909

119873with the

same idea determine weighting coefficients 119908(2)

119873119895 119895 =

Table 4 1198712and 119871

infinerror norms for ℎ = 002 Δ119905 = 001 and V =

001 (SFEM)


119871infin

times 103

2 00973818 028025263 01460008 041848724 01945704 055541215 02430873 069100626 02915506 082523127 03399602 095804338 03883156 108944139 04366131 1219411110 04848547 13479880

minus1 0 119873 + 1 we got the following algebraic equationsystem

[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11

2 14

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(2)

119873minus1

119908(2)

1198730

119908(2)

119873119873minus3

119908(2)

119873119873minus2

119908(2)

119873119873minus1

119908(2)

119873119873

119908(2)

119873119873+1

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[

[

0

0

0

12

ℎ2

minus12

ℎ2

minus12

ℎ2minus 119860119873

18

ℎ2minus 8119860119873

]]]]]]]]]]]]]]]]]]]]]

]

(44)

where 119860119873equals 119860

119873= 119908(2)

119873119873+2= 2119908(1)

119873119873+2(119908(1)

119873119873minus 1(119909

119873minus

119909119873+2

))

3 Test Problem and Experimental Results

In this section we obtained numerical solutions of the MBEby the subdomain finite element method and differential


Table 5 1198712and 119871

infinerror norms for V = 001 Δ119905 = 001 and ℎ = 002

Time QBDQM [ℎ = 002] Ramadan et al [13] [ℎ = 002]

1198712times 103

119871infin

times 103

1198712times 103

119871infin

times 103

2 07955855586 13795978925 07904296620 17030921188

3 06690533313 11943543646 06551928290 11832698216

4 05250528343 09764154381 05576794264 09964523368

5 04048512821 07849457015 05105617536 08561342445

6 03452210304 06374950443 05167229575 07610530060

7 03638648688 06705419608 05677438614 10654548090

8 04337013450 09863405006 06427542266 13581113635

9 05197862999 12551335234 07236430257 16048306653

10 06042925888 14747885309 08002564201 18023938553

Table 6 1198712and 119871

infinerror norms for V = 001Δ119905 = 001 and119873 = 81

at 0 le 119909 le 13

Time QBDQM1198712times 103

119871infin

times 103

2 07607107169 137041821953 06480181273 118549841904 05604986926 100524764525 04927784148 086540324196 04359075842 075315510237 03885737191 066013265128 03520185942 058333349709 03282544303 0520132366310 03187570280 04691560472

quadrature method The accuracy of the numerical methodis checked using the error norms 119871

2and 119871

infin respectively

1198712=10038171003817100381710038171003817119880

exactminus 119880119873

100381710038171003817100381710038172≃ radicℎ

119873

sum

119869=1

100381610038161003816100381610038161003816119880

exact119895

minus (119880119873)119895

100381610038161003816100381610038161003816

2

119871infin

=10038171003817100381710038171003817119880


10038171003817100381710038171003817infin≃ max119895

100381610038161003816100381610038161003816119880

exact119895

minus (119880119873)119895

100381610038161003816100381610038161003816

119895 = 1 2 119873 minus 1

(45)

All numerical calculations were computed in double pre-cision arithmetic on a Pentium4PCusing a Fortran compilerThe analytical solution of modified Burgersrsquo equation is givenin [33] as

119880 (119909 119905) =(119909119905)

1 + (radic1199051198880) exp (11990924V119905)

(46)

where 1198880is a constant and 0 lt 119888

0lt 1 For our numerical

calculation we take 1198880= 05 We use the initial condition for

(46) evaluating at 119905 = 1 and the boundary conditions aretaken as 119880(0 119905) = 119880

119909(0 119905) = 0 and 119880(1 119905) = 119880

119909(1 119905) = 0

31 Experimental Results for FEM For the numerical sim-ulation we have chosen the various viscosity parametersV = 001 0001 0005 space steps ℎ = 002 0005 and

0000

0005

0010

0015

0020

0025

U(xt)

00 02 04 06 08 10

x

t = 1

t = 2

t = 4

t = 6t = 8

Figure 1 Solution behavior of the equation with ℎ = 0005 119905 = 001and V = 001

time steps Δ119905 = 001 0001 over the interval 0 le 119909 le 1The computed values of the error norms 119871

2and 119871

infinare

presented at some selected times up to 119905 = 10 The obtainedresults are tabulated in Tables 1 2 3 and 4 It is clearly seenthat the results obtained by the SFEM are found to be moreacceptable Also we observe from these tables that if thevalue of viscosity decreases the value of the error norms willdecreaseWhen the value of viscosity parameter is smaller weget better resultsThe behaviors of the numerical solutions forviscosity V = 001 0005 0001 space steps ℎ = 002 0005and time steps Δ119905 = 001 0001 at times 119905 = 1 2 4 6 and 8

are shown in Figures 1 2 and 3 As seen in the figures the topcurve is at 119905 = 1 and the bottom curve is at 119905 = 8 Also wehave observed from the figures that as the time increases thecurve of the numerical solution decaysWith smaller viscosityvalue numerical solution decay gets faster

32 Experimental Results for QBDQM We calculate thenumerical rates of convergence (ROC) with the help of thefollowing formula

ROC asympln (119864 (119873

2) 119864 (119873

1))

ln (11987311198732)

(47)


Table 7 1198712and 119871


Time QBDQM Ramadan et al [13]1198712times 103

119871infin

times 103

1198712times 103

119871infin

times 103

2 01370706949 04453892504 01835491190 081852111123 01168507335 03842839811 01441424335 052348333464 01019761971 03258391192 01144110783 035635372075 00920706001 02816616769 00947865272 025497900586 00849484881 02484289381 00814174677 021348478357 00794570772 02225471690 00718977757 018800484328 00750035859 02019577762 00648368942 016826017709 00712618898 01851510002 00594114970 0152407496610 00680382860 01711033543 00551151456 01394312127

Table 8 Error norms and rate of convergence for various numbersof grid points at 119905 = 10

119873 1198712times 103 ROC(119871

2) 119871

infintimes 103 ROC(119871

infin)

11 043 mdash 098 mdash21 035 031 088 01631 022 119 052 13541 017 092 039 10251 014 088 030 12081 010 072 019 098

0000

0004

0006

0008

0010

0012

0002

U(xt)

00 02 04 06 08 10

x

t = 1

t = 2

t = 4

t = 6

t = 8

Figure 2 Solution behavior of the equation with ℎ = 0 005 119905 =

0 01 and V = 0001

Here 119864(119873119895) denotes either the 119871

2error norm or the 119871

infin

error norm in case of using 119873119895grid points Therefore some

further numerical runs for different numbers of space stepshave been performed Ultimately some computations havebeen made about the ROC by assuming that these methodsare algebraically convergent in space Namely we supposethat 119864(119873) sim 119873

119901 for some 119901 lt 0 when 119864(119873) denotes the1198712or the 119871

infinerror norms by using119873 subintervals

For the numerical treatment we have taken the differentviscosity parameters V = 001 0001 and time step Δ119905 = 001

over the intervals 0 le 119909 le 1 and 0 le 119909 le 13 As it

0000

0005

0010

0015

0020

0030

0025

0035

U(xt)

00 02 04 06 08 10

x

t = 1

t = 2

t = 4

t = 6

t = 8

Figure 3 Solution behavior of the equation with ℎ = 0 02 119905 = 0 01and V = 001

is seen from Figure 4 when we select the solution domain0 le 119909 le 1 the right part of the shock wave cannot be seenclearly By using the larger domain like 0 le 119909 le 13 asseen in Figure 5 solution is got better than narrow domain0 le 119909 le 1 shown in Figure 4 The computed values of theerror norms 119871

2and 119871

infinare presented at some selected times

up to 119905 = 10 The obtained results are recorded in Tables5 and 6 As it is seen from the tables the error norms 119871

2

and 119871infin

are sufficiently small and satisfactorily acceptableWe obtain better results if the value of viscosity parameter issmaller as shown in Table 7 The behaviors of the numericalsolutions for viscosity V = 001 and 0001 and time stepΔ119905 = 001 at times 119905 = 1 3 5 7 and 9 are shown in Figures4ndash6 It is observed from the figures that the top curve is at119905 = 1 and the bottom curve is at 119905 = 9 It is obviouslyseen that smaller viscosity value V in shock wave with smalleramplitude and propagation front becomes smoother Alsowe have seen from the figures that as the time increases thecurve of the numerical solution decaysWith smaller viscosityvalue numerical solution decay gets faster These numericalsolutions graphs also agree with published earlier work [13]Distributions of the error at time 119905 = 10 are drawn for solitarywaves Figures 7 and 8 from which maximum error happens


Table 9 Comparison of our results with earlier studies

Values and methods 1198712times 103

119871infin

times 103

1198712times 103

119871infin

times 103

119905 = 2 119905 = 2 119905 = 10 119905 = 10

V = 0005 Δ119905 = 0001 ℎ = 0005

SFEM 002469 008456 012326 041604[14] 025786 072264 018735 030006[17] SBCM1 022890 058623 014042 023019[17] SBCM2 023397 058424 013747 022626

V = 0001 Δ119905 = 001 ℎ = 0005

SFEM 000549 002820 002743 013913QBDQM 013707 044538 006803 017110[13] 018354 081852 005511 013943[14] 006703 027967 005010 012129[17] SBCM1 006843 026233 004080 010295[17] SBCM2 007220 025975 003871 009882[18] 006607 026186 004160 010470

V = 001 Δ119905 = 001 and ℎ = 0005

SFEM 009785 028062 048711 134692[14] 052308 121698 064007 128124[17] SBCM1 038489 082934 054826 128127[17] SBCM2 039078 082734 054612 128127[18] 037552 081766 019391 023074

V = 001 Δ119905 = 001 and ℎ = 002

SFEM 009738 028025 048485 134798QBDQM 079558 137959 060429 147478[13] 079042 170309 080025 180239[17] SBCM1 038474 082611 055985 128127[17] SBCM2 041321 081502 055095 128127

at the right hand boundary when greater value of viscosityV = 001 is used andwith smaller value of viscosity V = 0001maximum error is recorded around the location where shockwave has the highest amplitude The 119871

2and 119871

infinerror norms

and numerical rate of convergence analysis for V = 0001 andΔ119905 = 001 and different numbers of grid points are tabulatedin Table 8

Table 9 presents a comparison of the values of the errornorms obtained by the present methods with those obtainedby other methods [13 14 17 18] It is clearly seen from thetable that the error norm 119871

2obtained by the SFEM is smaller

than those given in [13 14 17 18] whereas the error norm 119871infin

is very close to those given in [14 17 18] The error norm 119871infin

is better than the paper [13] For the QBDQM both 1198712and

119871infin

are almost the same as these papers

4 Conclusion

In this paper SFEM and DQM based on quartic B-splineshave been set up to find the numerical solution of the MBE(2) The performance of the schemes has been consideredby studying the propagation of a single solitary wave Theefficiency and accuracy of the methods were shown bycalculating the error norms 119871

2and 119871

infin Stability analysis of

the approximation obtained by the schemes shows that the

000

001

002

003

004

005

U

00 02 04 06 08 10

x

t = 1

t = 5

t = 3 t = 9

t = 7

Figure 4 Solutions for V = 001 ℎ = 002 Δ119905 = 001 and 0 le 119909 le 1

methods are unconditionally stable An obvious conclusioncan be drawn from the numerical results that for the SFEM1198712error norm is found to be better than the methods cited in


000

001

002

003

004

005

U

00 02 04 06 08 141210

x

t = 1

t = 5

t = 3 t = 9

t = 7

Figure 5 Solutions for V = 001 ℎ = 002 Δ119905 = 001 and 0 le 119909 le

13

0000

0002

0004

0006

0008

0014

0012

0010

U

00 02 04 06 08 10

x

t = 1

t = 5

t = 3 t = 9

t = 7

Figure 6 Solutions for V = 0001 Δ119905 = 001119873 = 166 and 0 le 119909 le

1

[13 14 17 18] whereas119871infinerror norm is found to be very close

to values given in [13 14 17 18] The obtained results showthat our methods can be used to produce reasonable accuratenumerical solutions of modified Burgersrsquo equation So thesemethods are reliable for getting the numerical solutions of thephysically important nonlinear problems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

00 02 04 06 08 10

x

00000

00005

00010

00015

00020

Error

Figure 7 Errors for V = 001 Δ119905 = 001 ℎ = 002 and 0 le 119909 le 1

000000

000005

000010

000015

000020

00 02 04 06 08 10

x

Error

Figure 8 Errors for V = 0001 Δ119905 = 001 and119873 = 166 0 le 119909 le 1

References

[1] H Bateman ldquoSome recent researches on the motion of fluidsrdquoMonthly Weather Review vol 43 pp 163ndash170 1915

[2] J M Burgers ldquoA mathematical model illustrating the theory ofturbulancerdquo Advances in Applied Mechanics vol 1 pp 171ndash1991948

[3] E N Aksan A Ozdes and T Ozis ldquoA numerical solutionof Burgersrsquo equation based on least squares approximationrdquoApplied Mathematics and Computation vol 176 no 1 pp 270ndash279 2006

[4] J D Cole ldquoOn a quasi-linear parabolic equations occuring inaerodynamicsrdquo Quarterly of Applied Mathematics vol 9 pp225ndash236 1951

[5] J Caldwell and P Smith ldquoSolution of Burgersrsquo equation with alarge Reynolds numberrdquo Applied Mathematical Modelling vol6 no 5 pp 381ndash385 1982

[6] S Kutluay A R Bahadir and A Ozdes ldquoNumerical solu-tion of one-dimensional Burgers equation explicit and exact-explicit finite difference methodsrdquo Journal of Computationaland Applied Mathematics vol 103 no 2 pp 251ndash261 1999


[7] R C Mittal and P Singhal ldquoNumerical solution of Burgerrsquosequationrdquo Communications in Numerical Methods in Engineer-ing vol 9 no 5 pp 397ndash406 1993

[8] R C Mittal and P Singhal ldquoNumerical solution of periodicburgers equationrdquo Indian Journal of Pure and Applied Mathe-matics vol 27 no 7 pp 689ndash700 1996

[9] M B Abd-el-Malek and S M A El-Mansi ldquoGroup theoreticmethods applied to Burgersrsquo equationrdquo Journal of Computa-tional and Applied Mathematics vol 115 no 1-2 pp 1ndash12 2000

[10] T Ozis E N Aksan and A Ozdes ldquoA finite element approachfor solution of Burgersrsquo equationrdquo Applied Mathematics andComputation vol 139 no 2-3 pp 417ndash428 2003

[11] L R T Gardner G A Gardner and A Dogan ldquoA Petrov-Galerkin finite element scheme for Burgersrsquo equationrdquo ArabianJournal for Science and Engineering vol 22 no 2 pp 99ndash1091997

[12] S Kutluay A Esen and I Dag ldquoNumerical solutions ofthe Burgersrsquo equation by the least-squares quadratic B-splinefinite element methodrdquo Journal of Computational and AppliedMathematics vol 167 no 1 pp 21ndash33 2004

[13] M A Ramadan T S El-Danaf and F E I A Alaal ldquoAnumerical solution of the Burgersrsquo equation using septic B-splinesrdquo Chaos Solitons and Fractals vol 26 no 4 pp 1249ndash1258 2005

[14] M A Ramadan and T S El-Danaf ldquoNumerical treatment forthe modified burgers equationrdquoMathematics and Computers inSimulation vol 70 no 2 pp 90ndash98 2005

[15] Y Duan R Liu and Y Jiang ldquoLattice Boltzmann modelfor the modified Burgersrsquo equationrdquo Applied Mathematics andComputation vol 202 no 2 pp 489ndash497 2008

[16] I Dag B Saka and D Irk ldquoGalerkin method for the numericalsolution of the RLW equation using quintic B-splinesrdquo Journalof Computational and Applied Mathematics vol 190 no 1-2 pp532ndash547 2006

[17] D Irk ldquoSextic B-spline collocation method for the modifiedBurgersrsquo equationrdquo Kybernetes vol 38 no 9 pp 1599ndash16202009

[18] T Roshan and K S Bhamra ldquoNumerical solutions of the mod-ified Burgersrsquo equation by Petrov-Galerkin methodrdquo AppliedMathematics and Computation vol 218 no 7 pp 3673ndash36792011

[19] R P Zhang X J Yu and G Z Zhao ldquoModified Burgersrsquoequation by the local discontinuous Galerkin methodrdquo ChinesePhysics B vol 22 Article ID 030210 2013

[20] A G Bratsos ldquoA fourth-order numerical scheme for solving themodified Burgers equationrdquo Computers and Mathematics withApplications vol 60 no 5 pp 1393ndash1400 2010

[21] R Bellman BG Kashef and J Casti ldquoDifferential quadrature atechnique for the rapid solution of nonlinear partial differentialequationsrdquo Journal of Computational Physics vol 10 no 1 pp40ndash52 1972

[22] R Bellman B Kashef E S Lee and R Vasudevan ldquoDifferentialQuadrature and Splinesrdquo in Computers and Mathematics withApplications pp 371ndash376 Pergamon Oxford UK 1976

[23] J R Quan andC T Chang ldquoNew insights in solving distributedsystem equations by the quadraturemethodmdashIrdquoComputers andChemical Engineering vol 13 no 7 pp 779ndash788 1989

[24] J R Quan and C T Chang ldquoNew sightings in involvingdistributed system equations by the quadrature methodsmdashIIrdquoComputers and Chemical Engineering vol 13 pp 717ndash724 1989

[25] H Zhong ldquoSpline-based differential quadrature for fourthorder differential equations and its application to Kirchhoffplatesrdquo Applied Mathematical Modelling vol 28 no 4 pp 353ndash366 2004

[26] Q Guo and H Zhong ldquoNon-linear vibration analysis of beamsby a spline-based differential quadrature methodrdquo Journal ofSound and Vibration vol 269 no 1-2 pp 413ndash420 2004

[27] H Zhong and M Lan ldquoSolution of nonlinear initial-valueproblems by the spline-based differential quadrature methodrdquoJournal of Sound and Vibration vol 296 no 4-5 pp 908ndash9182006

[28] C Shu and Y L Wu ldquoIntegrated radial basis functions-based differential quadrature method and its performancerdquoInternational Journal for Numerical Methods in Fluids vol 53no 6 pp 969ndash984 2007

[29] A Korkmaz and I Dag ldquoPolynomial based differential quadra-ture method for numerical solution of nonlinear Burgersequationrdquo Journal of the Franklin Institute vol 348 no 10 pp2863ndash2875 2011

[30] PM Prenter Splines and VariationalMethods JohnWiley NewYork NY USA 1975

[31] A Korkmaz Numerical solutions of some one dimensional par-tial differential equations using B-spline differential quadraturemethod [Doctoral dissertation] Eskisehir Osmangazi Univer-sity 2010

[32] C Shu Differential Quadrature and Its Application in Engineer-ing Springer London UK 2000

[33] S E Harris ldquoSonic shocks governed by the modified Burgersrsquoequationrdquo European Journal of Applied Mathematics vol 6 pp75ndash107 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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based on fourth-order rational approximants to the matrix-exponential term in a two-time level recurrence relation forcalculating the numerical solution of the equation

Recently DQM has become a very efficient and effectivemethod to obtain the numerical solutions of various typesof partial differential equations In 1972 Bellman et al [21]first introduced differential quadrature method (DQM) forsolving partial differential equations The main idea behindthe method is to find out the weighting coefficients of thefunctional values at nodal points by using base functions ofwhich derivatives are already known at the same nodal pointsover the entire region Various researchers have developeddifferent types of DQMs by utilizing various test functionsBellman et al [22] have used Legendre polynomials andspline functions in order to get weighting coefficients Quanand Chang [23 24] have presented an explicit formulationfor determining the weighting coefficients using Lagrangeinterpolation polynomials Zhong [25] Guo and Zhong[26] and Zhong and Lan [27] have introduced anotherefficient DQM as spline based DQM and applied it todifferent problems Shu and Wu [28] have considered someof the implicit formulations of weighting coefficients with thehelp of radial basis functions Nonlinear Burgersrsquo equationis solved using polynomial based differential quadraturemethod by Korkmaz and Dag [29] The DQM has manyadvantages over the classical techniques mainly it preventslinearization and perturbation in order to find better solu-tions of given nonlinear equations Since QBDQM do notneed transforming for solving the equation the method hasbeen preferred

In the present work we have applied a subdomainfinite element method and a quartic B-spline differentialquadrature method to the MBE To show the performanceand accuracy of the methods and make comparisons ofnumerical solutions we have taken different values of V

2 Numerical Methods

To implement the numerical schemes the interval [119886 119887] issplitted up into uniformly sized intervals by the nodes 119909

119898

119898 = 1 2 119873 such that 119886 = 1199090lt 1199091sdot sdot sdot lt 119909

119873= 119887 where

ℎ = (119909119898+1

minus 119909119898)

21 Subdomain Finite Element Method (SFEM) We willconsider (2) with the boundary conditions chosen from

119880 (119886 119905) = 1205731 119880 (119887 119905) = 120573

2

119880119909(119886 119905) = 0 119880

119909(119887 119905) = 0

119880119909119909

(119886 119905) = 0 119880119909119909

(119887 119905) = 0 119905 gt 0

(3)

with the initial condition

119880 (119909 0) = 119891 (119909) 119886 le 119909 le 119887 (4)

where 1205731and 120573

2are constants The quartic B-splines 120601

119898(119909)

(119898 = minus2(1) 119873 + 1) at the knots 119909119898which form a basis over

the interval [119886 119887] are defined by the relationships [30]

120601119898(119909)

=1

ℎ4

(119909 minus 119909119898minus2

)4

119909 isin [119909119898minus2

119909119898minus1

]

(119909 minus 119909119898minus2

)4


)4

119909 isin [119909119898minus1

119909119898]

(119909 minus 119909119898minus2

)4


)4

+10(119909 minus 119909119898)4

119909 isin [119909

119898 119909119898+1

]

(119909119898+3

minus 119909)4

minus 5(119909119898+2

minus 119909)4

119909 isin [119909119898+1

119909119898+2

]

(119909119898+3

minus 119909)4

119909 isin [119909119898+2

119909119898+3

]

0 otherwise(5)

Our numerical treatment for solving the MBE using thesubdomain finite element method with quartic B-splines isto find a global approximation 119880

119873(119909 119905) to the exact solution

119880(119909 119905) that can be expressed in the following form

119880119873(119909 119905) =

119873+1

sum

119895=minus2

120575119895(119905) 120601119895(119909) (6)

where 120575119895are time-dependent parameters to be determined

from both boundary and weighted residual conditions Thenodal values 119880

119898 1198801015840119898 11988010158401015840119898 and 119880

101584010158401015840

119898at the knots 119909

119898can be

obtained from (5) and (6) in the following form

119880119898

= 119880 (119909119898) = 120575119898minus2

+ 11120575119898minus1

+ 11120575119898+ 120575119898+1

1198801015840

119898= 1198801015840(119909119898) =

4

ℎ(minus120575119898minus2


+ 3120575119898+ 120575119898+1

)

11988010158401015840

119898= 11988010158401015840(119909119898) =

12

ℎ2(120575119898minus2


minus 120575119898+ 120575119898+1

)

119880101584010158401015840

119898= 119880101584010158401015840

(119909119898) =

24

ℎ3(minus120575119898minus2

+ 3120575119898minus1

minus 3120575119898+ 120575119898+1

)

(7)

For each element using the local coordinate transformation

ℎ120585 = 119909 minus 119909119898 0 le 120585 le 1 (8)

a typical finite interval [119909119898 119909119898+1

] is mapped into the interval[0 1]Therefore the quartic B-spline shape functions over theelement [0 1] can be defined as

120601119890=

120601119898minus2

= 1 minus 4120585 + 61205852minus 41205853+ 1205854

120601119898minus1

= 11 minus 12120585 minus 61205852+ 121205853minus 1205854

120601119898

= 11 + 12120585 minus 61205852minus 121205853+ 1205854

120601119898+1

= 1 + 4120585 + 61205852+ 41205853minus 1205854

120601119898+2

= 1205854

(9)

All other splines apart from 120601119898minus2

(119909) 120601119898minus1

(119909) 120601119898(119909)

120601119898+1

(119909) and 120601119898+2

(119909) are zero over the element [0 1] So the



119880119873(120585 119905) =

119898+2

sum

119895=119898minus2

120575119895(119905) 120601119895(120585) (10)


120575119898minus1

120575119898 120575119898+1

and 120575119898+2


119898minus2(119909) 120601

119898minus1 120601119898 120601119898+1

and 120601119898+2


119882119898(119909) =

1 119909 isin [119909119898 119909119898+1

]

0 otherwise(11)


int

119909119898+1

119909119898

1 (119880119905+ 1198802119880119909minus V119880119909119909) 119889119909 = 0 (12)


ℎ

5( 120575119898minus2

+ 26 120575119898minus1

+ 66 120575119898+ 26 120575

119898+1+ 120575119898+2

)

+ 119885119898(minus120575119898minus2

minus 10120575119898minus1

+ 10120575119898+1

+ 120575119898+2

)

minus4Vℎ

(120575119898minus2

+ 2120575119898minus1

minus 6120575119898+ 2120575119898+1

+ 120575119898+2

) = 0

(13)


119885119898

= (120575119898minus2

+ 11120575119898minus1

+ 11120575119898+ 120575119898+1

)2

(14)


120575119898

=120575119899

119898+ 120575119899+1

119898

2 120575

119898=

120575119899+1

119898minus 120575119899

119898

Δ119905

(15)


119894and

120575119899

119894 for 119894 = 119898 minus 2119898 minus 1 119898 + 2

1205721198981

120575119899+1

119898minus2+ 1205721198982

120575119899+1

119898minus1+ 1205721198983

120575119899+1

119898+ 1205721198984

120575119899+1

119898+1

+ 1205721198985

120575119899+1

119898+2

= 1205721198986

120575119899

119898minus2+ 1205721198987

120575119899

119898minus1+ 1205721198988

120575119899

119898+ 1205721198989

120575119899

119898+1

+ 12057211989810

120575119899

119898+2

119898 = 0 1 119873 minus 1

(16)

where

1205721198981

= 1 minus 119864119885119898minus 119872 120572

1198982= 26 minus 10119864119885

119898minus 2119872

1205721198983

= 66 + 6119872 1205721198984

= 26 + 10119864119885119898minus 2119872

1205721198985

= 1 + 119864119885119898minus 119872 120572

1198986= 1 + 119864119885

119898+ 119872

1205721198987

= 26 + 10119864119885119898+ 2119872 120572

1198988= 66 minus 6119872

1205721198989

= 26 minus 10119864119885119898+ 2119872 120572

11989810= 1 minus 119864119885

119898+ 119872

119864 =5Δ119905

2ℎ 119872 =

20VΔ1199052ℎ2

(17)


minus2 120575minus1 120575



minus2 120575minus1 120575119873 and 120575



(1205750 1205751 120575


119860119889119899+1

= 119861119889119899 (18)


119898+ 119880119898+1

)24 as

119885119898

=1

4(120575119898minus2

+ 12120575119898minus1

+ 22120575119898+ 12120575

119898+1+ 120575119898+2

)2

(19)


lowast)119899+1

= 120575119899+

(12)(120575119899+1




1205741120575119899+1

119898minus2+ 1205742120575119899+1

119898minus1+ 1205743120575119899+1

119898+ 1205744120575119899+1

119898+1+ 1205745120575119899+1

119898+2

= 1205746120575119899

119898minus2+ 1205747120575119899

119898minus1+ 1205748120575119899

119898+ 1205749120575119899

119898+1+ 12057410120575119899

119898+2

(20)

where

1205741= 120572 minus 120573 minus 120582 120574

2= 26120572 minus 10120573 minus 2120582

1205743= 66120572 + 6120582 120574

4= 26120572 + 10120573 minus 2120582

1205745= 120572 + 120573 minus 120582 120574

6= 120572 + 120573 + 120582

1205747= 26120572 + 10120573 + 2120582 120574

8= 66120572 minus 6120582

1205749= 26120572 minus 10120573 + 2120582 120574

10= 120572 minus 120573 + 120582

120572 = 1 120573 = 119864119885119898 120582 = 119872

(21)



0 1205751 120575



1198801015840(119886 0) =

4

ℎ(minus1205750


minus1+ 31205750

0+ 1205750

1) = 0

11988010158401015840(119886 0) =

12

ℎ2(1205750

minus2minus 1205750

minus1minus 1205750

0+ 1205750

1) = 0

119880 (119909119898 0) = 120575

0

119898minus2+ 11120575

0

119898minus1+ 11120575

0

119898+ 1205750

119898+1= 119891 (119909)

119898 = 0 1 119873 minus 1

1198801015840(119887 0) =

4

ℎ(minus1205750

119873minus2minus 31205750

119873minus1+ 31205750

119873+ 1205750

119873+1) = 0

11988010158401015840(119887 0) =

12

ℎ2(1205750



119873+ 1205750

119873+1) = 0

(22)


minus2 1205750minus1 1205750119873 and 120575

0

119873+1


1198601205750= 119861 (23)

where 119860 is

119860 =

[[[[[

[

18 6

115 115 1

1 11 11 1

1 11 11 1

2 14 8

]]]]]

]

(24)

1205750

= [1205750

0 1205750

1 120575

0

119873minus1]119879 and 119861 = [119880(119909

0 0) 119880(119909

1 0)

119880(119909119873minus1


Thomas algorithm


2119880119909is


120575119899

119895= 120585119899119890119894119895119896ℎ

(25)


119892 =1198601+ 119894119887

1198602minus 119894119887

(26)


1198602= (120572 + 120582) cos (2119896ℎ) + (26120572 + 2120582) cos (119896ℎ) + 66 minus 6120582

119887 = sin (2119896ℎ) + 10 sin (119896ℎ)

(27)








119880(119903)

119909(119909119894) =

119889119880(119903)

119889119909(119903)|119909119894

=

119873

sum

119895=1

119908(119903)

119894119895119880(119909119895)

119894 = 1 2 119873 119903 = 1 2 119873 minus 1

(28)


119894119895repre-





(119903)





119889(119903)119876119898(119909119894)

119889119909(119903)=

119898+2

sum

119895=119898minus1

119908(119903)

119894119895119876119898(119909119895)

119898 = minus1 0 119873 + 2 119894 = 1 2 119873

(29)


119889119876119898(119909119894)

119889119909=

119898+2

sum

119895=119898minus1

119908(1)

119894119895119876119898(119909119895)

119898 = minus1 0 119873 + 1 119894 = 1 2 119873

(30)




119898 119898 =


119898test



[[[[[[[[[[[[[[[[[[[[[

[

1 11 11 1

1 11 11 1

d d d d

1 11 11 1

1 11 11 1

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(1)

1minus2

119908(1)

1minus1

119908(1)

10

119908(1)

11

119908(1)

12

119908(1)

1119873+2

119908(1)

1119873+3

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[[

[

minus4

ℎ

minus12

ℎ

12

ℎ

4

ℎ

0

0

0

]]]]]]]]]]]]]]]]]]]]]]

]

(31)




119889(2)

119876minus1

(1199091)

119889119909(2)=

1

sum

119895=minus2

119908(1)

11198951198761015840

minus1(119909119895)

119889(2)

119876119873+1

(1199091)

119889119909(2)=

119873+3

sum

119895=119873

119908(1)

11198951198761015840

119873+1(119909119895)

119889(3)

119876119873+1

(1199091)

120597119909(3)=

119873+3

sum

119895=119873

119908(1)

111989511987610158401015840

119873+1(119909119895)

(32)


[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11 1

2 23 23

6 18

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(1)

1minus1

119908(1)

10

119908(1)

11

119908(1)

12

119908(1)

13

119908(1)

1119873

119908(1)

1119873+1

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[

[

minus7

ℎ

minus12

ℎ

12

ℎ

4

ℎ0

0

0

]]]]]]]]]]]]]]]]]]]]]

]

(33)


119896119895 119895 =



[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11 1

2 23 23

6 18

]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

119908(1)

119896minus1

119908(1)

119896119896minus3

119908(1)

119896119896minus2

119908(1)

119896119896minus1

119908(1)

119896119896

119908(1)

119896119896+1

119908(1)

119896119896+2

119908(1)

119896119873+1

]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]


=

[[[[[[[[[[[[[[[[[[[[[[[[[[

[

0

0

minus4

ℎ

minus12

ℎ

12

ℎ

4

ℎ0

0

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

(34)




[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11 1

2 23 23

6 18

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(1)

119873minus1

119908(1)

1198730

119908(1)

119873119873minus3

119908(1)

119873119873minus2

119908(1)

119873119873minus1

119908(1)

119873119873

119908(1)

119873119873+1

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[

[

0

0

0

minus4

ℎ

minus12

ℎ

53

2ℎ

17

ℎ

]]]]]]]]]]]]]]]]]]]]]

]

(35)


1198892119876119898

1198891199092(119909119894) =

119898+2

sum

119895=119898minus1

119908(2)

119894119895119876119898(119909119895)

119898 = minus1 0 119873 + 1 119894 = 1 2 119873

(36)

Table 1 1198712and 119871


V = 0001 (SFEM)


119871infin

times 103

2 00054945 002820493 00082404 004224214 00109858 005622805 00137296 007015666 00164729 008404277 00192154 009789758 00219573 011169349 00246985 0125446610 00274379 01391304

where 119908(2)




119898119898 = minus1 0 119873 + 1 are used


[[[[[[[[[[[[[[[[[[[[[

[

1 11 11 1

1 11 11 1

d d d d

1 11 11 1

1 11 11 1

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(2)

1minus2

119908(2)

1minus1

119908(2)

10

119908(2)

11

119908(2)

12

119908(2)

1119873+2

119908(2)

1119873+3

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[

[

12

ℎ2

minus12

ℎ2

minus12

ℎ2

12

ℎ2

0

0

0

]]]]]]]]]]]]]]]]]]]]]

]

(37)


1198893119876minus1

(1199091)

1198891199093=

1

sum

119895=minus2

119908(1)

11198951198761015840

minus1(119909119895) (38)


1198893119876119873+1

(1199091)

1198891199093=

119873+3

sum

119895=119873

119908(1)

11198951198761015840

119873+1(119909119895) (39)


[[[[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11 1

2 14 8

]]]]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[[[

[

119908(2)

1minus1

119908(2)

10

119908(2)

11

119908(2)

12

119908(2)

13

119908(2)

1119873+1

119908(2)

1119873+2

]]]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[[

[

18

ℎ2

minus12

ℎ2

minus12

ℎ2

12

ℎ2

0

0

0

]]]]]]]]]]]]]]]]]]]]]]

]

(40)




119908(2)

119894119895= 2119908(1)

119894119895(119908(1)

119894119894minus

1

119909119894minus 119909119895

) 119894 = 119895 (41)

After we use (41)

1198601= 119908(2)

1119873+2= 2119908(1)

1119873+2(119908(1)

11minus

1

1199091minus 119909119873+2

)

[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11

2 14

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(2)

1minus1

119908(2)

10

119908(2)

11

119908(2)

12

119908(2)

13

119908(2)

1119873

119908(2)

1119873+1

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[

[

18

ℎ2

minus12

ℎ2

minus12

ℎ2

12

ℎ2

0

minus1198601

minus81198601

]]]]]]]]]]]]]]]]]]]]]

]

(42)



119896 2 le 119896 le


[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11

2 14

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

119908(2)

119896minus1

119908(2)

119896119896minus3

119908(2)

119896119896minus2

119908(2)

119896119896minus1

119908(2)

119896119896

119908(2)

119896119896+1

119908(2)

119896119896+2

119908(2)

119896119873minus1

119908(2)

119896119873

119908(2)

119896119873+1

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]


Table 2 1198712and 119871


V = 0005 (SFEM)


119871infin

times 103

2 00246966 008456893 00370384 012662224 00493707 016843625 00616997 021013196 00740253 025163927 00863444 029301788 00986573 033419229 01109636 0375245710 01232629 04160477

Table 3 1198712and 119871


V = 001 (SFEM)


119871infin

times 103

2 00978574 028062433 01467089 041859814 01955072 055502865 02442506 068987136 02929396 082386297 03415703 095666888 03901436 108812899 04386580 1218223110 04871136 13469237

=

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

0

12

ℎ2

minus12

ℎ2

minus12

ℎ2

12

ℎ2

0

0

minus119860119896

minus8119860119896

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

(43)

where119860119896equals119860

119896= 119908(2)

119896119873+2= 2119908(1)

119896119873+2(119908(1)

119896119896minus1(119909

119896minus119909119873+2


119873with the


119873119895 119895 =

Table 4 1198712and 119871


001 (SFEM)


119871infin

times 103

2 00973818 028025263 01460008 041848724 01945704 055541215 02430873 069100626 02915506 082523127 03399602 095804338 03883156 108944139 04366131 1219411110 04848547 13479880


[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11

2 14

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(2)

119873minus1

119908(2)

1198730

119908(2)

119873119873minus3

119908(2)

119873119873minus2

119908(2)

119873119873minus1

119908(2)

119873119873

119908(2)

119873119873+1

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[

[

0

0

0

12

ℎ2

minus12

ℎ2

minus12

ℎ2minus 119860119873

18

ℎ2minus 8119860119873

]]]]]]]]]]]]]]]]]]]]]

]

(44)

where 119860119873equals 119860

119873= 119908(2)

119873119873+2= 2119908(1)

119873119873+2(119908(1)

119873119873minus 1(119909

119873minus

119909119873+2

))




Table 5 1198712and 119871



1198712times 103

119871infin

times 103

1198712times 103

119871infin

times 103

2 07955855586 13795978925 07904296620 17030921188

3 06690533313 11943543646 06551928290 11832698216

4 05250528343 09764154381 05576794264 09964523368

5 04048512821 07849457015 05105617536 08561342445

6 03452210304 06374950443 05167229575 07610530060

7 03638648688 06705419608 05677438614 10654548090

8 04337013450 09863405006 06427542266 13581113635

9 05197862999 12551335234 07236430257 16048306653

10 06042925888 14747885309 08002564201 18023938553

Table 6 1198712and 119871


at 0 le 119909 le 13


119871infin

times 103

2 07607107169 137041821953 06480181273 118549841904 05604986926 100524764525 04927784148 086540324196 04359075842 075315510237 03885737191 066013265128 03520185942 058333349709 03282544303 0520132366310 03187570280 04691560472


2and 119871

infin respectively

1198712=10038171003817100381710038171003817119880


100381710038171003817100381710038172≃ radicℎ

119873

sum

119869=1

100381610038161003816100381610038161003816119880

exact119895

minus (119880119873)119895

100381610038161003816100381610038161003816

2

119871infin

=10038171003817100381710038171003817119880


10038171003817100381710038171003817infin≃ max119895

100381610038161003816100381610038161003816119880

exact119895

minus (119880119873)119895

100381610038161003816100381610038161003816

119895 = 1 2 119873 minus 1

(45)


119880 (119909 119905) =(119909119905)

1 + (radic1199051198880) exp (11990924V119905)

(46)





119909(0 119905) = 0 and 119880(1 119905) = 119880

119909(1 119905) = 0


0000

0005

0010

0015

0020

0025

U(xt)

00 02 04 06 08 10

x

t = 1

t = 2

t = 4

t = 6t = 8



2and 119871

infinare




ROC asympln (119864 (119873

2) 119864 (119873

1))

ln (11987311198732)

(47)


Table 7 1198712and 119871



119871infin

times 103

1198712times 103

119871infin

times 103

2 01370706949 04453892504 01835491190 081852111123 01168507335 03842839811 01441424335 052348333464 01019761971 03258391192 01144110783 035635372075 00920706001 02816616769 00947865272 025497900586 00849484881 02484289381 00814174677 021348478357 00794570772 02225471690 00718977757 018800484328 00750035859 02019577762 00648368942 016826017709 00712618898 01851510002 00594114970 0152407496610 00680382860 01711033543 00551151456 01394312127


119873 1198712times 103 ROC(119871

2) 119871


infin)

11 043 mdash 098 mdash21 035 031 088 01631 022 119 052 13541 017 092 039 10251 014 088 030 12081 010 072 019 098

0000

0004

0006

0008

0010

0012

0002

U(xt)

00 02 04 06 08 10

x

t = 1

t = 2

t = 4

t = 6

t = 8


0 01 and V = 0001



infin







0000

0005

0010

0015

0020

0030

0025

0035

U(xt)

00 02 04 06 08 10

x

t = 1

t = 2

t = 4

t = 6

t = 8



2and 119871



2

and 119871infin





119871infin

times 103

1198712times 103

119871infin

times 103

119905 = 2 119905 = 2 119905 = 10 119905 = 10

V = 0005 Δ119905 = 0001 ℎ = 0005

SFEM 002469 008456 012326 041604[14] 025786 072264 018735 030006[17] SBCM1 022890 058623 014042 023019[17] SBCM2 023397 058424 013747 022626

V = 0001 Δ119905 = 001 ℎ = 0005


V = 001 Δ119905 = 001 and ℎ = 0005

SFEM 009785 028062 048711 134692[14] 052308 121698 064007 128124[17] SBCM1 038489 082934 054826 128127[17] SBCM2 039078 082734 054612 128127[18] 037552 081766 019391 023074

V = 001 Δ119905 = 001 and ℎ = 002



2and 119871

infinerror norms







119871infin


4 Conclusion


2and 119871



000

001

002

003

004

005

U

00 02 04 06 08 10

x

t = 1

t = 5

t = 3 t = 9

t = 7




000

001

002

003

004

005

U

00 02 04 06 08 141210

x

t = 1

t = 5

t = 3 t = 9

t = 7


13

0000

0002

0004

0006

0008

0014

0012

0010

U

00 02 04 06 08 10

x

t = 1

t = 5

t = 3 t = 9

t = 7


1





00 02 04 06 08 10

x

00000

00005

00010

00015

00020

Error


000000

000005

000010

000015

000020

00 02 04 06 08 10

x

Error


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119880119873(120585 119905) =

119898+2

sum

119895=119898minus2

120575119895(119905) 120601119895(120585) (10)


120575119898minus1

120575119898 120575119898+1

and 120575119898+2


119898minus2(119909) 120601

119898minus1 120601119898 120601119898+1

and 120601119898+2


119882119898(119909) =

1 119909 isin [119909119898 119909119898+1

]

0 otherwise(11)


int

119909119898+1

119909119898

1 (119880119905+ 1198802119880119909minus V119880119909119909) 119889119909 = 0 (12)


ℎ

5( 120575119898minus2

+ 26 120575119898minus1

+ 66 120575119898+ 26 120575

119898+1+ 120575119898+2

)

+ 119885119898(minus120575119898minus2

minus 10120575119898minus1

+ 10120575119898+1

+ 120575119898+2

)

minus4Vℎ

(120575119898minus2

+ 2120575119898minus1

minus 6120575119898+ 2120575119898+1

+ 120575119898+2

) = 0

(13)


119885119898

= (120575119898minus2

+ 11120575119898minus1

+ 11120575119898+ 120575119898+1

)2

(14)


120575119898

=120575119899

119898+ 120575119899+1

119898

2 120575

119898=

120575119899+1

119898minus 120575119899

119898

Δ119905

(15)


119894and

120575119899

119894 for 119894 = 119898 minus 2119898 minus 1 119898 + 2

1205721198981

120575119899+1

119898minus2+ 1205721198982

120575119899+1

119898minus1+ 1205721198983

120575119899+1

119898+ 1205721198984

120575119899+1

119898+1

+ 1205721198985

120575119899+1

119898+2

= 1205721198986

120575119899

119898minus2+ 1205721198987

120575119899

119898minus1+ 1205721198988

120575119899

119898+ 1205721198989

120575119899

119898+1

+ 12057211989810

120575119899

119898+2

119898 = 0 1 119873 minus 1

(16)

where

1205721198981

= 1 minus 119864119885119898minus 119872 120572

1198982= 26 minus 10119864119885

119898minus 2119872

1205721198983

= 66 + 6119872 1205721198984

= 26 + 10119864119885119898minus 2119872

1205721198985

= 1 + 119864119885119898minus 119872 120572

1198986= 1 + 119864119885

119898+ 119872

1205721198987

= 26 + 10119864119885119898+ 2119872 120572

1198988= 66 minus 6119872

1205721198989

= 26 minus 10119864119885119898+ 2119872 120572

11989810= 1 minus 119864119885

119898+ 119872

119864 =5Δ119905

2ℎ 119872 =

20VΔ1199052ℎ2

(17)


minus2 120575minus1 120575



minus2 120575minus1 120575119873 and 120575



(1205750 1205751 120575


119860119889119899+1

= 119861119889119899 (18)


119898+ 119880119898+1

)24 as

119885119898

=1

4(120575119898minus2

+ 12120575119898minus1

+ 22120575119898+ 12120575

119898+1+ 120575119898+2

)2

(19)


lowast)119899+1

= 120575119899+

(12)(120575119899+1




1205741120575119899+1

119898minus2+ 1205742120575119899+1

119898minus1+ 1205743120575119899+1

119898+ 1205744120575119899+1

119898+1+ 1205745120575119899+1

119898+2

= 1205746120575119899

119898minus2+ 1205747120575119899

119898minus1+ 1205748120575119899

119898+ 1205749120575119899

119898+1+ 12057410120575119899

119898+2

(20)

where

1205741= 120572 minus 120573 minus 120582 120574

2= 26120572 minus 10120573 minus 2120582

1205743= 66120572 + 6120582 120574

4= 26120572 + 10120573 minus 2120582

1205745= 120572 + 120573 minus 120582 120574

6= 120572 + 120573 + 120582

1205747= 26120572 + 10120573 + 2120582 120574

8= 66120572 minus 6120582

1205749= 26120572 minus 10120573 + 2120582 120574

10= 120572 minus 120573 + 120582

120572 = 1 120573 = 119864119885119898 120582 = 119872

(21)



0 1205751 120575



1198801015840(119886 0) =

4

ℎ(minus1205750


minus1+ 31205750

0+ 1205750

1) = 0

11988010158401015840(119886 0) =

12

ℎ2(1205750

minus2minus 1205750

minus1minus 1205750

0+ 1205750

1) = 0

119880 (119909119898 0) = 120575

0

119898minus2+ 11120575

0

119898minus1+ 11120575

0

119898+ 1205750

119898+1= 119891 (119909)

119898 = 0 1 119873 minus 1

1198801015840(119887 0) =

4

ℎ(minus1205750

119873minus2minus 31205750

119873minus1+ 31205750

119873+ 1205750

119873+1) = 0

11988010158401015840(119887 0) =

12

ℎ2(1205750



119873+ 1205750

119873+1) = 0

(22)


minus2 1205750minus1 1205750119873 and 120575

0

119873+1


1198601205750= 119861 (23)

where 119860 is

119860 =

[[[[[

[

18 6

115 115 1

1 11 11 1

1 11 11 1

2 14 8

]]]]]

]

(24)

1205750

= [1205750

0 1205750

1 120575

0

119873minus1]119879 and 119861 = [119880(119909

0 0) 119880(119909

1 0)

119880(119909119873minus1


Thomas algorithm


2119880119909is


120575119899

119895= 120585119899119890119894119895119896ℎ

(25)


119892 =1198601+ 119894119887

1198602minus 119894119887

(26)


1198602= (120572 + 120582) cos (2119896ℎ) + (26120572 + 2120582) cos (119896ℎ) + 66 minus 6120582

119887 = sin (2119896ℎ) + 10 sin (119896ℎ)

(27)








119880(119903)

119909(119909119894) =

119889119880(119903)

119889119909(119903)|119909119894

=

119873

sum

119895=1

119908(119903)

119894119895119880(119909119895)

119894 = 1 2 119873 119903 = 1 2 119873 minus 1

(28)


119894119895repre-





(119903)





119889(119903)119876119898(119909119894)

119889119909(119903)=

119898+2

sum

119895=119898minus1

119908(119903)

119894119895119876119898(119909119895)

119898 = minus1 0 119873 + 2 119894 = 1 2 119873

(29)


119889119876119898(119909119894)

119889119909=

119898+2

sum

119895=119898minus1

119908(1)

119894119895119876119898(119909119895)

119898 = minus1 0 119873 + 1 119894 = 1 2 119873

(30)




119898 119898 =


119898test



[[[[[[[[[[[[[[[[[[[[[

[

1 11 11 1

1 11 11 1

d d d d

1 11 11 1

1 11 11 1

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(1)

1minus2

119908(1)

1minus1

119908(1)

10

119908(1)

11

119908(1)

12

119908(1)

1119873+2

119908(1)

1119873+3

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[[

[

minus4

ℎ

minus12

ℎ

12

ℎ

4

ℎ

0

0

0

]]]]]]]]]]]]]]]]]]]]]]

]

(31)




119889(2)

119876minus1

(1199091)

119889119909(2)=

1

sum

119895=minus2

119908(1)

11198951198761015840

minus1(119909119895)

119889(2)

119876119873+1

(1199091)

119889119909(2)=

119873+3

sum

119895=119873

119908(1)

11198951198761015840

119873+1(119909119895)

119889(3)

119876119873+1

(1199091)

120597119909(3)=

119873+3

sum

119895=119873

119908(1)

111989511987610158401015840

119873+1(119909119895)

(32)


[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11 1

2 23 23

6 18

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(1)

1minus1

119908(1)

10

119908(1)

11

119908(1)

12

119908(1)

13

119908(1)

1119873

119908(1)

1119873+1

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[

[

minus7

ℎ

minus12

ℎ

12

ℎ

4

ℎ0

0

0

]]]]]]]]]]]]]]]]]]]]]

]

(33)


119896119895 119895 =



[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11 1

2 23 23

6 18

]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

119908(1)

119896minus1

119908(1)

119896119896minus3

119908(1)

119896119896minus2

119908(1)

119896119896minus1

119908(1)

119896119896

119908(1)

119896119896+1

119908(1)

119896119896+2

119908(1)

119896119873+1

]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]


=

[[[[[[[[[[[[[[[[[[[[[[[[[[

[

0

0

minus4

ℎ

minus12

ℎ

12

ℎ

4

ℎ0

0

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

(34)




[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11 1

2 23 23

6 18

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(1)

119873minus1

119908(1)

1198730

119908(1)

119873119873minus3

119908(1)

119873119873minus2

119908(1)

119873119873minus1

119908(1)

119873119873

119908(1)

119873119873+1

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[

[

0

0

0

minus4

ℎ

minus12

ℎ

53

2ℎ

17

ℎ

]]]]]]]]]]]]]]]]]]]]]

]

(35)


1198892119876119898

1198891199092(119909119894) =

119898+2

sum

119895=119898minus1

119908(2)

119894119895119876119898(119909119895)

119898 = minus1 0 119873 + 1 119894 = 1 2 119873

(36)

Table 1 1198712and 119871


V = 0001 (SFEM)


119871infin

times 103

2 00054945 002820493 00082404 004224214 00109858 005622805 00137296 007015666 00164729 008404277 00192154 009789758 00219573 011169349 00246985 0125446610 00274379 01391304

where 119908(2)




119898119898 = minus1 0 119873 + 1 are used


[[[[[[[[[[[[[[[[[[[[[

[

1 11 11 1

1 11 11 1

d d d d

1 11 11 1

1 11 11 1

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(2)

1minus2

119908(2)

1minus1

119908(2)

10

119908(2)

11

119908(2)

12

119908(2)

1119873+2

119908(2)

1119873+3

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[

[

12

ℎ2

minus12

ℎ2

minus12

ℎ2

12

ℎ2

0

0

0

]]]]]]]]]]]]]]]]]]]]]

]

(37)


1198893119876minus1

(1199091)

1198891199093=

1

sum

119895=minus2

119908(1)

11198951198761015840

minus1(119909119895) (38)


1198893119876119873+1

(1199091)

1198891199093=

119873+3

sum

119895=119873

119908(1)

11198951198761015840

119873+1(119909119895) (39)


[[[[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11 1

2 14 8

]]]]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[[[

[

119908(2)

1minus1

119908(2)

10

119908(2)

11

119908(2)

12

119908(2)

13

119908(2)

1119873+1

119908(2)

1119873+2

]]]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[[

[

18

ℎ2

minus12

ℎ2

minus12

ℎ2

12

ℎ2

0

0

0

]]]]]]]]]]]]]]]]]]]]]]

]

(40)




119908(2)

119894119895= 2119908(1)

119894119895(119908(1)

119894119894minus

1

119909119894minus 119909119895

) 119894 = 119895 (41)

After we use (41)

1198601= 119908(2)

1119873+2= 2119908(1)

1119873+2(119908(1)

11minus

1

1199091minus 119909119873+2

)

[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11

2 14

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(2)

1minus1

119908(2)

10

119908(2)

11

119908(2)

12

119908(2)

13

119908(2)

1119873

119908(2)

1119873+1

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[

[

18

ℎ2

minus12

ℎ2

minus12

ℎ2

12

ℎ2

0

minus1198601

minus81198601

]]]]]]]]]]]]]]]]]]]]]

]

(42)



119896 2 le 119896 le


[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11

2 14

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

119908(2)

119896minus1

119908(2)

119896119896minus3

119908(2)

119896119896minus2

119908(2)

119896119896minus1

119908(2)

119896119896

119908(2)

119896119896+1

119908(2)

119896119896+2

119908(2)

119896119873minus1

119908(2)

119896119873

119908(2)

119896119873+1

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]


Table 2 1198712and 119871


V = 0005 (SFEM)


119871infin

times 103

2 00246966 008456893 00370384 012662224 00493707 016843625 00616997 021013196 00740253 025163927 00863444 029301788 00986573 033419229 01109636 0375245710 01232629 04160477

Table 3 1198712and 119871


V = 001 (SFEM)


119871infin

times 103

2 00978574 028062433 01467089 041859814 01955072 055502865 02442506 068987136 02929396 082386297 03415703 095666888 03901436 108812899 04386580 1218223110 04871136 13469237

=

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

0

12

ℎ2

minus12

ℎ2

minus12

ℎ2

12

ℎ2

0

0

minus119860119896

minus8119860119896

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

(43)

where119860119896equals119860

119896= 119908(2)

119896119873+2= 2119908(1)

119896119873+2(119908(1)

119896119896minus1(119909

119896minus119909119873+2


119873with the


119873119895 119895 =

Table 4 1198712and 119871


001 (SFEM)


119871infin

times 103

2 00973818 028025263 01460008 041848724 01945704 055541215 02430873 069100626 02915506 082523127 03399602 095804338 03883156 108944139 04366131 1219411110 04848547 13479880


[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11

2 14

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(2)

119873minus1

119908(2)

1198730

119908(2)

119873119873minus3

119908(2)

119873119873minus2

119908(2)

119873119873minus1

119908(2)

119873119873

119908(2)

119873119873+1

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[

[

0

0

0

12

ℎ2

minus12

ℎ2

minus12

ℎ2minus 119860119873

18

ℎ2minus 8119860119873

]]]]]]]]]]]]]]]]]]]]]

]

(44)

where 119860119873equals 119860

119873= 119908(2)

119873119873+2= 2119908(1)

119873119873+2(119908(1)

119873119873minus 1(119909

119873minus

119909119873+2

))




Table 5 1198712and 119871



1198712times 103

119871infin

times 103

1198712times 103

119871infin

times 103

2 07955855586 13795978925 07904296620 17030921188

3 06690533313 11943543646 06551928290 11832698216

4 05250528343 09764154381 05576794264 09964523368

5 04048512821 07849457015 05105617536 08561342445

6 03452210304 06374950443 05167229575 07610530060

7 03638648688 06705419608 05677438614 10654548090

8 04337013450 09863405006 06427542266 13581113635

9 05197862999 12551335234 07236430257 16048306653

10 06042925888 14747885309 08002564201 18023938553

Table 6 1198712and 119871


at 0 le 119909 le 13


119871infin

times 103

2 07607107169 137041821953 06480181273 118549841904 05604986926 100524764525 04927784148 086540324196 04359075842 075315510237 03885737191 066013265128 03520185942 058333349709 03282544303 0520132366310 03187570280 04691560472


2and 119871

infin respectively

1198712=10038171003817100381710038171003817119880


100381710038171003817100381710038172≃ radicℎ

119873

sum

119869=1

100381610038161003816100381610038161003816119880

exact119895

minus (119880119873)119895

100381610038161003816100381610038161003816

2

119871infin

=10038171003817100381710038171003817119880


10038171003817100381710038171003817infin≃ max119895

100381610038161003816100381610038161003816119880

exact119895

minus (119880119873)119895

100381610038161003816100381610038161003816

119895 = 1 2 119873 minus 1

(45)


119880 (119909 119905) =(119909119905)

1 + (radic1199051198880) exp (11990924V119905)

(46)





119909(0 119905) = 0 and 119880(1 119905) = 119880

119909(1 119905) = 0


0000

0005

0010

0015

0020

0025

U(xt)

00 02 04 06 08 10

x

t = 1

t = 2

t = 4

t = 6t = 8



2and 119871

infinare




ROC asympln (119864 (119873

2) 119864 (119873

1))

ln (11987311198732)

(47)


Table 7 1198712and 119871



119871infin

times 103

1198712times 103

119871infin

times 103

2 01370706949 04453892504 01835491190 081852111123 01168507335 03842839811 01441424335 052348333464 01019761971 03258391192 01144110783 035635372075 00920706001 02816616769 00947865272 025497900586 00849484881 02484289381 00814174677 021348478357 00794570772 02225471690 00718977757 018800484328 00750035859 02019577762 00648368942 016826017709 00712618898 01851510002 00594114970 0152407496610 00680382860 01711033543 00551151456 01394312127


119873 1198712times 103 ROC(119871

2) 119871


infin)

11 043 mdash 098 mdash21 035 031 088 01631 022 119 052 13541 017 092 039 10251 014 088 030 12081 010 072 019 098

0000

0004

0006

0008

0010

0012

0002

U(xt)

00 02 04 06 08 10

x

t = 1

t = 2

t = 4

t = 6

t = 8


0 01 and V = 0001



infin







0000

0005

0010

0015

0020

0030

0025

0035

U(xt)

00 02 04 06 08 10

x

t = 1

t = 2

t = 4

t = 6

t = 8



2and 119871



2

and 119871infin





119871infin

times 103

1198712times 103

119871infin

times 103

119905 = 2 119905 = 2 119905 = 10 119905 = 10

V = 0005 Δ119905 = 0001 ℎ = 0005

SFEM 002469 008456 012326 041604[14] 025786 072264 018735 030006[17] SBCM1 022890 058623 014042 023019[17] SBCM2 023397 058424 013747 022626

V = 0001 Δ119905 = 001 ℎ = 0005


V = 001 Δ119905 = 001 and ℎ = 0005

SFEM 009785 028062 048711 134692[14] 052308 121698 064007 128124[17] SBCM1 038489 082934 054826 128127[17] SBCM2 039078 082734 054612 128127[18] 037552 081766 019391 023074

V = 001 Δ119905 = 001 and ℎ = 002



2and 119871

infinerror norms







119871infin


4 Conclusion


2and 119871



000

001

002

003

004

005

U

00 02 04 06 08 10

x

t = 1

t = 5

t = 3 t = 9

t = 7




000

001

002

003

004

005

U

00 02 04 06 08 141210

x

t = 1

t = 5

t = 3 t = 9

t = 7


13

0000

0002

0004

0006

0008

0014

0012

0010

U

00 02 04 06 08 10

x

t = 1

t = 5

t = 3 t = 9

t = 7


1





00 02 04 06 08 10

x

00000

00005

00010

00015

00020

Error


000000

000005

000010

000015

000020

00 02 04 06 08 10

x

Error


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0 1205751 120575



1198801015840(119886 0) =

4

ℎ(minus1205750


minus1+ 31205750

0+ 1205750

1) = 0

11988010158401015840(119886 0) =

12

ℎ2(1205750

minus2minus 1205750

minus1minus 1205750

0+ 1205750

1) = 0

119880 (119909119898 0) = 120575

0

119898minus2+ 11120575

0

119898minus1+ 11120575

0

119898+ 1205750

119898+1= 119891 (119909)

119898 = 0 1 119873 minus 1

1198801015840(119887 0) =

4

ℎ(minus1205750

119873minus2minus 31205750

119873minus1+ 31205750

119873+ 1205750

119873+1) = 0

11988010158401015840(119887 0) =

12

ℎ2(1205750



119873+ 1205750

119873+1) = 0

(22)


minus2 1205750minus1 1205750119873 and 120575

0

119873+1


1198601205750= 119861 (23)

where 119860 is

119860 =

[[[[[

[

18 6

115 115 1

1 11 11 1

1 11 11 1

2 14 8

]]]]]

]

(24)

1205750

= [1205750

0 1205750

1 120575

0

119873minus1]119879 and 119861 = [119880(119909

0 0) 119880(119909

1 0)

119880(119909119873minus1


Thomas algorithm


2119880119909is


120575119899

119895= 120585119899119890119894119895119896ℎ

(25)


119892 =1198601+ 119894119887

1198602minus 119894119887

(26)


1198602= (120572 + 120582) cos (2119896ℎ) + (26120572 + 2120582) cos (119896ℎ) + 66 minus 6120582

119887 = sin (2119896ℎ) + 10 sin (119896ℎ)

(27)








119880(119903)

119909(119909119894) =

119889119880(119903)

119889119909(119903)|119909119894

=

119873

sum

119895=1

119908(119903)

119894119895119880(119909119895)

119894 = 1 2 119873 119903 = 1 2 119873 minus 1

(28)


119894119895repre-





(119903)





119889(119903)119876119898(119909119894)

119889119909(119903)=

119898+2

sum

119895=119898minus1

119908(119903)

119894119895119876119898(119909119895)

119898 = minus1 0 119873 + 2 119894 = 1 2 119873

(29)


119889119876119898(119909119894)

119889119909=

119898+2

sum

119895=119898minus1

119908(1)

119894119895119876119898(119909119895)

119898 = minus1 0 119873 + 1 119894 = 1 2 119873

(30)




119898 119898 =


119898test



[[[[[[[[[[[[[[[[[[[[[

[

1 11 11 1

1 11 11 1

d d d d

1 11 11 1

1 11 11 1

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(1)

1minus2

119908(1)

1minus1

119908(1)

10

119908(1)

11

119908(1)

12

119908(1)

1119873+2

119908(1)

1119873+3

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[[

[

minus4

ℎ

minus12

ℎ

12

ℎ

4

ℎ

0

0

0

]]]]]]]]]]]]]]]]]]]]]]

]

(31)




119889(2)

119876minus1

(1199091)

119889119909(2)=

1

sum

119895=minus2

119908(1)

11198951198761015840

minus1(119909119895)

119889(2)

119876119873+1

(1199091)

119889119909(2)=

119873+3

sum

119895=119873

119908(1)

11198951198761015840

119873+1(119909119895)

119889(3)

119876119873+1

(1199091)

120597119909(3)=

119873+3

sum

119895=119873

119908(1)

111989511987610158401015840

119873+1(119909119895)

(32)


[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11 1

2 23 23

6 18

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(1)

1minus1

119908(1)

10

119908(1)

11

119908(1)

12

119908(1)

13

119908(1)

1119873

119908(1)

1119873+1

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[

[

minus7

ℎ

minus12

ℎ

12

ℎ

4

ℎ0

0

0

]]]]]]]]]]]]]]]]]]]]]

]

(33)


119896119895 119895 =



[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11 1

2 23 23

6 18

]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

119908(1)

119896minus1

119908(1)

119896119896minus3

119908(1)

119896119896minus2

119908(1)

119896119896minus1

119908(1)

119896119896

119908(1)

119896119896+1

119908(1)

119896119896+2

119908(1)

119896119873+1

]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]


=

[[[[[[[[[[[[[[[[[[[[[[[[[[

[

0

0

minus4

ℎ

minus12

ℎ

12

ℎ

4

ℎ0

0

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

(34)




[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11 1

2 23 23

6 18

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(1)

119873minus1

119908(1)

1198730

119908(1)

119873119873minus3

119908(1)

119873119873minus2

119908(1)

119873119873minus1

119908(1)

119873119873

119908(1)

119873119873+1

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[

[

0

0

0

minus4

ℎ

minus12

ℎ

53

2ℎ

17

ℎ

]]]]]]]]]]]]]]]]]]]]]

]

(35)


1198892119876119898

1198891199092(119909119894) =

119898+2

sum

119895=119898minus1

119908(2)

119894119895119876119898(119909119895)

119898 = minus1 0 119873 + 1 119894 = 1 2 119873

(36)

Table 1 1198712and 119871


V = 0001 (SFEM)


119871infin

times 103

2 00054945 002820493 00082404 004224214 00109858 005622805 00137296 007015666 00164729 008404277 00192154 009789758 00219573 011169349 00246985 0125446610 00274379 01391304

where 119908(2)




119898119898 = minus1 0 119873 + 1 are used


[[[[[[[[[[[[[[[[[[[[[

[

1 11 11 1

1 11 11 1

d d d d

1 11 11 1

1 11 11 1

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(2)

1minus2

119908(2)

1minus1

119908(2)

10

119908(2)

11

119908(2)

12

119908(2)

1119873+2

119908(2)

1119873+3

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[

[

12

ℎ2

minus12

ℎ2

minus12

ℎ2

12

ℎ2

0

0

0

]]]]]]]]]]]]]]]]]]]]]

]

(37)


1198893119876minus1

(1199091)

1198891199093=

1

sum

119895=minus2

119908(1)

11198951198761015840

minus1(119909119895) (38)


1198893119876119873+1

(1199091)

1198891199093=

119873+3

sum

119895=119873

119908(1)

11198951198761015840

119873+1(119909119895) (39)


[[[[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11 1

2 14 8

]]]]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[[[

[

119908(2)

1minus1

119908(2)

10

119908(2)

11

119908(2)

12

119908(2)

13

119908(2)

1119873+1

119908(2)

1119873+2

]]]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[[

[

18

ℎ2

minus12

ℎ2

minus12

ℎ2

12

ℎ2

0

0

0

]]]]]]]]]]]]]]]]]]]]]]

]

(40)




119908(2)

119894119895= 2119908(1)

119894119895(119908(1)

119894119894minus

1

119909119894minus 119909119895

) 119894 = 119895 (41)

After we use (41)

1198601= 119908(2)

1119873+2= 2119908(1)

1119873+2(119908(1)

11minus

1

1199091minus 119909119873+2

)

[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11

2 14

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(2)

1minus1

119908(2)

10

119908(2)

11

119908(2)

12

119908(2)

13

119908(2)

1119873

119908(2)

1119873+1

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[

[

18

ℎ2

minus12

ℎ2

minus12

ℎ2

12

ℎ2

0

minus1198601

minus81198601

]]]]]]]]]]]]]]]]]]]]]

]

(42)



119896 2 le 119896 le


[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11

2 14

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

119908(2)

119896minus1

119908(2)

119896119896minus3

119908(2)

119896119896minus2

119908(2)

119896119896minus1

119908(2)

119896119896

119908(2)

119896119896+1

119908(2)

119896119896+2

119908(2)

119896119873minus1

119908(2)

119896119873

119908(2)

119896119873+1

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]


Table 2 1198712and 119871


V = 0005 (SFEM)


119871infin

times 103

2 00246966 008456893 00370384 012662224 00493707 016843625 00616997 021013196 00740253 025163927 00863444 029301788 00986573 033419229 01109636 0375245710 01232629 04160477

Table 3 1198712and 119871


V = 001 (SFEM)


119871infin

times 103

2 00978574 028062433 01467089 041859814 01955072 055502865 02442506 068987136 02929396 082386297 03415703 095666888 03901436 108812899 04386580 1218223110 04871136 13469237

=

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

0

12

ℎ2

minus12

ℎ2

minus12

ℎ2

12

ℎ2

0

0

minus119860119896

minus8119860119896

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

(43)

where119860119896equals119860

119896= 119908(2)

119896119873+2= 2119908(1)

119896119873+2(119908(1)

119896119896minus1(119909

119896minus119909119873+2


119873with the


119873119895 119895 =

Table 4 1198712and 119871


001 (SFEM)


119871infin

times 103

2 00973818 028025263 01460008 041848724 01945704 055541215 02430873 069100626 02915506 082523127 03399602 095804338 03883156 108944139 04366131 1219411110 04848547 13479880


[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11

2 14

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(2)

119873minus1

119908(2)

1198730

119908(2)

119873119873minus3

119908(2)

119873119873minus2

119908(2)

119873119873minus1

119908(2)

119873119873

119908(2)

119873119873+1

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[

[

0

0

0

12

ℎ2

minus12

ℎ2

minus12

ℎ2minus 119860119873

18

ℎ2minus 8119860119873

]]]]]]]]]]]]]]]]]]]]]

]

(44)

where 119860119873equals 119860

119873= 119908(2)

119873119873+2= 2119908(1)

119873119873+2(119908(1)

119873119873minus 1(119909

119873minus

119909119873+2

))




Table 5 1198712and 119871



1198712times 103

119871infin

times 103

1198712times 103

119871infin

times 103

2 07955855586 13795978925 07904296620 17030921188

3 06690533313 11943543646 06551928290 11832698216

4 05250528343 09764154381 05576794264 09964523368

5 04048512821 07849457015 05105617536 08561342445

6 03452210304 06374950443 05167229575 07610530060

7 03638648688 06705419608 05677438614 10654548090

8 04337013450 09863405006 06427542266 13581113635

9 05197862999 12551335234 07236430257 16048306653

10 06042925888 14747885309 08002564201 18023938553

Table 6 1198712and 119871


at 0 le 119909 le 13


119871infin

times 103

2 07607107169 137041821953 06480181273 118549841904 05604986926 100524764525 04927784148 086540324196 04359075842 075315510237 03885737191 066013265128 03520185942 058333349709 03282544303 0520132366310 03187570280 04691560472


2and 119871

infin respectively

1198712=10038171003817100381710038171003817119880


100381710038171003817100381710038172≃ radicℎ

119873

sum

119869=1

100381610038161003816100381610038161003816119880

exact119895

minus (119880119873)119895

100381610038161003816100381610038161003816

2

119871infin

=10038171003817100381710038171003817119880


10038171003817100381710038171003817infin≃ max119895

100381610038161003816100381610038161003816119880

exact119895

minus (119880119873)119895

100381610038161003816100381610038161003816

119895 = 1 2 119873 minus 1

(45)


119880 (119909 119905) =(119909119905)

1 + (radic1199051198880) exp (11990924V119905)

(46)





119909(0 119905) = 0 and 119880(1 119905) = 119880

119909(1 119905) = 0


0000

0005

0010

0015

0020

0025

U(xt)

00 02 04 06 08 10

x

t = 1

t = 2

t = 4

t = 6t = 8



2and 119871

infinare




ROC asympln (119864 (119873

2) 119864 (119873

1))

ln (11987311198732)

(47)


Table 7 1198712and 119871



119871infin

times 103

1198712times 103

119871infin

times 103

2 01370706949 04453892504 01835491190 081852111123 01168507335 03842839811 01441424335 052348333464 01019761971 03258391192 01144110783 035635372075 00920706001 02816616769 00947865272 025497900586 00849484881 02484289381 00814174677 021348478357 00794570772 02225471690 00718977757 018800484328 00750035859 02019577762 00648368942 016826017709 00712618898 01851510002 00594114970 0152407496610 00680382860 01711033543 00551151456 01394312127


119873 1198712times 103 ROC(119871

2) 119871


infin)

11 043 mdash 098 mdash21 035 031 088 01631 022 119 052 13541 017 092 039 10251 014 088 030 12081 010 072 019 098

0000

0004

0006

0008

0010

0012

0002

U(xt)

00 02 04 06 08 10

x

t = 1

t = 2

t = 4

t = 6

t = 8


0 01 and V = 0001



infin







0000

0005

0010

0015

0020

0030

0025

0035

U(xt)

00 02 04 06 08 10

x

t = 1

t = 2

t = 4

t = 6

t = 8



2and 119871



2

and 119871infin





119871infin

times 103

1198712times 103

119871infin

times 103

119905 = 2 119905 = 2 119905 = 10 119905 = 10

V = 0005 Δ119905 = 0001 ℎ = 0005

SFEM 002469 008456 012326 041604[14] 025786 072264 018735 030006[17] SBCM1 022890 058623 014042 023019[17] SBCM2 023397 058424 013747 022626

V = 0001 Δ119905 = 001 ℎ = 0005


V = 001 Δ119905 = 001 and ℎ = 0005

SFEM 009785 028062 048711 134692[14] 052308 121698 064007 128124[17] SBCM1 038489 082934 054826 128127[17] SBCM2 039078 082734 054612 128127[18] 037552 081766 019391 023074

V = 001 Δ119905 = 001 and ℎ = 002



2and 119871

infinerror norms







119871infin


4 Conclusion


2and 119871



000

001

002

003

004

005

U

00 02 04 06 08 10

x

t = 1

t = 5

t = 3 t = 9

t = 7




000

001

002

003

004

005

U

00 02 04 06 08 141210

x

t = 1

t = 5

t = 3 t = 9

t = 7


13

0000

0002

0004

0006

0008

0014

0012

0010

U

00 02 04 06 08 10

x

t = 1

t = 5

t = 3 t = 9

t = 7


1





00 02 04 06 08 10

x

00000

00005

00010

00015

00020

Error


000000

000005

000010

000015

000020

00 02 04 06 08 10

x

Error


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











[[[[[[[[[[[[[[[[[[[[[

[

1 11 11 1

1 11 11 1

d d d d

1 11 11 1

1 11 11 1

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(1)

1minus2

119908(1)

1minus1

119908(1)

10

119908(1)

11

119908(1)

12

119908(1)

1119873+2

119908(1)

1119873+3

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[[

[

minus4

ℎ

minus12

ℎ

12

ℎ

4

ℎ

0

0

0

]]]]]]]]]]]]]]]]]]]]]]

]

(31)




119889(2)

119876minus1

(1199091)

119889119909(2)=

1

sum

119895=minus2

119908(1)

11198951198761015840

minus1(119909119895)

119889(2)

119876119873+1

(1199091)

119889119909(2)=

119873+3

sum

119895=119873

119908(1)

11198951198761015840

119873+1(119909119895)

119889(3)

119876119873+1

(1199091)

120597119909(3)=

119873+3

sum

119895=119873

119908(1)

111989511987610158401015840

119873+1(119909119895)

(32)


[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11 1

2 23 23

6 18

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(1)

1minus1

119908(1)

10

119908(1)

11

119908(1)

12

119908(1)

13

119908(1)

1119873

119908(1)

1119873+1

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[

[

minus7

ℎ

minus12

ℎ

12

ℎ

4

ℎ0

0

0

]]]]]]]]]]]]]]]]]]]]]

]

(33)


119896119895 119895 =



[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11 1

2 23 23

6 18

]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

119908(1)

119896minus1

119908(1)

119896119896minus3

119908(1)

119896119896minus2

119908(1)

119896119896minus1

119908(1)

119896119896

119908(1)

119896119896+1

119908(1)

119896119896+2

119908(1)

119896119873+1

]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]


=

[[[[[[[[[[[[[[[[[[[[[[[[[[

[

0

0

minus4

ℎ

minus12

ℎ

12

ℎ

4

ℎ0

0

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

(34)




[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11 1

2 23 23

6 18

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(1)

119873minus1

119908(1)

1198730

119908(1)

119873119873minus3

119908(1)

119873119873minus2

119908(1)

119873119873minus1

119908(1)

119873119873

119908(1)

119873119873+1

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[

[

0

0

0

minus4

ℎ

minus12

ℎ

53

2ℎ

17

ℎ

]]]]]]]]]]]]]]]]]]]]]

]

(35)


1198892119876119898

1198891199092(119909119894) =

119898+2

sum

119895=119898minus1

119908(2)

119894119895119876119898(119909119895)

119898 = minus1 0 119873 + 1 119894 = 1 2 119873

(36)

Table 1 1198712and 119871


V = 0001 (SFEM)


119871infin

times 103

2 00054945 002820493 00082404 004224214 00109858 005622805 00137296 007015666 00164729 008404277 00192154 009789758 00219573 011169349 00246985 0125446610 00274379 01391304

where 119908(2)




119898119898 = minus1 0 119873 + 1 are used


[[[[[[[[[[[[[[[[[[[[[

[

1 11 11 1

1 11 11 1

d d d d

1 11 11 1

1 11 11 1

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(2)

1minus2

119908(2)

1minus1

119908(2)

10

119908(2)

11

119908(2)

12

119908(2)

1119873+2

119908(2)

1119873+3

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[

[

12

ℎ2

minus12

ℎ2

minus12

ℎ2

12

ℎ2

0

0

0

]]]]]]]]]]]]]]]]]]]]]

]

(37)


1198893119876minus1

(1199091)

1198891199093=

1

sum

119895=minus2

119908(1)

11198951198761015840

minus1(119909119895) (38)


1198893119876119873+1

(1199091)

1198891199093=

119873+3

sum

119895=119873

119908(1)

11198951198761015840

119873+1(119909119895) (39)


[[[[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11 1

2 14 8

]]]]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[[[

[

119908(2)

1minus1

119908(2)

10

119908(2)

11

119908(2)

12

119908(2)

13

119908(2)

1119873+1

119908(2)

1119873+2

]]]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[[

[

18

ℎ2

minus12

ℎ2

minus12

ℎ2

12

ℎ2

0

0

0

]]]]]]]]]]]]]]]]]]]]]]

]

(40)




119908(2)

119894119895= 2119908(1)

119894119895(119908(1)

119894119894minus

1

119909119894minus 119909119895

) 119894 = 119895 (41)

After we use (41)

1198601= 119908(2)

1119873+2= 2119908(1)

1119873+2(119908(1)

11minus

1

1199091minus 119909119873+2

)

[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11

2 14

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(2)

1minus1

119908(2)

10

119908(2)

11

119908(2)

12

119908(2)

13

119908(2)

1119873

119908(2)

1119873+1

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[

[

18

ℎ2

minus12

ℎ2

minus12

ℎ2

12

ℎ2

0

minus1198601

minus81198601

]]]]]]]]]]]]]]]]]]]]]

]

(42)



119896 2 le 119896 le


[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11

2 14

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

119908(2)

119896minus1

119908(2)

119896119896minus3

119908(2)

119896119896minus2

119908(2)

119896119896minus1

119908(2)

119896119896

119908(2)

119896119896+1

119908(2)

119896119896+2

119908(2)

119896119873minus1

119908(2)

119896119873

119908(2)

119896119873+1

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]


Table 2 1198712and 119871


V = 0005 (SFEM)


119871infin

times 103

2 00246966 008456893 00370384 012662224 00493707 016843625 00616997 021013196 00740253 025163927 00863444 029301788 00986573 033419229 01109636 0375245710 01232629 04160477

Table 3 1198712and 119871


V = 001 (SFEM)


119871infin

times 103

2 00978574 028062433 01467089 041859814 01955072 055502865 02442506 068987136 02929396 082386297 03415703 095666888 03901436 108812899 04386580 1218223110 04871136 13469237

=

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

0

12

ℎ2

minus12

ℎ2

minus12

ℎ2

12

ℎ2

0

0

minus119860119896

minus8119860119896

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

(43)

where119860119896equals119860

119896= 119908(2)

119896119873+2= 2119908(1)

119896119873+2(119908(1)

119896119896minus1(119909

119896minus119909119873+2


119873with the


119873119895 119895 =

Table 4 1198712and 119871


001 (SFEM)


119871infin

times 103

2 00973818 028025263 01460008 041848724 01945704 055541215 02430873 069100626 02915506 082523127 03399602 095804338 03883156 108944139 04366131 1219411110 04848547 13479880


[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11

2 14

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(2)

119873minus1

119908(2)

1198730

119908(2)

119873119873minus3

119908(2)

119873119873minus2

119908(2)

119873119873minus1

119908(2)

119873119873

119908(2)

119873119873+1

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[

[

0

0

0

12

ℎ2

minus12

ℎ2

minus12

ℎ2minus 119860119873

18

ℎ2minus 8119860119873

]]]]]]]]]]]]]]]]]]]]]

]

(44)

where 119860119873equals 119860

119873= 119908(2)

119873119873+2= 2119908(1)

119873119873+2(119908(1)

119873119873minus 1(119909

119873minus

119909119873+2

))




Table 5 1198712and 119871



1198712times 103

119871infin

times 103

1198712times 103

119871infin

times 103

2 07955855586 13795978925 07904296620 17030921188

3 06690533313 11943543646 06551928290 11832698216

4 05250528343 09764154381 05576794264 09964523368

5 04048512821 07849457015 05105617536 08561342445

6 03452210304 06374950443 05167229575 07610530060

7 03638648688 06705419608 05677438614 10654548090

8 04337013450 09863405006 06427542266 13581113635

9 05197862999 12551335234 07236430257 16048306653

10 06042925888 14747885309 08002564201 18023938553

Table 6 1198712and 119871


at 0 le 119909 le 13


119871infin

times 103

2 07607107169 137041821953 06480181273 118549841904 05604986926 100524764525 04927784148 086540324196 04359075842 075315510237 03885737191 066013265128 03520185942 058333349709 03282544303 0520132366310 03187570280 04691560472


2and 119871

infin respectively

1198712=10038171003817100381710038171003817119880


100381710038171003817100381710038172≃ radicℎ

119873

sum

119869=1

100381610038161003816100381610038161003816119880

exact119895

minus (119880119873)119895

100381610038161003816100381610038161003816

2

119871infin

=10038171003817100381710038171003817119880


10038171003817100381710038171003817infin≃ max119895

100381610038161003816100381610038161003816119880

exact119895

minus (119880119873)119895

100381610038161003816100381610038161003816

119895 = 1 2 119873 minus 1

(45)


119880 (119909 119905) =(119909119905)

1 + (radic1199051198880) exp (11990924V119905)

(46)





119909(0 119905) = 0 and 119880(1 119905) = 119880

119909(1 119905) = 0


0000

0005

0010

0015

0020

0025

U(xt)

00 02 04 06 08 10

x

t = 1

t = 2

t = 4

t = 6t = 8



2and 119871

infinare




ROC asympln (119864 (119873

2) 119864 (119873

1))

ln (11987311198732)

(47)


Table 7 1198712and 119871



119871infin

times 103

1198712times 103

119871infin

times 103

2 01370706949 04453892504 01835491190 081852111123 01168507335 03842839811 01441424335 052348333464 01019761971 03258391192 01144110783 035635372075 00920706001 02816616769 00947865272 025497900586 00849484881 02484289381 00814174677 021348478357 00794570772 02225471690 00718977757 018800484328 00750035859 02019577762 00648368942 016826017709 00712618898 01851510002 00594114970 0152407496610 00680382860 01711033543 00551151456 01394312127


119873 1198712times 103 ROC(119871

2) 119871


infin)

11 043 mdash 098 mdash21 035 031 088 01631 022 119 052 13541 017 092 039 10251 014 088 030 12081 010 072 019 098

0000

0004

0006

0008

0010

0012

0002

U(xt)

00 02 04 06 08 10

x

t = 1

t = 2

t = 4

t = 6

t = 8


0 01 and V = 0001



infin







0000

0005

0010

0015

0020

0030

0025

0035

U(xt)

00 02 04 06 08 10

x

t = 1

t = 2

t = 4

t = 6

t = 8



2and 119871



2

and 119871infin





119871infin

times 103

1198712times 103

119871infin

times 103

119905 = 2 119905 = 2 119905 = 10 119905 = 10

V = 0005 Δ119905 = 0001 ℎ = 0005

SFEM 002469 008456 012326 041604[14] 025786 072264 018735 030006[17] SBCM1 022890 058623 014042 023019[17] SBCM2 023397 058424 013747 022626

V = 0001 Δ119905 = 001 ℎ = 0005


V = 001 Δ119905 = 001 and ℎ = 0005

SFEM 009785 028062 048711 134692[14] 052308 121698 064007 128124[17] SBCM1 038489 082934 054826 128127[17] SBCM2 039078 082734 054612 128127[18] 037552 081766 019391 023074

V = 001 Δ119905 = 001 and ℎ = 002



2and 119871

infinerror norms







119871infin


4 Conclusion


2and 119871



000

001

002

003

004

005

U

00 02 04 06 08 10

x

t = 1

t = 5

t = 3 t = 9

t = 7




000

001

002

003

004

005

U

00 02 04 06 08 141210

x

t = 1

t = 5

t = 3 t = 9

t = 7


13

0000

0002

0004

0006

0008

0014

0012

0010

U

00 02 04 06 08 10

x

t = 1

t = 5

t = 3 t = 9

t = 7


1





00 02 04 06 08 10

x

00000

00005

00010

00015

00020

Error


000000

000005

000010

000015

000020

00 02 04 06 08 10

x

Error


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










=

[[[[[[[[[[[[[[[[[[[[[[[[[[

[

0

0

minus4

ℎ

minus12

ℎ

12

ℎ

4

ℎ0

0

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

(34)




[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11 1

2 23 23

6 18

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(1)

119873minus1

119908(1)

1198730

119908(1)

119873119873minus3

119908(1)

119873119873minus2

119908(1)

119873119873minus1

119908(1)

119873119873

119908(1)

119873119873+1

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[

[

0

0

0

minus4

ℎ

minus12

ℎ

53

2ℎ

17

ℎ

]]]]]]]]]]]]]]]]]]]]]

]

(35)


1198892119876119898

1198891199092(119909119894) =

119898+2

sum

119895=119898minus1

119908(2)

119894119895119876119898(119909119895)

119898 = minus1 0 119873 + 1 119894 = 1 2 119873

(36)

Table 1 1198712and 119871


V = 0001 (SFEM)


119871infin

times 103

2 00054945 002820493 00082404 004224214 00109858 005622805 00137296 007015666 00164729 008404277 00192154 009789758 00219573 011169349 00246985 0125446610 00274379 01391304

where 119908(2)




119898119898 = minus1 0 119873 + 1 are used


[[[[[[[[[[[[[[[[[[[[[

[

1 11 11 1

1 11 11 1

d d d d

1 11 11 1

1 11 11 1

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(2)

1minus2

119908(2)

1minus1

119908(2)

10

119908(2)

11

119908(2)

12

119908(2)

1119873+2

119908(2)

1119873+3

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[

[

12

ℎ2

minus12

ℎ2

minus12

ℎ2

12

ℎ2

0

0

0

]]]]]]]]]]]]]]]]]]]]]

]

(37)


1198893119876minus1

(1199091)

1198891199093=

1

sum

119895=minus2

119908(1)

11198951198761015840

minus1(119909119895) (38)


1198893119876119873+1

(1199091)

1198891199093=

119873+3

sum

119895=119873

119908(1)

11198951198761015840

119873+1(119909119895) (39)


[[[[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11 1

2 14 8

]]]]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[[[

[

119908(2)

1minus1

119908(2)

10

119908(2)

11

119908(2)

12

119908(2)

13

119908(2)

1119873+1

119908(2)

1119873+2

]]]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[[

[

18

ℎ2

minus12

ℎ2

minus12

ℎ2

12

ℎ2

0

0

0

]]]]]]]]]]]]]]]]]]]]]]

]

(40)




119908(2)

119894119895= 2119908(1)

119894119895(119908(1)

119894119894minus

1

119909119894minus 119909119895

) 119894 = 119895 (41)

After we use (41)

1198601= 119908(2)

1119873+2= 2119908(1)

1119873+2(119908(1)

11minus

1

1199091minus 119909119873+2

)

[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11

2 14

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(2)

1minus1

119908(2)

10

119908(2)

11

119908(2)

12

119908(2)

13

119908(2)

1119873

119908(2)

1119873+1

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[

[

18

ℎ2

minus12

ℎ2

minus12

ℎ2

12

ℎ2

0

minus1198601

minus81198601

]]]]]]]]]]]]]]]]]]]]]

]

(42)



119896 2 le 119896 le


[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11

2 14

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

119908(2)

119896minus1

119908(2)

119896119896minus3

119908(2)

119896119896minus2

119908(2)

119896119896minus1

119908(2)

119896119896

119908(2)

119896119896+1

119908(2)

119896119896+2

119908(2)

119896119873minus1

119908(2)

119896119873

119908(2)

119896119873+1

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]


Table 2 1198712and 119871


V = 0005 (SFEM)


119871infin

times 103

2 00246966 008456893 00370384 012662224 00493707 016843625 00616997 021013196 00740253 025163927 00863444 029301788 00986573 033419229 01109636 0375245710 01232629 04160477

Table 3 1198712and 119871


V = 001 (SFEM)


119871infin

times 103

2 00978574 028062433 01467089 041859814 01955072 055502865 02442506 068987136 02929396 082386297 03415703 095666888 03901436 108812899 04386580 1218223110 04871136 13469237

=

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

0

12

ℎ2

minus12

ℎ2

minus12

ℎ2

12

ℎ2

0

0

minus119860119896

minus8119860119896

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

(43)

where119860119896equals119860

119896= 119908(2)

119896119873+2= 2119908(1)

119896119873+2(119908(1)

119896119896minus1(119909

119896minus119909119873+2


119873with the


119873119895 119895 =

Table 4 1198712and 119871


001 (SFEM)


119871infin

times 103

2 00973818 028025263 01460008 041848724 01945704 055541215 02430873 069100626 02915506 082523127 03399602 095804338 03883156 108944139 04366131 1219411110 04848547 13479880


[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11

2 14

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(2)

119873minus1

119908(2)

1198730

119908(2)

119873119873minus3

119908(2)

119873119873minus2

119908(2)

119873119873minus1

119908(2)

119873119873

119908(2)

119873119873+1

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[

[

0

0

0

12

ℎ2

minus12

ℎ2

minus12

ℎ2minus 119860119873

18

ℎ2minus 8119860119873

]]]]]]]]]]]]]]]]]]]]]

]

(44)

where 119860119873equals 119860

119873= 119908(2)

119873119873+2= 2119908(1)

119873119873+2(119908(1)

119873119873minus 1(119909

119873minus

119909119873+2

))




Table 5 1198712and 119871



1198712times 103

119871infin

times 103

1198712times 103

119871infin

times 103

2 07955855586 13795978925 07904296620 17030921188

3 06690533313 11943543646 06551928290 11832698216

4 05250528343 09764154381 05576794264 09964523368

5 04048512821 07849457015 05105617536 08561342445

6 03452210304 06374950443 05167229575 07610530060

7 03638648688 06705419608 05677438614 10654548090

8 04337013450 09863405006 06427542266 13581113635

9 05197862999 12551335234 07236430257 16048306653

10 06042925888 14747885309 08002564201 18023938553

Table 6 1198712and 119871


at 0 le 119909 le 13


119871infin

times 103

2 07607107169 137041821953 06480181273 118549841904 05604986926 100524764525 04927784148 086540324196 04359075842 075315510237 03885737191 066013265128 03520185942 058333349709 03282544303 0520132366310 03187570280 04691560472


2and 119871

infin respectively

1198712=10038171003817100381710038171003817119880


100381710038171003817100381710038172≃ radicℎ

119873

sum

119869=1

100381610038161003816100381610038161003816119880

exact119895

minus (119880119873)119895

100381610038161003816100381610038161003816

2

119871infin

=10038171003817100381710038171003817119880


10038171003817100381710038171003817infin≃ max119895

100381610038161003816100381610038161003816119880

exact119895

minus (119880119873)119895

100381610038161003816100381610038161003816

119895 = 1 2 119873 minus 1

(45)


119880 (119909 119905) =(119909119905)

1 + (radic1199051198880) exp (11990924V119905)

(46)





119909(0 119905) = 0 and 119880(1 119905) = 119880

119909(1 119905) = 0


0000

0005

0010

0015

0020

0025

U(xt)

00 02 04 06 08 10

x

t = 1

t = 2

t = 4

t = 6t = 8



2and 119871

infinare




ROC asympln (119864 (119873

2) 119864 (119873

1))

ln (11987311198732)

(47)


Table 7 1198712and 119871



119871infin

times 103

1198712times 103

119871infin

times 103

2 01370706949 04453892504 01835491190 081852111123 01168507335 03842839811 01441424335 052348333464 01019761971 03258391192 01144110783 035635372075 00920706001 02816616769 00947865272 025497900586 00849484881 02484289381 00814174677 021348478357 00794570772 02225471690 00718977757 018800484328 00750035859 02019577762 00648368942 016826017709 00712618898 01851510002 00594114970 0152407496610 00680382860 01711033543 00551151456 01394312127


119873 1198712times 103 ROC(119871

2) 119871


infin)

11 043 mdash 098 mdash21 035 031 088 01631 022 119 052 13541 017 092 039 10251 014 088 030 12081 010 072 019 098

0000

0004

0006

0008

0010

0012

0002

U(xt)

00 02 04 06 08 10

x

t = 1

t = 2

t = 4

t = 6

t = 8


0 01 and V = 0001



infin







0000

0005

0010

0015

0020

0030

0025

0035

U(xt)

00 02 04 06 08 10

x

t = 1

t = 2

t = 4

t = 6

t = 8



2and 119871



2

and 119871infin





119871infin

times 103

1198712times 103

119871infin

times 103

119905 = 2 119905 = 2 119905 = 10 119905 = 10

V = 0005 Δ119905 = 0001 ℎ = 0005

SFEM 002469 008456 012326 041604[14] 025786 072264 018735 030006[17] SBCM1 022890 058623 014042 023019[17] SBCM2 023397 058424 013747 022626

V = 0001 Δ119905 = 001 ℎ = 0005


V = 001 Δ119905 = 001 and ℎ = 0005

SFEM 009785 028062 048711 134692[14] 052308 121698 064007 128124[17] SBCM1 038489 082934 054826 128127[17] SBCM2 039078 082734 054612 128127[18] 037552 081766 019391 023074

V = 001 Δ119905 = 001 and ℎ = 002



2and 119871

infinerror norms







119871infin


4 Conclusion


2and 119871



000

001

002

003

004

005

U

00 02 04 06 08 10

x

t = 1

t = 5

t = 3 t = 9

t = 7




000

001

002

003

004

005

U

00 02 04 06 08 141210

x

t = 1

t = 5

t = 3 t = 9

t = 7


13

0000

0002

0004

0006

0008

0014

0012

0010

U

00 02 04 06 08 10

x

t = 1

t = 5

t = 3 t = 9

t = 7


1





00 02 04 06 08 10

x

00000

00005

00010

00015

00020

Error


000000

000005

000010

000015

000020

00 02 04 06 08 10

x

Error


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1198893119876119873+1

(1199091)

1198891199093=

119873+3

sum

119895=119873

119908(1)

11198951198761015840

119873+1(119909119895) (39)


[[[[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11 1

2 14 8

]]]]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[[[

[

119908(2)

1minus1

119908(2)

10

119908(2)

11

119908(2)

12

119908(2)

13

119908(2)

1119873+1

119908(2)

1119873+2

]]]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[[

[

18

ℎ2

minus12

ℎ2

minus12

ℎ2

12

ℎ2

0

0

0

]]]]]]]]]]]]]]]]]]]]]]

]

(40)




119908(2)

119894119895= 2119908(1)

119894119895(119908(1)

119894119894minus

1

119909119894minus 119909119895

) 119894 = 119895 (41)

After we use (41)

1198601= 119908(2)

1119873+2= 2119908(1)

1119873+2(119908(1)

11minus

1

1199091minus 119909119873+2

)

[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11

2 14

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(2)

1minus1

119908(2)

10

119908(2)

11

119908(2)

12

119908(2)

13

119908(2)

1119873

119908(2)

1119873+1

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[

[

18

ℎ2

minus12

ℎ2

minus12

ℎ2

12

ℎ2

0

minus1198601

minus81198601

]]]]]]]]]]]]]]]]]]]]]

]

(42)



119896 2 le 119896 le


[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11

2 14

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

119908(2)

119896minus1

119908(2)

119896119896minus3

119908(2)

119896119896minus2

119908(2)

119896119896minus1

119908(2)

119896119896

119908(2)

119896119896+1

119908(2)

119896119896+2

119908(2)

119896119873minus1

119908(2)

119896119873

119908(2)

119896119873+1

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]


Table 2 1198712and 119871


V = 0005 (SFEM)


119871infin

times 103

2 00246966 008456893 00370384 012662224 00493707 016843625 00616997 021013196 00740253 025163927 00863444 029301788 00986573 033419229 01109636 0375245710 01232629 04160477

Table 3 1198712and 119871


V = 001 (SFEM)


119871infin

times 103

2 00978574 028062433 01467089 041859814 01955072 055502865 02442506 068987136 02929396 082386297 03415703 095666888 03901436 108812899 04386580 1218223110 04871136 13469237

=

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

0

12

ℎ2

minus12

ℎ2

minus12

ℎ2

12

ℎ2

0

0

minus119860119896

minus8119860119896

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

(43)

where119860119896equals119860

119896= 119908(2)

119896119873+2= 2119908(1)

119896119873+2(119908(1)

119896119896minus1(119909

119896minus119909119873+2


119873with the


119873119895 119895 =

Table 4 1198712and 119871


001 (SFEM)


119871infin

times 103

2 00973818 028025263 01460008 041848724 01945704 055541215 02430873 069100626 02915506 082523127 03399602 095804338 03883156 108944139 04366131 1219411110 04848547 13479880


[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11

2 14

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(2)

119873minus1

119908(2)

1198730

119908(2)

119873119873minus3

119908(2)

119873119873minus2

119908(2)

119873119873minus1

119908(2)

119873119873

119908(2)

119873119873+1

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[

[

0

0

0

12

ℎ2

minus12

ℎ2

minus12

ℎ2minus 119860119873

18

ℎ2minus 8119860119873

]]]]]]]]]]]]]]]]]]]]]

]

(44)

where 119860119873equals 119860

119873= 119908(2)

119873119873+2= 2119908(1)

119873119873+2(119908(1)

119873119873minus 1(119909

119873minus

119909119873+2

))




Table 5 1198712and 119871



1198712times 103

119871infin

times 103

1198712times 103

119871infin

times 103

2 07955855586 13795978925 07904296620 17030921188

3 06690533313 11943543646 06551928290 11832698216

4 05250528343 09764154381 05576794264 09964523368

5 04048512821 07849457015 05105617536 08561342445

6 03452210304 06374950443 05167229575 07610530060

7 03638648688 06705419608 05677438614 10654548090

8 04337013450 09863405006 06427542266 13581113635

9 05197862999 12551335234 07236430257 16048306653

10 06042925888 14747885309 08002564201 18023938553

Table 6 1198712and 119871


at 0 le 119909 le 13


119871infin

times 103

2 07607107169 137041821953 06480181273 118549841904 05604986926 100524764525 04927784148 086540324196 04359075842 075315510237 03885737191 066013265128 03520185942 058333349709 03282544303 0520132366310 03187570280 04691560472


2and 119871

infin respectively

1198712=10038171003817100381710038171003817119880


100381710038171003817100381710038172≃ radicℎ

119873

sum

119869=1

100381610038161003816100381610038161003816119880

exact119895

minus (119880119873)119895

100381610038161003816100381610038161003816

2

119871infin

=10038171003817100381710038171003817119880


10038171003817100381710038171003817infin≃ max119895

100381610038161003816100381610038161003816119880

exact119895

minus (119880119873)119895

100381610038161003816100381610038161003816

119895 = 1 2 119873 minus 1

(45)


119880 (119909 119905) =(119909119905)

1 + (radic1199051198880) exp (11990924V119905)

(46)





119909(0 119905) = 0 and 119880(1 119905) = 119880

119909(1 119905) = 0


0000

0005

0010

0015

0020

0025

U(xt)

00 02 04 06 08 10

x

t = 1

t = 2

t = 4

t = 6t = 8



2and 119871

infinare




ROC asympln (119864 (119873

2) 119864 (119873

1))

ln (11987311198732)

(47)


Table 7 1198712and 119871



119871infin

times 103

1198712times 103

119871infin

times 103

2 01370706949 04453892504 01835491190 081852111123 01168507335 03842839811 01441424335 052348333464 01019761971 03258391192 01144110783 035635372075 00920706001 02816616769 00947865272 025497900586 00849484881 02484289381 00814174677 021348478357 00794570772 02225471690 00718977757 018800484328 00750035859 02019577762 00648368942 016826017709 00712618898 01851510002 00594114970 0152407496610 00680382860 01711033543 00551151456 01394312127


119873 1198712times 103 ROC(119871

2) 119871


infin)

11 043 mdash 098 mdash21 035 031 088 01631 022 119 052 13541 017 092 039 10251 014 088 030 12081 010 072 019 098

0000

0004

0006

0008

0010

0012

0002

U(xt)

00 02 04 06 08 10

x

t = 1

t = 2

t = 4

t = 6

t = 8


0 01 and V = 0001



infin







0000

0005

0010

0015

0020

0030

0025

0035

U(xt)

00 02 04 06 08 10

x

t = 1

t = 2

t = 4

t = 6

t = 8



2and 119871



2

and 119871infin





119871infin

times 103

1198712times 103

119871infin

times 103

119905 = 2 119905 = 2 119905 = 10 119905 = 10

V = 0005 Δ119905 = 0001 ℎ = 0005

SFEM 002469 008456 012326 041604[14] 025786 072264 018735 030006[17] SBCM1 022890 058623 014042 023019[17] SBCM2 023397 058424 013747 022626

V = 0001 Δ119905 = 001 ℎ = 0005


V = 001 Δ119905 = 001 and ℎ = 0005

SFEM 009785 028062 048711 134692[14] 052308 121698 064007 128124[17] SBCM1 038489 082934 054826 128127[17] SBCM2 039078 082734 054612 128127[18] 037552 081766 019391 023074

V = 001 Δ119905 = 001 and ℎ = 002



2and 119871

infinerror norms







119871infin


4 Conclusion


2and 119871



000

001

002

003

004

005

U

00 02 04 06 08 10

x

t = 1

t = 5

t = 3 t = 9

t = 7




000

001

002

003

004

005

U

00 02 04 06 08 141210

x

t = 1

t = 5

t = 3 t = 9

t = 7


13

0000

0002

0004

0006

0008

0014

0012

0010

U

00 02 04 06 08 10

x

t = 1

t = 5

t = 3 t = 9

t = 7


1





00 02 04 06 08 10

x

00000

00005

00010

00015

00020

Error


000000

000005

000010

000015

000020

00 02 04 06 08 10

x

Error


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 2 1198712and 119871


V = 0005 (SFEM)


119871infin

times 103

2 00246966 008456893 00370384 012662224 00493707 016843625 00616997 021013196 00740253 025163927 00863444 029301788 00986573 033419229 01109636 0375245710 01232629 04160477

Table 3 1198712and 119871


V = 001 (SFEM)


119871infin

times 103

2 00978574 028062433 01467089 041859814 01955072 055502865 02442506 068987136 02929396 082386297 03415703 095666888 03901436 108812899 04386580 1218223110 04871136 13469237

=

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

0

12

ℎ2

minus12

ℎ2

minus12

ℎ2

12

ℎ2

0

0

minus119860119896

minus8119860119896

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

(43)

where119860119896equals119860

119896= 119908(2)

119896119873+2= 2119908(1)

119896119873+2(119908(1)

119896119896minus1(119909

119896minus119909119873+2


119873with the


119873119895 119895 =

Table 4 1198712and 119871


001 (SFEM)


119871infin

times 103

2 00973818 028025263 01460008 041848724 01945704 055541215 02430873 069100626 02915506 082523127 03399602 095804338 03883156 108944139 04366131 1219411110 04848547 13479880


[[[[[[[[[[[[[[[[[[[[[

[

8 14 2

1 11 11 1

d d d d

1 11 11

2 14

]]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[[[[

[

119908(2)

119873minus1

119908(2)

1198730

119908(2)

119873119873minus3

119908(2)

119873119873minus2

119908(2)

119873119873minus1

119908(2)

119873119873

119908(2)

119873119873+1

]]]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[[

[

0

0

0

12

ℎ2

minus12

ℎ2

minus12

ℎ2minus 119860119873

18

ℎ2minus 8119860119873

]]]]]]]]]]]]]]]]]]]]]

]

(44)

where 119860119873equals 119860

119873= 119908(2)

119873119873+2= 2119908(1)

119873119873+2(119908(1)

119873119873minus 1(119909

119873minus

119909119873+2

))




Table 5 1198712and 119871



1198712times 103

119871infin

times 103

1198712times 103

119871infin

times 103

2 07955855586 13795978925 07904296620 17030921188

3 06690533313 11943543646 06551928290 11832698216

4 05250528343 09764154381 05576794264 09964523368

5 04048512821 07849457015 05105617536 08561342445

6 03452210304 06374950443 05167229575 07610530060

7 03638648688 06705419608 05677438614 10654548090

8 04337013450 09863405006 06427542266 13581113635

9 05197862999 12551335234 07236430257 16048306653

10 06042925888 14747885309 08002564201 18023938553

Table 6 1198712and 119871


at 0 le 119909 le 13


119871infin

times 103

2 07607107169 137041821953 06480181273 118549841904 05604986926 100524764525 04927784148 086540324196 04359075842 075315510237 03885737191 066013265128 03520185942 058333349709 03282544303 0520132366310 03187570280 04691560472


2and 119871

infin respectively

1198712=10038171003817100381710038171003817119880


100381710038171003817100381710038172≃ radicℎ

119873

sum

119869=1

100381610038161003816100381610038161003816119880

exact119895

minus (119880119873)119895

100381610038161003816100381610038161003816

2

119871infin

=10038171003817100381710038171003817119880


10038171003817100381710038171003817infin≃ max119895

100381610038161003816100381610038161003816119880

exact119895

minus (119880119873)119895

100381610038161003816100381610038161003816

119895 = 1 2 119873 minus 1

(45)


119880 (119909 119905) =(119909119905)

1 + (radic1199051198880) exp (11990924V119905)

(46)





119909(0 119905) = 0 and 119880(1 119905) = 119880

119909(1 119905) = 0


0000

0005

0010

0015

0020

0025

U(xt)

00 02 04 06 08 10

x

t = 1

t = 2

t = 4

t = 6t = 8



2and 119871

infinare




ROC asympln (119864 (119873

2) 119864 (119873

1))

ln (11987311198732)

(47)


Table 7 1198712and 119871



119871infin

times 103

1198712times 103

119871infin

times 103

2 01370706949 04453892504 01835491190 081852111123 01168507335 03842839811 01441424335 052348333464 01019761971 03258391192 01144110783 035635372075 00920706001 02816616769 00947865272 025497900586 00849484881 02484289381 00814174677 021348478357 00794570772 02225471690 00718977757 018800484328 00750035859 02019577762 00648368942 016826017709 00712618898 01851510002 00594114970 0152407496610 00680382860 01711033543 00551151456 01394312127


119873 1198712times 103 ROC(119871

2) 119871


infin)

11 043 mdash 098 mdash21 035 031 088 01631 022 119 052 13541 017 092 039 10251 014 088 030 12081 010 072 019 098

0000

0004

0006

0008

0010

0012

0002

U(xt)

00 02 04 06 08 10

x

t = 1

t = 2

t = 4

t = 6

t = 8


0 01 and V = 0001



infin







0000

0005

0010

0015

0020

0030

0025

0035

U(xt)

00 02 04 06 08 10

x

t = 1

t = 2

t = 4

t = 6

t = 8



2and 119871



2

and 119871infin





119871infin

times 103

1198712times 103

119871infin

times 103

119905 = 2 119905 = 2 119905 = 10 119905 = 10

V = 0005 Δ119905 = 0001 ℎ = 0005

SFEM 002469 008456 012326 041604[14] 025786 072264 018735 030006[17] SBCM1 022890 058623 014042 023019[17] SBCM2 023397 058424 013747 022626

V = 0001 Δ119905 = 001 ℎ = 0005


V = 001 Δ119905 = 001 and ℎ = 0005

SFEM 009785 028062 048711 134692[14] 052308 121698 064007 128124[17] SBCM1 038489 082934 054826 128127[17] SBCM2 039078 082734 054612 128127[18] 037552 081766 019391 023074

V = 001 Δ119905 = 001 and ℎ = 002



2and 119871

infinerror norms







119871infin


4 Conclusion


2and 119871



000

001

002

003

004

005

U

00 02 04 06 08 10

x

t = 1

t = 5

t = 3 t = 9

t = 7




000

001

002

003

004

005

U

00 02 04 06 08 141210

x

t = 1

t = 5

t = 3 t = 9

t = 7


13

0000

0002

0004

0006

0008

0014

0012

0010

U

00 02 04 06 08 10

x

t = 1

t = 5

t = 3 t = 9

t = 7


1





00 02 04 06 08 10

x

00000

00005

00010

00015

00020

Error


000000

000005

000010

000015

000020

00 02 04 06 08 10

x

Error


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 5 1198712and 119871



1198712times 103

119871infin

times 103

1198712times 103

119871infin

times 103

2 07955855586 13795978925 07904296620 17030921188

3 06690533313 11943543646 06551928290 11832698216

4 05250528343 09764154381 05576794264 09964523368

5 04048512821 07849457015 05105617536 08561342445

6 03452210304 06374950443 05167229575 07610530060

7 03638648688 06705419608 05677438614 10654548090

8 04337013450 09863405006 06427542266 13581113635

9 05197862999 12551335234 07236430257 16048306653

10 06042925888 14747885309 08002564201 18023938553

Table 6 1198712and 119871


at 0 le 119909 le 13


119871infin

times 103

2 07607107169 137041821953 06480181273 118549841904 05604986926 100524764525 04927784148 086540324196 04359075842 075315510237 03885737191 066013265128 03520185942 058333349709 03282544303 0520132366310 03187570280 04691560472


2and 119871

infin respectively

1198712=10038171003817100381710038171003817119880


100381710038171003817100381710038172≃ radicℎ

119873

sum

119869=1

100381610038161003816100381610038161003816119880

exact119895

minus (119880119873)119895

100381610038161003816100381610038161003816

2

119871infin

=10038171003817100381710038171003817119880


10038171003817100381710038171003817infin≃ max119895

100381610038161003816100381610038161003816119880

exact119895

minus (119880119873)119895

100381610038161003816100381610038161003816

119895 = 1 2 119873 minus 1

(45)


119880 (119909 119905) =(119909119905)

1 + (radic1199051198880) exp (11990924V119905)

(46)





119909(0 119905) = 0 and 119880(1 119905) = 119880

119909(1 119905) = 0


0000

0005

0010

0015

0020

0025

U(xt)

00 02 04 06 08 10

x

t = 1

t = 2

t = 4

t = 6t = 8



2and 119871

infinare




ROC asympln (119864 (119873

2) 119864 (119873

1))

ln (11987311198732)

(47)


Table 7 1198712and 119871



119871infin

times 103

1198712times 103

119871infin

times 103

2 01370706949 04453892504 01835491190 081852111123 01168507335 03842839811 01441424335 052348333464 01019761971 03258391192 01144110783 035635372075 00920706001 02816616769 00947865272 025497900586 00849484881 02484289381 00814174677 021348478357 00794570772 02225471690 00718977757 018800484328 00750035859 02019577762 00648368942 016826017709 00712618898 01851510002 00594114970 0152407496610 00680382860 01711033543 00551151456 01394312127


119873 1198712times 103 ROC(119871

2) 119871


infin)

11 043 mdash 098 mdash21 035 031 088 01631 022 119 052 13541 017 092 039 10251 014 088 030 12081 010 072 019 098

0000

0004

0006

0008

0010

0012

0002

U(xt)

00 02 04 06 08 10

x

t = 1

t = 2

t = 4

t = 6

t = 8


0 01 and V = 0001



infin







0000

0005

0010

0015

0020

0030

0025

0035

U(xt)

00 02 04 06 08 10

x

t = 1

t = 2

t = 4

t = 6

t = 8



2and 119871



2

and 119871infin





119871infin

times 103

1198712times 103

119871infin

times 103

119905 = 2 119905 = 2 119905 = 10 119905 = 10

V = 0005 Δ119905 = 0001 ℎ = 0005

SFEM 002469 008456 012326 041604[14] 025786 072264 018735 030006[17] SBCM1 022890 058623 014042 023019[17] SBCM2 023397 058424 013747 022626

V = 0001 Δ119905 = 001 ℎ = 0005


V = 001 Δ119905 = 001 and ℎ = 0005

SFEM 009785 028062 048711 134692[14] 052308 121698 064007 128124[17] SBCM1 038489 082934 054826 128127[17] SBCM2 039078 082734 054612 128127[18] 037552 081766 019391 023074

V = 001 Δ119905 = 001 and ℎ = 002



2and 119871

infinerror norms







119871infin


4 Conclusion


2and 119871



000

001

002

003

004

005

U

00 02 04 06 08 10

x

t = 1

t = 5

t = 3 t = 9

t = 7




000

001

002

003

004

005

U

00 02 04 06 08 141210

x

t = 1

t = 5

t = 3 t = 9

t = 7


13

0000

0002

0004

0006

0008

0014

0012

0010

U

00 02 04 06 08 10

x

t = 1

t = 5

t = 3 t = 9

t = 7


1





00 02 04 06 08 10

x

00000

00005

00010

00015

00020

Error


000000

000005

000010

000015

000020

00 02 04 06 08 10

x

Error


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 7 1198712and 119871



119871infin

times 103

1198712times 103

119871infin

times 103

2 01370706949 04453892504 01835491190 081852111123 01168507335 03842839811 01441424335 052348333464 01019761971 03258391192 01144110783 035635372075 00920706001 02816616769 00947865272 025497900586 00849484881 02484289381 00814174677 021348478357 00794570772 02225471690 00718977757 018800484328 00750035859 02019577762 00648368942 016826017709 00712618898 01851510002 00594114970 0152407496610 00680382860 01711033543 00551151456 01394312127


119873 1198712times 103 ROC(119871

2) 119871


infin)

11 043 mdash 098 mdash21 035 031 088 01631 022 119 052 13541 017 092 039 10251 014 088 030 12081 010 072 019 098

0000

0004

0006

0008

0010

0012

0002

U(xt)

00 02 04 06 08 10

x

t = 1

t = 2

t = 4

t = 6

t = 8


0 01 and V = 0001



infin







0000

0005

0010

0015

0020

0030

0025

0035

U(xt)

00 02 04 06 08 10

x

t = 1

t = 2

t = 4

t = 6

t = 8



2and 119871



2

and 119871infin





119871infin

times 103

1198712times 103

119871infin

times 103

119905 = 2 119905 = 2 119905 = 10 119905 = 10

V = 0005 Δ119905 = 0001 ℎ = 0005

SFEM 002469 008456 012326 041604[14] 025786 072264 018735 030006[17] SBCM1 022890 058623 014042 023019[17] SBCM2 023397 058424 013747 022626

V = 0001 Δ119905 = 001 ℎ = 0005


V = 001 Δ119905 = 001 and ℎ = 0005

SFEM 009785 028062 048711 134692[14] 052308 121698 064007 128124[17] SBCM1 038489 082934 054826 128127[17] SBCM2 039078 082734 054612 128127[18] 037552 081766 019391 023074

V = 001 Δ119905 = 001 and ℎ = 002



2and 119871

infinerror norms







119871infin


4 Conclusion


2and 119871



000

001

002

003

004

005

U

00 02 04 06 08 10

x

t = 1

t = 5

t = 3 t = 9

t = 7




000

001

002

003

004

005

U

00 02 04 06 08 141210

x

t = 1

t = 5

t = 3 t = 9

t = 7


13

0000

0002

0004

0006

0008

0014

0012

0010

U

00 02 04 06 08 10

x

t = 1

t = 5

t = 3 t = 9

t = 7


1





00 02 04 06 08 10

x

00000

00005

00010

00015

00020

Error


000000

000005

000010

000015

000020

00 02 04 06 08 10

x

Error


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119871infin

times 103

1198712times 103

119871infin

times 103

119905 = 2 119905 = 2 119905 = 10 119905 = 10

V = 0005 Δ119905 = 0001 ℎ = 0005

SFEM 002469 008456 012326 041604[14] 025786 072264 018735 030006[17] SBCM1 022890 058623 014042 023019[17] SBCM2 023397 058424 013747 022626

V = 0001 Δ119905 = 001 ℎ = 0005


V = 001 Δ119905 = 001 and ℎ = 0005

SFEM 009785 028062 048711 134692[14] 052308 121698 064007 128124[17] SBCM1 038489 082934 054826 128127[17] SBCM2 039078 082734 054612 128127[18] 037552 081766 019391 023074

V = 001 Δ119905 = 001 and ℎ = 002



2and 119871

infinerror norms







119871infin


4 Conclusion


2and 119871



000

001

002

003

004

005

U

00 02 04 06 08 10

x

t = 1

t = 5

t = 3 t = 9

t = 7




000

001

002

003

004

005

U

00 02 04 06 08 141210

x

t = 1

t = 5

t = 3 t = 9

t = 7


13

0000

0002

0004

0006

0008

0014

0012

0010

U

00 02 04 06 08 10

x

t = 1

t = 5

t = 3 t = 9

t = 7


1





00 02 04 06 08 10

x

00000

00005

00010

00015

00020

Error


000000

000005

000010

000015

000020

00 02 04 06 08 10

x

Error


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










000

001

002

003

004

005

U

00 02 04 06 08 141210

x

t = 1

t = 5

t = 3 t = 9

t = 7


13

0000

0002

0004

0006

0008

0014

0012

0010

U

00 02 04 06 08 10

x

t = 1

t = 5

t = 3 t = 9

t = 7


1





00 02 04 06 08 10

x

00000

00005

00010

00015

00020

Error


000000

000005

000010

000015

000020

00 02 04 06 08 10

x

Error


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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