Research ArticleSolving Operator Equation Based on Expansion Approach

A Aminataei1 S Ahmadi-Asl2 and M Pakbaz2

1 Department of Applied Mathematics Faculty of Mathematics K N Toosi University of Technology PO Box 16315-1618 Tehran Iran2Department of Mathematics Ilam University Ilam Iran

Correspondence should be addressed to A Aminataei ataeikntuacir

Received 19 May 2014 Accepted 21 August 2014 Published 7 September 2014

Academic Editor Yaohang Li

Copyright copy 2014 A Aminataei et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

To date researchers usually use spectral and pseudospectral methods for only numerical approximation of ordinary and partialdifferential equations and also based on polynomial basis But the principal importance of this paper is to develop the expansionapproach based on general basis functions (in particular case polynomial basis) for solving general operator equations wherein theparticular cases of our development are integral equations ordinary differential equations difference equations partial differentialequations and fractional differential equations In other words this paper presents the expansion approach for solving generaloperator equations in the form L119906 + N119906 = 119892(119909) 119909 isin Γ with respect to boundary condition B119906 = 120582 where L N and B arelinear nonlinear and boundary operators respectively related to a suitable Hilbert space Γ is the domain of approximation 120582 isan arbitrary constant and 119892(119909) isin 119871

2(Γ) is an arbitrary function Also the other importance of this paper is to introduce the general

version of pseudospectral method based on general interpolation problem Finally some experiments show the accuracy of ourdevelopment and the error analysis is presented in 119871

2(Γ) norm

1 Introduction

Approximation theory is an important field of mathemat-ics which has pure and applied aspects This field triesto approximate the complicated functions with simplifiedrepresentments Some important branches of this field areinterpolation extrapolation best approximation and soforth Also expansion series have high applications in thisfield (approximation theory) Expansion series try to rep-resent a function using suitable basis functions such asstandard polynomials (nonperiodic case) and trigonometricpolynomials (periodic case) Taylor series and Fourier seriesin classical and nonclassical forms are two high applicableversions of expansions These series are used in the numer-ical solution of ordinary partial differential and integralequations Weighted residual methods are classes of methodswhich use expansion series for solving ordinary partialdifferential and integral equations In these methods bysubstituting a suitable expansion of exact solution of ordinarypartial differential and integral equations in its equation theresiduals are obtained and then the residuals are minimizedin a certain wayThis minimization leads to specific methods

such as Galerkin collocation and Tau formulations Inthis paper using Fourier series and interpolation cases asexpansions we implement the spectral and pseudospectralmethods for solving the general operator equation L119906 +

N119906 = 119892(119909) 119909 isin Γ with respect to boundary conditionB119906 =

120582The remainder of the paper proceeds as follows In Sections11 to 14 we introduce the preliminaries and notations thegeneral interpolation problem spectral and pseudospectralmethods and operator equations In Section 2 we implementspectral method for general operator equation Section 3 isdevoted to the implementation of pseudospectral method forgeneral operator equation with error analysis In Section 4we present some test experiments to show the accuracyand validity of our approach Finally in the last section wemonitor a brief conclusion

11 Preliminaries and Notations First we introduce somenotations which we use in the following

Definition 1 A mapping from a linear space 119883 into 1198771

is called a functional Also the set of all bounded linear

Hindawi Publishing CorporationInternational Journal of Computational MathematicsVolume 2014 Article ID 671965 9 pageshttpdxdoiorg1011552014671965

2 International Journal of Computational Mathematics

functionals on the linear space119883 is itself a linear space calledthe algebraic conjugate or dual-space on119883 and is denoted by119883lowast [1]

Definition 2 A projection is a linear transformation 119875 froma vector space to itself such that 1198752 = 119875 It leaves its imageunchanged Though abstract this definition of projectionformalizes and generalizes the idea of graphical projection [1]

Definition 3 Suppose 119883 is a linear space and 119883lowast is its dual-

space then 1198711 1198712 119871

119899isin 119883lowast are linearly independent if

and only if

12057211198711+ 12057221198712+ sdot sdot sdot + 120572

119899119871119899= 0

997904rArr 1205721= 1205722= sdot sdot sdot = 120572

119899= 0

(1)

Theorem 4 Let 119883 be an 119899-dimensional space and let1206011(119909) 1206012(119909) 120601

119899(119909) isin 119883 be linearly independent Then

1198711 1198712 119871

119899isin 119883lowast are linearly independent if and only if

det (119871119894(120601119895)) =

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198711(1206011) sdot sdot sdot 119871

1(120601119899)

sdot sdot sdot

119871119899(1206011) sdot sdot sdot 119871

119899(120601119899)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (2)

Proof See [1]

Remark 5 Suppose 119910 isin 1198712(Γ) then the operators 119871

1(119910) =

119910 1198712(119910) = int 119910119889119910 (integral operator) 119871

3(119910) = 119910

1015840 (derivativeoperator) and 119871

ℎ(119910(119909)) = 119910(119909 minus ℎ) (shift operator) ℎ isin 119877

are independent [1]

12 General Interpolation Problem Thegeneral interpolationproblem is stated as follows Let119883 be a space of dimension 119899and let 119871

1 1198712 119871

119899be given linear functionals on119883

lowast Findan 120601(119909) isin 119883 such that

119871119894(120601 (119909)) = 119908

119894 119894 = 1 119899 (3)

where the 119908119894119899

119894=1are given This form of interpolation is a

generalization of the classical polynomial interpolation Tomake the connection we take P

119899minus1as 119883 and define the

functional by

119871119894(120601 (119909)) = 120601 (119905

119894) 119894 = 1 119899 (4)

where the 119905119894119899

119894=1are a set of distinct points The interpolation

problem is then to find a polynomial 119875119899minus1

(119909) taking onpreassigned values of119908

1 1199082 119908

119899at the points 119905

1 1199052 119905

119899

This type of interpolation is called Lagrange interpolationThree important subjects in the generalized interpolationare existence uniqueness and accuracy of correspondinginterpolation problems respectively wherein the first twosubjects are discussed in [1] and the third one is much moredifficult and we are able to give useful answer only for certaintypes of polynomial interpolation Formore review literaturesabout this subject see [1]Throughout this paper we considera linear space119883 as a Hilbert space 1198712[119886 119887]

13 Spectral and Pseudospectral Methods In the last twodecades spectral and in particular pseudospectral methodshave emerged as intriguing alternatives in many situationsand as superior ones in several areas [2ndash7] Spectral andpseudospectralmethods are known as highly accurate solversfor ordinary partial differential and integral equations Thebasic idea of spectral methods is using general Fourier seriesas an approximate solution with unknown coefficients [2]Three most widely used spectral versions are the Galerkincollocation and Tau methods [2] Their utility is based onthe fact that if the solution sought is smooth usually onlya few terms in an expansion of global basis functions areneeded to represent it to high accuracy It is well known thatspectral methods converge to the solution of the continuousproblems faster than any finite power of 1119873 where 119873

is the dimension of the reduced order model for smoothsolutions [2] Spectral methods in the context of numericalschemes for differential equations belong to the family ofweighted residual methods which are traditionally regardedas the foundation of many numerical methods such asfinite element spectral finite volume and boundary elementmethods Weighted residual methods represent a particulargroupof approximation techniques inwhich the residuals areminimized in a certain way and this leads to specificmethodsincluding Galerkin collocation and Tau formulations Alsothe basic idea of pseudospectral methods (see eg [6ndash10])is based on interpolating an unknown function (the exactsolution of our problem) in suitable points and obtaining theunknown values In other words using some basis functions120601119894 119894 = 1 119873 such as polynomials to represent an

unknown function (the approximate solution of the operatorequation) via

119906119873(119909) =

119873

sum

119894=0

119886119894120601119894(119909) 119909 isin R (5)

An important feature of pseudospectral methods is the factthat one usually is content with obtaining an approximationto the solution on a discrete set of grid points 119909

119894 119894 =

1 119873 One of several ways to implement the pseudospec-tral method is via some matrix operations For instance fordifferential operator (L119906 = 119906

1015840) we have useful matrix which

is denoted by 119863 and is named as differentiation matrix thatis one finds a matrix 119863 such that at the grid points 119909

119894 119894 =

1 119873 we have

1199061015840

119873= 119863119906119873 (6)

where

119906119873= [119906119873(1199090) 119906119873(1199091) 119906

119873(119909119873)] (7)

is the vector of values of 119906119873at the grid points For every oper-

ator also we can define its corresponding operational matrixFrequently orthogonal polynomials such as Chebyshev poly-nomials Jacobi polynomials and Hermite polynomials areused as basis functions and the grid points are correspondingto Chebyshev Jacobi and Hermite points respectively In theChebyshev polynomials case the entries of the differentiation

International Journal of Computational Mathematics 3

matrix are explicitly known (see eg [7]) Using such anapproach for solving boundary value problems involves thesolution of linear systems of equations which are knownto be very ill-conditioned for example for methods basedon orthogonal polynomials the condition number of theapproximate first order operator grows like 119873

2 while thecondition number of second order operator in general scaledwith 119873

4 For more review literatures about this method see[6] Precondition procedure is the method of reducing theobtained ill-condition system from pseudospectral methods[6] This method is used for the numerical solution ofintegral equations [11] partial differential equations [8 10]and ordinary differential equations [9 10]

14 Operator Equations Many important engineering prob-lems fall into the category of being operator equations withboundary operator conditions such as integral differenceand partial differential equations The general form of theseequations is in the form

L119906 +N119906 = 119892 (119909) 119909 isin Γ (8)

with respect to boundary condition B119906 = 120582 where L NandB are linear nonlinear and boundary operators respec-tively related to a suitable Hilbert space Γ is the domain ofapproximation and 119892(119909) isin 119871

2(Γ) is an arbitrary function

The necessary and sufficient conditions for existence anduniqueness of (8) can be seen in [12] Obtaining the explicitsolutions of operator equation (8) in general is a difficultproblem and it depends on the properties of operators LN and B Also some numerical methods are presented forsolving such operator problems as iterativemethodsmethodof paper [13] and so on Simplification of the nonlinear termN in the operator equation (8) to the linear operators for easyand efficient implementation is an important subject Somemethods exist for this simplification such as using Taylorseries and linearization method [13] In Section 2 we explainthis subject very well

2 Spectral Method for GeneralOperator Equation

In this section we describe the spectral methods for solvinggeneral operator equation L119906 + N119906 = 119892(119909) 119909 isin Γ withrespect to boundary conditionB119906 = 120582 For this purpose firstconsider suitable basis set of functions such as 120601

119894(119909)119899

119894=0 and

expand a function 119891(119909) (exact solution of operator equation(8)) based on these basis functions as

119891 (119909) ≃ 119891119899(119909) =

119899

sum

119894=0

119886119894120601119894(119909) (9)

From basis property of 120601119894(119909)119899

119894=0 we can conclude

119891 (119909) =

infin

sum

119894=0

119886119894120601119894(119909) (10)

Now we substitute the expansion series (9) in operatorequation (8) and its boundary conditions as the following

L (119891119899(119909)) +N (119891

119899(119909)) ≃ 119892 (119909) 119909 isin Γ

B (119891119899(119909)) ≃ 120582

(11)

If the basis functions 120601119894(119909)119899

119894=0 automatically satisfy bound-

ary conditions then they will be called Galerkin basisfunctions From (11) we must simplify two terms L(119891

119899(119909))

andN(119891119899(119909)) so we perform this simplification in two cases

But simplification of the first case L(119891119899(119909)) is easy because

by using the linear property of operatorL we obtain

L (119891119899(119909)) = L(

119899

sum

119894=0

119886119894120601119894(119909)) =

119899

sum

119894=0

119886119894L (120601119894(119909)) (12)

But the nonlinear case N(119891119899(119909)) is almost cumbersome

and some ideas exist for its simplification Now we presentthe idea for simplification of the nonlinear term N(119891

119899(119909))

For this goal based on basis functions 120601119894(119909)119899

119894=1 and using

Gram-Schmidt algorithm we must obtain an orthogonalsystem 120593

119895(119909)119899

119895=1 Then it is not difficult to see that

N (119891119899(119909)) ≃

119899

sum

119894=0

119887119894120593119894(119909) (13)

119887119894=

(N (119891119899(119909)) 120593

119894)

(120593119894 120593119894)

(14)

Now from (11) and using (13) we obtain

119899

sum

119894=0

119886119894L (120601119894(119909)) +

119899

sum

119894=0

119887119894120593119894(119909) ≃ 119892 (119909) 119909 isin Γ

B (119891119899(119909)) ≃ 120582

(15)

By using some project operators on (15) as 1198751 119875

119899defined

on119883 we obtain

119875119894(

119899

sum

119894=0

119886119894L (120601119894(119909))) + 119875

119894(

119899

sum

119894=0

119887119894120593119894(119909)) ≃ 119875

119894(119892 (119909))

119909 isin Γ 119894 = 0 119899

B (119891119899(119909)) ≃ 120582

(16)

If we suppose that the boundary conditions give us 119898

equations then from (16) we obtain 119899 + 119898 + 1 equations for119899 + 1 unknown coefficients So for obtaining the unknownswe must eliminate 119898 equations for satisfying the boundaryconditions and the obtained system must be solved viasuitable iteration methods such as Newton method

Remark 6 In (16) if119875119894(119891(119909)) = int

119887

119886119891(119909)119908

119894(119909)119889119909where119908

119894(119909)

is a weight function the method is named as Tau methodAlso if119875

119894= int

119887

119886120575(119909minus119909

119894)119889119909 where 119909

119894119899

119894=0are the set of distinct

points and 120575(119909) is a dirac function then themethod is namedas collocation method

4 International Journal of Computational Mathematics

3 The General Pseudospectral Method forGeneral Operator Equation

In this section we describe the general pseudospectralmethod for numerical solution of the general operatorequations For this goal first we introduce the generalpseudospectral method Let 119871

1 1198712 119871

119899minus119898isin 119883lowast be some

functionals and scalars 1199081 1199082 119908

119899minus119898 wherein

119871119894(119906) = 119908

119894 119894 = 0 119899 minus 119898 (17)

in other words from (17) we have a generalized interpolationproblem with respect to 119906 Also we suppose that a boundaryconditionB119906 = 120582 can be written in the following interpola-tion form

119871119861

119894(119906) = 119888

119894 119894 = 119899 minus 119898 + 1 119899 (18)

where 119871119861

119899minus119898+1(119906) 119871119861

119899minus119898+2(119906) 119871

119861

119899(119906) are functionals on

119883lowastNow solving simultaneous interpolation problems (17)

and (18) is a principal part of this paper The next theoremgives the explicit form of the interpolation problems (17) and(18)

Theorem 7 Suppose 119883 is a Hilbert space and 119883119899

=

1205930 1205931 120593

119899 isin 119883 is a finite dimensional subspace of 119883

with dimension 119899 then I119899(119906) the generalized interpolation

problem from (17) and (18) is of the form

I119899(119906) =

119899

sum

119894=0

119888119894120593119894(119909) +

119899

sum

119894=0

119908119894120601119894(119909) (19)

where 119871119894 119871119861

119894isin 119883lowast are functionals of119883lowast wherein

119871119894(120593119895) = 120575119894119895 119894 = 1 119899 minus 119898 forall119895

119871119861

119894(120593119895) = 120575119894119895 119894 = 119899 minus 119898 + 1 119899 forall119895

(20)

119871119894(120601119895) = 120575119894119895 119894 = 1 119899 minus 119898 forall119895

119871119861

119894(120601119895) = 120575119894119895 119894 = 119899 minus 119898 + 1 119899 forall119895

(21)

Proof By imposing functionals 119871119894and 119871

119861

119894on both sides of

(19) and using properties (20) and (21) it is easy to show that(19) satisfies conditions (17) and (18)

Remark 8 The particular cases of interpolation (19) areLagrange Barycentric Hermite and Birkhoff interpolations

The important point inTheorem 7 is the existence of basisfunctions 120593

119895(119909)119899

119895=1and 120601

119895(119909)119899

119895=1which satisfy (20) and

(21) respectively We can answer this question by the conceptof linear independency of operators We perform this for120593119895(119909)119899

119895=1 and in similarmannerwe can obtain the necessary

and sufficient conditions of existence of 120601119895(119909)119899

119895=1 Now we

consider an arbitrary basis function 120595119895(119909)119899

119895=1of 119883119899and we

construct new basis functions which satisfy (20) and (21)

For this goal suppose

120593119895(119909) =

119895

sum

119894=1

119886119895119894120595119894 119895 = 1 119899 (22)

then we must obtain the 119899(119899 + 1)2 unknown coefficients119886119895119894 119894 = 1 119899 We start from 120593

0(119909) (119895 = 0) this means that

1205930(119909) = 119886

001205950(119909) (23)

from (23) wemust obtain one unknown coefficient 11988600

and itis sufficient for imposing the arbitrary functionals 119871

119894or 119871119861119894isin

119883lowast on (22) for obtaining the unknown coefficient 119886

00 so

1198710(1205930(119909)) = 119886

001198710(1205950(119909)) (24)

thus

11988600

=

1198710(1205930(119909))

1198710(1205950(119909))

1198710(1205950(119909)) = 0 (25)

Also for 1205931(119909)(119895 = 1) we have

1205931(119909) = 119886

101205950(119909) + 119886

111205951(119909) (26)

for unknown coefficients 11988610

and 11988611 If we impose two

functionals 1198710and 119871

1isin 119883lowast on (26) for obtaining unknown

coefficients 11988610

and 11988611

we can obtain

1198710(1205931(119909)) = 119886

101198710(1205950(119909)) + 119886

111198710(1205951(119909))

1198711(1205931(119909)) = 119886

101198711(1205950(119909)) + 119886

111198711(1205951(119909))

(27)

the unknown coefficients 11988610

and 11988611

exist if and only if10038161003816100381610038161003816100381610038161003816

1198710(1205950(119909)) 119871

0(1205951(119909))

1198711(1205950(119909)) 119871

1(1205951(119909))

10038161003816100381610038161003816100381610038161003816

= 0 (28)

By repeating this procedure we can obtain the result in whichthe unknown coefficients 119886

1198990 1198861198991 119886

119899119899exist if and only if

det (119871119894(120595119895)) =

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198710(1205951(119909)) sdot sdot sdot 119871

119899(120595119899(119909))

sdot sdot sdot

119871119899(1205951(119909)) sdot sdot sdot 119871

119899(120595119899(119909))

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (29)

and this result is equivalent to the independence of function-als 1198711 1198712 119871

119899minus119898 119871119861

119899minus119898+1 119871

119861

119899isin 119883lowast

Now for implementing the general pseudospectralmethod for numerical solution of general operator equation(8) with respect to its boundary condition first we substitutea general interpolation operator (19) in (8) The generalinterpolation function 119906(119909) (the exact solution of (8) withrespect to its boundary condition) fromTheorem 7 is

119906 (119909) ≃ I119899(119906) (119909) =

119899

sum

119894=0

119888119894120593119894(119909) +

119899

sum

119894=0

119908119894120593119894(119909) (30)

where119908119894and 119888119894are defined in (17) and (18) Also we consider

the general interpolation problem of the function 119892(119909) (thefunction of the right-hand side of (8)) as

1198711

119894(119892) = 119887

119894 119894 = 0 119899 (31)

International Journal of Computational Mathematics 5

where 11987110(119892) 119871

1

119899(119892) isin 119883

lowast are functionals and 1198870 1198871 119887

119899

are constants then we have

119892 (119909) ≃ I119899(119892) (119909) =

119899

sum

119894=0

119887119894120593119894(119909) (32)

Now by substituting (19) and (32) in (8) we obtain

L (I119899(119906) (119909)) +N (I

119899(119906) (119909)) ≃ I

119899(119892) (119909) (33)

then using some project operators on (33) as 1198751 119875

119899

defined on119883 we obtain

119875119894(L (I

119899(119906) (119909)) +N (I

119899(119906) (119909))) ≃ 119875

119894(I119899(119892) (119909))

119894 = 1 119899

(34)

From (34) we obtain 119899 algebraic equations Simplificationof the left-hand side of (34) is an important subject and weexpress this subject for linear and nonlinear operators (I)

and (N)

(i) Linear Case In the linear case if we suppose that theprojection operators are linear then from property of linearoperatorL we have

119875119905L (I

119899(119906)) = 119875

119905L(

119899

sum

119894=0

119888119894120593119894+

119899

sum

119894=0

119908119894120593119894)

=

119899

sum

119894=0

119888119894119875119905L (120593119894) +

119899

sum

119894=0

119908119894119875119905L (120593119894)

(35)

so for obtaining a matrix relation between the values of119875119905L(I119899(119906)) and 119888

119894and 119908

119894 it is sufficient to calculate the

119875119905L(120593119894) values as the following

[

[

[

[

[

[

1198750L (I

119899(119906))

119875119899L (I

119899(119906))

]

]

]

]

]

]

=

[

[

[

[

[

[

11989600

sdot sdot sdot sdot sdot sdot 1198960119899

sdot sdot sdot sdot sdot sdot

sdot sdot sdot sdot sdot sdot

1198961198990

sdot sdot sdot sdot sdot sdot 119896119899119899

]

]

]

]

]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

(

[

[

[

[

[

[

1198880

119888119899

]

]

]

]

]

]

+

[

[

[

[

[

[

1199080

119908119899

]

]

]

]

]

]

)

(36)

where

119896119894119895

= 119875119894L (120593119895) 119894 119895 = 0 119899 (37)

The previous matrix (under bracket) is a square matrix of(119873+ 1) times (119873+ 1) dimensions and has high application in thepseudospectral method In particular case when the linearoperator is a derivative this matrix is named as derivativematrix and is very ill conditioned

(ii) Nonlinear Case The nonlinear case is almost cumber-some and different ideas can be used The first idea is to

approximate the nonlinear operator N with some linearoperator F and using the method of part (i) But thisapproach is very complicated and approximating a nonlinearoperator with linear operators in general is difficult and wasdown only for particular operators [13] The second ideais based on approximating a nonlinear operator via Taylorseries [14] Also another approach for simplifying nonlinearoperator is based on orthogonal expansions (general Fourierseries) of N(I

119899(119906)) For this goal we must obtain an

orthogonal system 120601119894(119909)119899

119894=1from basis functions 120593

119895(119909)119899

119895=1

using Gram-Schmidt algorithmThen it is not difficult to seethat

I119899(119906) (119909) =

119899

sum

119894=0

120582119894120593119894(119909) +

119899

sum

119894=0

119888119894120593119894(119909) ≃

119873

sum

119894=0

119886119894120601119894(119909) (38)

where

119886119894=

(I119873(119906) 120601

119894)

(120601119894 120601119894)

(39)

Also forN(I119899(119906)(119909)) we have

N (I119899(119906) (119909)) ≃

119899

sum

119894=0

119887119894120601119894(119909) (40)

where

119887119894=

(N (I119899(119906)) 120601

119894)

(120601119894 120601119894)

(41)

We must note that the 119886119894and 119887119894coefficients are functions of

119871119894and 119871

119861

119894

Now for case study let119883 be the space of polynomials withreal coefficients wherein it is a Hilbert space with respect to1198712 norm and119883

119899is a subspace of119883 with dimension 119899

Also suppose that the functionals1198711 1198712 119871

119899in (9) are

defined as

119871119894(120593) = 119888

119894 (42)

where 119888119894are also functionals of 120593

119895(119909)119899

119895=1 For instance

the well known functionals with high applications are 119888119894=

int 119905119894120593119889119909 (moment operator) 119888

119894= (120593)

(119894)(119909) (ith-derivative

operator) 119888119894= 120593(119909 minus ℎ

119894) ℎ119894isin 119877 (shift operator) and 119888

119894=

120593(119905119894) (119905119894are distinct points and in particular case the roots

of shifted orthogonal polynomials as Hermite and Laguerrepolynomials in unbounded domains and shifted Chebyshevand Legendre and in general case Jacobi polynomials inbounded domains [15ndash17]) Also in the higher dimensionswe can define the suitable functionals

31 Error Analysis The error analysis of pseudospectralmethod in the general case is very difficult and we canperform this analysis only in polynomial case For this goalat first we introduce some useful notations and theorems thatare used in this paper

Definition 9 The Jacobi polynomials [18] 119875(120572120573)

119894(119909) are

defined as the orthogonal polynomials with respect to the

6 International Journal of Computational Mathematics

weight function119908120572120573

(119909) = (1minus119909)120572(1 +119909)

120573 (120572 gt minus1 120573 gt minus1)

on (minus1 1) An explicit formula for these polynomials is givenby

119875(120572120573)

119899(119909)

=

1

2119899

119899

sum

119894=0

(

119899 + 120572

119894)(

119899 + 120573

119899 minus 119894) (119909 minus 1)

119899minus119894(119909 + 1)

119894

(43)

Further we have

119889

119889119909

119875(120572120573)

119899(119909) =

1

2

(119899 + 120572 + 120573 + 1) 119875(120572+1120573+1)

119899minus1(119909) (44)

Jacobi polynomials have an important application in severalfields of numerical analysis such as quadratures formula andspectral methods Also the shifted Jacobi polynomials aredefined as orthogonal polynomials in the interval [119886 119887] withrespect to weight function 119908

120572120573(119909) = (119887 minus 119909)

120572(119886 minus 119909)

120573 (120572 gt

minus1 120573 gt minus1) with change of variable 119910 = (2(119887 minus 119886))119909 +

((119886 + 119887)(119886 minus 119887)) For more properties and applications ofthese polynomials see [15ndash17] Now let Λ = [minus1 1] and let119875(120572120573)

119873+1(119909) be the Jacobi polynomial of degree119873+1with respect

to weight function 119908(120572120573)

(119909) = (1 minus 119909)120572(1 + 119909)

120573 (120572 120573 gt minus1)

and let P119873be the polynomials spaces of degree less than or

equal to119873

Remark 10 The best choice of points for polynomial inter-polation are the roots of orthogonal polynomials and in par-ticular case Jacobi polynomials due to their small Lebesgueconstants [6]

The symbol 1198712

119908(120572120573)(Λ) shows the space of measurable

functions whose square is Lebesgue integrable in Λ relativeto the weight function119908

(120572120573) The inner product and norm of1198712

119908(120572120573)(Λ) are defined by

(119906 V)119908(120572120573)Λ

= int

Λ

119906 (119909) V (119909) 119908(120572120573) (119909) 119889119909

119906119908(120572120573)Λ

= radic(119906 119906)119908(120572120573)

forall119906 V isin 1198712

119908(120572120573) (Λ)

(45)

Also another useful space is

119867119898

119908(120572120573) = V V

119898119908(120572120573) lt infin (46)

with seminorm and norms respectively Consider

|V|119898119908(120572120573) =

1003817100381710038171003817120597119898

119909V1003817100381710038171003817119908(120572120573)

V119898119908(120572120573) = (

119898

sum

119894=0

|V|2119894119908(120572120573))

12

(47)

But an applicable norm that appears in bounding of errors ofthe spectral method is

|V|119867119898119908(120572120573) = (

119898

sum

119896=119890

10038171003817100381710038171003817120597119896

119909V10038171003817100381710038171003817

2

119908(120572120573)

)

12

119890 = min (119898119873 + 1)

(48)

Now we present a useful theorem

Theorem 11 If ℎ isin 119867119898

119908(120572120573)(Λ)119898 ge 1 then one has

1003817100381710038171003817ℎ minusI

119873(ℎ)

1003817100381710038171003817119908(120572120573) le 119862119873

minus11989810038161003816100381610038161198911003816100381610038161003816119867119898

119908(120572120573)

(49)

where I119873(ℎ) is a Lagrange interpolation in distinct points

119909(120572120573)

119894(roots of Jacobi polynomials)

Proof See [4 5]

Noting to Theorem 11 we can obtain an error boundfor any classes of polynomial interpolation such as Hermiteinterpolation Birkhoff interpolation and other classes Thismeans that if we have an arbitrary polynomial interpolationI1119873(ℎ) thenwe can transform the interpolationI1

119873(ℎ) to the

Lagrange interpolation I119873(ℎ) and then we use Theorem 11

for obtaining an error bound Now by using Theorem 11 inthe next theorem we obtain an upper bound for the error ofour approximate solution of (8)

Theorem 12 Suppose L and N are the linear and nonlinearoperators with respect to (8) wherein L has inversion and119891 119892 isin 119867

119898

119908(120572120573)(Λ) (119898 ge 1) are satisfied in (8) then

10038171003817100381710038171003817119891 minusI

1

119873(119891)

10038171003817100381710038171003817119908(120572120573)

le 119862119873minus119898

(

10038161003816100381610038161003816Lminus1N (119891)

10038161003816100381610038161003816119867119898

119908(120572120573)

+

10038161003816100381610038161003816Lminus1119892

10038161003816100381610038161003816119867119898

119908(120572120573)

)

(50)

whereI1119873(119891) is an arbitrary polynomial interpolation satisfied

in interpolation conditions

Proof First we must note that I1119873(119891) can be expressed via

Lagrange interpolation and for this goal it is sufficient tointerpolate function I

119873(119891) in 119873 distinct points 119909

119895 119895 =

1 119873 From (8) and invertibility ofL we have

119891 +Lminus1N119891 = L

minus1(119892) (51)

also due to pseudospectral method we obtain

119891 (119909119895) +L

minus1N119891 (119909

119895) = L

minus1119892 (119909119895) (52)

by producing both sides of (52) with 119871119895(119909) (Lagrange inter-

polation basis) and sum from 119894 = 0 119873 we obtain

I119873(119891) +I

119873(Lminus1N (119891)) = I

119873(Lminus1

(119892)) (53)

Now by subtracting (51) from (53) we get

I119873(119891) = L

minus1N (119891) minusI

119873(Lminus1N (119891))

+Lminus1 (119892) minusI119873(Lminus1 (119892))

(54)

International Journal of Computational Mathematics 7

therefore we have10038171003817100381710038171003817119891 minusI

1

119873(119891)

10038171003817100381710038171003817119908(120572120573)

=1003817100381710038171003817119891 minusI

119873(119891)

1003817100381710038171003817119908(120572120573)

le

10038171003817100381710038171003817Lminus1N (119891) minusI

119873(Lminus1N (119891))

10038171003817100381710038171003817119908(120572120573)

+

10038171003817100381710038171003817Lminus1

(119892) minusI119873(Lminus1

(119892))

10038171003817100381710038171003817119908(120572120573)

(55)

Finally by usingTheorem 11 we obtain10038171003817100381710038171003817119891 minusI

1

119873(119891)

10038171003817100381710038171003817119908(120572120573)

=1003817100381710038171003817119891 minusI

119873(119891)

1003817100381710038171003817119908(120572120573)

le 119862119873minus119898

(

10038161003816100381610038161003816Lminus1N119891

10038161003816100381610038161003816119867119898119873

119908(120572120573)

+

10038161003816100381610038161003816Lminus1119892

10038161003816100381610038161003816119867119898119873

119908(120572120573)

)

(56)

therefore the proof is completed

4 The Test Experiments

Experiment 1 Let us consider the following nonlinear differ-ential equation problem [19]

11991010158401015840= minus (1 + 001119910

2) 119910 + 001cos3 (119909)

119910 (minus1) = cos (minus1)

119910 (1) = cos (1)

(57)

with the exact solution 119910(119909) = cos(119909)By simplifying (57) we have

11991010158401015840= minus 119910 minus 001119910

3+ 001cos3 (119909)

119910 (minus1) = cos (minus1)

119910 (1) = cos (1)

(58)

thus the linear and nonlinear parts of our ordinary differen-tial equation respectively are

L (119910) =

1

sum

119894=0

L119894(119910)

N (119910) = 0011199103(119909)

(59)

where

L0(119910) = 119910

10158401015840(119909) L

1(119910) = 119910 (60)

Now we consider 119873 + 1 collocation points 1199090

=

minus1 119909(120572120573)

119894119873minus1

119894=1 119909119873

= 1 where 119909(120572120573)

119894119873minus1

119894=1are the roots of

Jacobi polynomials of degree 119873 minus 1 then we consider thisinterpolation problem

119910 (minus1) = cos (minus1)

119910 (119909(120572120573)

119894) = 119875119873+1

(119909(120572120573)

119894) 119894 = 0 119873 minus 1

119910 (1) = cos (1)

(61)

Table 1 Comparison between the exact and approximate solutionsfor 120572 = 0 120573 = 0 and119873 = 5 of Experiment 1

119909Exact

solutionPseudospectral

method Error

minus10 054030231 054030231 000000000minus080 069670671 069670667 000000004minus060 082533561 082533555 000000006minus040 092106099 092106045 000000054minus020 098006658 098006645 000000013000 100000000 099999922 000000078020 098006658 098006633 000000025040 092106099 092106056 000000043060 082533561 082533545 000000016080 069670671 069670656 00000001510 054030231 054030231 000000000

where 119875119873+1

is the corresponding polynomial interpolationwith respect to problem (61) wherein interpolate the bound-ary conditions automatically and 119910(119909

(120572120573)

119894) 119894 = 0 119873 minus

1 are the unknown coefficients In Table 1 we see thecomparison between the exact and approximate solutionsfrom pseudospectral method for 120572 = 0 120573 = 0 and119873 = 5

Experiment 2 Consider the second-order difference equa-tion problem [20]

119910 (119909 + 1) minus 119910 (119909) = 119890119909 119910 (

1

2

) = 1 (62)

The exact solution is 119910(119909) = (119890119909+119890minusradic119890minus1)(119890minus1) We have

L (119910) = 119910 (119909 + 1) minus 119910 (119909) (63)

and this equation has no nonlinear operator part Theobtained results between the exact and approximate solutionsare shown in Table 2 and Figure 1

Experiment 3 Consider the Fredholm integrodifferentialequation [21]

1199101015840(119909) = int

1

0

119890119909119905119910 (119905) 119889119905 + 119910 (119909) +

1 minus 119890119909+1

119909 + 1

0 le 119909 119910 le 1

(64)

under the initial condition

119910 (0) = 1 (65)

with the exact solution 119910(119909) = 119890119909 In this case we have

L (119910) =

2

sum

119894=0

L119894(119910) (66)

where

L0(119910) = int

1

0

119890119909119905119910 (119905) 119889119905

L1(119910) = 119910

1015840(119909) L

2(119910) = 119910 (119909)

(67)

8 International Journal of Computational Mathematics

28

26

24

22

2

18

16

14

12

1

0

y

x

01 02 03 04 05 06 07 08 09 1

Exact solutionApproximate solution

Figure 1 The obtained results from approximation solution and exact solution for 120572 = minus05 120573 = 07 and 119873 = 5 of Experiment 2

Table 2 Comparison between the exact and approximate solutionsfor 120572 = minus05 120573 = 07 and119873 = 5 of Experiment 2

119909Exact

solutionPseudospectral

method Error

00 062245933 062245922 00000001101 068366636 068366643 minus00000000702 075131058 075131034 00000002403 082606901 082606923 minus00000002204 090868985 090868925 00000006005 100000000 100000000 0000000006 110091330 110091120 00000002107 121243980 121243880 00000001008 133569560 133569450 00000001109 147191430 147191350 00000000810 162245930 162245560 000000037

Table 3 Comparison between the exact and approximate solutionsfor 120572 = minus05 120573 = 07 and119873 = 5 of Experiment 3

119909Exact

solutionPseudospectral

method Error

00 10000000 10000000 00000000001 11051709 11051701 00000000802 12214028 12214011 00000001703 13498588 13498533 00000005504 14918247 14918232 00000001505 16487213 16487211 00000000206 18221188 18221166 00000002207 20137527 20137511 00000001508 22255409 22255401 00000000709 24596031 24596030 00000000110 27182818 27182810 000000008

Now by using the pseudospectral method for 120572 =

minus05 120573 = 07 and 119873 = 5 we obtain the following resultsthat are shown in Table 3

Experiment 4 We consider the linear partial differentialequation problem [22]

119906119905119905= 119906119909119909

+ 6 119906119905(119909 0) = 4119909 119906 (119909 0) = 119909

2 (68)

with the exact solution 119906(119909 119910) = (119909 + 2119905)2

In this case we have

L (119906) = 119906119905119905minus 119906119909119909 (69)

We obtain the approximate solution of (68) on the domain[minus1 1] times [minus1 1] For this purpose we consider the discretepoints (119909(120572120573)

119894 119905(120572120573)

119895)119873

119894119895=0and the general Lagrange interpola-

tion in the discrete points (119909(120572120573)119894

119905(120572120573)

119895)119873

119894119895=0in the following

form (two-dimensional case)

119871 (119909 119905) =

119873

sum

119894=0

119873

sum

119895=0

119891 (119909(120572120573)

119894 119905(120572120573)

119895) 119871119894(119909) 119871119895(119905) (70)

where 119871119894(119909) and 119871

119895(119905) are ordinary Lagrange interpolations

By using collocation method for arbitrary parameters 120572 120573and119873 = 5 we obtain the exact solution

Experiment 5 Finally let us consider the linear fractionaldifferential equation problem [23]

11986352

119910 (119909) + 3119910 (119909) = 31199093+

6

Γ (15)

11990912

119910 (0) = 0 1199101015840(0) = 0 119910

10158401015840(0) = 0 119909 isin [minus1 1]

(71)

This equation has the exact solution 119910(119909) = 1199093

In this case we have

L (119910 (119909)) = 11986352

(119910 (119909)) + 3119910 (119909) (72)

which is a linear operator Using the presented method forevery 120572 120573 and119873 = 3 we obtain the exact solution

International Journal of Computational Mathematics 9

5 Conclusion

In this paper we have developed the expansion approach forsolving some operator equations of the form L119906 + N119906 =

119892(119909) 119909 isin Γ with respect to boundary condition B119906 = 120582For this the principal importance is the development of theexpansion approach for solving general operator equationwherein the particular cases are in the solution of integralequations (Experiment 3) ordinary differential equations(Experiment 1) difference equations (Experiment 2) partialdifferential equations (Experiment 4) and fractional differ-ential equations (Experiment 5) Also the error analysis ispresented in 119871

2(Γ) norm Using the roots of another orthog-

onal polynomials (Jacobi polynomials) rather than classicalorthogonal polynomials such as Chebyshev polynomialsand Legendre polynomials for collocation points (with freeparameters 120572 120573) in pseudospectral method is an importantsubject which is performed (Experiments 2 and 3)

Conflict of Interests

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

Acknowledgments

The authors wish to thank the anonymous referees for theirvaluable comments and suggestions Also they would like toappreciate the Editor Dr Yaohanz Li who gives them theopportunity to review the paper

References

[1] P LinzTheoretical Numerical Analysis JohnWiley amp Sons NewYork NY USA 1979

[2] D Gottlieb and S A Orszag Numerical Analysis of SpectralMethods Theory and Applications SIAM-CBMS PhiladelphiaPa USA 1977

[3] G Ben-Yu The State of Art in Spectral Methods Hong KongUniversity 1996

[4] C Canuto M Y Hussaini A Quarteroni and T A ZhangSpectral Methods Fundamentals in Single Domains Springer2006

[5] C Canutuo M Y Hussaini A Quarteroni and T A ZhangSpectral Method in Fluid Dynamics Prentice Hall EngelwoodCliffs NJ USA 1984

[6] B FornbergAPractical Quide to PseudospectralMethods Cam-bridgeMonographs on Applied and Computational Mathemat-ics Cambridge University Press Cambridge UK 1st edition1996

[7] L N Trefethen Spectral Methods in Matlab SIAM Philadel-phia Pa USA 2000

[8] J S Hesthaven S Gottlieb and D Gottlieb Spectral Methodsfor Time-Dependent Problems vol 21 ofCambridgeMonographson Applied and Computational Mathematics Cambridge Uni-versity Press Cambridge UK 2007

[9] M Deville and E Mund ldquoChebyshev pseudospectral solutionof second-order elliptic equations with finite element precon-ditioningrdquo Journal of Computational Physics vol 60 no 3 pp517ndash533 1985

[10] J P Boyd ldquoMultipole expansions and pseudospectral cardinalfunctions a new generalization of the fast Fourier transformrdquoJournal of Computational Physics vol 103 no 1 pp 184ndash1861992

[11] G N Elnagar and M Razzaghi ldquoA pseudospectral method forHammerstein equationsrdquo Journal of Mathematical Analysis andApplications vol 199 no 2 pp 579ndash591 1996

[12] M Meehan and D OrsquoRegan ldquoExistence principles for nonreso-nant operator and integral equationsrdquo Computers ampMathemat-ics with Applications vol 35 no 9 pp 79ndash87 1998

[13] A Chaillou and M Suri ldquoA posteriori estimation of thelinearization error for strongly monotone nonlinear operatorsrdquoJournal of Computational andAppliedMathematics vol 205 no1 pp 72ndash87 2007

[14] R P Kanwal and K C Liu ldquoA Taylor expansion approach forsolving integral equationsrdquo International Journal of Mathemat-ical Education in Science and Technology vol 20 pp 411ndash4141989

[15] M R Eslahchi M Dehghan and S Ahmadi Asl ldquoThe generalJacobi matrix method for solving some nonlinear ordinarydifferential equationsrdquoAppliedMathematical Modelling vol 36no 8 pp 3387ndash3398 2012

[16] A Imani A Aminataei and A Imani ldquoCollocation method viaJacobi polynomials for solving nonlinear ordinary differentialequationsrdquo International Journal of Mathematics and Mathe-matical Sciences vol 2011 Article ID 673085 11 pages 2011

[17] M R Eslahchi and M Dehghan ldquoApplication of Taylor seriesin obtaining the orthogonal operational matrixrdquo Computers ampMathematics with Applications vol 61 no 9 pp 2596ndash26042011

[18] D Funaro Polynomial Approximation of Differential Equationsvol 8 of Lecture Notes in Physics Monographs Springer BerlinGermany 1992

[19] D F Griffiths and D J Higham Numerical Methods forOrdinaryDifferential Equations SpringerUndergraduateMath-ematics Series Springer London UK 2010

[20] M Gulsu M Sezer and B Tanay ldquoA matrix method for solvinghigh-order linear difference equations with mixed argumentusing hybrid Legendre and Taylor polynomialsrdquo Journal of theFranklin Institute vol 343 no 6 pp 647ndash659 2006

[21] N Kurt and M Sezer ldquoPolynomial solution of high-orderlinear Fredholm integro-differential equations with constantcoefficientsrdquo Journal of the Franklin Institute vol 345 no 8 pp839ndash850 2008

[22] V S Vladimirov A Collection of Problems on the Equations ofMathematical Physics Mir Publishers Moscow Russia 1968

[23] A H Bhrawy M M Alghamdi and T M Taha ldquoA newmodified generalized Laguerre operational matrix of fractionalintegration for solving fractional differential equations on thehalf linerdquoAdvances in Difference Equations vol 2012 article 1792012

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Stochastic AnalysisInternational Journal of

2 International Journal of Computational Mathematics

functionals on the linear space119883 is itself a linear space calledthe algebraic conjugate or dual-space on119883 and is denoted by119883lowast [1]

Definition 2 A projection is a linear transformation 119875 froma vector space to itself such that 1198752 = 119875 It leaves its imageunchanged Though abstract this definition of projectionformalizes and generalizes the idea of graphical projection [1]

Definition 3 Suppose 119883 is a linear space and 119883lowast is its dual-

space then 1198711 1198712 119871

119899isin 119883lowast are linearly independent if

and only if

12057211198711+ 12057221198712+ sdot sdot sdot + 120572

119899119871119899= 0

997904rArr 1205721= 1205722= sdot sdot sdot = 120572

119899= 0

(1)

Theorem 4 Let 119883 be an 119899-dimensional space and let1206011(119909) 1206012(119909) 120601

119899(119909) isin 119883 be linearly independent Then

1198711 1198712 119871

119899isin 119883lowast are linearly independent if and only if

det (119871119894(120601119895)) =

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198711(1206011) sdot sdot sdot 119871

1(120601119899)

sdot sdot sdot

119871119899(1206011) sdot sdot sdot 119871

119899(120601119899)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (2)

Proof See [1]

Remark 5 Suppose 119910 isin 1198712(Γ) then the operators 119871

1(119910) =

119910 1198712(119910) = int 119910119889119910 (integral operator) 119871

3(119910) = 119910

1015840 (derivativeoperator) and 119871

ℎ(119910(119909)) = 119910(119909 minus ℎ) (shift operator) ℎ isin 119877

are independent [1]

12 General Interpolation Problem Thegeneral interpolationproblem is stated as follows Let119883 be a space of dimension 119899and let 119871

1 1198712 119871

119899be given linear functionals on119883

lowast Findan 120601(119909) isin 119883 such that

119871119894(120601 (119909)) = 119908

119894 119894 = 1 119899 (3)

where the 119908119894119899

119894=1are given This form of interpolation is a

generalization of the classical polynomial interpolation Tomake the connection we take P

119899minus1as 119883 and define the

functional by

119871119894(120601 (119909)) = 120601 (119905

119894) 119894 = 1 119899 (4)

where the 119905119894119899

119894=1are a set of distinct points The interpolation

problem is then to find a polynomial 119875119899minus1

(119909) taking onpreassigned values of119908

1 1199082 119908

119899at the points 119905

1 1199052 119905

119899

This type of interpolation is called Lagrange interpolationThree important subjects in the generalized interpolationare existence uniqueness and accuracy of correspondinginterpolation problems respectively wherein the first twosubjects are discussed in [1] and the third one is much moredifficult and we are able to give useful answer only for certaintypes of polynomial interpolation Formore review literaturesabout this subject see [1]Throughout this paper we considera linear space119883 as a Hilbert space 1198712[119886 119887]

13 Spectral and Pseudospectral Methods In the last twodecades spectral and in particular pseudospectral methodshave emerged as intriguing alternatives in many situationsand as superior ones in several areas [2ndash7] Spectral andpseudospectralmethods are known as highly accurate solversfor ordinary partial differential and integral equations Thebasic idea of spectral methods is using general Fourier seriesas an approximate solution with unknown coefficients [2]Three most widely used spectral versions are the Galerkincollocation and Tau methods [2] Their utility is based onthe fact that if the solution sought is smooth usually onlya few terms in an expansion of global basis functions areneeded to represent it to high accuracy It is well known thatspectral methods converge to the solution of the continuousproblems faster than any finite power of 1119873 where 119873

is the dimension of the reduced order model for smoothsolutions [2] Spectral methods in the context of numericalschemes for differential equations belong to the family ofweighted residual methods which are traditionally regardedas the foundation of many numerical methods such asfinite element spectral finite volume and boundary elementmethods Weighted residual methods represent a particulargroupof approximation techniques inwhich the residuals areminimized in a certain way and this leads to specificmethodsincluding Galerkin collocation and Tau formulations Alsothe basic idea of pseudospectral methods (see eg [6ndash10])is based on interpolating an unknown function (the exactsolution of our problem) in suitable points and obtaining theunknown values In other words using some basis functions120601119894 119894 = 1 119873 such as polynomials to represent an

unknown function (the approximate solution of the operatorequation) via

119906119873(119909) =

119873

sum

119894=0

119886119894120601119894(119909) 119909 isin R (5)

An important feature of pseudospectral methods is the factthat one usually is content with obtaining an approximationto the solution on a discrete set of grid points 119909

119894 119894 =

1 119873 One of several ways to implement the pseudospec-tral method is via some matrix operations For instance fordifferential operator (L119906 = 119906

1015840) we have useful matrix which

is denoted by 119863 and is named as differentiation matrix thatis one finds a matrix 119863 such that at the grid points 119909

119894 119894 =

1 119873 we have

1199061015840

119873= 119863119906119873 (6)

where

119906119873= [119906119873(1199090) 119906119873(1199091) 119906

119873(119909119873)] (7)

is the vector of values of 119906119873at the grid points For every oper-

ator also we can define its corresponding operational matrixFrequently orthogonal polynomials such as Chebyshev poly-nomials Jacobi polynomials and Hermite polynomials areused as basis functions and the grid points are correspondingto Chebyshev Jacobi and Hermite points respectively In theChebyshev polynomials case the entries of the differentiation

International Journal of Computational Mathematics 3

matrix are explicitly known (see eg [7]) Using such anapproach for solving boundary value problems involves thesolution of linear systems of equations which are knownto be very ill-conditioned for example for methods basedon orthogonal polynomials the condition number of theapproximate first order operator grows like 119873

2 while thecondition number of second order operator in general scaledwith 119873

4 For more review literatures about this method see[6] Precondition procedure is the method of reducing theobtained ill-condition system from pseudospectral methods[6] This method is used for the numerical solution ofintegral equations [11] partial differential equations [8 10]and ordinary differential equations [9 10]

14 Operator Equations Many important engineering prob-lems fall into the category of being operator equations withboundary operator conditions such as integral differenceand partial differential equations The general form of theseequations is in the form

L119906 +N119906 = 119892 (119909) 119909 isin Γ (8)

with respect to boundary condition B119906 = 120582 where L NandB are linear nonlinear and boundary operators respec-tively related to a suitable Hilbert space Γ is the domain ofapproximation and 119892(119909) isin 119871

2(Γ) is an arbitrary function

The necessary and sufficient conditions for existence anduniqueness of (8) can be seen in [12] Obtaining the explicitsolutions of operator equation (8) in general is a difficultproblem and it depends on the properties of operators LN and B Also some numerical methods are presented forsolving such operator problems as iterativemethodsmethodof paper [13] and so on Simplification of the nonlinear termN in the operator equation (8) to the linear operators for easyand efficient implementation is an important subject Somemethods exist for this simplification such as using Taylorseries and linearization method [13] In Section 2 we explainthis subject very well

2 Spectral Method for GeneralOperator Equation

In this section we describe the spectral methods for solvinggeneral operator equation L119906 + N119906 = 119892(119909) 119909 isin Γ withrespect to boundary conditionB119906 = 120582 For this purpose firstconsider suitable basis set of functions such as 120601

119894(119909)119899

119894=0 and

expand a function 119891(119909) (exact solution of operator equation(8)) based on these basis functions as

119891 (119909) ≃ 119891119899(119909) =

119899

sum

119894=0

119886119894120601119894(119909) (9)

From basis property of 120601119894(119909)119899

119894=0 we can conclude

119891 (119909) =

infin

sum

119894=0

119886119894120601119894(119909) (10)

Now we substitute the expansion series (9) in operatorequation (8) and its boundary conditions as the following

L (119891119899(119909)) +N (119891

119899(119909)) ≃ 119892 (119909) 119909 isin Γ

B (119891119899(119909)) ≃ 120582

(11)

If the basis functions 120601119894(119909)119899

119894=0 automatically satisfy bound-

ary conditions then they will be called Galerkin basisfunctions From (11) we must simplify two terms L(119891

119899(119909))

andN(119891119899(119909)) so we perform this simplification in two cases

But simplification of the first case L(119891119899(119909)) is easy because

by using the linear property of operatorL we obtain

L (119891119899(119909)) = L(

119899

sum

119894=0

119886119894120601119894(119909)) =

119899

sum

119894=0

119886119894L (120601119894(119909)) (12)

But the nonlinear case N(119891119899(119909)) is almost cumbersome

and some ideas exist for its simplification Now we presentthe idea for simplification of the nonlinear term N(119891

119899(119909))

For this goal based on basis functions 120601119894(119909)119899

119894=1 and using

Gram-Schmidt algorithm we must obtain an orthogonalsystem 120593

119895(119909)119899

119895=1 Then it is not difficult to see that

N (119891119899(119909)) ≃

119899

sum

119894=0

119887119894120593119894(119909) (13)

119887119894=

(N (119891119899(119909)) 120593

119894)

(120593119894 120593119894)

(14)

Now from (11) and using (13) we obtain

119899

sum

119894=0

119886119894L (120601119894(119909)) +

119899

sum

119894=0

119887119894120593119894(119909) ≃ 119892 (119909) 119909 isin Γ

B (119891119899(119909)) ≃ 120582

(15)

By using some project operators on (15) as 1198751 119875

119899defined

on119883 we obtain

119875119894(

119899

sum

119894=0

119886119894L (120601119894(119909))) + 119875

119894(

119899

sum

119894=0

119887119894120593119894(119909)) ≃ 119875

119894(119892 (119909))

119909 isin Γ 119894 = 0 119899

B (119891119899(119909)) ≃ 120582

(16)

If we suppose that the boundary conditions give us 119898

equations then from (16) we obtain 119899 + 119898 + 1 equations for119899 + 1 unknown coefficients So for obtaining the unknownswe must eliminate 119898 equations for satisfying the boundaryconditions and the obtained system must be solved viasuitable iteration methods such as Newton method

Remark 6 In (16) if119875119894(119891(119909)) = int

119887

119886119891(119909)119908

119894(119909)119889119909where119908

119894(119909)

is a weight function the method is named as Tau methodAlso if119875

119894= int

119887

119886120575(119909minus119909

119894)119889119909 where 119909

119894119899

119894=0are the set of distinct

points and 120575(119909) is a dirac function then themethod is namedas collocation method

4 International Journal of Computational Mathematics

3 The General Pseudospectral Method forGeneral Operator Equation

In this section we describe the general pseudospectralmethod for numerical solution of the general operatorequations For this goal first we introduce the generalpseudospectral method Let 119871

1 1198712 119871

119899minus119898isin 119883lowast be some

functionals and scalars 1199081 1199082 119908

119899minus119898 wherein

119871119894(119906) = 119908

119894 119894 = 0 119899 minus 119898 (17)

in other words from (17) we have a generalized interpolationproblem with respect to 119906 Also we suppose that a boundaryconditionB119906 = 120582 can be written in the following interpola-tion form

119871119861

119894(119906) = 119888

119894 119894 = 119899 minus 119898 + 1 119899 (18)

where 119871119861

119899minus119898+1(119906) 119871119861

119899minus119898+2(119906) 119871

119861

119899(119906) are functionals on

119883lowastNow solving simultaneous interpolation problems (17)

and (18) is a principal part of this paper The next theoremgives the explicit form of the interpolation problems (17) and(18)

Theorem 7 Suppose 119883 is a Hilbert space and 119883119899

=

1205930 1205931 120593

119899 isin 119883 is a finite dimensional subspace of 119883

with dimension 119899 then I119899(119906) the generalized interpolation

problem from (17) and (18) is of the form

I119899(119906) =

119899

sum

119894=0

119888119894120593119894(119909) +

119899

sum

119894=0

119908119894120601119894(119909) (19)

where 119871119894 119871119861

119894isin 119883lowast are functionals of119883lowast wherein

119871119894(120593119895) = 120575119894119895 119894 = 1 119899 minus 119898 forall119895

119871119861

119894(120593119895) = 120575119894119895 119894 = 119899 minus 119898 + 1 119899 forall119895

(20)

119871119894(120601119895) = 120575119894119895 119894 = 1 119899 minus 119898 forall119895

119871119861

119894(120601119895) = 120575119894119895 119894 = 119899 minus 119898 + 1 119899 forall119895

(21)

Proof By imposing functionals 119871119894and 119871

119861

119894on both sides of

(19) and using properties (20) and (21) it is easy to show that(19) satisfies conditions (17) and (18)

Remark 8 The particular cases of interpolation (19) areLagrange Barycentric Hermite and Birkhoff interpolations

The important point inTheorem 7 is the existence of basisfunctions 120593

119895(119909)119899

119895=1and 120601

119895(119909)119899

119895=1which satisfy (20) and

(21) respectively We can answer this question by the conceptof linear independency of operators We perform this for120593119895(119909)119899

119895=1 and in similarmannerwe can obtain the necessary

and sufficient conditions of existence of 120601119895(119909)119899

119895=1 Now we

consider an arbitrary basis function 120595119895(119909)119899

119895=1of 119883119899and we

construct new basis functions which satisfy (20) and (21)

For this goal suppose

120593119895(119909) =

119895

sum

119894=1

119886119895119894120595119894 119895 = 1 119899 (22)

then we must obtain the 119899(119899 + 1)2 unknown coefficients119886119895119894 119894 = 1 119899 We start from 120593

0(119909) (119895 = 0) this means that

1205930(119909) = 119886

001205950(119909) (23)

from (23) wemust obtain one unknown coefficient 11988600

and itis sufficient for imposing the arbitrary functionals 119871

119894or 119871119861119894isin

119883lowast on (22) for obtaining the unknown coefficient 119886

00 so

1198710(1205930(119909)) = 119886

001198710(1205950(119909)) (24)

thus

11988600

=

1198710(1205930(119909))

1198710(1205950(119909))

1198710(1205950(119909)) = 0 (25)

Also for 1205931(119909)(119895 = 1) we have

1205931(119909) = 119886

101205950(119909) + 119886

111205951(119909) (26)

for unknown coefficients 11988610

and 11988611 If we impose two

functionals 1198710and 119871

1isin 119883lowast on (26) for obtaining unknown

coefficients 11988610

and 11988611

we can obtain

1198710(1205931(119909)) = 119886

101198710(1205950(119909)) + 119886

111198710(1205951(119909))

1198711(1205931(119909)) = 119886

101198711(1205950(119909)) + 119886

111198711(1205951(119909))

(27)

the unknown coefficients 11988610

and 11988611

exist if and only if10038161003816100381610038161003816100381610038161003816

1198710(1205950(119909)) 119871

0(1205951(119909))

1198711(1205950(119909)) 119871

1(1205951(119909))

10038161003816100381610038161003816100381610038161003816

= 0 (28)

By repeating this procedure we can obtain the result in whichthe unknown coefficients 119886

1198990 1198861198991 119886

119899119899exist if and only if

det (119871119894(120595119895)) =

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198710(1205951(119909)) sdot sdot sdot 119871

119899(120595119899(119909))

sdot sdot sdot

119871119899(1205951(119909)) sdot sdot sdot 119871

119899(120595119899(119909))

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (29)

and this result is equivalent to the independence of function-als 1198711 1198712 119871

119899minus119898 119871119861

119899minus119898+1 119871

119861

119899isin 119883lowast

Now for implementing the general pseudospectralmethod for numerical solution of general operator equation(8) with respect to its boundary condition first we substitutea general interpolation operator (19) in (8) The generalinterpolation function 119906(119909) (the exact solution of (8) withrespect to its boundary condition) fromTheorem 7 is

119906 (119909) ≃ I119899(119906) (119909) =

119899

sum

119894=0

119888119894120593119894(119909) +

119899

sum

119894=0

119908119894120593119894(119909) (30)

where119908119894and 119888119894are defined in (17) and (18) Also we consider

the general interpolation problem of the function 119892(119909) (thefunction of the right-hand side of (8)) as

1198711

119894(119892) = 119887

119894 119894 = 0 119899 (31)

International Journal of Computational Mathematics 5

where 11987110(119892) 119871

1

119899(119892) isin 119883

lowast are functionals and 1198870 1198871 119887

119899

are constants then we have

119892 (119909) ≃ I119899(119892) (119909) =

119899

sum

119894=0

119887119894120593119894(119909) (32)

Now by substituting (19) and (32) in (8) we obtain

L (I119899(119906) (119909)) +N (I

119899(119906) (119909)) ≃ I

119899(119892) (119909) (33)

then using some project operators on (33) as 1198751 119875

119899

defined on119883 we obtain

119875119894(L (I

119899(119906) (119909)) +N (I

119899(119906) (119909))) ≃ 119875

119894(I119899(119892) (119909))

119894 = 1 119899

(34)

From (34) we obtain 119899 algebraic equations Simplificationof the left-hand side of (34) is an important subject and weexpress this subject for linear and nonlinear operators (I)

and (N)

(i) Linear Case In the linear case if we suppose that theprojection operators are linear then from property of linearoperatorL we have

119875119905L (I

119899(119906)) = 119875

119905L(

119899

sum

119894=0

119888119894120593119894+

119899

sum

119894=0

119908119894120593119894)

=

119899

sum

119894=0

119888119894119875119905L (120593119894) +

119899

sum

119894=0

119908119894119875119905L (120593119894)

(35)

so for obtaining a matrix relation between the values of119875119905L(I119899(119906)) and 119888

119894and 119908

119894 it is sufficient to calculate the

119875119905L(120593119894) values as the following

[

[

[

[

[

[

1198750L (I

119899(119906))

119875119899L (I

119899(119906))

]

]

]

]

]

]

=

[

[

[

[

[

[

11989600

sdot sdot sdot sdot sdot sdot 1198960119899

sdot sdot sdot sdot sdot sdot

sdot sdot sdot sdot sdot sdot

1198961198990

sdot sdot sdot sdot sdot sdot 119896119899119899

]

]

]

]

]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

(

[

[

[

[

[

[

1198880

119888119899

]

]

]

]

]

]

+

[

[

[

[

[

[

1199080

119908119899

]

]

]

]

]

]

)

(36)

where

119896119894119895

= 119875119894L (120593119895) 119894 119895 = 0 119899 (37)

The previous matrix (under bracket) is a square matrix of(119873+ 1) times (119873+ 1) dimensions and has high application in thepseudospectral method In particular case when the linearoperator is a derivative this matrix is named as derivativematrix and is very ill conditioned

(ii) Nonlinear Case The nonlinear case is almost cumber-some and different ideas can be used The first idea is to

approximate the nonlinear operator N with some linearoperator F and using the method of part (i) But thisapproach is very complicated and approximating a nonlinearoperator with linear operators in general is difficult and wasdown only for particular operators [13] The second ideais based on approximating a nonlinear operator via Taylorseries [14] Also another approach for simplifying nonlinearoperator is based on orthogonal expansions (general Fourierseries) of N(I

119899(119906)) For this goal we must obtain an

orthogonal system 120601119894(119909)119899

119894=1from basis functions 120593

119895(119909)119899

119895=1

using Gram-Schmidt algorithmThen it is not difficult to seethat

I119899(119906) (119909) =

119899

sum

119894=0

120582119894120593119894(119909) +

119899

sum

119894=0

119888119894120593119894(119909) ≃

119873

sum

119894=0

119886119894120601119894(119909) (38)

where

119886119894=

(I119873(119906) 120601

119894)

(120601119894 120601119894)

(39)

Also forN(I119899(119906)(119909)) we have

N (I119899(119906) (119909)) ≃

119899

sum

119894=0

119887119894120601119894(119909) (40)

where

119887119894=

(N (I119899(119906)) 120601

119894)

(120601119894 120601119894)

(41)

We must note that the 119886119894and 119887119894coefficients are functions of

119871119894and 119871

119861

119894

Now for case study let119883 be the space of polynomials withreal coefficients wherein it is a Hilbert space with respect to1198712 norm and119883

119899is a subspace of119883 with dimension 119899

Also suppose that the functionals1198711 1198712 119871

119899in (9) are

defined as

119871119894(120593) = 119888

119894 (42)

where 119888119894are also functionals of 120593

119895(119909)119899

119895=1 For instance

the well known functionals with high applications are 119888119894=

int 119905119894120593119889119909 (moment operator) 119888

119894= (120593)

(119894)(119909) (ith-derivative

operator) 119888119894= 120593(119909 minus ℎ

119894) ℎ119894isin 119877 (shift operator) and 119888

119894=

120593(119905119894) (119905119894are distinct points and in particular case the roots

of shifted orthogonal polynomials as Hermite and Laguerrepolynomials in unbounded domains and shifted Chebyshevand Legendre and in general case Jacobi polynomials inbounded domains [15ndash17]) Also in the higher dimensionswe can define the suitable functionals

31 Error Analysis The error analysis of pseudospectralmethod in the general case is very difficult and we canperform this analysis only in polynomial case For this goalat first we introduce some useful notations and theorems thatare used in this paper

Definition 9 The Jacobi polynomials [18] 119875(120572120573)

119894(119909) are

defined as the orthogonal polynomials with respect to the

6 International Journal of Computational Mathematics

weight function119908120572120573

(119909) = (1minus119909)120572(1 +119909)

120573 (120572 gt minus1 120573 gt minus1)

on (minus1 1) An explicit formula for these polynomials is givenby

119875(120572120573)

119899(119909)

=

1

2119899

119899

sum

119894=0

(

119899 + 120572

119894)(

119899 + 120573

119899 minus 119894) (119909 minus 1)

119899minus119894(119909 + 1)

119894

(43)

Further we have

119889

119889119909

119875(120572120573)

119899(119909) =

1

2

(119899 + 120572 + 120573 + 1) 119875(120572+1120573+1)

119899minus1(119909) (44)

Jacobi polynomials have an important application in severalfields of numerical analysis such as quadratures formula andspectral methods Also the shifted Jacobi polynomials aredefined as orthogonal polynomials in the interval [119886 119887] withrespect to weight function 119908

120572120573(119909) = (119887 minus 119909)

120572(119886 minus 119909)

120573 (120572 gt

minus1 120573 gt minus1) with change of variable 119910 = (2(119887 minus 119886))119909 +

((119886 + 119887)(119886 minus 119887)) For more properties and applications ofthese polynomials see [15ndash17] Now let Λ = [minus1 1] and let119875(120572120573)

119873+1(119909) be the Jacobi polynomial of degree119873+1with respect

to weight function 119908(120572120573)

(119909) = (1 minus 119909)120572(1 + 119909)

120573 (120572 120573 gt minus1)

and let P119873be the polynomials spaces of degree less than or

equal to119873

Remark 10 The best choice of points for polynomial inter-polation are the roots of orthogonal polynomials and in par-ticular case Jacobi polynomials due to their small Lebesgueconstants [6]

The symbol 1198712

119908(120572120573)(Λ) shows the space of measurable

functions whose square is Lebesgue integrable in Λ relativeto the weight function119908

(120572120573) The inner product and norm of1198712

119908(120572120573)(Λ) are defined by

(119906 V)119908(120572120573)Λ

= int

Λ

119906 (119909) V (119909) 119908(120572120573) (119909) 119889119909

119906119908(120572120573)Λ

= radic(119906 119906)119908(120572120573)

forall119906 V isin 1198712

119908(120572120573) (Λ)

(45)

Also another useful space is

119867119898

119908(120572120573) = V V

119898119908(120572120573) lt infin (46)

with seminorm and norms respectively Consider

|V|119898119908(120572120573) =

1003817100381710038171003817120597119898

119909V1003817100381710038171003817119908(120572120573)

V119898119908(120572120573) = (

119898

sum

119894=0

|V|2119894119908(120572120573))

12

(47)

But an applicable norm that appears in bounding of errors ofthe spectral method is

|V|119867119898119908(120572120573) = (

119898

sum

119896=119890

10038171003817100381710038171003817120597119896

119909V10038171003817100381710038171003817

2

119908(120572120573)

)

12

119890 = min (119898119873 + 1)

(48)

Now we present a useful theorem

Theorem 11 If ℎ isin 119867119898

119908(120572120573)(Λ)119898 ge 1 then one has

1003817100381710038171003817ℎ minusI

119873(ℎ)

1003817100381710038171003817119908(120572120573) le 119862119873

minus11989810038161003816100381610038161198911003816100381610038161003816119867119898

119908(120572120573)

(49)

where I119873(ℎ) is a Lagrange interpolation in distinct points

119909(120572120573)

119894(roots of Jacobi polynomials)

Proof See [4 5]

Noting to Theorem 11 we can obtain an error boundfor any classes of polynomial interpolation such as Hermiteinterpolation Birkhoff interpolation and other classes Thismeans that if we have an arbitrary polynomial interpolationI1119873(ℎ) thenwe can transform the interpolationI1

119873(ℎ) to the

Lagrange interpolation I119873(ℎ) and then we use Theorem 11

for obtaining an error bound Now by using Theorem 11 inthe next theorem we obtain an upper bound for the error ofour approximate solution of (8)

Theorem 12 Suppose L and N are the linear and nonlinearoperators with respect to (8) wherein L has inversion and119891 119892 isin 119867

119898

119908(120572120573)(Λ) (119898 ge 1) are satisfied in (8) then

10038171003817100381710038171003817119891 minusI

1

119873(119891)

10038171003817100381710038171003817119908(120572120573)

le 119862119873minus119898

(

10038161003816100381610038161003816Lminus1N (119891)

10038161003816100381610038161003816119867119898

119908(120572120573)

+

10038161003816100381610038161003816Lminus1119892

10038161003816100381610038161003816119867119898

119908(120572120573)

)

(50)

whereI1119873(119891) is an arbitrary polynomial interpolation satisfied

in interpolation conditions

Proof First we must note that I1119873(119891) can be expressed via

Lagrange interpolation and for this goal it is sufficient tointerpolate function I

119873(119891) in 119873 distinct points 119909

119895 119895 =

1 119873 From (8) and invertibility ofL we have

119891 +Lminus1N119891 = L

minus1(119892) (51)

also due to pseudospectral method we obtain

119891 (119909119895) +L

minus1N119891 (119909

119895) = L

minus1119892 (119909119895) (52)

by producing both sides of (52) with 119871119895(119909) (Lagrange inter-

polation basis) and sum from 119894 = 0 119873 we obtain

I119873(119891) +I

119873(Lminus1N (119891)) = I

119873(Lminus1

(119892)) (53)

Now by subtracting (51) from (53) we get

I119873(119891) = L

minus1N (119891) minusI

119873(Lminus1N (119891))

+Lminus1 (119892) minusI119873(Lminus1 (119892))

(54)

International Journal of Computational Mathematics 7

therefore we have10038171003817100381710038171003817119891 minusI

1

119873(119891)

10038171003817100381710038171003817119908(120572120573)

=1003817100381710038171003817119891 minusI

119873(119891)

1003817100381710038171003817119908(120572120573)

le

10038171003817100381710038171003817Lminus1N (119891) minusI

119873(Lminus1N (119891))

10038171003817100381710038171003817119908(120572120573)

+

10038171003817100381710038171003817Lminus1

(119892) minusI119873(Lminus1

(119892))

10038171003817100381710038171003817119908(120572120573)

(55)

Finally by usingTheorem 11 we obtain10038171003817100381710038171003817119891 minusI

1

119873(119891)

10038171003817100381710038171003817119908(120572120573)

=1003817100381710038171003817119891 minusI

119873(119891)

1003817100381710038171003817119908(120572120573)

le 119862119873minus119898

(

10038161003816100381610038161003816Lminus1N119891

10038161003816100381610038161003816119867119898119873

119908(120572120573)

+

10038161003816100381610038161003816Lminus1119892

10038161003816100381610038161003816119867119898119873

119908(120572120573)

)

(56)

therefore the proof is completed

4 The Test Experiments

Experiment 1 Let us consider the following nonlinear differ-ential equation problem [19]

11991010158401015840= minus (1 + 001119910

2) 119910 + 001cos3 (119909)

119910 (minus1) = cos (minus1)

119910 (1) = cos (1)

(57)

with the exact solution 119910(119909) = cos(119909)By simplifying (57) we have

11991010158401015840= minus 119910 minus 001119910

3+ 001cos3 (119909)

119910 (minus1) = cos (minus1)

119910 (1) = cos (1)

(58)

thus the linear and nonlinear parts of our ordinary differen-tial equation respectively are

L (119910) =

1

sum

119894=0

L119894(119910)

N (119910) = 0011199103(119909)

(59)

where

L0(119910) = 119910

10158401015840(119909) L

1(119910) = 119910 (60)

Now we consider 119873 + 1 collocation points 1199090

=

minus1 119909(120572120573)

119894119873minus1

119894=1 119909119873

= 1 where 119909(120572120573)

119894119873minus1

119894=1are the roots of

Jacobi polynomials of degree 119873 minus 1 then we consider thisinterpolation problem

119910 (minus1) = cos (minus1)

119910 (119909(120572120573)

119894) = 119875119873+1

(119909(120572120573)

119894) 119894 = 0 119873 minus 1

119910 (1) = cos (1)

(61)

Table 1 Comparison between the exact and approximate solutionsfor 120572 = 0 120573 = 0 and119873 = 5 of Experiment 1

119909Exact

solutionPseudospectral

method Error

minus10 054030231 054030231 000000000minus080 069670671 069670667 000000004minus060 082533561 082533555 000000006minus040 092106099 092106045 000000054minus020 098006658 098006645 000000013000 100000000 099999922 000000078020 098006658 098006633 000000025040 092106099 092106056 000000043060 082533561 082533545 000000016080 069670671 069670656 00000001510 054030231 054030231 000000000

where 119875119873+1

is the corresponding polynomial interpolationwith respect to problem (61) wherein interpolate the bound-ary conditions automatically and 119910(119909

(120572120573)

119894) 119894 = 0 119873 minus

1 are the unknown coefficients In Table 1 we see thecomparison between the exact and approximate solutionsfrom pseudospectral method for 120572 = 0 120573 = 0 and119873 = 5

Experiment 2 Consider the second-order difference equa-tion problem [20]

119910 (119909 + 1) minus 119910 (119909) = 119890119909 119910 (

1

2

) = 1 (62)

The exact solution is 119910(119909) = (119890119909+119890minusradic119890minus1)(119890minus1) We have

L (119910) = 119910 (119909 + 1) minus 119910 (119909) (63)

and this equation has no nonlinear operator part Theobtained results between the exact and approximate solutionsare shown in Table 2 and Figure 1

Experiment 3 Consider the Fredholm integrodifferentialequation [21]

1199101015840(119909) = int

1

0

119890119909119905119910 (119905) 119889119905 + 119910 (119909) +

1 minus 119890119909+1

119909 + 1

0 le 119909 119910 le 1

(64)

under the initial condition

119910 (0) = 1 (65)

with the exact solution 119910(119909) = 119890119909 In this case we have

L (119910) =

2

sum

119894=0

L119894(119910) (66)

where

L0(119910) = int

1

0

119890119909119905119910 (119905) 119889119905

L1(119910) = 119910

1015840(119909) L

2(119910) = 119910 (119909)

(67)

8 International Journal of Computational Mathematics

28

26

24

22

2

18

16

14

12

1

0

y

x

01 02 03 04 05 06 07 08 09 1

Exact solutionApproximate solution

Figure 1 The obtained results from approximation solution and exact solution for 120572 = minus05 120573 = 07 and 119873 = 5 of Experiment 2

Table 2 Comparison between the exact and approximate solutionsfor 120572 = minus05 120573 = 07 and119873 = 5 of Experiment 2

119909Exact

solutionPseudospectral

method Error

00 062245933 062245922 00000001101 068366636 068366643 minus00000000702 075131058 075131034 00000002403 082606901 082606923 minus00000002204 090868985 090868925 00000006005 100000000 100000000 0000000006 110091330 110091120 00000002107 121243980 121243880 00000001008 133569560 133569450 00000001109 147191430 147191350 00000000810 162245930 162245560 000000037

Table 3 Comparison between the exact and approximate solutionsfor 120572 = minus05 120573 = 07 and119873 = 5 of Experiment 3

119909Exact

solutionPseudospectral

method Error

00 10000000 10000000 00000000001 11051709 11051701 00000000802 12214028 12214011 00000001703 13498588 13498533 00000005504 14918247 14918232 00000001505 16487213 16487211 00000000206 18221188 18221166 00000002207 20137527 20137511 00000001508 22255409 22255401 00000000709 24596031 24596030 00000000110 27182818 27182810 000000008

Now by using the pseudospectral method for 120572 =

minus05 120573 = 07 and 119873 = 5 we obtain the following resultsthat are shown in Table 3

Experiment 4 We consider the linear partial differentialequation problem [22]

119906119905119905= 119906119909119909

+ 6 119906119905(119909 0) = 4119909 119906 (119909 0) = 119909

2 (68)

with the exact solution 119906(119909 119910) = (119909 + 2119905)2

In this case we have

L (119906) = 119906119905119905minus 119906119909119909 (69)

We obtain the approximate solution of (68) on the domain[minus1 1] times [minus1 1] For this purpose we consider the discretepoints (119909(120572120573)

119894 119905(120572120573)

119895)119873

119894119895=0and the general Lagrange interpola-

tion in the discrete points (119909(120572120573)119894

119905(120572120573)

119895)119873

119894119895=0in the following

form (two-dimensional case)

119871 (119909 119905) =

119873

sum

119894=0

119873

sum

119895=0

119891 (119909(120572120573)

119894 119905(120572120573)

119895) 119871119894(119909) 119871119895(119905) (70)

where 119871119894(119909) and 119871

119895(119905) are ordinary Lagrange interpolations

By using collocation method for arbitrary parameters 120572 120573and119873 = 5 we obtain the exact solution

Experiment 5 Finally let us consider the linear fractionaldifferential equation problem [23]

11986352

119910 (119909) + 3119910 (119909) = 31199093+

6

Γ (15)

11990912

119910 (0) = 0 1199101015840(0) = 0 119910

10158401015840(0) = 0 119909 isin [minus1 1]

(71)

This equation has the exact solution 119910(119909) = 1199093

In this case we have

L (119910 (119909)) = 11986352

(119910 (119909)) + 3119910 (119909) (72)

which is a linear operator Using the presented method forevery 120572 120573 and119873 = 3 we obtain the exact solution

International Journal of Computational Mathematics 9

5 Conclusion

In this paper we have developed the expansion approach forsolving some operator equations of the form L119906 + N119906 =

119892(119909) 119909 isin Γ with respect to boundary condition B119906 = 120582For this the principal importance is the development of theexpansion approach for solving general operator equationwherein the particular cases are in the solution of integralequations (Experiment 3) ordinary differential equations(Experiment 1) difference equations (Experiment 2) partialdifferential equations (Experiment 4) and fractional differ-ential equations (Experiment 5) Also the error analysis ispresented in 119871

2(Γ) norm Using the roots of another orthog-

onal polynomials (Jacobi polynomials) rather than classicalorthogonal polynomials such as Chebyshev polynomialsand Legendre polynomials for collocation points (with freeparameters 120572 120573) in pseudospectral method is an importantsubject which is performed (Experiments 2 and 3)

Conflict of Interests

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

Acknowledgments

The authors wish to thank the anonymous referees for theirvaluable comments and suggestions Also they would like toappreciate the Editor Dr Yaohanz Li who gives them theopportunity to review the paper

References

[1] P LinzTheoretical Numerical Analysis JohnWiley amp Sons NewYork NY USA 1979

[2] D Gottlieb and S A Orszag Numerical Analysis of SpectralMethods Theory and Applications SIAM-CBMS PhiladelphiaPa USA 1977

[3] G Ben-Yu The State of Art in Spectral Methods Hong KongUniversity 1996

[4] C Canuto M Y Hussaini A Quarteroni and T A ZhangSpectral Methods Fundamentals in Single Domains Springer2006

[5] C Canutuo M Y Hussaini A Quarteroni and T A ZhangSpectral Method in Fluid Dynamics Prentice Hall EngelwoodCliffs NJ USA 1984

[6] B FornbergAPractical Quide to PseudospectralMethods Cam-bridgeMonographs on Applied and Computational Mathemat-ics Cambridge University Press Cambridge UK 1st edition1996

[7] L N Trefethen Spectral Methods in Matlab SIAM Philadel-phia Pa USA 2000

[8] J S Hesthaven S Gottlieb and D Gottlieb Spectral Methodsfor Time-Dependent Problems vol 21 ofCambridgeMonographson Applied and Computational Mathematics Cambridge Uni-versity Press Cambridge UK 2007

[9] M Deville and E Mund ldquoChebyshev pseudospectral solutionof second-order elliptic equations with finite element precon-ditioningrdquo Journal of Computational Physics vol 60 no 3 pp517ndash533 1985

[10] J P Boyd ldquoMultipole expansions and pseudospectral cardinalfunctions a new generalization of the fast Fourier transformrdquoJournal of Computational Physics vol 103 no 1 pp 184ndash1861992

[11] G N Elnagar and M Razzaghi ldquoA pseudospectral method forHammerstein equationsrdquo Journal of Mathematical Analysis andApplications vol 199 no 2 pp 579ndash591 1996

[12] M Meehan and D OrsquoRegan ldquoExistence principles for nonreso-nant operator and integral equationsrdquo Computers ampMathemat-ics with Applications vol 35 no 9 pp 79ndash87 1998

[13] A Chaillou and M Suri ldquoA posteriori estimation of thelinearization error for strongly monotone nonlinear operatorsrdquoJournal of Computational andAppliedMathematics vol 205 no1 pp 72ndash87 2007

[14] R P Kanwal and K C Liu ldquoA Taylor expansion approach forsolving integral equationsrdquo International Journal of Mathemat-ical Education in Science and Technology vol 20 pp 411ndash4141989

[15] M R Eslahchi M Dehghan and S Ahmadi Asl ldquoThe generalJacobi matrix method for solving some nonlinear ordinarydifferential equationsrdquoAppliedMathematical Modelling vol 36no 8 pp 3387ndash3398 2012

[16] A Imani A Aminataei and A Imani ldquoCollocation method viaJacobi polynomials for solving nonlinear ordinary differentialequationsrdquo International Journal of Mathematics and Mathe-matical Sciences vol 2011 Article ID 673085 11 pages 2011

[17] M R Eslahchi and M Dehghan ldquoApplication of Taylor seriesin obtaining the orthogonal operational matrixrdquo Computers ampMathematics with Applications vol 61 no 9 pp 2596ndash26042011

[18] D Funaro Polynomial Approximation of Differential Equationsvol 8 of Lecture Notes in Physics Monographs Springer BerlinGermany 1992

[19] D F Griffiths and D J Higham Numerical Methods forOrdinaryDifferential Equations SpringerUndergraduateMath-ematics Series Springer London UK 2010

[20] M Gulsu M Sezer and B Tanay ldquoA matrix method for solvinghigh-order linear difference equations with mixed argumentusing hybrid Legendre and Taylor polynomialsrdquo Journal of theFranklin Institute vol 343 no 6 pp 647ndash659 2006

[21] N Kurt and M Sezer ldquoPolynomial solution of high-orderlinear Fredholm integro-differential equations with constantcoefficientsrdquo Journal of the Franklin Institute vol 345 no 8 pp839ndash850 2008

[22] V S Vladimirov A Collection of Problems on the Equations ofMathematical Physics Mir Publishers Moscow Russia 1968

[23] A H Bhrawy M M Alghamdi and T M Taha ldquoA newmodified generalized Laguerre operational matrix of fractionalintegration for solving fractional differential equations on thehalf linerdquoAdvances in Difference Equations vol 2012 article 1792012

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Computational Mathematics 3

matrix are explicitly known (see eg [7]) Using such anapproach for solving boundary value problems involves thesolution of linear systems of equations which are knownto be very ill-conditioned for example for methods basedon orthogonal polynomials the condition number of theapproximate first order operator grows like 119873

2 while thecondition number of second order operator in general scaledwith 119873

4 For more review literatures about this method see[6] Precondition procedure is the method of reducing theobtained ill-condition system from pseudospectral methods[6] This method is used for the numerical solution ofintegral equations [11] partial differential equations [8 10]and ordinary differential equations [9 10]

14 Operator Equations Many important engineering prob-lems fall into the category of being operator equations withboundary operator conditions such as integral differenceand partial differential equations The general form of theseequations is in the form

L119906 +N119906 = 119892 (119909) 119909 isin Γ (8)

with respect to boundary condition B119906 = 120582 where L NandB are linear nonlinear and boundary operators respec-tively related to a suitable Hilbert space Γ is the domain ofapproximation and 119892(119909) isin 119871

2(Γ) is an arbitrary function

The necessary and sufficient conditions for existence anduniqueness of (8) can be seen in [12] Obtaining the explicitsolutions of operator equation (8) in general is a difficultproblem and it depends on the properties of operators LN and B Also some numerical methods are presented forsolving such operator problems as iterativemethodsmethodof paper [13] and so on Simplification of the nonlinear termN in the operator equation (8) to the linear operators for easyand efficient implementation is an important subject Somemethods exist for this simplification such as using Taylorseries and linearization method [13] In Section 2 we explainthis subject very well

2 Spectral Method for GeneralOperator Equation

In this section we describe the spectral methods for solvinggeneral operator equation L119906 + N119906 = 119892(119909) 119909 isin Γ withrespect to boundary conditionB119906 = 120582 For this purpose firstconsider suitable basis set of functions such as 120601

119894(119909)119899

119894=0 and

expand a function 119891(119909) (exact solution of operator equation(8)) based on these basis functions as

119891 (119909) ≃ 119891119899(119909) =

119899

sum

119894=0

119886119894120601119894(119909) (9)

From basis property of 120601119894(119909)119899

119894=0 we can conclude

119891 (119909) =

infin

sum

119894=0

119886119894120601119894(119909) (10)

Now we substitute the expansion series (9) in operatorequation (8) and its boundary conditions as the following

L (119891119899(119909)) +N (119891

119899(119909)) ≃ 119892 (119909) 119909 isin Γ

B (119891119899(119909)) ≃ 120582

(11)

If the basis functions 120601119894(119909)119899

119894=0 automatically satisfy bound-

ary conditions then they will be called Galerkin basisfunctions From (11) we must simplify two terms L(119891

119899(119909))

andN(119891119899(119909)) so we perform this simplification in two cases

But simplification of the first case L(119891119899(119909)) is easy because

by using the linear property of operatorL we obtain

L (119891119899(119909)) = L(

119899

sum

119894=0

119886119894120601119894(119909)) =

119899

sum

119894=0

119886119894L (120601119894(119909)) (12)

But the nonlinear case N(119891119899(119909)) is almost cumbersome

and some ideas exist for its simplification Now we presentthe idea for simplification of the nonlinear term N(119891

119899(119909))

For this goal based on basis functions 120601119894(119909)119899

119894=1 and using

Gram-Schmidt algorithm we must obtain an orthogonalsystem 120593

119895(119909)119899

119895=1 Then it is not difficult to see that

N (119891119899(119909)) ≃

119899

sum

119894=0

119887119894120593119894(119909) (13)

119887119894=

(N (119891119899(119909)) 120593

119894)

(120593119894 120593119894)

(14)

Now from (11) and using (13) we obtain

119899

sum

119894=0

119886119894L (120601119894(119909)) +

119899

sum

119894=0

119887119894120593119894(119909) ≃ 119892 (119909) 119909 isin Γ

B (119891119899(119909)) ≃ 120582

(15)

By using some project operators on (15) as 1198751 119875

119899defined

on119883 we obtain

119875119894(

119899

sum

119894=0

119886119894L (120601119894(119909))) + 119875

119894(

119899

sum

119894=0

119887119894120593119894(119909)) ≃ 119875

119894(119892 (119909))

119909 isin Γ 119894 = 0 119899

B (119891119899(119909)) ≃ 120582

(16)

If we suppose that the boundary conditions give us 119898

equations then from (16) we obtain 119899 + 119898 + 1 equations for119899 + 1 unknown coefficients So for obtaining the unknownswe must eliminate 119898 equations for satisfying the boundaryconditions and the obtained system must be solved viasuitable iteration methods such as Newton method

Remark 6 In (16) if119875119894(119891(119909)) = int

119887

119886119891(119909)119908

119894(119909)119889119909where119908

119894(119909)

is a weight function the method is named as Tau methodAlso if119875

119894= int

119887

119886120575(119909minus119909

119894)119889119909 where 119909

119894119899

119894=0are the set of distinct

points and 120575(119909) is a dirac function then themethod is namedas collocation method

4 International Journal of Computational Mathematics

3 The General Pseudospectral Method forGeneral Operator Equation

In this section we describe the general pseudospectralmethod for numerical solution of the general operatorequations For this goal first we introduce the generalpseudospectral method Let 119871

1 1198712 119871

119899minus119898isin 119883lowast be some

functionals and scalars 1199081 1199082 119908

119899minus119898 wherein

119871119894(119906) = 119908

119894 119894 = 0 119899 minus 119898 (17)

in other words from (17) we have a generalized interpolationproblem with respect to 119906 Also we suppose that a boundaryconditionB119906 = 120582 can be written in the following interpola-tion form

119871119861

119894(119906) = 119888

119894 119894 = 119899 minus 119898 + 1 119899 (18)

where 119871119861

119899minus119898+1(119906) 119871119861

119899minus119898+2(119906) 119871

119861

119899(119906) are functionals on

119883lowastNow solving simultaneous interpolation problems (17)

and (18) is a principal part of this paper The next theoremgives the explicit form of the interpolation problems (17) and(18)

Theorem 7 Suppose 119883 is a Hilbert space and 119883119899

=

1205930 1205931 120593

119899 isin 119883 is a finite dimensional subspace of 119883

with dimension 119899 then I119899(119906) the generalized interpolation

problem from (17) and (18) is of the form

I119899(119906) =

119899

sum

119894=0

119888119894120593119894(119909) +

119899

sum

119894=0

119908119894120601119894(119909) (19)

where 119871119894 119871119861

119894isin 119883lowast are functionals of119883lowast wherein

119871119894(120593119895) = 120575119894119895 119894 = 1 119899 minus 119898 forall119895

119871119861

119894(120593119895) = 120575119894119895 119894 = 119899 minus 119898 + 1 119899 forall119895

(20)

119871119894(120601119895) = 120575119894119895 119894 = 1 119899 minus 119898 forall119895

119871119861

119894(120601119895) = 120575119894119895 119894 = 119899 minus 119898 + 1 119899 forall119895

(21)

Proof By imposing functionals 119871119894and 119871

119861

119894on both sides of

(19) and using properties (20) and (21) it is easy to show that(19) satisfies conditions (17) and (18)

Remark 8 The particular cases of interpolation (19) areLagrange Barycentric Hermite and Birkhoff interpolations

The important point inTheorem 7 is the existence of basisfunctions 120593

119895(119909)119899

119895=1and 120601

119895(119909)119899

119895=1which satisfy (20) and

(21) respectively We can answer this question by the conceptof linear independency of operators We perform this for120593119895(119909)119899

119895=1 and in similarmannerwe can obtain the necessary

and sufficient conditions of existence of 120601119895(119909)119899

119895=1 Now we

consider an arbitrary basis function 120595119895(119909)119899

119895=1of 119883119899and we

construct new basis functions which satisfy (20) and (21)

For this goal suppose

120593119895(119909) =

119895

sum

119894=1

119886119895119894120595119894 119895 = 1 119899 (22)

then we must obtain the 119899(119899 + 1)2 unknown coefficients119886119895119894 119894 = 1 119899 We start from 120593

0(119909) (119895 = 0) this means that

1205930(119909) = 119886

001205950(119909) (23)

from (23) wemust obtain one unknown coefficient 11988600

and itis sufficient for imposing the arbitrary functionals 119871

119894or 119871119861119894isin

119883lowast on (22) for obtaining the unknown coefficient 119886

00 so

1198710(1205930(119909)) = 119886

001198710(1205950(119909)) (24)

thus

11988600

=

1198710(1205930(119909))

1198710(1205950(119909))

1198710(1205950(119909)) = 0 (25)

Also for 1205931(119909)(119895 = 1) we have

1205931(119909) = 119886

101205950(119909) + 119886

111205951(119909) (26)

for unknown coefficients 11988610

and 11988611 If we impose two

functionals 1198710and 119871

1isin 119883lowast on (26) for obtaining unknown

coefficients 11988610

and 11988611

we can obtain

1198710(1205931(119909)) = 119886

101198710(1205950(119909)) + 119886

111198710(1205951(119909))

1198711(1205931(119909)) = 119886

101198711(1205950(119909)) + 119886

111198711(1205951(119909))

(27)

the unknown coefficients 11988610

and 11988611

exist if and only if10038161003816100381610038161003816100381610038161003816

1198710(1205950(119909)) 119871

0(1205951(119909))

1198711(1205950(119909)) 119871

1(1205951(119909))

10038161003816100381610038161003816100381610038161003816

= 0 (28)

By repeating this procedure we can obtain the result in whichthe unknown coefficients 119886

1198990 1198861198991 119886

119899119899exist if and only if

det (119871119894(120595119895)) =

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198710(1205951(119909)) sdot sdot sdot 119871

119899(120595119899(119909))

sdot sdot sdot

119871119899(1205951(119909)) sdot sdot sdot 119871

119899(120595119899(119909))

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (29)

and this result is equivalent to the independence of function-als 1198711 1198712 119871

119899minus119898 119871119861

119899minus119898+1 119871

119861

119899isin 119883lowast

Now for implementing the general pseudospectralmethod for numerical solution of general operator equation(8) with respect to its boundary condition first we substitutea general interpolation operator (19) in (8) The generalinterpolation function 119906(119909) (the exact solution of (8) withrespect to its boundary condition) fromTheorem 7 is

119906 (119909) ≃ I119899(119906) (119909) =

119899

sum

119894=0

119888119894120593119894(119909) +

119899

sum

119894=0

119908119894120593119894(119909) (30)

where119908119894and 119888119894are defined in (17) and (18) Also we consider

the general interpolation problem of the function 119892(119909) (thefunction of the right-hand side of (8)) as

1198711

119894(119892) = 119887

119894 119894 = 0 119899 (31)

International Journal of Computational Mathematics 5

where 11987110(119892) 119871

1

119899(119892) isin 119883

lowast are functionals and 1198870 1198871 119887

119899

are constants then we have

119892 (119909) ≃ I119899(119892) (119909) =

119899

sum

119894=0

119887119894120593119894(119909) (32)

Now by substituting (19) and (32) in (8) we obtain

L (I119899(119906) (119909)) +N (I

119899(119906) (119909)) ≃ I

119899(119892) (119909) (33)

then using some project operators on (33) as 1198751 119875

119899

defined on119883 we obtain

119875119894(L (I

119899(119906) (119909)) +N (I

119899(119906) (119909))) ≃ 119875

119894(I119899(119892) (119909))

119894 = 1 119899

(34)

From (34) we obtain 119899 algebraic equations Simplificationof the left-hand side of (34) is an important subject and weexpress this subject for linear and nonlinear operators (I)

and (N)

(i) Linear Case In the linear case if we suppose that theprojection operators are linear then from property of linearoperatorL we have

119875119905L (I

119899(119906)) = 119875

119905L(

119899

sum

119894=0

119888119894120593119894+

119899

sum

119894=0

119908119894120593119894)

=

119899

sum

119894=0

119888119894119875119905L (120593119894) +

119899

sum

119894=0

119908119894119875119905L (120593119894)

(35)

so for obtaining a matrix relation between the values of119875119905L(I119899(119906)) and 119888

119894and 119908

119894 it is sufficient to calculate the

119875119905L(120593119894) values as the following

[

[

[

[

[

[

1198750L (I

119899(119906))

119875119899L (I

119899(119906))

]

]

]

]

]

]

=

[

[

[

[

[

[

11989600

sdot sdot sdot sdot sdot sdot 1198960119899

sdot sdot sdot sdot sdot sdot

sdot sdot sdot sdot sdot sdot

1198961198990

sdot sdot sdot sdot sdot sdot 119896119899119899

]

]

]

]

]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

(

[

[

[

[

[

[

1198880

119888119899

]

]

]

]

]

]

+

[

[

[

[

[

[

1199080

119908119899

]

]

]

]

]

]

)

(36)

where

119896119894119895

= 119875119894L (120593119895) 119894 119895 = 0 119899 (37)

The previous matrix (under bracket) is a square matrix of(119873+ 1) times (119873+ 1) dimensions and has high application in thepseudospectral method In particular case when the linearoperator is a derivative this matrix is named as derivativematrix and is very ill conditioned

(ii) Nonlinear Case The nonlinear case is almost cumber-some and different ideas can be used The first idea is to

approximate the nonlinear operator N with some linearoperator F and using the method of part (i) But thisapproach is very complicated and approximating a nonlinearoperator with linear operators in general is difficult and wasdown only for particular operators [13] The second ideais based on approximating a nonlinear operator via Taylorseries [14] Also another approach for simplifying nonlinearoperator is based on orthogonal expansions (general Fourierseries) of N(I

119899(119906)) For this goal we must obtain an

orthogonal system 120601119894(119909)119899

119894=1from basis functions 120593

119895(119909)119899

119895=1

using Gram-Schmidt algorithmThen it is not difficult to seethat

I119899(119906) (119909) =

119899

sum

119894=0

120582119894120593119894(119909) +

119899

sum

119894=0

119888119894120593119894(119909) ≃

119873

sum

119894=0

119886119894120601119894(119909) (38)

where

119886119894=

(I119873(119906) 120601

119894)

(120601119894 120601119894)

(39)

Also forN(I119899(119906)(119909)) we have

N (I119899(119906) (119909)) ≃

119899

sum

119894=0

119887119894120601119894(119909) (40)

where

119887119894=

(N (I119899(119906)) 120601

119894)

(120601119894 120601119894)

(41)

We must note that the 119886119894and 119887119894coefficients are functions of

119871119894and 119871

119861

119894

Now for case study let119883 be the space of polynomials withreal coefficients wherein it is a Hilbert space with respect to1198712 norm and119883

119899is a subspace of119883 with dimension 119899

Also suppose that the functionals1198711 1198712 119871

119899in (9) are

defined as

119871119894(120593) = 119888

119894 (42)

where 119888119894are also functionals of 120593

119895(119909)119899

119895=1 For instance

the well known functionals with high applications are 119888119894=

int 119905119894120593119889119909 (moment operator) 119888

119894= (120593)

(119894)(119909) (ith-derivative

operator) 119888119894= 120593(119909 minus ℎ

119894) ℎ119894isin 119877 (shift operator) and 119888

119894=

120593(119905119894) (119905119894are distinct points and in particular case the roots

of shifted orthogonal polynomials as Hermite and Laguerrepolynomials in unbounded domains and shifted Chebyshevand Legendre and in general case Jacobi polynomials inbounded domains [15ndash17]) Also in the higher dimensionswe can define the suitable functionals

31 Error Analysis The error analysis of pseudospectralmethod in the general case is very difficult and we canperform this analysis only in polynomial case For this goalat first we introduce some useful notations and theorems thatare used in this paper

Definition 9 The Jacobi polynomials [18] 119875(120572120573)

119894(119909) are

defined as the orthogonal polynomials with respect to the

6 International Journal of Computational Mathematics

weight function119908120572120573

(119909) = (1minus119909)120572(1 +119909)

120573 (120572 gt minus1 120573 gt minus1)

on (minus1 1) An explicit formula for these polynomials is givenby

119875(120572120573)

119899(119909)

=

1

2119899

119899

sum

119894=0

(

119899 + 120572

119894)(

119899 + 120573

119899 minus 119894) (119909 minus 1)

119899minus119894(119909 + 1)

119894

(43)

Further we have

119889

119889119909

119875(120572120573)

119899(119909) =

1

2

(119899 + 120572 + 120573 + 1) 119875(120572+1120573+1)

119899minus1(119909) (44)

Jacobi polynomials have an important application in severalfields of numerical analysis such as quadratures formula andspectral methods Also the shifted Jacobi polynomials aredefined as orthogonal polynomials in the interval [119886 119887] withrespect to weight function 119908

120572120573(119909) = (119887 minus 119909)

120572(119886 minus 119909)

120573 (120572 gt

minus1 120573 gt minus1) with change of variable 119910 = (2(119887 minus 119886))119909 +

((119886 + 119887)(119886 minus 119887)) For more properties and applications ofthese polynomials see [15ndash17] Now let Λ = [minus1 1] and let119875(120572120573)

119873+1(119909) be the Jacobi polynomial of degree119873+1with respect

to weight function 119908(120572120573)

(119909) = (1 minus 119909)120572(1 + 119909)

120573 (120572 120573 gt minus1)

and let P119873be the polynomials spaces of degree less than or

equal to119873

Remark 10 The best choice of points for polynomial inter-polation are the roots of orthogonal polynomials and in par-ticular case Jacobi polynomials due to their small Lebesgueconstants [6]

The symbol 1198712

119908(120572120573)(Λ) shows the space of measurable

functions whose square is Lebesgue integrable in Λ relativeto the weight function119908

(120572120573) The inner product and norm of1198712

119908(120572120573)(Λ) are defined by

(119906 V)119908(120572120573)Λ

= int

Λ

119906 (119909) V (119909) 119908(120572120573) (119909) 119889119909

119906119908(120572120573)Λ

= radic(119906 119906)119908(120572120573)

forall119906 V isin 1198712

119908(120572120573) (Λ)

(45)

Also another useful space is

119867119898

119908(120572120573) = V V

119898119908(120572120573) lt infin (46)

with seminorm and norms respectively Consider

|V|119898119908(120572120573) =

1003817100381710038171003817120597119898

119909V1003817100381710038171003817119908(120572120573)

V119898119908(120572120573) = (

119898

sum

119894=0

|V|2119894119908(120572120573))

12

(47)

But an applicable norm that appears in bounding of errors ofthe spectral method is

|V|119867119898119908(120572120573) = (

119898

sum

119896=119890

10038171003817100381710038171003817120597119896

119909V10038171003817100381710038171003817

2

119908(120572120573)

)

12

119890 = min (119898119873 + 1)

(48)

Now we present a useful theorem

Theorem 11 If ℎ isin 119867119898

119908(120572120573)(Λ)119898 ge 1 then one has

1003817100381710038171003817ℎ minusI

119873(ℎ)

1003817100381710038171003817119908(120572120573) le 119862119873

minus11989810038161003816100381610038161198911003816100381610038161003816119867119898

119908(120572120573)

(49)

where I119873(ℎ) is a Lagrange interpolation in distinct points

119909(120572120573)

119894(roots of Jacobi polynomials)

Proof See [4 5]

Noting to Theorem 11 we can obtain an error boundfor any classes of polynomial interpolation such as Hermiteinterpolation Birkhoff interpolation and other classes Thismeans that if we have an arbitrary polynomial interpolationI1119873(ℎ) thenwe can transform the interpolationI1

119873(ℎ) to the

Lagrange interpolation I119873(ℎ) and then we use Theorem 11

for obtaining an error bound Now by using Theorem 11 inthe next theorem we obtain an upper bound for the error ofour approximate solution of (8)

Theorem 12 Suppose L and N are the linear and nonlinearoperators with respect to (8) wherein L has inversion and119891 119892 isin 119867

119898

119908(120572120573)(Λ) (119898 ge 1) are satisfied in (8) then

10038171003817100381710038171003817119891 minusI

1

119873(119891)

10038171003817100381710038171003817119908(120572120573)

le 119862119873minus119898

(

10038161003816100381610038161003816Lminus1N (119891)

10038161003816100381610038161003816119867119898

119908(120572120573)

+

10038161003816100381610038161003816Lminus1119892

10038161003816100381610038161003816119867119898

119908(120572120573)

)

(50)

whereI1119873(119891) is an arbitrary polynomial interpolation satisfied

in interpolation conditions

Proof First we must note that I1119873(119891) can be expressed via

Lagrange interpolation and for this goal it is sufficient tointerpolate function I

119873(119891) in 119873 distinct points 119909

119895 119895 =

1 119873 From (8) and invertibility ofL we have

119891 +Lminus1N119891 = L

minus1(119892) (51)

also due to pseudospectral method we obtain

119891 (119909119895) +L

minus1N119891 (119909

119895) = L

minus1119892 (119909119895) (52)

by producing both sides of (52) with 119871119895(119909) (Lagrange inter-

polation basis) and sum from 119894 = 0 119873 we obtain

I119873(119891) +I

119873(Lminus1N (119891)) = I

119873(Lminus1

(119892)) (53)

Now by subtracting (51) from (53) we get

I119873(119891) = L

minus1N (119891) minusI

119873(Lminus1N (119891))

+Lminus1 (119892) minusI119873(Lminus1 (119892))

(54)

International Journal of Computational Mathematics 7

therefore we have10038171003817100381710038171003817119891 minusI

1

119873(119891)

10038171003817100381710038171003817119908(120572120573)

=1003817100381710038171003817119891 minusI

119873(119891)

1003817100381710038171003817119908(120572120573)

le

10038171003817100381710038171003817Lminus1N (119891) minusI

119873(Lminus1N (119891))

10038171003817100381710038171003817119908(120572120573)

+

10038171003817100381710038171003817Lminus1

(119892) minusI119873(Lminus1

(119892))

10038171003817100381710038171003817119908(120572120573)

(55)

Finally by usingTheorem 11 we obtain10038171003817100381710038171003817119891 minusI

1

119873(119891)

10038171003817100381710038171003817119908(120572120573)

=1003817100381710038171003817119891 minusI

119873(119891)

1003817100381710038171003817119908(120572120573)

le 119862119873minus119898

(

10038161003816100381610038161003816Lminus1N119891

10038161003816100381610038161003816119867119898119873

119908(120572120573)

+

10038161003816100381610038161003816Lminus1119892

10038161003816100381610038161003816119867119898119873

119908(120572120573)

)

(56)

therefore the proof is completed

4 The Test Experiments

Experiment 1 Let us consider the following nonlinear differ-ential equation problem [19]

11991010158401015840= minus (1 + 001119910

2) 119910 + 001cos3 (119909)

119910 (minus1) = cos (minus1)

119910 (1) = cos (1)

(57)

with the exact solution 119910(119909) = cos(119909)By simplifying (57) we have

11991010158401015840= minus 119910 minus 001119910

3+ 001cos3 (119909)

119910 (minus1) = cos (minus1)

119910 (1) = cos (1)

(58)

thus the linear and nonlinear parts of our ordinary differen-tial equation respectively are

L (119910) =

1

sum

119894=0

L119894(119910)

N (119910) = 0011199103(119909)

(59)

where

L0(119910) = 119910

10158401015840(119909) L

1(119910) = 119910 (60)

Now we consider 119873 + 1 collocation points 1199090

=

minus1 119909(120572120573)

119894119873minus1

119894=1 119909119873

= 1 where 119909(120572120573)

119894119873minus1

119894=1are the roots of

Jacobi polynomials of degree 119873 minus 1 then we consider thisinterpolation problem

119910 (minus1) = cos (minus1)

119910 (119909(120572120573)

119894) = 119875119873+1

(119909(120572120573)

119894) 119894 = 0 119873 minus 1

119910 (1) = cos (1)

(61)

Table 1 Comparison between the exact and approximate solutionsfor 120572 = 0 120573 = 0 and119873 = 5 of Experiment 1

119909Exact

solutionPseudospectral

method Error

minus10 054030231 054030231 000000000minus080 069670671 069670667 000000004minus060 082533561 082533555 000000006minus040 092106099 092106045 000000054minus020 098006658 098006645 000000013000 100000000 099999922 000000078020 098006658 098006633 000000025040 092106099 092106056 000000043060 082533561 082533545 000000016080 069670671 069670656 00000001510 054030231 054030231 000000000

where 119875119873+1

is the corresponding polynomial interpolationwith respect to problem (61) wherein interpolate the bound-ary conditions automatically and 119910(119909

(120572120573)

119894) 119894 = 0 119873 minus

1 are the unknown coefficients In Table 1 we see thecomparison between the exact and approximate solutionsfrom pseudospectral method for 120572 = 0 120573 = 0 and119873 = 5

Experiment 2 Consider the second-order difference equa-tion problem [20]

119910 (119909 + 1) minus 119910 (119909) = 119890119909 119910 (

1

2

) = 1 (62)

The exact solution is 119910(119909) = (119890119909+119890minusradic119890minus1)(119890minus1) We have

L (119910) = 119910 (119909 + 1) minus 119910 (119909) (63)

and this equation has no nonlinear operator part Theobtained results between the exact and approximate solutionsare shown in Table 2 and Figure 1

Experiment 3 Consider the Fredholm integrodifferentialequation [21]

1199101015840(119909) = int

1

0

119890119909119905119910 (119905) 119889119905 + 119910 (119909) +

1 minus 119890119909+1

119909 + 1

0 le 119909 119910 le 1

(64)

under the initial condition

119910 (0) = 1 (65)

with the exact solution 119910(119909) = 119890119909 In this case we have

L (119910) =

2

sum

119894=0

L119894(119910) (66)

where

L0(119910) = int

1

0

119890119909119905119910 (119905) 119889119905

L1(119910) = 119910

1015840(119909) L

2(119910) = 119910 (119909)

(67)

8 International Journal of Computational Mathematics

28

26

24

22

2

18

16

14

12

1

0

y

x

01 02 03 04 05 06 07 08 09 1

Exact solutionApproximate solution

Figure 1 The obtained results from approximation solution and exact solution for 120572 = minus05 120573 = 07 and 119873 = 5 of Experiment 2

Table 2 Comparison between the exact and approximate solutionsfor 120572 = minus05 120573 = 07 and119873 = 5 of Experiment 2

119909Exact

solutionPseudospectral

method Error

00 062245933 062245922 00000001101 068366636 068366643 minus00000000702 075131058 075131034 00000002403 082606901 082606923 minus00000002204 090868985 090868925 00000006005 100000000 100000000 0000000006 110091330 110091120 00000002107 121243980 121243880 00000001008 133569560 133569450 00000001109 147191430 147191350 00000000810 162245930 162245560 000000037

Table 3 Comparison between the exact and approximate solutionsfor 120572 = minus05 120573 = 07 and119873 = 5 of Experiment 3

119909Exact

solutionPseudospectral

method Error

00 10000000 10000000 00000000001 11051709 11051701 00000000802 12214028 12214011 00000001703 13498588 13498533 00000005504 14918247 14918232 00000001505 16487213 16487211 00000000206 18221188 18221166 00000002207 20137527 20137511 00000001508 22255409 22255401 00000000709 24596031 24596030 00000000110 27182818 27182810 000000008

Now by using the pseudospectral method for 120572 =

minus05 120573 = 07 and 119873 = 5 we obtain the following resultsthat are shown in Table 3

Experiment 4 We consider the linear partial differentialequation problem [22]

119906119905119905= 119906119909119909

+ 6 119906119905(119909 0) = 4119909 119906 (119909 0) = 119909

2 (68)

with the exact solution 119906(119909 119910) = (119909 + 2119905)2

In this case we have

L (119906) = 119906119905119905minus 119906119909119909 (69)

We obtain the approximate solution of (68) on the domain[minus1 1] times [minus1 1] For this purpose we consider the discretepoints (119909(120572120573)

119894 119905(120572120573)

119895)119873

119894119895=0and the general Lagrange interpola-

tion in the discrete points (119909(120572120573)119894

119905(120572120573)

119895)119873

119894119895=0in the following

form (two-dimensional case)

119871 (119909 119905) =

119873

sum

119894=0

119873

sum

119895=0

119891 (119909(120572120573)

119894 119905(120572120573)

119895) 119871119894(119909) 119871119895(119905) (70)

where 119871119894(119909) and 119871

119895(119905) are ordinary Lagrange interpolations

By using collocation method for arbitrary parameters 120572 120573and119873 = 5 we obtain the exact solution

Experiment 5 Finally let us consider the linear fractionaldifferential equation problem [23]

11986352

119910 (119909) + 3119910 (119909) = 31199093+

6

Γ (15)

11990912

119910 (0) = 0 1199101015840(0) = 0 119910

10158401015840(0) = 0 119909 isin [minus1 1]

(71)

This equation has the exact solution 119910(119909) = 1199093

In this case we have

L (119910 (119909)) = 11986352

(119910 (119909)) + 3119910 (119909) (72)

which is a linear operator Using the presented method forevery 120572 120573 and119873 = 3 we obtain the exact solution

International Journal of Computational Mathematics 9

5 Conclusion

In this paper we have developed the expansion approach forsolving some operator equations of the form L119906 + N119906 =

119892(119909) 119909 isin Γ with respect to boundary condition B119906 = 120582For this the principal importance is the development of theexpansion approach for solving general operator equationwherein the particular cases are in the solution of integralequations (Experiment 3) ordinary differential equations(Experiment 1) difference equations (Experiment 2) partialdifferential equations (Experiment 4) and fractional differ-ential equations (Experiment 5) Also the error analysis ispresented in 119871

2(Γ) norm Using the roots of another orthog-

onal polynomials (Jacobi polynomials) rather than classicalorthogonal polynomials such as Chebyshev polynomialsand Legendre polynomials for collocation points (with freeparameters 120572 120573) in pseudospectral method is an importantsubject which is performed (Experiments 2 and 3)

Conflict of Interests

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

Acknowledgments

The authors wish to thank the anonymous referees for theirvaluable comments and suggestions Also they would like toappreciate the Editor Dr Yaohanz Li who gives them theopportunity to review the paper

References

[1] P LinzTheoretical Numerical Analysis JohnWiley amp Sons NewYork NY USA 1979

[2] D Gottlieb and S A Orszag Numerical Analysis of SpectralMethods Theory and Applications SIAM-CBMS PhiladelphiaPa USA 1977

[3] G Ben-Yu The State of Art in Spectral Methods Hong KongUniversity 1996

[4] C Canuto M Y Hussaini A Quarteroni and T A ZhangSpectral Methods Fundamentals in Single Domains Springer2006

[5] C Canutuo M Y Hussaini A Quarteroni and T A ZhangSpectral Method in Fluid Dynamics Prentice Hall EngelwoodCliffs NJ USA 1984

[6] B FornbergAPractical Quide to PseudospectralMethods Cam-bridgeMonographs on Applied and Computational Mathemat-ics Cambridge University Press Cambridge UK 1st edition1996

[7] L N Trefethen Spectral Methods in Matlab SIAM Philadel-phia Pa USA 2000

[8] J S Hesthaven S Gottlieb and D Gottlieb Spectral Methodsfor Time-Dependent Problems vol 21 ofCambridgeMonographson Applied and Computational Mathematics Cambridge Uni-versity Press Cambridge UK 2007

[9] M Deville and E Mund ldquoChebyshev pseudospectral solutionof second-order elliptic equations with finite element precon-ditioningrdquo Journal of Computational Physics vol 60 no 3 pp517ndash533 1985

[10] J P Boyd ldquoMultipole expansions and pseudospectral cardinalfunctions a new generalization of the fast Fourier transformrdquoJournal of Computational Physics vol 103 no 1 pp 184ndash1861992

[11] G N Elnagar and M Razzaghi ldquoA pseudospectral method forHammerstein equationsrdquo Journal of Mathematical Analysis andApplications vol 199 no 2 pp 579ndash591 1996

[12] M Meehan and D OrsquoRegan ldquoExistence principles for nonreso-nant operator and integral equationsrdquo Computers ampMathemat-ics with Applications vol 35 no 9 pp 79ndash87 1998

[13] A Chaillou and M Suri ldquoA posteriori estimation of thelinearization error for strongly monotone nonlinear operatorsrdquoJournal of Computational andAppliedMathematics vol 205 no1 pp 72ndash87 2007

[14] R P Kanwal and K C Liu ldquoA Taylor expansion approach forsolving integral equationsrdquo International Journal of Mathemat-ical Education in Science and Technology vol 20 pp 411ndash4141989

[15] M R Eslahchi M Dehghan and S Ahmadi Asl ldquoThe generalJacobi matrix method for solving some nonlinear ordinarydifferential equationsrdquoAppliedMathematical Modelling vol 36no 8 pp 3387ndash3398 2012

[16] A Imani A Aminataei and A Imani ldquoCollocation method viaJacobi polynomials for solving nonlinear ordinary differentialequationsrdquo International Journal of Mathematics and Mathe-matical Sciences vol 2011 Article ID 673085 11 pages 2011

[17] M R Eslahchi and M Dehghan ldquoApplication of Taylor seriesin obtaining the orthogonal operational matrixrdquo Computers ampMathematics with Applications vol 61 no 9 pp 2596ndash26042011

[18] D Funaro Polynomial Approximation of Differential Equationsvol 8 of Lecture Notes in Physics Monographs Springer BerlinGermany 1992

[19] D F Griffiths and D J Higham Numerical Methods forOrdinaryDifferential Equations SpringerUndergraduateMath-ematics Series Springer London UK 2010

[20] M Gulsu M Sezer and B Tanay ldquoA matrix method for solvinghigh-order linear difference equations with mixed argumentusing hybrid Legendre and Taylor polynomialsrdquo Journal of theFranklin Institute vol 343 no 6 pp 647ndash659 2006

[21] N Kurt and M Sezer ldquoPolynomial solution of high-orderlinear Fredholm integro-differential equations with constantcoefficientsrdquo Journal of the Franklin Institute vol 345 no 8 pp839ndash850 2008

[22] V S Vladimirov A Collection of Problems on the Equations ofMathematical Physics Mir Publishers Moscow Russia 1968

[23] A H Bhrawy M M Alghamdi and T M Taha ldquoA newmodified generalized Laguerre operational matrix of fractionalintegration for solving fractional differential equations on thehalf linerdquoAdvances in Difference Equations vol 2012 article 1792012

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Stochastic AnalysisInternational Journal of

4 International Journal of Computational Mathematics

3 The General Pseudospectral Method forGeneral Operator Equation

In this section we describe the general pseudospectralmethod for numerical solution of the general operatorequations For this goal first we introduce the generalpseudospectral method Let 119871

1 1198712 119871

119899minus119898isin 119883lowast be some

functionals and scalars 1199081 1199082 119908

119899minus119898 wherein

119871119894(119906) = 119908

119894 119894 = 0 119899 minus 119898 (17)

in other words from (17) we have a generalized interpolationproblem with respect to 119906 Also we suppose that a boundaryconditionB119906 = 120582 can be written in the following interpola-tion form

119871119861

119894(119906) = 119888

119894 119894 = 119899 minus 119898 + 1 119899 (18)

where 119871119861

119899minus119898+1(119906) 119871119861

119899minus119898+2(119906) 119871

119861

119899(119906) are functionals on

119883lowastNow solving simultaneous interpolation problems (17)

and (18) is a principal part of this paper The next theoremgives the explicit form of the interpolation problems (17) and(18)

Theorem 7 Suppose 119883 is a Hilbert space and 119883119899

=

1205930 1205931 120593

119899 isin 119883 is a finite dimensional subspace of 119883

with dimension 119899 then I119899(119906) the generalized interpolation

problem from (17) and (18) is of the form

I119899(119906) =

119899

sum

119894=0

119888119894120593119894(119909) +

119899

sum

119894=0

119908119894120601119894(119909) (19)

where 119871119894 119871119861

119894isin 119883lowast are functionals of119883lowast wherein

119871119894(120593119895) = 120575119894119895 119894 = 1 119899 minus 119898 forall119895

119871119861

119894(120593119895) = 120575119894119895 119894 = 119899 minus 119898 + 1 119899 forall119895

(20)

119871119894(120601119895) = 120575119894119895 119894 = 1 119899 minus 119898 forall119895

119871119861

119894(120601119895) = 120575119894119895 119894 = 119899 minus 119898 + 1 119899 forall119895

(21)

Proof By imposing functionals 119871119894and 119871

119861

119894on both sides of

(19) and using properties (20) and (21) it is easy to show that(19) satisfies conditions (17) and (18)

Remark 8 The particular cases of interpolation (19) areLagrange Barycentric Hermite and Birkhoff interpolations

The important point inTheorem 7 is the existence of basisfunctions 120593

119895(119909)119899

119895=1and 120601

119895(119909)119899

119895=1which satisfy (20) and

(21) respectively We can answer this question by the conceptof linear independency of operators We perform this for120593119895(119909)119899

119895=1 and in similarmannerwe can obtain the necessary

and sufficient conditions of existence of 120601119895(119909)119899

119895=1 Now we

consider an arbitrary basis function 120595119895(119909)119899

119895=1of 119883119899and we

construct new basis functions which satisfy (20) and (21)

For this goal suppose

120593119895(119909) =

119895

sum

119894=1

119886119895119894120595119894 119895 = 1 119899 (22)

then we must obtain the 119899(119899 + 1)2 unknown coefficients119886119895119894 119894 = 1 119899 We start from 120593

0(119909) (119895 = 0) this means that

1205930(119909) = 119886

001205950(119909) (23)

from (23) wemust obtain one unknown coefficient 11988600

and itis sufficient for imposing the arbitrary functionals 119871

119894or 119871119861119894isin

119883lowast on (22) for obtaining the unknown coefficient 119886

00 so

1198710(1205930(119909)) = 119886

001198710(1205950(119909)) (24)

thus

11988600

=

1198710(1205930(119909))

1198710(1205950(119909))

1198710(1205950(119909)) = 0 (25)

Also for 1205931(119909)(119895 = 1) we have

1205931(119909) = 119886

101205950(119909) + 119886

111205951(119909) (26)

for unknown coefficients 11988610

and 11988611 If we impose two

functionals 1198710and 119871

1isin 119883lowast on (26) for obtaining unknown

coefficients 11988610

and 11988611

we can obtain

1198710(1205931(119909)) = 119886

101198710(1205950(119909)) + 119886

111198710(1205951(119909))

1198711(1205931(119909)) = 119886

101198711(1205950(119909)) + 119886

111198711(1205951(119909))

(27)

the unknown coefficients 11988610

and 11988611

exist if and only if10038161003816100381610038161003816100381610038161003816

1198710(1205950(119909)) 119871

0(1205951(119909))

1198711(1205950(119909)) 119871

1(1205951(119909))

10038161003816100381610038161003816100381610038161003816

= 0 (28)

By repeating this procedure we can obtain the result in whichthe unknown coefficients 119886

1198990 1198861198991 119886

119899119899exist if and only if

det (119871119894(120595119895)) =

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198710(1205951(119909)) sdot sdot sdot 119871

119899(120595119899(119909))

sdot sdot sdot

119871119899(1205951(119909)) sdot sdot sdot 119871

119899(120595119899(119909))

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (29)

and this result is equivalent to the independence of function-als 1198711 1198712 119871

119899minus119898 119871119861

119899minus119898+1 119871

119861

119899isin 119883lowast

Now for implementing the general pseudospectralmethod for numerical solution of general operator equation(8) with respect to its boundary condition first we substitutea general interpolation operator (19) in (8) The generalinterpolation function 119906(119909) (the exact solution of (8) withrespect to its boundary condition) fromTheorem 7 is

119906 (119909) ≃ I119899(119906) (119909) =

119899

sum

119894=0

119888119894120593119894(119909) +

119899

sum

119894=0

119908119894120593119894(119909) (30)

where119908119894and 119888119894are defined in (17) and (18) Also we consider

the general interpolation problem of the function 119892(119909) (thefunction of the right-hand side of (8)) as

1198711

119894(119892) = 119887

119894 119894 = 0 119899 (31)

International Journal of Computational Mathematics 5

where 11987110(119892) 119871

1

119899(119892) isin 119883

lowast are functionals and 1198870 1198871 119887

119899

are constants then we have

119892 (119909) ≃ I119899(119892) (119909) =

119899

sum

119894=0

119887119894120593119894(119909) (32)

Now by substituting (19) and (32) in (8) we obtain

L (I119899(119906) (119909)) +N (I

119899(119906) (119909)) ≃ I

119899(119892) (119909) (33)

then using some project operators on (33) as 1198751 119875

119899

defined on119883 we obtain

119875119894(L (I

119899(119906) (119909)) +N (I

119899(119906) (119909))) ≃ 119875

119894(I119899(119892) (119909))

119894 = 1 119899

(34)

From (34) we obtain 119899 algebraic equations Simplificationof the left-hand side of (34) is an important subject and weexpress this subject for linear and nonlinear operators (I)

and (N)

(i) Linear Case In the linear case if we suppose that theprojection operators are linear then from property of linearoperatorL we have

119875119905L (I

119899(119906)) = 119875

119905L(

119899

sum

119894=0

119888119894120593119894+

119899

sum

119894=0

119908119894120593119894)

=

119899

sum

119894=0

119888119894119875119905L (120593119894) +

119899

sum

119894=0

119908119894119875119905L (120593119894)

(35)

so for obtaining a matrix relation between the values of119875119905L(I119899(119906)) and 119888

119894and 119908

119894 it is sufficient to calculate the

119875119905L(120593119894) values as the following

[

[

[

[

[

[

1198750L (I

119899(119906))

119875119899L (I

119899(119906))

]

]

]

]

]

]

=

[

[

[

[

[

[

11989600

sdot sdot sdot sdot sdot sdot 1198960119899

sdot sdot sdot sdot sdot sdot

sdot sdot sdot sdot sdot sdot

1198961198990

sdot sdot sdot sdot sdot sdot 119896119899119899

]

]

]

]

]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

(

[

[

[

[

[

[

1198880

119888119899

]

]

]

]

]

]

+

[

[

[

[

[

[

1199080

119908119899

]

]

]

]

]

]

)

(36)

where

119896119894119895

= 119875119894L (120593119895) 119894 119895 = 0 119899 (37)

The previous matrix (under bracket) is a square matrix of(119873+ 1) times (119873+ 1) dimensions and has high application in thepseudospectral method In particular case when the linearoperator is a derivative this matrix is named as derivativematrix and is very ill conditioned

(ii) Nonlinear Case The nonlinear case is almost cumber-some and different ideas can be used The first idea is to

approximate the nonlinear operator N with some linearoperator F and using the method of part (i) But thisapproach is very complicated and approximating a nonlinearoperator with linear operators in general is difficult and wasdown only for particular operators [13] The second ideais based on approximating a nonlinear operator via Taylorseries [14] Also another approach for simplifying nonlinearoperator is based on orthogonal expansions (general Fourierseries) of N(I

119899(119906)) For this goal we must obtain an

orthogonal system 120601119894(119909)119899

119894=1from basis functions 120593

119895(119909)119899

119895=1

using Gram-Schmidt algorithmThen it is not difficult to seethat

I119899(119906) (119909) =

119899

sum

119894=0

120582119894120593119894(119909) +

119899

sum

119894=0

119888119894120593119894(119909) ≃

119873

sum

119894=0

119886119894120601119894(119909) (38)

where

119886119894=

(I119873(119906) 120601

119894)

(120601119894 120601119894)

(39)

Also forN(I119899(119906)(119909)) we have

N (I119899(119906) (119909)) ≃

119899

sum

119894=0

119887119894120601119894(119909) (40)

where

119887119894=

(N (I119899(119906)) 120601

119894)

(120601119894 120601119894)

(41)

We must note that the 119886119894and 119887119894coefficients are functions of

119871119894and 119871

119861

119894

Now for case study let119883 be the space of polynomials withreal coefficients wherein it is a Hilbert space with respect to1198712 norm and119883

119899is a subspace of119883 with dimension 119899

Also suppose that the functionals1198711 1198712 119871

119899in (9) are

defined as

119871119894(120593) = 119888

119894 (42)

where 119888119894are also functionals of 120593

119895(119909)119899

119895=1 For instance

the well known functionals with high applications are 119888119894=

int 119905119894120593119889119909 (moment operator) 119888

119894= (120593)

(119894)(119909) (ith-derivative

operator) 119888119894= 120593(119909 minus ℎ

119894) ℎ119894isin 119877 (shift operator) and 119888

119894=

120593(119905119894) (119905119894are distinct points and in particular case the roots

of shifted orthogonal polynomials as Hermite and Laguerrepolynomials in unbounded domains and shifted Chebyshevand Legendre and in general case Jacobi polynomials inbounded domains [15ndash17]) Also in the higher dimensionswe can define the suitable functionals

31 Error Analysis The error analysis of pseudospectralmethod in the general case is very difficult and we canperform this analysis only in polynomial case For this goalat first we introduce some useful notations and theorems thatare used in this paper

Definition 9 The Jacobi polynomials [18] 119875(120572120573)

119894(119909) are

defined as the orthogonal polynomials with respect to the

6 International Journal of Computational Mathematics

weight function119908120572120573

(119909) = (1minus119909)120572(1 +119909)

120573 (120572 gt minus1 120573 gt minus1)

on (minus1 1) An explicit formula for these polynomials is givenby

119875(120572120573)

119899(119909)

=

1

2119899

119899

sum

119894=0

(

119899 + 120572

119894)(

119899 + 120573

119899 minus 119894) (119909 minus 1)

119899minus119894(119909 + 1)

119894

(43)

Further we have

119889

119889119909

119875(120572120573)

119899(119909) =

1

2

(119899 + 120572 + 120573 + 1) 119875(120572+1120573+1)

119899minus1(119909) (44)

Jacobi polynomials have an important application in severalfields of numerical analysis such as quadratures formula andspectral methods Also the shifted Jacobi polynomials aredefined as orthogonal polynomials in the interval [119886 119887] withrespect to weight function 119908

120572120573(119909) = (119887 minus 119909)

120572(119886 minus 119909)

120573 (120572 gt

minus1 120573 gt minus1) with change of variable 119910 = (2(119887 minus 119886))119909 +

((119886 + 119887)(119886 minus 119887)) For more properties and applications ofthese polynomials see [15ndash17] Now let Λ = [minus1 1] and let119875(120572120573)

119873+1(119909) be the Jacobi polynomial of degree119873+1with respect

to weight function 119908(120572120573)

(119909) = (1 minus 119909)120572(1 + 119909)

120573 (120572 120573 gt minus1)

and let P119873be the polynomials spaces of degree less than or

equal to119873

Remark 10 The best choice of points for polynomial inter-polation are the roots of orthogonal polynomials and in par-ticular case Jacobi polynomials due to their small Lebesgueconstants [6]

The symbol 1198712

119908(120572120573)(Λ) shows the space of measurable

functions whose square is Lebesgue integrable in Λ relativeto the weight function119908

(120572120573) The inner product and norm of1198712

119908(120572120573)(Λ) are defined by

(119906 V)119908(120572120573)Λ

= int

Λ

119906 (119909) V (119909) 119908(120572120573) (119909) 119889119909

119906119908(120572120573)Λ

= radic(119906 119906)119908(120572120573)

forall119906 V isin 1198712

119908(120572120573) (Λ)

(45)

Also another useful space is

119867119898

119908(120572120573) = V V

119898119908(120572120573) lt infin (46)

with seminorm and norms respectively Consider

|V|119898119908(120572120573) =

1003817100381710038171003817120597119898

119909V1003817100381710038171003817119908(120572120573)

V119898119908(120572120573) = (

119898

sum

119894=0

|V|2119894119908(120572120573))

12

(47)

But an applicable norm that appears in bounding of errors ofthe spectral method is

|V|119867119898119908(120572120573) = (

119898

sum

119896=119890

10038171003817100381710038171003817120597119896

119909V10038171003817100381710038171003817

2

119908(120572120573)

)

12

119890 = min (119898119873 + 1)

(48)

Now we present a useful theorem

Theorem 11 If ℎ isin 119867119898

119908(120572120573)(Λ)119898 ge 1 then one has

1003817100381710038171003817ℎ minusI

119873(ℎ)

1003817100381710038171003817119908(120572120573) le 119862119873

minus11989810038161003816100381610038161198911003816100381610038161003816119867119898

119908(120572120573)

(49)

where I119873(ℎ) is a Lagrange interpolation in distinct points

119909(120572120573)

119894(roots of Jacobi polynomials)

Proof See [4 5]

Noting to Theorem 11 we can obtain an error boundfor any classes of polynomial interpolation such as Hermiteinterpolation Birkhoff interpolation and other classes Thismeans that if we have an arbitrary polynomial interpolationI1119873(ℎ) thenwe can transform the interpolationI1

119873(ℎ) to the

Lagrange interpolation I119873(ℎ) and then we use Theorem 11

for obtaining an error bound Now by using Theorem 11 inthe next theorem we obtain an upper bound for the error ofour approximate solution of (8)

Theorem 12 Suppose L and N are the linear and nonlinearoperators with respect to (8) wherein L has inversion and119891 119892 isin 119867

119898

119908(120572120573)(Λ) (119898 ge 1) are satisfied in (8) then

10038171003817100381710038171003817119891 minusI

1

119873(119891)

10038171003817100381710038171003817119908(120572120573)

le 119862119873minus119898

(

10038161003816100381610038161003816Lminus1N (119891)

10038161003816100381610038161003816119867119898

119908(120572120573)

+

10038161003816100381610038161003816Lminus1119892

10038161003816100381610038161003816119867119898

119908(120572120573)

)

(50)

whereI1119873(119891) is an arbitrary polynomial interpolation satisfied

in interpolation conditions

Proof First we must note that I1119873(119891) can be expressed via

Lagrange interpolation and for this goal it is sufficient tointerpolate function I

119873(119891) in 119873 distinct points 119909

119895 119895 =

1 119873 From (8) and invertibility ofL we have

119891 +Lminus1N119891 = L

minus1(119892) (51)

also due to pseudospectral method we obtain

119891 (119909119895) +L

minus1N119891 (119909

119895) = L

minus1119892 (119909119895) (52)

by producing both sides of (52) with 119871119895(119909) (Lagrange inter-

polation basis) and sum from 119894 = 0 119873 we obtain

I119873(119891) +I

119873(Lminus1N (119891)) = I

119873(Lminus1

(119892)) (53)

Now by subtracting (51) from (53) we get

I119873(119891) = L

minus1N (119891) minusI

119873(Lminus1N (119891))

+Lminus1 (119892) minusI119873(Lminus1 (119892))

(54)

International Journal of Computational Mathematics 7

therefore we have10038171003817100381710038171003817119891 minusI

1

119873(119891)

10038171003817100381710038171003817119908(120572120573)

=1003817100381710038171003817119891 minusI

119873(119891)

1003817100381710038171003817119908(120572120573)

le

10038171003817100381710038171003817Lminus1N (119891) minusI

119873(Lminus1N (119891))

10038171003817100381710038171003817119908(120572120573)

+

10038171003817100381710038171003817Lminus1

(119892) minusI119873(Lminus1

(119892))

10038171003817100381710038171003817119908(120572120573)

(55)

Finally by usingTheorem 11 we obtain10038171003817100381710038171003817119891 minusI

1

119873(119891)

10038171003817100381710038171003817119908(120572120573)

=1003817100381710038171003817119891 minusI

119873(119891)

1003817100381710038171003817119908(120572120573)

le 119862119873minus119898

(

10038161003816100381610038161003816Lminus1N119891

10038161003816100381610038161003816119867119898119873

119908(120572120573)

+

10038161003816100381610038161003816Lminus1119892

10038161003816100381610038161003816119867119898119873

119908(120572120573)

)

(56)

therefore the proof is completed

4 The Test Experiments

Experiment 1 Let us consider the following nonlinear differ-ential equation problem [19]

11991010158401015840= minus (1 + 001119910

2) 119910 + 001cos3 (119909)

119910 (minus1) = cos (minus1)

119910 (1) = cos (1)

(57)

with the exact solution 119910(119909) = cos(119909)By simplifying (57) we have

11991010158401015840= minus 119910 minus 001119910

3+ 001cos3 (119909)

119910 (minus1) = cos (minus1)

119910 (1) = cos (1)

(58)

thus the linear and nonlinear parts of our ordinary differen-tial equation respectively are

L (119910) =

1

sum

119894=0

L119894(119910)

N (119910) = 0011199103(119909)

(59)

where

L0(119910) = 119910

10158401015840(119909) L

1(119910) = 119910 (60)

Now we consider 119873 + 1 collocation points 1199090

=

minus1 119909(120572120573)

119894119873minus1

119894=1 119909119873

= 1 where 119909(120572120573)

119894119873minus1

119894=1are the roots of

Jacobi polynomials of degree 119873 minus 1 then we consider thisinterpolation problem

119910 (minus1) = cos (minus1)

119910 (119909(120572120573)

119894) = 119875119873+1

(119909(120572120573)

119894) 119894 = 0 119873 minus 1

119910 (1) = cos (1)

(61)

Table 1 Comparison between the exact and approximate solutionsfor 120572 = 0 120573 = 0 and119873 = 5 of Experiment 1

119909Exact

solutionPseudospectral

method Error

minus10 054030231 054030231 000000000minus080 069670671 069670667 000000004minus060 082533561 082533555 000000006minus040 092106099 092106045 000000054minus020 098006658 098006645 000000013000 100000000 099999922 000000078020 098006658 098006633 000000025040 092106099 092106056 000000043060 082533561 082533545 000000016080 069670671 069670656 00000001510 054030231 054030231 000000000

where 119875119873+1

is the corresponding polynomial interpolationwith respect to problem (61) wherein interpolate the bound-ary conditions automatically and 119910(119909

(120572120573)

119894) 119894 = 0 119873 minus

1 are the unknown coefficients In Table 1 we see thecomparison between the exact and approximate solutionsfrom pseudospectral method for 120572 = 0 120573 = 0 and119873 = 5

Experiment 2 Consider the second-order difference equa-tion problem [20]

119910 (119909 + 1) minus 119910 (119909) = 119890119909 119910 (

1

2

) = 1 (62)

The exact solution is 119910(119909) = (119890119909+119890minusradic119890minus1)(119890minus1) We have

L (119910) = 119910 (119909 + 1) minus 119910 (119909) (63)

and this equation has no nonlinear operator part Theobtained results between the exact and approximate solutionsare shown in Table 2 and Figure 1

Experiment 3 Consider the Fredholm integrodifferentialequation [21]

1199101015840(119909) = int

1

0

119890119909119905119910 (119905) 119889119905 + 119910 (119909) +

1 minus 119890119909+1

119909 + 1

0 le 119909 119910 le 1

(64)

under the initial condition

119910 (0) = 1 (65)

with the exact solution 119910(119909) = 119890119909 In this case we have

L (119910) =

2

sum

119894=0

L119894(119910) (66)

where

L0(119910) = int

1

0

119890119909119905119910 (119905) 119889119905

L1(119910) = 119910

1015840(119909) L

2(119910) = 119910 (119909)

(67)

8 International Journal of Computational Mathematics

28

26

24

22

2

18

16

14

12

1

0

y

x

01 02 03 04 05 06 07 08 09 1

Exact solutionApproximate solution

Figure 1 The obtained results from approximation solution and exact solution for 120572 = minus05 120573 = 07 and 119873 = 5 of Experiment 2

Table 2 Comparison between the exact and approximate solutionsfor 120572 = minus05 120573 = 07 and119873 = 5 of Experiment 2

119909Exact

solutionPseudospectral

method Error

00 062245933 062245922 00000001101 068366636 068366643 minus00000000702 075131058 075131034 00000002403 082606901 082606923 minus00000002204 090868985 090868925 00000006005 100000000 100000000 0000000006 110091330 110091120 00000002107 121243980 121243880 00000001008 133569560 133569450 00000001109 147191430 147191350 00000000810 162245930 162245560 000000037

Table 3 Comparison between the exact and approximate solutionsfor 120572 = minus05 120573 = 07 and119873 = 5 of Experiment 3

119909Exact

solutionPseudospectral

method Error

00 10000000 10000000 00000000001 11051709 11051701 00000000802 12214028 12214011 00000001703 13498588 13498533 00000005504 14918247 14918232 00000001505 16487213 16487211 00000000206 18221188 18221166 00000002207 20137527 20137511 00000001508 22255409 22255401 00000000709 24596031 24596030 00000000110 27182818 27182810 000000008

Now by using the pseudospectral method for 120572 =

minus05 120573 = 07 and 119873 = 5 we obtain the following resultsthat are shown in Table 3

Experiment 4 We consider the linear partial differentialequation problem [22]

119906119905119905= 119906119909119909

+ 6 119906119905(119909 0) = 4119909 119906 (119909 0) = 119909

2 (68)

with the exact solution 119906(119909 119910) = (119909 + 2119905)2

In this case we have

L (119906) = 119906119905119905minus 119906119909119909 (69)

We obtain the approximate solution of (68) on the domain[minus1 1] times [minus1 1] For this purpose we consider the discretepoints (119909(120572120573)

119894 119905(120572120573)

119895)119873

119894119895=0and the general Lagrange interpola-

tion in the discrete points (119909(120572120573)119894

119905(120572120573)

119895)119873

119894119895=0in the following

form (two-dimensional case)

119871 (119909 119905) =

119873

sum

119894=0

119873

sum

119895=0

119891 (119909(120572120573)

119894 119905(120572120573)

119895) 119871119894(119909) 119871119895(119905) (70)

where 119871119894(119909) and 119871

119895(119905) are ordinary Lagrange interpolations

By using collocation method for arbitrary parameters 120572 120573and119873 = 5 we obtain the exact solution

Experiment 5 Finally let us consider the linear fractionaldifferential equation problem [23]

11986352

119910 (119909) + 3119910 (119909) = 31199093+

6

Γ (15)

11990912

119910 (0) = 0 1199101015840(0) = 0 119910

10158401015840(0) = 0 119909 isin [minus1 1]

(71)

This equation has the exact solution 119910(119909) = 1199093

In this case we have

L (119910 (119909)) = 11986352

(119910 (119909)) + 3119910 (119909) (72)

which is a linear operator Using the presented method forevery 120572 120573 and119873 = 3 we obtain the exact solution

International Journal of Computational Mathematics 9

5 Conclusion

In this paper we have developed the expansion approach forsolving some operator equations of the form L119906 + N119906 =

119892(119909) 119909 isin Γ with respect to boundary condition B119906 = 120582For this the principal importance is the development of theexpansion approach for solving general operator equationwherein the particular cases are in the solution of integralequations (Experiment 3) ordinary differential equations(Experiment 1) difference equations (Experiment 2) partialdifferential equations (Experiment 4) and fractional differ-ential equations (Experiment 5) Also the error analysis ispresented in 119871

2(Γ) norm Using the roots of another orthog-

onal polynomials (Jacobi polynomials) rather than classicalorthogonal polynomials such as Chebyshev polynomialsand Legendre polynomials for collocation points (with freeparameters 120572 120573) in pseudospectral method is an importantsubject which is performed (Experiments 2 and 3)

Conflict of Interests

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

Acknowledgments

The authors wish to thank the anonymous referees for theirvaluable comments and suggestions Also they would like toappreciate the Editor Dr Yaohanz Li who gives them theopportunity to review the paper

References

[1] P LinzTheoretical Numerical Analysis JohnWiley amp Sons NewYork NY USA 1979

[2] D Gottlieb and S A Orszag Numerical Analysis of SpectralMethods Theory and Applications SIAM-CBMS PhiladelphiaPa USA 1977

[3] G Ben-Yu The State of Art in Spectral Methods Hong KongUniversity 1996

[4] C Canuto M Y Hussaini A Quarteroni and T A ZhangSpectral Methods Fundamentals in Single Domains Springer2006

[5] C Canutuo M Y Hussaini A Quarteroni and T A ZhangSpectral Method in Fluid Dynamics Prentice Hall EngelwoodCliffs NJ USA 1984

[6] B FornbergAPractical Quide to PseudospectralMethods Cam-bridgeMonographs on Applied and Computational Mathemat-ics Cambridge University Press Cambridge UK 1st edition1996

[7] L N Trefethen Spectral Methods in Matlab SIAM Philadel-phia Pa USA 2000

[8] J S Hesthaven S Gottlieb and D Gottlieb Spectral Methodsfor Time-Dependent Problems vol 21 ofCambridgeMonographson Applied and Computational Mathematics Cambridge Uni-versity Press Cambridge UK 2007

[9] M Deville and E Mund ldquoChebyshev pseudospectral solutionof second-order elliptic equations with finite element precon-ditioningrdquo Journal of Computational Physics vol 60 no 3 pp517ndash533 1985

[10] J P Boyd ldquoMultipole expansions and pseudospectral cardinalfunctions a new generalization of the fast Fourier transformrdquoJournal of Computational Physics vol 103 no 1 pp 184ndash1861992

[11] G N Elnagar and M Razzaghi ldquoA pseudospectral method forHammerstein equationsrdquo Journal of Mathematical Analysis andApplications vol 199 no 2 pp 579ndash591 1996

[12] M Meehan and D OrsquoRegan ldquoExistence principles for nonreso-nant operator and integral equationsrdquo Computers ampMathemat-ics with Applications vol 35 no 9 pp 79ndash87 1998

[13] A Chaillou and M Suri ldquoA posteriori estimation of thelinearization error for strongly monotone nonlinear operatorsrdquoJournal of Computational andAppliedMathematics vol 205 no1 pp 72ndash87 2007

[14] R P Kanwal and K C Liu ldquoA Taylor expansion approach forsolving integral equationsrdquo International Journal of Mathemat-ical Education in Science and Technology vol 20 pp 411ndash4141989

[15] M R Eslahchi M Dehghan and S Ahmadi Asl ldquoThe generalJacobi matrix method for solving some nonlinear ordinarydifferential equationsrdquoAppliedMathematical Modelling vol 36no 8 pp 3387ndash3398 2012

[16] A Imani A Aminataei and A Imani ldquoCollocation method viaJacobi polynomials for solving nonlinear ordinary differentialequationsrdquo International Journal of Mathematics and Mathe-matical Sciences vol 2011 Article ID 673085 11 pages 2011

[17] M R Eslahchi and M Dehghan ldquoApplication of Taylor seriesin obtaining the orthogonal operational matrixrdquo Computers ampMathematics with Applications vol 61 no 9 pp 2596ndash26042011

[18] D Funaro Polynomial Approximation of Differential Equationsvol 8 of Lecture Notes in Physics Monographs Springer BerlinGermany 1992

[19] D F Griffiths and D J Higham Numerical Methods forOrdinaryDifferential Equations SpringerUndergraduateMath-ematics Series Springer London UK 2010

[20] M Gulsu M Sezer and B Tanay ldquoA matrix method for solvinghigh-order linear difference equations with mixed argumentusing hybrid Legendre and Taylor polynomialsrdquo Journal of theFranklin Institute vol 343 no 6 pp 647ndash659 2006

[21] N Kurt and M Sezer ldquoPolynomial solution of high-orderlinear Fredholm integro-differential equations with constantcoefficientsrdquo Journal of the Franklin Institute vol 345 no 8 pp839ndash850 2008

[22] V S Vladimirov A Collection of Problems on the Equations ofMathematical Physics Mir Publishers Moscow Russia 1968

[23] A H Bhrawy M M Alghamdi and T M Taha ldquoA newmodified generalized Laguerre operational matrix of fractionalintegration for solving fractional differential equations on thehalf linerdquoAdvances in Difference Equations vol 2012 article 1792012

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Computational Mathematics 5

where 11987110(119892) 119871

1

119899(119892) isin 119883

lowast are functionals and 1198870 1198871 119887

119899

are constants then we have

119892 (119909) ≃ I119899(119892) (119909) =

119899

sum

119894=0

119887119894120593119894(119909) (32)

Now by substituting (19) and (32) in (8) we obtain

L (I119899(119906) (119909)) +N (I

119899(119906) (119909)) ≃ I

119899(119892) (119909) (33)

then using some project operators on (33) as 1198751 119875

119899

defined on119883 we obtain

119875119894(L (I

119899(119906) (119909)) +N (I

119899(119906) (119909))) ≃ 119875

119894(I119899(119892) (119909))

119894 = 1 119899

(34)

From (34) we obtain 119899 algebraic equations Simplificationof the left-hand side of (34) is an important subject and weexpress this subject for linear and nonlinear operators (I)

and (N)

(i) Linear Case In the linear case if we suppose that theprojection operators are linear then from property of linearoperatorL we have

119875119905L (I

119899(119906)) = 119875

119905L(

119899

sum

119894=0

119888119894120593119894+

119899

sum

119894=0

119908119894120593119894)

=

119899

sum

119894=0

119888119894119875119905L (120593119894) +

119899

sum

119894=0

119908119894119875119905L (120593119894)

(35)

so for obtaining a matrix relation between the values of119875119905L(I119899(119906)) and 119888

119894and 119908

119894 it is sufficient to calculate the

119875119905L(120593119894) values as the following

[

[

[

[

[

[

1198750L (I

119899(119906))

119875119899L (I

119899(119906))

]

]

]

]

]

]

=

[

[

[

[

[

[

11989600

sdot sdot sdot sdot sdot sdot 1198960119899

sdot sdot sdot sdot sdot sdot

sdot sdot sdot sdot sdot sdot

1198961198990

sdot sdot sdot sdot sdot sdot 119896119899119899

]

]

]

]

]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

(

[

[

[

[

[

[

1198880

119888119899

]

]

]

]

]

]

+

[

[

[

[

[

[

1199080

119908119899

]

]

]

]

]

]

)

(36)

where

119896119894119895

= 119875119894L (120593119895) 119894 119895 = 0 119899 (37)

The previous matrix (under bracket) is a square matrix of(119873+ 1) times (119873+ 1) dimensions and has high application in thepseudospectral method In particular case when the linearoperator is a derivative this matrix is named as derivativematrix and is very ill conditioned

(ii) Nonlinear Case The nonlinear case is almost cumber-some and different ideas can be used The first idea is to

approximate the nonlinear operator N with some linearoperator F and using the method of part (i) But thisapproach is very complicated and approximating a nonlinearoperator with linear operators in general is difficult and wasdown only for particular operators [13] The second ideais based on approximating a nonlinear operator via Taylorseries [14] Also another approach for simplifying nonlinearoperator is based on orthogonal expansions (general Fourierseries) of N(I

119899(119906)) For this goal we must obtain an

orthogonal system 120601119894(119909)119899

119894=1from basis functions 120593

119895(119909)119899

119895=1

using Gram-Schmidt algorithmThen it is not difficult to seethat

I119899(119906) (119909) =

119899

sum

119894=0

120582119894120593119894(119909) +

119899

sum

119894=0

119888119894120593119894(119909) ≃

119873

sum

119894=0

119886119894120601119894(119909) (38)

where

119886119894=

(I119873(119906) 120601

119894)

(120601119894 120601119894)

(39)

Also forN(I119899(119906)(119909)) we have

N (I119899(119906) (119909)) ≃

119899

sum

119894=0

119887119894120601119894(119909) (40)

where

119887119894=

(N (I119899(119906)) 120601

119894)

(120601119894 120601119894)

(41)

We must note that the 119886119894and 119887119894coefficients are functions of

119871119894and 119871

119861

119894

Now for case study let119883 be the space of polynomials withreal coefficients wherein it is a Hilbert space with respect to1198712 norm and119883

119899is a subspace of119883 with dimension 119899

Also suppose that the functionals1198711 1198712 119871

119899in (9) are

defined as

119871119894(120593) = 119888

119894 (42)

where 119888119894are also functionals of 120593

119895(119909)119899

119895=1 For instance

the well known functionals with high applications are 119888119894=

int 119905119894120593119889119909 (moment operator) 119888

119894= (120593)

(119894)(119909) (ith-derivative

operator) 119888119894= 120593(119909 minus ℎ

119894) ℎ119894isin 119877 (shift operator) and 119888

119894=

120593(119905119894) (119905119894are distinct points and in particular case the roots

of shifted orthogonal polynomials as Hermite and Laguerrepolynomials in unbounded domains and shifted Chebyshevand Legendre and in general case Jacobi polynomials inbounded domains [15ndash17]) Also in the higher dimensionswe can define the suitable functionals

31 Error Analysis The error analysis of pseudospectralmethod in the general case is very difficult and we canperform this analysis only in polynomial case For this goalat first we introduce some useful notations and theorems thatare used in this paper

Definition 9 The Jacobi polynomials [18] 119875(120572120573)

119894(119909) are

defined as the orthogonal polynomials with respect to the

6 International Journal of Computational Mathematics

weight function119908120572120573

(119909) = (1minus119909)120572(1 +119909)

120573 (120572 gt minus1 120573 gt minus1)

on (minus1 1) An explicit formula for these polynomials is givenby

119875(120572120573)

119899(119909)

=

1

2119899

119899

sum

119894=0

(

119899 + 120572

119894)(

119899 + 120573

119899 minus 119894) (119909 minus 1)

119899minus119894(119909 + 1)

119894

(43)

Further we have

119889

119889119909

119875(120572120573)

119899(119909) =

1

2

(119899 + 120572 + 120573 + 1) 119875(120572+1120573+1)

119899minus1(119909) (44)

Jacobi polynomials have an important application in severalfields of numerical analysis such as quadratures formula andspectral methods Also the shifted Jacobi polynomials aredefined as orthogonal polynomials in the interval [119886 119887] withrespect to weight function 119908

120572120573(119909) = (119887 minus 119909)

120572(119886 minus 119909)

120573 (120572 gt

minus1 120573 gt minus1) with change of variable 119910 = (2(119887 minus 119886))119909 +

((119886 + 119887)(119886 minus 119887)) For more properties and applications ofthese polynomials see [15ndash17] Now let Λ = [minus1 1] and let119875(120572120573)

119873+1(119909) be the Jacobi polynomial of degree119873+1with respect

to weight function 119908(120572120573)

(119909) = (1 minus 119909)120572(1 + 119909)

120573 (120572 120573 gt minus1)

and let P119873be the polynomials spaces of degree less than or

equal to119873

Remark 10 The best choice of points for polynomial inter-polation are the roots of orthogonal polynomials and in par-ticular case Jacobi polynomials due to their small Lebesgueconstants [6]

The symbol 1198712

119908(120572120573)(Λ) shows the space of measurable

functions whose square is Lebesgue integrable in Λ relativeto the weight function119908

(120572120573) The inner product and norm of1198712

119908(120572120573)(Λ) are defined by

(119906 V)119908(120572120573)Λ

= int

Λ

119906 (119909) V (119909) 119908(120572120573) (119909) 119889119909

119906119908(120572120573)Λ

= radic(119906 119906)119908(120572120573)

forall119906 V isin 1198712

119908(120572120573) (Λ)

(45)

Also another useful space is

119867119898

119908(120572120573) = V V

119898119908(120572120573) lt infin (46)

with seminorm and norms respectively Consider

|V|119898119908(120572120573) =

1003817100381710038171003817120597119898

119909V1003817100381710038171003817119908(120572120573)

V119898119908(120572120573) = (

119898

sum

119894=0

|V|2119894119908(120572120573))

12

(47)

But an applicable norm that appears in bounding of errors ofthe spectral method is

|V|119867119898119908(120572120573) = (

119898

sum

119896=119890

10038171003817100381710038171003817120597119896

119909V10038171003817100381710038171003817

2

119908(120572120573)

)

12

119890 = min (119898119873 + 1)

(48)

Now we present a useful theorem

Theorem 11 If ℎ isin 119867119898

119908(120572120573)(Λ)119898 ge 1 then one has

1003817100381710038171003817ℎ minusI

119873(ℎ)

1003817100381710038171003817119908(120572120573) le 119862119873

minus11989810038161003816100381610038161198911003816100381610038161003816119867119898

119908(120572120573)

(49)

where I119873(ℎ) is a Lagrange interpolation in distinct points

119909(120572120573)

119894(roots of Jacobi polynomials)

Proof See [4 5]

Noting to Theorem 11 we can obtain an error boundfor any classes of polynomial interpolation such as Hermiteinterpolation Birkhoff interpolation and other classes Thismeans that if we have an arbitrary polynomial interpolationI1119873(ℎ) thenwe can transform the interpolationI1

119873(ℎ) to the

Lagrange interpolation I119873(ℎ) and then we use Theorem 11

for obtaining an error bound Now by using Theorem 11 inthe next theorem we obtain an upper bound for the error ofour approximate solution of (8)

Theorem 12 Suppose L and N are the linear and nonlinearoperators with respect to (8) wherein L has inversion and119891 119892 isin 119867

119898

119908(120572120573)(Λ) (119898 ge 1) are satisfied in (8) then

10038171003817100381710038171003817119891 minusI

1

119873(119891)

10038171003817100381710038171003817119908(120572120573)

le 119862119873minus119898

(

10038161003816100381610038161003816Lminus1N (119891)

10038161003816100381610038161003816119867119898

119908(120572120573)

+

10038161003816100381610038161003816Lminus1119892

10038161003816100381610038161003816119867119898

119908(120572120573)

)

(50)

whereI1119873(119891) is an arbitrary polynomial interpolation satisfied

in interpolation conditions

Proof First we must note that I1119873(119891) can be expressed via

Lagrange interpolation and for this goal it is sufficient tointerpolate function I

119873(119891) in 119873 distinct points 119909

119895 119895 =

1 119873 From (8) and invertibility ofL we have

119891 +Lminus1N119891 = L

minus1(119892) (51)

also due to pseudospectral method we obtain

119891 (119909119895) +L

minus1N119891 (119909

119895) = L

minus1119892 (119909119895) (52)

by producing both sides of (52) with 119871119895(119909) (Lagrange inter-

polation basis) and sum from 119894 = 0 119873 we obtain

I119873(119891) +I

119873(Lminus1N (119891)) = I

119873(Lminus1

(119892)) (53)

Now by subtracting (51) from (53) we get

I119873(119891) = L

minus1N (119891) minusI

119873(Lminus1N (119891))

+Lminus1 (119892) minusI119873(Lminus1 (119892))

(54)

International Journal of Computational Mathematics 7

therefore we have10038171003817100381710038171003817119891 minusI

1

119873(119891)

10038171003817100381710038171003817119908(120572120573)

=1003817100381710038171003817119891 minusI

119873(119891)

1003817100381710038171003817119908(120572120573)

le

10038171003817100381710038171003817Lminus1N (119891) minusI

119873(Lminus1N (119891))

10038171003817100381710038171003817119908(120572120573)

+

10038171003817100381710038171003817Lminus1

(119892) minusI119873(Lminus1

(119892))

10038171003817100381710038171003817119908(120572120573)

(55)

Finally by usingTheorem 11 we obtain10038171003817100381710038171003817119891 minusI

1

119873(119891)

10038171003817100381710038171003817119908(120572120573)

=1003817100381710038171003817119891 minusI

119873(119891)

1003817100381710038171003817119908(120572120573)

le 119862119873minus119898

(

10038161003816100381610038161003816Lminus1N119891

10038161003816100381610038161003816119867119898119873

119908(120572120573)

+

10038161003816100381610038161003816Lminus1119892

10038161003816100381610038161003816119867119898119873

119908(120572120573)

)

(56)

therefore the proof is completed

4 The Test Experiments

Experiment 1 Let us consider the following nonlinear differ-ential equation problem [19]

11991010158401015840= minus (1 + 001119910

2) 119910 + 001cos3 (119909)

119910 (minus1) = cos (minus1)

119910 (1) = cos (1)

(57)

with the exact solution 119910(119909) = cos(119909)By simplifying (57) we have

11991010158401015840= minus 119910 minus 001119910

3+ 001cos3 (119909)

119910 (minus1) = cos (minus1)

119910 (1) = cos (1)

(58)

thus the linear and nonlinear parts of our ordinary differen-tial equation respectively are

L (119910) =

1

sum

119894=0

L119894(119910)

N (119910) = 0011199103(119909)

(59)

where

L0(119910) = 119910

10158401015840(119909) L

1(119910) = 119910 (60)

Now we consider 119873 + 1 collocation points 1199090

=

minus1 119909(120572120573)

119894119873minus1

119894=1 119909119873

= 1 where 119909(120572120573)

119894119873minus1

119894=1are the roots of

Jacobi polynomials of degree 119873 minus 1 then we consider thisinterpolation problem

119910 (minus1) = cos (minus1)

119910 (119909(120572120573)

119894) = 119875119873+1

(119909(120572120573)

119894) 119894 = 0 119873 minus 1

119910 (1) = cos (1)

(61)

Table 1 Comparison between the exact and approximate solutionsfor 120572 = 0 120573 = 0 and119873 = 5 of Experiment 1

119909Exact

solutionPseudospectral

method Error

minus10 054030231 054030231 000000000minus080 069670671 069670667 000000004minus060 082533561 082533555 000000006minus040 092106099 092106045 000000054minus020 098006658 098006645 000000013000 100000000 099999922 000000078020 098006658 098006633 000000025040 092106099 092106056 000000043060 082533561 082533545 000000016080 069670671 069670656 00000001510 054030231 054030231 000000000

where 119875119873+1

is the corresponding polynomial interpolationwith respect to problem (61) wherein interpolate the bound-ary conditions automatically and 119910(119909

(120572120573)

119894) 119894 = 0 119873 minus

1 are the unknown coefficients In Table 1 we see thecomparison between the exact and approximate solutionsfrom pseudospectral method for 120572 = 0 120573 = 0 and119873 = 5

Experiment 2 Consider the second-order difference equa-tion problem [20]

119910 (119909 + 1) minus 119910 (119909) = 119890119909 119910 (

1

2

) = 1 (62)

The exact solution is 119910(119909) = (119890119909+119890minusradic119890minus1)(119890minus1) We have

L (119910) = 119910 (119909 + 1) minus 119910 (119909) (63)

and this equation has no nonlinear operator part Theobtained results between the exact and approximate solutionsare shown in Table 2 and Figure 1

Experiment 3 Consider the Fredholm integrodifferentialequation [21]

1199101015840(119909) = int

1

0

119890119909119905119910 (119905) 119889119905 + 119910 (119909) +

1 minus 119890119909+1

119909 + 1

0 le 119909 119910 le 1

(64)

under the initial condition

119910 (0) = 1 (65)

with the exact solution 119910(119909) = 119890119909 In this case we have

L (119910) =

2

sum

119894=0

L119894(119910) (66)

where

L0(119910) = int

1

0

119890119909119905119910 (119905) 119889119905

L1(119910) = 119910

1015840(119909) L

2(119910) = 119910 (119909)

(67)

8 International Journal of Computational Mathematics

28

26

24

22

2

18

16

14

12

1

0

y

x

01 02 03 04 05 06 07 08 09 1

Exact solutionApproximate solution

Figure 1 The obtained results from approximation solution and exact solution for 120572 = minus05 120573 = 07 and 119873 = 5 of Experiment 2

Table 2 Comparison between the exact and approximate solutionsfor 120572 = minus05 120573 = 07 and119873 = 5 of Experiment 2

119909Exact

solutionPseudospectral

method Error

00 062245933 062245922 00000001101 068366636 068366643 minus00000000702 075131058 075131034 00000002403 082606901 082606923 minus00000002204 090868985 090868925 00000006005 100000000 100000000 0000000006 110091330 110091120 00000002107 121243980 121243880 00000001008 133569560 133569450 00000001109 147191430 147191350 00000000810 162245930 162245560 000000037

Table 3 Comparison between the exact and approximate solutionsfor 120572 = minus05 120573 = 07 and119873 = 5 of Experiment 3

119909Exact

solutionPseudospectral

method Error

00 10000000 10000000 00000000001 11051709 11051701 00000000802 12214028 12214011 00000001703 13498588 13498533 00000005504 14918247 14918232 00000001505 16487213 16487211 00000000206 18221188 18221166 00000002207 20137527 20137511 00000001508 22255409 22255401 00000000709 24596031 24596030 00000000110 27182818 27182810 000000008

Now by using the pseudospectral method for 120572 =

minus05 120573 = 07 and 119873 = 5 we obtain the following resultsthat are shown in Table 3

Experiment 4 We consider the linear partial differentialequation problem [22]

119906119905119905= 119906119909119909

+ 6 119906119905(119909 0) = 4119909 119906 (119909 0) = 119909

2 (68)

with the exact solution 119906(119909 119910) = (119909 + 2119905)2

In this case we have

L (119906) = 119906119905119905minus 119906119909119909 (69)

We obtain the approximate solution of (68) on the domain[minus1 1] times [minus1 1] For this purpose we consider the discretepoints (119909(120572120573)

119894 119905(120572120573)

119895)119873

119894119895=0and the general Lagrange interpola-

tion in the discrete points (119909(120572120573)119894

119905(120572120573)

119895)119873

119894119895=0in the following

form (two-dimensional case)

119871 (119909 119905) =

119873

sum

119894=0

119873

sum

119895=0

119891 (119909(120572120573)

119894 119905(120572120573)

119895) 119871119894(119909) 119871119895(119905) (70)

where 119871119894(119909) and 119871

119895(119905) are ordinary Lagrange interpolations

By using collocation method for arbitrary parameters 120572 120573and119873 = 5 we obtain the exact solution

Experiment 5 Finally let us consider the linear fractionaldifferential equation problem [23]

11986352

119910 (119909) + 3119910 (119909) = 31199093+

6

Γ (15)

11990912

119910 (0) = 0 1199101015840(0) = 0 119910

10158401015840(0) = 0 119909 isin [minus1 1]

(71)

This equation has the exact solution 119910(119909) = 1199093

In this case we have

L (119910 (119909)) = 11986352

(119910 (119909)) + 3119910 (119909) (72)

which is a linear operator Using the presented method forevery 120572 120573 and119873 = 3 we obtain the exact solution

International Journal of Computational Mathematics 9

5 Conclusion

In this paper we have developed the expansion approach forsolving some operator equations of the form L119906 + N119906 =

119892(119909) 119909 isin Γ with respect to boundary condition B119906 = 120582For this the principal importance is the development of theexpansion approach for solving general operator equationwherein the particular cases are in the solution of integralequations (Experiment 3) ordinary differential equations(Experiment 1) difference equations (Experiment 2) partialdifferential equations (Experiment 4) and fractional differ-ential equations (Experiment 5) Also the error analysis ispresented in 119871

2(Γ) norm Using the roots of another orthog-

onal polynomials (Jacobi polynomials) rather than classicalorthogonal polynomials such as Chebyshev polynomialsand Legendre polynomials for collocation points (with freeparameters 120572 120573) in pseudospectral method is an importantsubject which is performed (Experiments 2 and 3)

Conflict of Interests

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

Acknowledgments

The authors wish to thank the anonymous referees for theirvaluable comments and suggestions Also they would like toappreciate the Editor Dr Yaohanz Li who gives them theopportunity to review the paper

References

[1] P LinzTheoretical Numerical Analysis JohnWiley amp Sons NewYork NY USA 1979

[2] D Gottlieb and S A Orszag Numerical Analysis of SpectralMethods Theory and Applications SIAM-CBMS PhiladelphiaPa USA 1977

[3] G Ben-Yu The State of Art in Spectral Methods Hong KongUniversity 1996

[4] C Canuto M Y Hussaini A Quarteroni and T A ZhangSpectral Methods Fundamentals in Single Domains Springer2006

[5] C Canutuo M Y Hussaini A Quarteroni and T A ZhangSpectral Method in Fluid Dynamics Prentice Hall EngelwoodCliffs NJ USA 1984

[6] B FornbergAPractical Quide to PseudospectralMethods Cam-bridgeMonographs on Applied and Computational Mathemat-ics Cambridge University Press Cambridge UK 1st edition1996

[7] L N Trefethen Spectral Methods in Matlab SIAM Philadel-phia Pa USA 2000

[8] J S Hesthaven S Gottlieb and D Gottlieb Spectral Methodsfor Time-Dependent Problems vol 21 ofCambridgeMonographson Applied and Computational Mathematics Cambridge Uni-versity Press Cambridge UK 2007

[9] M Deville and E Mund ldquoChebyshev pseudospectral solutionof second-order elliptic equations with finite element precon-ditioningrdquo Journal of Computational Physics vol 60 no 3 pp517ndash533 1985

[10] J P Boyd ldquoMultipole expansions and pseudospectral cardinalfunctions a new generalization of the fast Fourier transformrdquoJournal of Computational Physics vol 103 no 1 pp 184ndash1861992

[11] G N Elnagar and M Razzaghi ldquoA pseudospectral method forHammerstein equationsrdquo Journal of Mathematical Analysis andApplications vol 199 no 2 pp 579ndash591 1996

[12] M Meehan and D OrsquoRegan ldquoExistence principles for nonreso-nant operator and integral equationsrdquo Computers ampMathemat-ics with Applications vol 35 no 9 pp 79ndash87 1998

[13] A Chaillou and M Suri ldquoA posteriori estimation of thelinearization error for strongly monotone nonlinear operatorsrdquoJournal of Computational andAppliedMathematics vol 205 no1 pp 72ndash87 2007

[14] R P Kanwal and K C Liu ldquoA Taylor expansion approach forsolving integral equationsrdquo International Journal of Mathemat-ical Education in Science and Technology vol 20 pp 411ndash4141989

[15] M R Eslahchi M Dehghan and S Ahmadi Asl ldquoThe generalJacobi matrix method for solving some nonlinear ordinarydifferential equationsrdquoAppliedMathematical Modelling vol 36no 8 pp 3387ndash3398 2012

[16] A Imani A Aminataei and A Imani ldquoCollocation method viaJacobi polynomials for solving nonlinear ordinary differentialequationsrdquo International Journal of Mathematics and Mathe-matical Sciences vol 2011 Article ID 673085 11 pages 2011

[17] M R Eslahchi and M Dehghan ldquoApplication of Taylor seriesin obtaining the orthogonal operational matrixrdquo Computers ampMathematics with Applications vol 61 no 9 pp 2596ndash26042011

[18] D Funaro Polynomial Approximation of Differential Equationsvol 8 of Lecture Notes in Physics Monographs Springer BerlinGermany 1992

[19] D F Griffiths and D J Higham Numerical Methods forOrdinaryDifferential Equations SpringerUndergraduateMath-ematics Series Springer London UK 2010

[20] M Gulsu M Sezer and B Tanay ldquoA matrix method for solvinghigh-order linear difference equations with mixed argumentusing hybrid Legendre and Taylor polynomialsrdquo Journal of theFranklin Institute vol 343 no 6 pp 647ndash659 2006

[21] N Kurt and M Sezer ldquoPolynomial solution of high-orderlinear Fredholm integro-differential equations with constantcoefficientsrdquo Journal of the Franklin Institute vol 345 no 8 pp839ndash850 2008

[22] V S Vladimirov A Collection of Problems on the Equations ofMathematical Physics Mir Publishers Moscow Russia 1968

[23] A H Bhrawy M M Alghamdi and T M Taha ldquoA newmodified generalized Laguerre operational matrix of fractionalintegration for solving fractional differential equations on thehalf linerdquoAdvances in Difference Equations vol 2012 article 1792012

Submit your manuscripts athttpwwwhindawicom

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 International Journal of Computational Mathematics

weight function119908120572120573

(119909) = (1minus119909)120572(1 +119909)

120573 (120572 gt minus1 120573 gt minus1)

on (minus1 1) An explicit formula for these polynomials is givenby

119875(120572120573)

119899(119909)

=

1

2119899

119899

sum

119894=0

(

119899 + 120572

119894)(

119899 + 120573

119899 minus 119894) (119909 minus 1)

119899minus119894(119909 + 1)

119894

(43)

Further we have

119889

119889119909

119875(120572120573)

119899(119909) =

1

2

(119899 + 120572 + 120573 + 1) 119875(120572+1120573+1)

119899minus1(119909) (44)

Jacobi polynomials have an important application in severalfields of numerical analysis such as quadratures formula andspectral methods Also the shifted Jacobi polynomials aredefined as orthogonal polynomials in the interval [119886 119887] withrespect to weight function 119908

120572120573(119909) = (119887 minus 119909)

120572(119886 minus 119909)

120573 (120572 gt

minus1 120573 gt minus1) with change of variable 119910 = (2(119887 minus 119886))119909 +

((119886 + 119887)(119886 minus 119887)) For more properties and applications ofthese polynomials see [15ndash17] Now let Λ = [minus1 1] and let119875(120572120573)

119873+1(119909) be the Jacobi polynomial of degree119873+1with respect

to weight function 119908(120572120573)

(119909) = (1 minus 119909)120572(1 + 119909)

120573 (120572 120573 gt minus1)

and let P119873be the polynomials spaces of degree less than or

equal to119873

Remark 10 The best choice of points for polynomial inter-polation are the roots of orthogonal polynomials and in par-ticular case Jacobi polynomials due to their small Lebesgueconstants [6]

The symbol 1198712

119908(120572120573)(Λ) shows the space of measurable

functions whose square is Lebesgue integrable in Λ relativeto the weight function119908

(120572120573) The inner product and norm of1198712

119908(120572120573)(Λ) are defined by

(119906 V)119908(120572120573)Λ

= int

Λ

119906 (119909) V (119909) 119908(120572120573) (119909) 119889119909

119906119908(120572120573)Λ

= radic(119906 119906)119908(120572120573)

forall119906 V isin 1198712

119908(120572120573) (Λ)

(45)

Also another useful space is

119867119898

119908(120572120573) = V V

119898119908(120572120573) lt infin (46)

with seminorm and norms respectively Consider

|V|119898119908(120572120573) =

1003817100381710038171003817120597119898

119909V1003817100381710038171003817119908(120572120573)

V119898119908(120572120573) = (

119898

sum

119894=0

|V|2119894119908(120572120573))

12

(47)

But an applicable norm that appears in bounding of errors ofthe spectral method is

|V|119867119898119908(120572120573) = (

119898

sum

119896=119890

10038171003817100381710038171003817120597119896

119909V10038171003817100381710038171003817

2

119908(120572120573)

)

12

119890 = min (119898119873 + 1)

(48)

Now we present a useful theorem

Theorem 11 If ℎ isin 119867119898

119908(120572120573)(Λ)119898 ge 1 then one has

1003817100381710038171003817ℎ minusI

119873(ℎ)

1003817100381710038171003817119908(120572120573) le 119862119873

minus11989810038161003816100381610038161198911003816100381610038161003816119867119898

119908(120572120573)

(49)

where I119873(ℎ) is a Lagrange interpolation in distinct points

119909(120572120573)

119894(roots of Jacobi polynomials)

Proof See [4 5]

Noting to Theorem 11 we can obtain an error boundfor any classes of polynomial interpolation such as Hermiteinterpolation Birkhoff interpolation and other classes Thismeans that if we have an arbitrary polynomial interpolationI1119873(ℎ) thenwe can transform the interpolationI1

119873(ℎ) to the

Lagrange interpolation I119873(ℎ) and then we use Theorem 11

for obtaining an error bound Now by using Theorem 11 inthe next theorem we obtain an upper bound for the error ofour approximate solution of (8)

Theorem 12 Suppose L and N are the linear and nonlinearoperators with respect to (8) wherein L has inversion and119891 119892 isin 119867

119898

119908(120572120573)(Λ) (119898 ge 1) are satisfied in (8) then

10038171003817100381710038171003817119891 minusI

1

119873(119891)

10038171003817100381710038171003817119908(120572120573)

le 119862119873minus119898

(

10038161003816100381610038161003816Lminus1N (119891)

10038161003816100381610038161003816119867119898

119908(120572120573)

+

10038161003816100381610038161003816Lminus1119892

10038161003816100381610038161003816119867119898

119908(120572120573)

)

(50)

whereI1119873(119891) is an arbitrary polynomial interpolation satisfied

in interpolation conditions

Proof First we must note that I1119873(119891) can be expressed via

Lagrange interpolation and for this goal it is sufficient tointerpolate function I

119873(119891) in 119873 distinct points 119909

119895 119895 =

1 119873 From (8) and invertibility ofL we have

119891 +Lminus1N119891 = L

minus1(119892) (51)

also due to pseudospectral method we obtain

119891 (119909119895) +L

minus1N119891 (119909

119895) = L

minus1119892 (119909119895) (52)

by producing both sides of (52) with 119871119895(119909) (Lagrange inter-

polation basis) and sum from 119894 = 0 119873 we obtain

I119873(119891) +I

119873(Lminus1N (119891)) = I

119873(Lminus1

(119892)) (53)

Now by subtracting (51) from (53) we get

I119873(119891) = L

minus1N (119891) minusI

119873(Lminus1N (119891))

+Lminus1 (119892) minusI119873(Lminus1 (119892))

(54)

International Journal of Computational Mathematics 7

therefore we have10038171003817100381710038171003817119891 minusI

1

119873(119891)

10038171003817100381710038171003817119908(120572120573)

=1003817100381710038171003817119891 minusI

119873(119891)

1003817100381710038171003817119908(120572120573)

le

10038171003817100381710038171003817Lminus1N (119891) minusI

119873(Lminus1N (119891))

10038171003817100381710038171003817119908(120572120573)

+

10038171003817100381710038171003817Lminus1

(119892) minusI119873(Lminus1

(119892))

10038171003817100381710038171003817119908(120572120573)

(55)

Finally by usingTheorem 11 we obtain10038171003817100381710038171003817119891 minusI

1

119873(119891)

10038171003817100381710038171003817119908(120572120573)

=1003817100381710038171003817119891 minusI

119873(119891)

1003817100381710038171003817119908(120572120573)

le 119862119873minus119898

(

10038161003816100381610038161003816Lminus1N119891

10038161003816100381610038161003816119867119898119873

119908(120572120573)

+

10038161003816100381610038161003816Lminus1119892

10038161003816100381610038161003816119867119898119873

119908(120572120573)

)

(56)

therefore the proof is completed

4 The Test Experiments

Experiment 1 Let us consider the following nonlinear differ-ential equation problem [19]

11991010158401015840= minus (1 + 001119910

2) 119910 + 001cos3 (119909)

119910 (minus1) = cos (minus1)

119910 (1) = cos (1)

(57)

with the exact solution 119910(119909) = cos(119909)By simplifying (57) we have

11991010158401015840= minus 119910 minus 001119910

3+ 001cos3 (119909)

119910 (minus1) = cos (minus1)

119910 (1) = cos (1)

(58)

thus the linear and nonlinear parts of our ordinary differen-tial equation respectively are

L (119910) =

1

sum

119894=0

L119894(119910)

N (119910) = 0011199103(119909)

(59)

where

L0(119910) = 119910

10158401015840(119909) L

1(119910) = 119910 (60)

Now we consider 119873 + 1 collocation points 1199090

=

minus1 119909(120572120573)

119894119873minus1

119894=1 119909119873

= 1 where 119909(120572120573)

119894119873minus1

119894=1are the roots of

Jacobi polynomials of degree 119873 minus 1 then we consider thisinterpolation problem

119910 (minus1) = cos (minus1)

119910 (119909(120572120573)

119894) = 119875119873+1

(119909(120572120573)

119894) 119894 = 0 119873 minus 1

119910 (1) = cos (1)

(61)

Table 1 Comparison between the exact and approximate solutionsfor 120572 = 0 120573 = 0 and119873 = 5 of Experiment 1

119909Exact

solutionPseudospectral

method Error

minus10 054030231 054030231 000000000minus080 069670671 069670667 000000004minus060 082533561 082533555 000000006minus040 092106099 092106045 000000054minus020 098006658 098006645 000000013000 100000000 099999922 000000078020 098006658 098006633 000000025040 092106099 092106056 000000043060 082533561 082533545 000000016080 069670671 069670656 00000001510 054030231 054030231 000000000

where 119875119873+1

is the corresponding polynomial interpolationwith respect to problem (61) wherein interpolate the bound-ary conditions automatically and 119910(119909

(120572120573)

119894) 119894 = 0 119873 minus

1 are the unknown coefficients In Table 1 we see thecomparison between the exact and approximate solutionsfrom pseudospectral method for 120572 = 0 120573 = 0 and119873 = 5

Experiment 2 Consider the second-order difference equa-tion problem [20]

119910 (119909 + 1) minus 119910 (119909) = 119890119909 119910 (

1

2

) = 1 (62)

The exact solution is 119910(119909) = (119890119909+119890minusradic119890minus1)(119890minus1) We have

L (119910) = 119910 (119909 + 1) minus 119910 (119909) (63)

and this equation has no nonlinear operator part Theobtained results between the exact and approximate solutionsare shown in Table 2 and Figure 1

Experiment 3 Consider the Fredholm integrodifferentialequation [21]

1199101015840(119909) = int

1

0

119890119909119905119910 (119905) 119889119905 + 119910 (119909) +

1 minus 119890119909+1

119909 + 1

0 le 119909 119910 le 1

(64)

under the initial condition

119910 (0) = 1 (65)

with the exact solution 119910(119909) = 119890119909 In this case we have

L (119910) =

2

sum

119894=0

L119894(119910) (66)

where

L0(119910) = int

1

0

119890119909119905119910 (119905) 119889119905

L1(119910) = 119910

1015840(119909) L

2(119910) = 119910 (119909)

(67)

8 International Journal of Computational Mathematics

28

26

24

22

2

18

16

14

12

1

0

y

x

01 02 03 04 05 06 07 08 09 1

Exact solutionApproximate solution

Figure 1 The obtained results from approximation solution and exact solution for 120572 = minus05 120573 = 07 and 119873 = 5 of Experiment 2

Table 2 Comparison between the exact and approximate solutionsfor 120572 = minus05 120573 = 07 and119873 = 5 of Experiment 2

119909Exact

solutionPseudospectral

method Error

00 062245933 062245922 00000001101 068366636 068366643 minus00000000702 075131058 075131034 00000002403 082606901 082606923 minus00000002204 090868985 090868925 00000006005 100000000 100000000 0000000006 110091330 110091120 00000002107 121243980 121243880 00000001008 133569560 133569450 00000001109 147191430 147191350 00000000810 162245930 162245560 000000037

Table 3 Comparison between the exact and approximate solutionsfor 120572 = minus05 120573 = 07 and119873 = 5 of Experiment 3

119909Exact

solutionPseudospectral

method Error

00 10000000 10000000 00000000001 11051709 11051701 00000000802 12214028 12214011 00000001703 13498588 13498533 00000005504 14918247 14918232 00000001505 16487213 16487211 00000000206 18221188 18221166 00000002207 20137527 20137511 00000001508 22255409 22255401 00000000709 24596031 24596030 00000000110 27182818 27182810 000000008

Now by using the pseudospectral method for 120572 =

minus05 120573 = 07 and 119873 = 5 we obtain the following resultsthat are shown in Table 3

Experiment 4 We consider the linear partial differentialequation problem [22]

119906119905119905= 119906119909119909

+ 6 119906119905(119909 0) = 4119909 119906 (119909 0) = 119909

2 (68)

with the exact solution 119906(119909 119910) = (119909 + 2119905)2

In this case we have

L (119906) = 119906119905119905minus 119906119909119909 (69)

We obtain the approximate solution of (68) on the domain[minus1 1] times [minus1 1] For this purpose we consider the discretepoints (119909(120572120573)

119894 119905(120572120573)

119895)119873

119894119895=0and the general Lagrange interpola-

tion in the discrete points (119909(120572120573)119894

119905(120572120573)

119895)119873

119894119895=0in the following

form (two-dimensional case)

119871 (119909 119905) =

119873

sum

119894=0

119873

sum

119895=0

119891 (119909(120572120573)

119894 119905(120572120573)

119895) 119871119894(119909) 119871119895(119905) (70)

where 119871119894(119909) and 119871

119895(119905) are ordinary Lagrange interpolations

By using collocation method for arbitrary parameters 120572 120573and119873 = 5 we obtain the exact solution

Experiment 5 Finally let us consider the linear fractionaldifferential equation problem [23]

11986352

119910 (119909) + 3119910 (119909) = 31199093+

6

Γ (15)

11990912

119910 (0) = 0 1199101015840(0) = 0 119910

10158401015840(0) = 0 119909 isin [minus1 1]

(71)

This equation has the exact solution 119910(119909) = 1199093

In this case we have

L (119910 (119909)) = 11986352

(119910 (119909)) + 3119910 (119909) (72)

which is a linear operator Using the presented method forevery 120572 120573 and119873 = 3 we obtain the exact solution

International Journal of Computational Mathematics 9

5 Conclusion

In this paper we have developed the expansion approach forsolving some operator equations of the form L119906 + N119906 =

119892(119909) 119909 isin Γ with respect to boundary condition B119906 = 120582For this the principal importance is the development of theexpansion approach for solving general operator equationwherein the particular cases are in the solution of integralequations (Experiment 3) ordinary differential equations(Experiment 1) difference equations (Experiment 2) partialdifferential equations (Experiment 4) and fractional differ-ential equations (Experiment 5) Also the error analysis ispresented in 119871

2(Γ) norm Using the roots of another orthog-

onal polynomials (Jacobi polynomials) rather than classicalorthogonal polynomials such as Chebyshev polynomialsand Legendre polynomials for collocation points (with freeparameters 120572 120573) in pseudospectral method is an importantsubject which is performed (Experiments 2 and 3)

Conflict of Interests

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

Acknowledgments

The authors wish to thank the anonymous referees for theirvaluable comments and suggestions Also they would like toappreciate the Editor Dr Yaohanz Li who gives them theopportunity to review the paper

References

[1] P LinzTheoretical Numerical Analysis JohnWiley amp Sons NewYork NY USA 1979

[2] D Gottlieb and S A Orszag Numerical Analysis of SpectralMethods Theory and Applications SIAM-CBMS PhiladelphiaPa USA 1977

[3] G Ben-Yu The State of Art in Spectral Methods Hong KongUniversity 1996

[4] C Canuto M Y Hussaini A Quarteroni and T A ZhangSpectral Methods Fundamentals in Single Domains Springer2006

[5] C Canutuo M Y Hussaini A Quarteroni and T A ZhangSpectral Method in Fluid Dynamics Prentice Hall EngelwoodCliffs NJ USA 1984

[6] B FornbergAPractical Quide to PseudospectralMethods Cam-bridgeMonographs on Applied and Computational Mathemat-ics Cambridge University Press Cambridge UK 1st edition1996

[7] L N Trefethen Spectral Methods in Matlab SIAM Philadel-phia Pa USA 2000

[8] J S Hesthaven S Gottlieb and D Gottlieb Spectral Methodsfor Time-Dependent Problems vol 21 ofCambridgeMonographson Applied and Computational Mathematics Cambridge Uni-versity Press Cambridge UK 2007

[9] M Deville and E Mund ldquoChebyshev pseudospectral solutionof second-order elliptic equations with finite element precon-ditioningrdquo Journal of Computational Physics vol 60 no 3 pp517ndash533 1985

[10] J P Boyd ldquoMultipole expansions and pseudospectral cardinalfunctions a new generalization of the fast Fourier transformrdquoJournal of Computational Physics vol 103 no 1 pp 184ndash1861992

[11] G N Elnagar and M Razzaghi ldquoA pseudospectral method forHammerstein equationsrdquo Journal of Mathematical Analysis andApplications vol 199 no 2 pp 579ndash591 1996

[12] M Meehan and D OrsquoRegan ldquoExistence principles for nonreso-nant operator and integral equationsrdquo Computers ampMathemat-ics with Applications vol 35 no 9 pp 79ndash87 1998

[13] A Chaillou and M Suri ldquoA posteriori estimation of thelinearization error for strongly monotone nonlinear operatorsrdquoJournal of Computational andAppliedMathematics vol 205 no1 pp 72ndash87 2007

[14] R P Kanwal and K C Liu ldquoA Taylor expansion approach forsolving integral equationsrdquo International Journal of Mathemat-ical Education in Science and Technology vol 20 pp 411ndash4141989

[15] M R Eslahchi M Dehghan and S Ahmadi Asl ldquoThe generalJacobi matrix method for solving some nonlinear ordinarydifferential equationsrdquoAppliedMathematical Modelling vol 36no 8 pp 3387ndash3398 2012

[16] A Imani A Aminataei and A Imani ldquoCollocation method viaJacobi polynomials for solving nonlinear ordinary differentialequationsrdquo International Journal of Mathematics and Mathe-matical Sciences vol 2011 Article ID 673085 11 pages 2011

[17] M R Eslahchi and M Dehghan ldquoApplication of Taylor seriesin obtaining the orthogonal operational matrixrdquo Computers ampMathematics with Applications vol 61 no 9 pp 2596ndash26042011

[18] D Funaro Polynomial Approximation of Differential Equationsvol 8 of Lecture Notes in Physics Monographs Springer BerlinGermany 1992

[19] D F Griffiths and D J Higham Numerical Methods forOrdinaryDifferential Equations SpringerUndergraduateMath-ematics Series Springer London UK 2010

[20] M Gulsu M Sezer and B Tanay ldquoA matrix method for solvinghigh-order linear difference equations with mixed argumentusing hybrid Legendre and Taylor polynomialsrdquo Journal of theFranklin Institute vol 343 no 6 pp 647ndash659 2006

[21] N Kurt and M Sezer ldquoPolynomial solution of high-orderlinear Fredholm integro-differential equations with constantcoefficientsrdquo Journal of the Franklin Institute vol 345 no 8 pp839ndash850 2008

[22] V S Vladimirov A Collection of Problems on the Equations ofMathematical Physics Mir Publishers Moscow Russia 1968

[23] A H Bhrawy M M Alghamdi and T M Taha ldquoA newmodified generalized Laguerre operational matrix of fractionalintegration for solving fractional differential equations on thehalf linerdquoAdvances in Difference Equations vol 2012 article 1792012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Computational Mathematics 7

therefore we have10038171003817100381710038171003817119891 minusI

1

119873(119891)

10038171003817100381710038171003817119908(120572120573)

=1003817100381710038171003817119891 minusI

119873(119891)

1003817100381710038171003817119908(120572120573)

le

10038171003817100381710038171003817Lminus1N (119891) minusI

119873(Lminus1N (119891))

10038171003817100381710038171003817119908(120572120573)

+

10038171003817100381710038171003817Lminus1

(119892) minusI119873(Lminus1

(119892))

10038171003817100381710038171003817119908(120572120573)

(55)

Finally by usingTheorem 11 we obtain10038171003817100381710038171003817119891 minusI

1

119873(119891)

10038171003817100381710038171003817119908(120572120573)

=1003817100381710038171003817119891 minusI

119873(119891)

1003817100381710038171003817119908(120572120573)

le 119862119873minus119898

(

10038161003816100381610038161003816Lminus1N119891

10038161003816100381610038161003816119867119898119873

119908(120572120573)

+

10038161003816100381610038161003816Lminus1119892

10038161003816100381610038161003816119867119898119873

119908(120572120573)

)

(56)

therefore the proof is completed

4 The Test Experiments

Experiment 1 Let us consider the following nonlinear differ-ential equation problem [19]

11991010158401015840= minus (1 + 001119910

2) 119910 + 001cos3 (119909)

119910 (minus1) = cos (minus1)

119910 (1) = cos (1)

(57)

with the exact solution 119910(119909) = cos(119909)By simplifying (57) we have

11991010158401015840= minus 119910 minus 001119910

3+ 001cos3 (119909)

119910 (minus1) = cos (minus1)

119910 (1) = cos (1)

(58)

thus the linear and nonlinear parts of our ordinary differen-tial equation respectively are

L (119910) =

1

sum

119894=0

L119894(119910)

N (119910) = 0011199103(119909)

(59)

where

L0(119910) = 119910

10158401015840(119909) L

1(119910) = 119910 (60)

Now we consider 119873 + 1 collocation points 1199090

=

minus1 119909(120572120573)

119894119873minus1

119894=1 119909119873

= 1 where 119909(120572120573)

119894119873minus1

119894=1are the roots of

Jacobi polynomials of degree 119873 minus 1 then we consider thisinterpolation problem

119910 (minus1) = cos (minus1)

119910 (119909(120572120573)

119894) = 119875119873+1

(119909(120572120573)

119894) 119894 = 0 119873 minus 1

119910 (1) = cos (1)

(61)

Table 1 Comparison between the exact and approximate solutionsfor 120572 = 0 120573 = 0 and119873 = 5 of Experiment 1

119909Exact

solutionPseudospectral

method Error

minus10 054030231 054030231 000000000minus080 069670671 069670667 000000004minus060 082533561 082533555 000000006minus040 092106099 092106045 000000054minus020 098006658 098006645 000000013000 100000000 099999922 000000078020 098006658 098006633 000000025040 092106099 092106056 000000043060 082533561 082533545 000000016080 069670671 069670656 00000001510 054030231 054030231 000000000

where 119875119873+1

is the corresponding polynomial interpolationwith respect to problem (61) wherein interpolate the bound-ary conditions automatically and 119910(119909

(120572120573)

119894) 119894 = 0 119873 minus

1 are the unknown coefficients In Table 1 we see thecomparison between the exact and approximate solutionsfrom pseudospectral method for 120572 = 0 120573 = 0 and119873 = 5

Experiment 2 Consider the second-order difference equa-tion problem [20]

119910 (119909 + 1) minus 119910 (119909) = 119890119909 119910 (

1

2

) = 1 (62)

The exact solution is 119910(119909) = (119890119909+119890minusradic119890minus1)(119890minus1) We have

L (119910) = 119910 (119909 + 1) minus 119910 (119909) (63)

and this equation has no nonlinear operator part Theobtained results between the exact and approximate solutionsare shown in Table 2 and Figure 1

Experiment 3 Consider the Fredholm integrodifferentialequation [21]

1199101015840(119909) = int

1

0

119890119909119905119910 (119905) 119889119905 + 119910 (119909) +

1 minus 119890119909+1

119909 + 1

0 le 119909 119910 le 1

(64)

under the initial condition

119910 (0) = 1 (65)

with the exact solution 119910(119909) = 119890119909 In this case we have

L (119910) =

2

sum

119894=0

L119894(119910) (66)

where

L0(119910) = int

1

0

119890119909119905119910 (119905) 119889119905

L1(119910) = 119910

1015840(119909) L

2(119910) = 119910 (119909)

(67)

8 International Journal of Computational Mathematics

28

26

24

22

2

18

16

14

12

1

0

y

x

01 02 03 04 05 06 07 08 09 1

Exact solutionApproximate solution

Figure 1 The obtained results from approximation solution and exact solution for 120572 = minus05 120573 = 07 and 119873 = 5 of Experiment 2

Table 2 Comparison between the exact and approximate solutionsfor 120572 = minus05 120573 = 07 and119873 = 5 of Experiment 2

119909Exact

solutionPseudospectral

method Error

00 062245933 062245922 00000001101 068366636 068366643 minus00000000702 075131058 075131034 00000002403 082606901 082606923 minus00000002204 090868985 090868925 00000006005 100000000 100000000 0000000006 110091330 110091120 00000002107 121243980 121243880 00000001008 133569560 133569450 00000001109 147191430 147191350 00000000810 162245930 162245560 000000037

Table 3 Comparison between the exact and approximate solutionsfor 120572 = minus05 120573 = 07 and119873 = 5 of Experiment 3

119909Exact

solutionPseudospectral

method Error

00 10000000 10000000 00000000001 11051709 11051701 00000000802 12214028 12214011 00000001703 13498588 13498533 00000005504 14918247 14918232 00000001505 16487213 16487211 00000000206 18221188 18221166 00000002207 20137527 20137511 00000001508 22255409 22255401 00000000709 24596031 24596030 00000000110 27182818 27182810 000000008

Now by using the pseudospectral method for 120572 =

minus05 120573 = 07 and 119873 = 5 we obtain the following resultsthat are shown in Table 3

Experiment 4 We consider the linear partial differentialequation problem [22]

119906119905119905= 119906119909119909

+ 6 119906119905(119909 0) = 4119909 119906 (119909 0) = 119909

2 (68)

with the exact solution 119906(119909 119910) = (119909 + 2119905)2

In this case we have

L (119906) = 119906119905119905minus 119906119909119909 (69)

We obtain the approximate solution of (68) on the domain[minus1 1] times [minus1 1] For this purpose we consider the discretepoints (119909(120572120573)

119894 119905(120572120573)

119895)119873

119894119895=0and the general Lagrange interpola-

tion in the discrete points (119909(120572120573)119894

119905(120572120573)

119895)119873

119894119895=0in the following

form (two-dimensional case)

119871 (119909 119905) =

119873

sum

119894=0

119873

sum

119895=0

119891 (119909(120572120573)

119894 119905(120572120573)

119895) 119871119894(119909) 119871119895(119905) (70)

where 119871119894(119909) and 119871

119895(119905) are ordinary Lagrange interpolations

By using collocation method for arbitrary parameters 120572 120573and119873 = 5 we obtain the exact solution

Experiment 5 Finally let us consider the linear fractionaldifferential equation problem [23]

11986352

119910 (119909) + 3119910 (119909) = 31199093+

6

Γ (15)

11990912

119910 (0) = 0 1199101015840(0) = 0 119910

10158401015840(0) = 0 119909 isin [minus1 1]

(71)

This equation has the exact solution 119910(119909) = 1199093

In this case we have

L (119910 (119909)) = 11986352

(119910 (119909)) + 3119910 (119909) (72)

which is a linear operator Using the presented method forevery 120572 120573 and119873 = 3 we obtain the exact solution

International Journal of Computational Mathematics 9

5 Conclusion

In this paper we have developed the expansion approach forsolving some operator equations of the form L119906 + N119906 =

119892(119909) 119909 isin Γ with respect to boundary condition B119906 = 120582For this the principal importance is the development of theexpansion approach for solving general operator equationwherein the particular cases are in the solution of integralequations (Experiment 3) ordinary differential equations(Experiment 1) difference equations (Experiment 2) partialdifferential equations (Experiment 4) and fractional differ-ential equations (Experiment 5) Also the error analysis ispresented in 119871

2(Γ) norm Using the roots of another orthog-

onal polynomials (Jacobi polynomials) rather than classicalorthogonal polynomials such as Chebyshev polynomialsand Legendre polynomials for collocation points (with freeparameters 120572 120573) in pseudospectral method is an importantsubject which is performed (Experiments 2 and 3)

Conflict of Interests

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

Acknowledgments

The authors wish to thank the anonymous referees for theirvaluable comments and suggestions Also they would like toappreciate the Editor Dr Yaohanz Li who gives them theopportunity to review the paper

References

[1] P LinzTheoretical Numerical Analysis JohnWiley amp Sons NewYork NY USA 1979

[2] D Gottlieb and S A Orszag Numerical Analysis of SpectralMethods Theory and Applications SIAM-CBMS PhiladelphiaPa USA 1977

[3] G Ben-Yu The State of Art in Spectral Methods Hong KongUniversity 1996

[4] C Canuto M Y Hussaini A Quarteroni and T A ZhangSpectral Methods Fundamentals in Single Domains Springer2006

[5] C Canutuo M Y Hussaini A Quarteroni and T A ZhangSpectral Method in Fluid Dynamics Prentice Hall EngelwoodCliffs NJ USA 1984

[6] B FornbergAPractical Quide to PseudospectralMethods Cam-bridgeMonographs on Applied and Computational Mathemat-ics Cambridge University Press Cambridge UK 1st edition1996

[7] L N Trefethen Spectral Methods in Matlab SIAM Philadel-phia Pa USA 2000

[8] J S Hesthaven S Gottlieb and D Gottlieb Spectral Methodsfor Time-Dependent Problems vol 21 ofCambridgeMonographson Applied and Computational Mathematics Cambridge Uni-versity Press Cambridge UK 2007

[9] M Deville and E Mund ldquoChebyshev pseudospectral solutionof second-order elliptic equations with finite element precon-ditioningrdquo Journal of Computational Physics vol 60 no 3 pp517ndash533 1985

[10] J P Boyd ldquoMultipole expansions and pseudospectral cardinalfunctions a new generalization of the fast Fourier transformrdquoJournal of Computational Physics vol 103 no 1 pp 184ndash1861992

[11] G N Elnagar and M Razzaghi ldquoA pseudospectral method forHammerstein equationsrdquo Journal of Mathematical Analysis andApplications vol 199 no 2 pp 579ndash591 1996

[12] M Meehan and D OrsquoRegan ldquoExistence principles for nonreso-nant operator and integral equationsrdquo Computers ampMathemat-ics with Applications vol 35 no 9 pp 79ndash87 1998

[13] A Chaillou and M Suri ldquoA posteriori estimation of thelinearization error for strongly monotone nonlinear operatorsrdquoJournal of Computational andAppliedMathematics vol 205 no1 pp 72ndash87 2007

[14] R P Kanwal and K C Liu ldquoA Taylor expansion approach forsolving integral equationsrdquo International Journal of Mathemat-ical Education in Science and Technology vol 20 pp 411ndash4141989

[15] M R Eslahchi M Dehghan and S Ahmadi Asl ldquoThe generalJacobi matrix method for solving some nonlinear ordinarydifferential equationsrdquoAppliedMathematical Modelling vol 36no 8 pp 3387ndash3398 2012

[16] A Imani A Aminataei and A Imani ldquoCollocation method viaJacobi polynomials for solving nonlinear ordinary differentialequationsrdquo International Journal of Mathematics and Mathe-matical Sciences vol 2011 Article ID 673085 11 pages 2011

[17] M R Eslahchi and M Dehghan ldquoApplication of Taylor seriesin obtaining the orthogonal operational matrixrdquo Computers ampMathematics with Applications vol 61 no 9 pp 2596ndash26042011

[18] D Funaro Polynomial Approximation of Differential Equationsvol 8 of Lecture Notes in Physics Monographs Springer BerlinGermany 1992

[19] D F Griffiths and D J Higham Numerical Methods forOrdinaryDifferential Equations SpringerUndergraduateMath-ematics Series Springer London UK 2010

[20] M Gulsu M Sezer and B Tanay ldquoA matrix method for solvinghigh-order linear difference equations with mixed argumentusing hybrid Legendre and Taylor polynomialsrdquo Journal of theFranklin Institute vol 343 no 6 pp 647ndash659 2006

[21] N Kurt and M Sezer ldquoPolynomial solution of high-orderlinear Fredholm integro-differential equations with constantcoefficientsrdquo Journal of the Franklin Institute vol 345 no 8 pp839ndash850 2008

[22] V S Vladimirov A Collection of Problems on the Equations ofMathematical Physics Mir Publishers Moscow Russia 1968

[23] A H Bhrawy M M Alghamdi and T M Taha ldquoA newmodified generalized Laguerre operational matrix of fractionalintegration for solving fractional differential equations on thehalf linerdquoAdvances in Difference Equations vol 2012 article 1792012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 International Journal of Computational Mathematics

28

26

24

22

2

18

16

14

12

1

0

y

x

01 02 03 04 05 06 07 08 09 1

Exact solutionApproximate solution

Figure 1 The obtained results from approximation solution and exact solution for 120572 = minus05 120573 = 07 and 119873 = 5 of Experiment 2

Table 2 Comparison between the exact and approximate solutionsfor 120572 = minus05 120573 = 07 and119873 = 5 of Experiment 2

119909Exact

solutionPseudospectral

method Error

00 062245933 062245922 00000001101 068366636 068366643 minus00000000702 075131058 075131034 00000002403 082606901 082606923 minus00000002204 090868985 090868925 00000006005 100000000 100000000 0000000006 110091330 110091120 00000002107 121243980 121243880 00000001008 133569560 133569450 00000001109 147191430 147191350 00000000810 162245930 162245560 000000037

Table 3 Comparison between the exact and approximate solutionsfor 120572 = minus05 120573 = 07 and119873 = 5 of Experiment 3

119909Exact

solutionPseudospectral

method Error

00 10000000 10000000 00000000001 11051709 11051701 00000000802 12214028 12214011 00000001703 13498588 13498533 00000005504 14918247 14918232 00000001505 16487213 16487211 00000000206 18221188 18221166 00000002207 20137527 20137511 00000001508 22255409 22255401 00000000709 24596031 24596030 00000000110 27182818 27182810 000000008

Now by using the pseudospectral method for 120572 =

minus05 120573 = 07 and 119873 = 5 we obtain the following resultsthat are shown in Table 3

Experiment 4 We consider the linear partial differentialequation problem [22]

119906119905119905= 119906119909119909

+ 6 119906119905(119909 0) = 4119909 119906 (119909 0) = 119909

2 (68)

with the exact solution 119906(119909 119910) = (119909 + 2119905)2

In this case we have

L (119906) = 119906119905119905minus 119906119909119909 (69)

We obtain the approximate solution of (68) on the domain[minus1 1] times [minus1 1] For this purpose we consider the discretepoints (119909(120572120573)

119894 119905(120572120573)

119895)119873

119894119895=0and the general Lagrange interpola-

tion in the discrete points (119909(120572120573)119894

119905(120572120573)

119895)119873

119894119895=0in the following

form (two-dimensional case)

119871 (119909 119905) =

119873

sum

119894=0

119873

sum

119895=0

119891 (119909(120572120573)

119894 119905(120572120573)

119895) 119871119894(119909) 119871119895(119905) (70)

where 119871119894(119909) and 119871

119895(119905) are ordinary Lagrange interpolations

By using collocation method for arbitrary parameters 120572 120573and119873 = 5 we obtain the exact solution

Experiment 5 Finally let us consider the linear fractionaldifferential equation problem [23]

11986352

119910 (119909) + 3119910 (119909) = 31199093+

6

Γ (15)

11990912

119910 (0) = 0 1199101015840(0) = 0 119910

10158401015840(0) = 0 119909 isin [minus1 1]

(71)

This equation has the exact solution 119910(119909) = 1199093

In this case we have

L (119910 (119909)) = 11986352

(119910 (119909)) + 3119910 (119909) (72)

which is a linear operator Using the presented method forevery 120572 120573 and119873 = 3 we obtain the exact solution

International Journal of Computational Mathematics 9

5 Conclusion

In this paper we have developed the expansion approach forsolving some operator equations of the form L119906 + N119906 =

119892(119909) 119909 isin Γ with respect to boundary condition B119906 = 120582For this the principal importance is the development of theexpansion approach for solving general operator equationwherein the particular cases are in the solution of integralequations (Experiment 3) ordinary differential equations(Experiment 1) difference equations (Experiment 2) partialdifferential equations (Experiment 4) and fractional differ-ential equations (Experiment 5) Also the error analysis ispresented in 119871

2(Γ) norm Using the roots of another orthog-

onal polynomials (Jacobi polynomials) rather than classicalorthogonal polynomials such as Chebyshev polynomialsand Legendre polynomials for collocation points (with freeparameters 120572 120573) in pseudospectral method is an importantsubject which is performed (Experiments 2 and 3)

Conflict of Interests

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

Acknowledgments

The authors wish to thank the anonymous referees for theirvaluable comments and suggestions Also they would like toappreciate the Editor Dr Yaohanz Li who gives them theopportunity to review the paper

References

[1] P LinzTheoretical Numerical Analysis JohnWiley amp Sons NewYork NY USA 1979

[2] D Gottlieb and S A Orszag Numerical Analysis of SpectralMethods Theory and Applications SIAM-CBMS PhiladelphiaPa USA 1977

[3] G Ben-Yu The State of Art in Spectral Methods Hong KongUniversity 1996

[4] C Canuto M Y Hussaini A Quarteroni and T A ZhangSpectral Methods Fundamentals in Single Domains Springer2006

[5] C Canutuo M Y Hussaini A Quarteroni and T A ZhangSpectral Method in Fluid Dynamics Prentice Hall EngelwoodCliffs NJ USA 1984

[6] B FornbergAPractical Quide to PseudospectralMethods Cam-bridgeMonographs on Applied and Computational Mathemat-ics Cambridge University Press Cambridge UK 1st edition1996

[7] L N Trefethen Spectral Methods in Matlab SIAM Philadel-phia Pa USA 2000

[8] J S Hesthaven S Gottlieb and D Gottlieb Spectral Methodsfor Time-Dependent Problems vol 21 ofCambridgeMonographson Applied and Computational Mathematics Cambridge Uni-versity Press Cambridge UK 2007

[9] M Deville and E Mund ldquoChebyshev pseudospectral solutionof second-order elliptic equations with finite element precon-ditioningrdquo Journal of Computational Physics vol 60 no 3 pp517ndash533 1985

[10] J P Boyd ldquoMultipole expansions and pseudospectral cardinalfunctions a new generalization of the fast Fourier transformrdquoJournal of Computational Physics vol 103 no 1 pp 184ndash1861992

[11] G N Elnagar and M Razzaghi ldquoA pseudospectral method forHammerstein equationsrdquo Journal of Mathematical Analysis andApplications vol 199 no 2 pp 579ndash591 1996

[12] M Meehan and D OrsquoRegan ldquoExistence principles for nonreso-nant operator and integral equationsrdquo Computers ampMathemat-ics with Applications vol 35 no 9 pp 79ndash87 1998

[13] A Chaillou and M Suri ldquoA posteriori estimation of thelinearization error for strongly monotone nonlinear operatorsrdquoJournal of Computational andAppliedMathematics vol 205 no1 pp 72ndash87 2007

[14] R P Kanwal and K C Liu ldquoA Taylor expansion approach forsolving integral equationsrdquo International Journal of Mathemat-ical Education in Science and Technology vol 20 pp 411ndash4141989

[15] M R Eslahchi M Dehghan and S Ahmadi Asl ldquoThe generalJacobi matrix method for solving some nonlinear ordinarydifferential equationsrdquoAppliedMathematical Modelling vol 36no 8 pp 3387ndash3398 2012

[16] A Imani A Aminataei and A Imani ldquoCollocation method viaJacobi polynomials for solving nonlinear ordinary differentialequationsrdquo International Journal of Mathematics and Mathe-matical Sciences vol 2011 Article ID 673085 11 pages 2011

[17] M R Eslahchi and M Dehghan ldquoApplication of Taylor seriesin obtaining the orthogonal operational matrixrdquo Computers ampMathematics with Applications vol 61 no 9 pp 2596ndash26042011

[18] D Funaro Polynomial Approximation of Differential Equationsvol 8 of Lecture Notes in Physics Monographs Springer BerlinGermany 1992

[19] D F Griffiths and D J Higham Numerical Methods forOrdinaryDifferential Equations SpringerUndergraduateMath-ematics Series Springer London UK 2010

[20] M Gulsu M Sezer and B Tanay ldquoA matrix method for solvinghigh-order linear difference equations with mixed argumentusing hybrid Legendre and Taylor polynomialsrdquo Journal of theFranklin Institute vol 343 no 6 pp 647ndash659 2006

[21] N Kurt and M Sezer ldquoPolynomial solution of high-orderlinear Fredholm integro-differential equations with constantcoefficientsrdquo Journal of the Franklin Institute vol 345 no 8 pp839ndash850 2008

[22] V S Vladimirov A Collection of Problems on the Equations ofMathematical Physics Mir Publishers Moscow Russia 1968

[23] A H Bhrawy M M Alghamdi and T M Taha ldquoA newmodified generalized Laguerre operational matrix of fractionalintegration for solving fractional differential equations on thehalf linerdquoAdvances in Difference Equations vol 2012 article 1792012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Computational Mathematics 9

5 Conclusion

In this paper we have developed the expansion approach forsolving some operator equations of the form L119906 + N119906 =

119892(119909) 119909 isin Γ with respect to boundary condition B119906 = 120582For this the principal importance is the development of theexpansion approach for solving general operator equationwherein the particular cases are in the solution of integralequations (Experiment 3) ordinary differential equations(Experiment 1) difference equations (Experiment 2) partialdifferential equations (Experiment 4) and fractional differ-ential equations (Experiment 5) Also the error analysis ispresented in 119871

2(Γ) norm Using the roots of another orthog-

onal polynomials (Jacobi polynomials) rather than classicalorthogonal polynomials such as Chebyshev polynomialsand Legendre polynomials for collocation points (with freeparameters 120572 120573) in pseudospectral method is an importantsubject which is performed (Experiments 2 and 3)

Conflict of Interests

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

Acknowledgments

The authors wish to thank the anonymous referees for theirvaluable comments and suggestions Also they would like toappreciate the Editor Dr Yaohanz Li who gives them theopportunity to review the paper

References

[1] P LinzTheoretical Numerical Analysis JohnWiley amp Sons NewYork NY USA 1979

[2] D Gottlieb and S A Orszag Numerical Analysis of SpectralMethods Theory and Applications SIAM-CBMS PhiladelphiaPa USA 1977

[3] G Ben-Yu The State of Art in Spectral Methods Hong KongUniversity 1996

[4] C Canuto M Y Hussaini A Quarteroni and T A ZhangSpectral Methods Fundamentals in Single Domains Springer2006

[5] C Canutuo M Y Hussaini A Quarteroni and T A ZhangSpectral Method in Fluid Dynamics Prentice Hall EngelwoodCliffs NJ USA 1984

[6] B FornbergAPractical Quide to PseudospectralMethods Cam-bridgeMonographs on Applied and Computational Mathemat-ics Cambridge University Press Cambridge UK 1st edition1996

[7] L N Trefethen Spectral Methods in Matlab SIAM Philadel-phia Pa USA 2000

[8] J S Hesthaven S Gottlieb and D Gottlieb Spectral Methodsfor Time-Dependent Problems vol 21 ofCambridgeMonographson Applied and Computational Mathematics Cambridge Uni-versity Press Cambridge UK 2007

[9] M Deville and E Mund ldquoChebyshev pseudospectral solutionof second-order elliptic equations with finite element precon-ditioningrdquo Journal of Computational Physics vol 60 no 3 pp517ndash533 1985

[10] J P Boyd ldquoMultipole expansions and pseudospectral cardinalfunctions a new generalization of the fast Fourier transformrdquoJournal of Computational Physics vol 103 no 1 pp 184ndash1861992

[11] G N Elnagar and M Razzaghi ldquoA pseudospectral method forHammerstein equationsrdquo Journal of Mathematical Analysis andApplications vol 199 no 2 pp 579ndash591 1996

[12] M Meehan and D OrsquoRegan ldquoExistence principles for nonreso-nant operator and integral equationsrdquo Computers ampMathemat-ics with Applications vol 35 no 9 pp 79ndash87 1998

[13] A Chaillou and M Suri ldquoA posteriori estimation of thelinearization error for strongly monotone nonlinear operatorsrdquoJournal of Computational andAppliedMathematics vol 205 no1 pp 72ndash87 2007

[14] R P Kanwal and K C Liu ldquoA Taylor expansion approach forsolving integral equationsrdquo International Journal of Mathemat-ical Education in Science and Technology vol 20 pp 411ndash4141989

[15] M R Eslahchi M Dehghan and S Ahmadi Asl ldquoThe generalJacobi matrix method for solving some nonlinear ordinarydifferential equationsrdquoAppliedMathematical Modelling vol 36no 8 pp 3387ndash3398 2012

[16] A Imani A Aminataei and A Imani ldquoCollocation method viaJacobi polynomials for solving nonlinear ordinary differentialequationsrdquo International Journal of Mathematics and Mathe-matical Sciences vol 2011 Article ID 673085 11 pages 2011

[17] M R Eslahchi and M Dehghan ldquoApplication of Taylor seriesin obtaining the orthogonal operational matrixrdquo Computers ampMathematics with Applications vol 61 no 9 pp 2596ndash26042011

[18] D Funaro Polynomial Approximation of Differential Equationsvol 8 of Lecture Notes in Physics Monographs Springer BerlinGermany 1992

[19] D F Griffiths and D J Higham Numerical Methods forOrdinaryDifferential Equations SpringerUndergraduateMath-ematics Series Springer London UK 2010

[20] M Gulsu M Sezer and B Tanay ldquoA matrix method for solvinghigh-order linear difference equations with mixed argumentusing hybrid Legendre and Taylor polynomialsrdquo Journal of theFranklin Institute vol 343 no 6 pp 647ndash659 2006

[21] N Kurt and M Sezer ldquoPolynomial solution of high-orderlinear Fredholm integro-differential equations with constantcoefficientsrdquo Journal of the Franklin Institute vol 345 no 8 pp839ndash850 2008

[22] V S Vladimirov A Collection of Problems on the Equations ofMathematical Physics Mir Publishers Moscow Russia 1968

[23] A H Bhrawy M M Alghamdi and T M Taha ldquoA newmodified generalized Laguerre operational matrix of fractionalintegration for solving fractional differential equations on thehalf linerdquoAdvances in Difference Equations vol 2012 article 1792012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of