Research Article Solving a System of Linear Volterra...

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 196308 5 pageshttpdxdoiorg1011552013196308

Research ArticleSolving a System of Linear Volterra Integral Equations Usingthe Modified Reproducing Kernel Method

Li-Hong Yang Hong-Ying Li and Jing-Ran Wang

College of Science Harbin Engineering University Heilongjiang 150001 China

Correspondence should be addressed to Li-Hong Yang lihongyanghrbeueducn

Received 16 May 2013 Accepted 30 September 2013

Academic Editor Rodrigo Lopez Pouso

Copyright copy 2013 Li-Hong Yang 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

A numerical technique based on reproducing kernel methods for the exact solution of linear Volterra integral equations systemof the second kind is given The traditional reproducing kernel method requests that operator a satisfied linear operator equation119860119906 = 119891 is bounded and its image space is the reproducing kernel space 119882

1

2[119886 119887] It limits its application Now we modify the

reproducing kernelmethod such that it can bemorewidely applicableThe n-term approximation solution obtained by themodifiedmethod is of high accuracyThenumerical example comparedwith othermethods shows that themodifiedmethod ismore efficient

1 Introduction

Thepurpose of this paper is to solve a systemof linearVolterraintegral equations

119865 (119904) = 119866119904 + int

119887

119886

119870 (119904 119905) 119865 (119905) 119889119905 119904 isin [0 1] (1)

where

119865 (119904) = [1198911(119904) 1198912(119904) 119891

119899(119904)]119879

119866 (119904) = [1198921(119904) 1198922(119904) 119892

119899(119904)]119879

119870 (119904 119905) = [119896119894119895] 119894 119895 = 1 2 119899

(2)

In (1) the functions119870 and 119866 are given and 119865 is the solu-tion to be determined We assume that (1) has a unique solu-tion Volterra integral equation arises in many physical appli-cations for example potential theory and Dirichlet prob-lems electrostatics mathematical problems of radiative equi-librium the particle transport problems of astrophysics andreactor theory and radiative heat transfer problems [1ndash5]Several valid methods for solving Volterra integral equationhave been developed in recent years including power seriesmethod [6] Adomainrsquos decomposition method [7] homo-topy perturbation method [8 9] block by block method [10]and expansion method [11]

Since the reproducing kernel space 1198821

2[119886 119887] which is a

special Hilbert space is constructed in 1986 [12] the repro-ducing kernel theory has been applied successfully to manylinear and nonlinear problems such as differential equationpopulation model and many other equations appearing inphysics and engineering [12ndash21] The traditional reproducingkernel method is limited because it requires that the imagespace of operator 119860 in linear operator equation 119860119906 = 119891 is1198821

2[119886 119887] and operator119860must be bounded In order to enlarge

its application range the MRKM removes the boundednessof 119860 and weakens its image space to 119871

2[119886 119887] Subsequently

we apply the MRKM to obtain the series expression of theexact solution for (1) The 119899-term approximation solutionis provided by truncating the series The final numericalcomparisons between our method and other methods showthe efficiency of the proposed method It is worth to mentionthat the MRKM can be generalized to solve other system oflinear equations

2 Preliminaries

21 The Reproducing Kernel Space11988212[0 1] The reproducing

kernel space1198821

2[0 1] consists of all absolute continuous real-

valued functions which defined on the closed interval [0 1]and the first derivative functions belong to 119871

2[0 1]

2 Abstract and Applied Analysis

The inner product and the norm are equipped with

(119906 V)1199081

2

= 119906 (0) V (0) + int

1

0

1199061015840(119909) V1015840 (119909) 119889119909 forall119906 V isin 119908

1

2

11990611988212

= radic(119906 V)1199081

2

(3)

Theorem 1 1198821

2[0 1] is a reproducing kernel space with repro-

ducing kernel [22]

119877119909(119910) =

1 + 119910 119910 le 119909

1 + 119909 119910 gt 119909(4)

that is for every 119909 isin [0 1] and 119906 isin 1198821

2 it follows that

(119906 (119910) 119877119909(119910))1199081

2

= 119906 (119909) (5)

22The Reproducing Kernel Space11988222[0 1] The reproducing

kernel space 1198822

2[0 1] consists of all real-valued functions in

which the first derivative functions are absolute continuouson the closed interval [0 1] and the second derivative func-tions belong to 119871

2[0 1]


(119906 V)1198822

2

=

1

sum

119896=0

119906(119896)

(0) V(119896) (0)

+ int

1

0

11990610158401015840(119909) V10158401015840 (119909) 119889119909 forall119906 V isin 119882

2

2[0 1]

1199061198822

2

= radic(119906 119906)1199082

2

(6)

Theorem2 1198822

2[0 1] is a reproducing kernel spacewith repro-

ducing kernel [22]

119876 (119909 119910) =

1 + 119909 times 119910 +119909 times 1199102

2minus

1199103

6119910 le 119909

1 + 119909 times 119910 +1199092times 119910

2minus

1199093

6 119910 gt 119909

(7)


2 it follows that

(119906 (119910) 119876 (119909 119910))1199082

2

= 119906 (119909) (8)

The proof of Theorems 1 and 2 can be found in [23]

23 Hilbert Space 119864 Hilbert space 119864 is defined by

119864 =

119899

⨁

119894=1

1198821

2= (1199061 119906

119899)119879

| 119906119894isin 1199081

2 119894 = 1 119899 (9)

The inner product and the norm are given by

(119906 V)119864=

119899

sum

119894=1

(119906119894 V119894)1199081

2

119906119864 = radic(119906 119906)119864

(10)

It is easy to prove that 119864 is a Hilbert space

3 The Exact Solution of (1)31 Identical Transformation of (1) Consider the ith equationof (1)

119891119894(119904) minus

119899

sum

119895=1

int

119904

0

119870119894119895(119904 119905) 119891

119895(119905) 119889119905 = 119892

119894(119904) (11)

Define operator 119860119894119895

1198821

2rarr 1198712[0 1] 119895 = 1 119899

119860119894119895

=

119906 (119904) minus int1

0119896119894119895(119904 119905) 119906 (119905) 119889119905 119895 = 119894

minusint

119904

0

119896119894119895(119904 119905) 119906 (119905) 119889119905 119895 = 119894

(12)

where 119906 isin 1198821

2 Then (1) can be turned into

119860111198911+ 119860121198912+ sdot sdot sdot + 119860

11198991198911119899

= 1198921(119904)

119860211198911+ 119860221198912+ sdot sdot sdot + 119860

21198991198911119899

= 1198922(119904)

11986011989911198911+ 11986011989921198912+ sdot sdot sdot + 119860

1198991198991198911119899

= 119892119899(119904)

(13)

where 119865(119904) = [1198911(119904) 1198912(119904) 119891

119899(119904)]119879

isin 119864

32 The Exact solution of (1) Let 119909119894infin

119894=1be a dense subset of

interval [0 1] and define

Ψ119894119895(119909) = (119860

1198951119910119877119909(119910)

10038161003816100381610038161003816119910=119909119894

1198601198952119910119877119909(119910)

10038161003816100381610038161003816119910=119909119894

119860119895119899119910119877119909(119910)

10038161003816100381610038161003816119910=119909119894

)

119879(14)

for every 119895 = 1 2 119899 119894 = 1 2 the subscript 119910 of 119860119894119895119910

means that the operator119860119894119895acts on the function of119910 It is easy

to prove that Ψ119894119895

isin 119864

Theorem 3 Ψ1198941 Ψ1198942 Ψ

119894119899infin

119894=1is complete in 119864

Proof Take 119906 = (1199061 1199062 119906

119899)119879

isin 119864 such that (119906(119909) Ψ119894119895(119909))

= 0 for every 119895 = 1 2 119899 119894 = 1 2 From this fact it holds that

(119906 (119909) Ψ119894119895(119909))

= ((1199061 1199062 119906

119899)119879

(1198601198951119910

119877119909(119910)

10038161003816100381610038161003816119910=119909119894

1198601198952119910

119877119909(119910)

10038161003816100381610038161003816119910=119909119894

119860119895119899119910

119877119909(119910)

100381610038161003816100381610038161003816119910=119909119894

)

119879

)

=

119899

sum

119896=1

119860119895119896119910

(119906119896(119909) 119877

119909(119910))1199081

2

100381610038161003816100381610038161003816119910=119909119894

=

119899

sum

119896=1

119860119895119896119906119896(119909119894) = 0

(15)

Abstract and Applied Analysis 3

for every 119895 = 1 2 119899 The dense 119909119894infin

119894=1assumes that

119860111199061+ 119860121199062+ sdot sdot sdot + 119860

1119899119906119899= 0

119860211199061+ 119860221199062+ sdot sdot sdot + 119860

2119899119906119899= 0

11986011989911199061+ 11986011989921199062+ sdot sdot sdot + 119860

119899119899119906119899= 0

(16)

Since (16) has a unique solution it follows that 119906 =

1199061 1199062 119906

119899

119879= 0 This completes the proof

We arrangeΨ11Ψ12 Ψ

1119899Ψ21Ψ22 Ψ

2119899 Ψ

1198941 Ψ1198942

Ψ119894119899 denoted by 119903

119894infin

119894=1 that is 119903

1= Ψ11 1199032

= Ψ12

119903119899

= Ψ1119899 119903119899+1

= Ψ21 119903119899+2

= Ψ22 119903

119899+119899= Ψ2119899

In a general way 119903(119894minus1)119899+119895

= Ψ119894119895 119894 = 1 2 3 119895 = 1

2 119899 The orthogonal basis 119903119894infin

119894=1in 119864 from Gram-Schmidt

orthogonalization of 119903119894infin

119894=1is as follows

119903119894=

119894

sum

119896=1

120573119894119896119903119896 119894 = 1 2 (17)

Theorem 4 The exact solution of (1) can be expressed by

119865 (119909) =

infin

sum

119894=1

119894

sum

119896=1

120573119894119896120588119896119903119894(119909) (18)

where 120588119896= (119865(119909) 119903

119896)119864 if 119903119896= Ψ119895119897 then 120588

119896= 119892119897(119909119895)

Proof Assume that 119865(119909) is the exact solution of (1) 119865(119909) canbe expanded to Fourier series in terms of normal orthogonalbasis 119903

119894(119909)infin

119894=1in 119864

119865 (119909) =

infin

sum

119894=1

(119865 119903119894)119864119903119894(119909) =

infin

sum

119894=1

119894

sum

119896=1

120573119894119896(119865 119903119896)119864119903119894(119909) (19)

if 120588119896= (119865 119903

119896)119864 then

119865 (119909) =

infin

sum

119894=1

119894

sum

119896=1

120573119894119895120588119896119903119894(119909) (20)

When 119903119896= Ψ119895119897 it holds that

120588119896= (119865Ψ

119895119897) =

119899

sum

119896=1

119860119897119896119906119896(119909119895) = 119892119897(119909119895) (21)

Corollary 5 The approximate solution of (1) is

119865119898

(119909) =

119898

sum

119894=1

119894

sum

119896=1

120573119894119896120588119896119903119894(119909) = (119891

1119898 1198912119898

119891119899119898

)119879

(22)

and 119891119894119898

(119909) converges uniformly to 119891119894(119909) on [0 1] as 119898 rarr infin

for every 119894 = 1 2 119899

Proof Obviously 119865119898

minus 1198652

119864rarr 0 holds as 119898 rarr infin that is

119865119898(119909) is the approximate solution of (1)

Note that sum119899

119894=1119891119894119898

minus 1198911198942

1198821

2

= 119865119898

minus 1198652

119864rarr 0 Com-

bining with the expression of 119877119909(119910) we have

1003816100381610038161003816119891119894119898 minus 119891119894

1003816100381610038161003816 =100381610038161003816100381610038161003816(119891119894119898

(119910) minus 119891119894(119910) 119877

119909(119910))1198821

2

100381610038161003816100381610038161003816

le1003817100381710038171003817119891119894119898 minus 119891

119894

100381710038171003817100381711988212

sdot1003817100381710038171003817119877119909 (119910)

100381710038171003817100381711988212

=1003817100381710038171003817119891119894119898 minus 119891

119894

100381710038171003817100381711988212

radic119877119909(119909)

le radic21003817100381710038171003817119891119894119898 minus 119891

119894

100381710038171003817100381711988212

forall119909 isin [0 1]

(23)

It shows that 119891119894119898

converges uniformly to 119891119894on [0 1] as119898 rarr

infin for every 119894 = 1 2 119899 So the proof is complete

Remark 6 If 119896119894119895(119904 119905) isin 119862([0 1] times [0 1]) and 119892

119894isin 1198822

2in (1)

then it is reasonable to regard the unknown functions as theelements of 1198822

2

4 Numerical Examples

Taking nodes 119909119894= (119894minus1)(119873minus1)

119873

119894=1119891119894119873

is the approximatesolutions of 119891

119894 and 119890(119891

119894119873) denotes the absolute errors of

119891119894 119894 = 1 2 119899 According to Remark 6 we solve the

following two examples appearing in [11] in 1198822

2

Example 7 Consider the following systemofVolterra integralequations of the second kind [11]

1198911(119904) = 119892

1(119904) + int

119904

0

(119904 minus 119905)31198911(119905) 119889119905 + int

119904

0

(119904 minus 119905)21198912(119905) 119889119905

1198912(119904) = 119892

2(119904) + int

119904

0

(119904 minus 119905)41198911(119905) 119889119905

+ int

119904

0

(119904 minus 119905)31198912(119905) 119889119905

(24)

where 1198921(119904) and 119892

2(119904) are chosen such that the exact solution

is 1198911(119904) = 1 + 119904

2 1198912(119904) = 1 + 119904 minus 119904

3 The numerical resultsobtained by using the present method are compared with [11]in Table 1

Example 8 Consider the following system of linear Volterraintegral equations of the second kind [11]

1198911(119904) = 119892

1(119904) + int

119904

0

(sin (119904 minus 119905) minus 1) 1198911(119905) 119889119905

+ int

119904

0

(1 minus 119905 cos 119904) 1198912(119905) 119889119905

1198912(119904) = 119892

2(119904) + int

119904

0

(1198911(119905)) 119889119905 + int

119904

0

(119904 minus 119905) 1198912(119905) 119889119905

(25)

where 1198921(119904) and 119892


is 1198911(119904) = cos 119904 119891

2(119904) = sin 119904 The numerical results

obtained by using the present method are compared with [11]in Table 2


Table 1 Absolute errors for Example 7

Nodes 119909119894

Errors 119890(1198911) [11] Errors 119890(119891

1100) Errors 119890(119891

2) [11] Errors 119890(119891

2100) [11]

00 0 158309119864 minus 10 0 398245119864 minus 10

01 263472119864 minus 7 392220119864 minus 12 211685119864 minus 8 394493119864 minus 10











Nodes 119909119894

Errors 119890(1198911) [11] Errors 119890(119891

1100) Errors 119890(119891

2) [11] Errors 119890(119891

2100) [11]

00 0 693348119864 minus 11 0 360316119864 minus 11











5 Conclusion

In this paper we modify the traditional reproducing kernelmethod to enlarge its application range The new methodnamed MRKM is applied successfully to solve a system oflinear Volterra integral equationsThenumerical results showthat our method is effective It is worth to be pointed out thatthe MRKM is still suitable for solving other systems of linearequations

Acknowledgments

The research was supported by the Fundamental ResearchFunds for the Central Universities

References

[1] Y Ren B Zhang and H Qiao ldquoA simple Taylor-series expan-sion method for a class of second kind integral equationsrdquoJournal of Computational and Applied Mathematics vol 110 no1 pp 15ndash24 1999

[2] KMaleknejad and YMahmoudi ldquoNumerical solution of linearFredholm integral equation by using hybrid Taylor and block-pulse functionsrdquo Applied Mathematics and Computation vol149 no 3 pp 799ndash806 2004

[3] S Yalcınbas and M Sezer ldquoThe approximate solution of high-order linear Volterra-Fredholm integro-differential equations

in terms of Taylor polynomialsrdquoAppliedMathematics and Com-putation vol 112 no 2-3 pp 291ndash308 2000

[4] S Yalcinbas ldquoTaylor polynomial solutions of nonlinear Volter-ra-Fredholm integral equationsrdquo Applied Mathematics andComputation vol 127 no 2-3 pp 195ndash206 2002

[5] E Deeba S A Khuri and S Xie ldquoAn algorithm for solvinga nonlinear integro-differential equationrdquo Applied Mathematicsand Computation vol 115 no 2-3 pp 123ndash131 2000

[6] A Tahmasbi and O S Fard ldquoNumerical solution of linear Vol-terra integral equations system of the second kindrdquo AppliedMathematics and Computation vol 201 no 1-2 pp 547ndash5522008

[7] E Babolian J Biazar and A R Vahidi ldquoOn the decompositionmethod for system of linear equations and system of linearVolterra integral equationsrdquo Applied Mathematics and Compu-tation vol 147 no 1 pp 19ndash27 2004

[8] J Biazar andHGhazvini ldquoHersquos homotopy perturbationmethodfor solving systems of Volterra integral equations of the secondkindrdquo Chaos Solitons amp Fractals vol 39 no 2 pp 770ndash7772009

[9] E Yusufoglu (Agadjanov) ldquoAhomotopy perturbation algorithmto solve a system of Fredholm-Volterra type integral equationsrdquoMathematical and Computer Modelling vol 47 no 11-12 pp1099ndash1107 2008

[10] R Katani and S Shahmorad ldquoBlock by block method for thesystems of nonlinearVolterra integral equationsrdquoAppliedMath-ematical Modelling vol 34 no 2 pp 400ndash406 2010


[11] M Rabbani K Maleknejad and N Aghazadeh ldquoNumericalcomputational solution of the Volterra integral equations sys-tem of the second kind by using an expansion methodrdquoAppliedMathematics and Computation vol 187 no 2 pp 1143ndash11462007

[12] MGCui andZXDeng ldquoOn the best operator of interpolationin1198821

2[119886 119887]rdquoMathematica Numerica Sinica vol 2 pp 209ndash216

1986[13] M Cui and F Geng ldquoA computational method for solving one-

dimensional variable-coefficient Burgers equationrdquo AppliedMathematics and Computation vol 188 no 2 pp 1389ndash14012007

[14] MCui and FGeng ldquoA computationalmethod for solving third-order singularly perturbed boundary-value problemsrdquo AppliedMathematics and Computation vol 198 no 2 pp 896ndash9032008

[15] M Cui and Z Chen ldquoThe exact solution of nonlinear age-structured population modelrdquo Nonlinear Analysis Real WorldApplications vol 8 no 4 pp 1096ndash1112 2007

[16] F Z Geng andMG Cui ldquoSolving singular nonlinear two-pointboundary value problems in the reproducing kernel spacerdquoJournal of the Korean Mathematical Society vol 45 no 3 pp631ndash644 2008

[17] M G Cui and Y B Yan ldquoThe representation of the solution of akind operator equation119860119906 = 119891rdquo Journal of Chinese Universitiesvol 3 pp 82ndash86 1995

[18] O Abu Arqub M Al-Smadi and S Momani ldquoApplication ofreproducing kernel method for solving nonlinear Fredholm-Volterra integrodifferential equationsrdquo Abstract and AppliedAnalysis vol 2012 Article ID 839836 16 pages 2012

[19] O A Arqub M Al-Smadi and N Shawagfeh ldquoSolving Fred-holm integro-differential equations using reproducing kernelHilbert space methodrdquo Applied Mathematics and Computationvol 219 no 17 pp 8938ndash8948 2013

[20] M Al-Smadi O Abu Arqub and S Momani ldquoA computationalmethod for two-point boundary value problems of fourth-ordermixed integrodifferential equationsrdquoMathematical Problems inEngineering vol 2013 Article ID 832074 10 pages 2013

[21] M Al-Smadi O Abu Arqub and N Shawagfeh ldquoApproximatesolution of BVPs for 4th-order IDEs by using RKHS methodrdquoAppliedMathematical Sciences vol 6 no 49ndash52 pp 2453ndash24642012

[22] Z Chen and Y Lin ldquoThe exact solution of a linear integralequation with weakly singular kernelrdquo Journal of MathematicalAnalysis and Applications vol 344 no 2 pp 726ndash734 2008

[23] M Cui and Y Lin Nonlinear Numerical Analysis in the Repro-ducing Kernel Space Nova Science Publishers New York NYUSA 2009

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Stochastic AnalysisInternational Journal of



(119906 V)1199081

2

= 119906 (0) V (0) + int

1

0

1199061015840(119909) V1015840 (119909) 119889119909 forall119906 V isin 119908

1

2

11990611988212

= radic(119906 V)1199081

2

(3)

Theorem 1 1198821

2[0 1] is a reproducing kernel space with repro-

ducing kernel [22]

119877119909(119910) =

1 + 119910 119910 le 119909

1 + 119909 119910 gt 119909(4)


2 it follows that

(119906 (119910) 119877119909(119910))1199081

2

= 119906 (119909) (5)

22The Reproducing Kernel Space11988222[0 1] The reproducing

kernel space 1198822

2[0 1] consists of all real-valued functions in

which the first derivative functions are absolute continuouson the closed interval [0 1] and the second derivative func-tions belong to 119871

2[0 1]


(119906 V)1198822

2

=

1

sum

119896=0

119906(119896)

(0) V(119896) (0)

+ int

1

0

11990610158401015840(119909) V10158401015840 (119909) 119889119909 forall119906 V isin 119882

2

2[0 1]

1199061198822

2

= radic(119906 119906)1199082

2

(6)

Theorem2 1198822

2[0 1] is a reproducing kernel spacewith repro-

ducing kernel [22]

119876 (119909 119910) =

1 + 119909 times 119910 +119909 times 1199102

2minus

1199103

6119910 le 119909

1 + 119909 times 119910 +1199092times 119910

2minus

1199093

6 119910 gt 119909

(7)


2 it follows that

(119906 (119910) 119876 (119909 119910))1199082

2

= 119906 (119909) (8)

The proof of Theorems 1 and 2 can be found in [23]

23 Hilbert Space 119864 Hilbert space 119864 is defined by

119864 =

119899

⨁

119894=1

1198821

2= (1199061 119906

119899)119879

| 119906119894isin 1199081

2 119894 = 1 119899 (9)

The inner product and the norm are given by

(119906 V)119864=

119899

sum

119894=1

(119906119894 V119894)1199081

2

119906119864 = radic(119906 119906)119864

(10)

It is easy to prove that 119864 is a Hilbert space

3 The Exact Solution of (1)31 Identical Transformation of (1) Consider the ith equationof (1)

119891119894(119904) minus

119899

sum

119895=1

int

119904

0

119870119894119895(119904 119905) 119891

119895(119905) 119889119905 = 119892

119894(119904) (11)

Define operator 119860119894119895

1198821

2rarr 1198712[0 1] 119895 = 1 119899

119860119894119895

=

119906 (119904) minus int1

0119896119894119895(119904 119905) 119906 (119905) 119889119905 119895 = 119894

minusint

119904

0

119896119894119895(119904 119905) 119906 (119905) 119889119905 119895 = 119894

(12)

where 119906 isin 1198821

2 Then (1) can be turned into

119860111198911+ 119860121198912+ sdot sdot sdot + 119860

11198991198911119899

= 1198921(119904)

119860211198911+ 119860221198912+ sdot sdot sdot + 119860

21198991198911119899

= 1198922(119904)

11986011989911198911+ 11986011989921198912+ sdot sdot sdot + 119860

1198991198991198911119899

= 119892119899(119904)

(13)

where 119865(119904) = [1198911(119904) 1198912(119904) 119891

119899(119904)]119879

isin 119864

32 The Exact solution of (1) Let 119909119894infin

119894=1be a dense subset of

interval [0 1] and define

Ψ119894119895(119909) = (119860

1198951119910119877119909(119910)

10038161003816100381610038161003816119910=119909119894

1198601198952119910119877119909(119910)

10038161003816100381610038161003816119910=119909119894

119860119895119899119910119877119909(119910)

10038161003816100381610038161003816119910=119909119894

)

119879(14)

for every 119895 = 1 2 119899 119894 = 1 2 the subscript 119910 of 119860119894119895119910

means that the operator119860119894119895acts on the function of119910 It is easy

to prove that Ψ119894119895

isin 119864

Theorem 3 Ψ1198941 Ψ1198942 Ψ

119894119899infin

119894=1is complete in 119864

Proof Take 119906 = (1199061 1199062 119906

119899)119879

isin 119864 such that (119906(119909) Ψ119894119895(119909))

= 0 for every 119895 = 1 2 119899 119894 = 1 2 From this fact it holds that

(119906 (119909) Ψ119894119895(119909))

= ((1199061 1199062 119906

119899)119879

(1198601198951119910

119877119909(119910)

10038161003816100381610038161003816119910=119909119894

1198601198952119910

119877119909(119910)

10038161003816100381610038161003816119910=119909119894

119860119895119899119910

119877119909(119910)

100381610038161003816100381610038161003816119910=119909119894

)

119879

)

=

119899

sum

119896=1

119860119895119896119910

(119906119896(119909) 119877

119909(119910))1199081

2

100381610038161003816100381610038161003816119910=119909119894

=

119899

sum

119896=1

119860119895119896119906119896(119909119894) = 0

(15)




119860111199061+ 119860121199062+ sdot sdot sdot + 119860

1119899119906119899= 0

119860211199061+ 119860221199062+ sdot sdot sdot + 119860

2119899119906119899= 0

11986011989911199061+ 11986011989921199062+ sdot sdot sdot + 119860

119899119899119906119899= 0

(16)


1199061 1199062 119906

119899



1119899Ψ21Ψ22 Ψ

2119899 Ψ

1198941 Ψ1198942

Ψ119894119899 denoted by 119903

119894infin

119894=1 that is 119903

1= Ψ11 1199032

= Ψ12

119903119899

= Ψ1119899 119903119899+1

= Ψ21 119903119899+2

= Ψ22 119903

119899+119899= Ψ2119899


= Ψ119894119895 119894 = 1 2 3 119895 = 1





119903119894=

119894

sum

119896=1

120573119894119896119903119896 119894 = 1 2 (17)


119865 (119909) =

infin

sum

119894=1

119894

sum

119896=1

120573119894119896120588119896119903119894(119909) (18)

where 120588119896= (119865(119909) 119903

119896)119864 if 119903119896= Ψ119895119897 then 120588

119896= 119892119897(119909119895)


119894(119909)infin

119894=1in 119864

119865 (119909) =

infin

sum

119894=1

(119865 119903119894)119864119903119894(119909) =

infin

sum

119894=1

119894

sum

119896=1

120573119894119896(119865 119903119896)119864119903119894(119909) (19)

if 120588119896= (119865 119903

119896)119864 then

119865 (119909) =

infin

sum

119894=1

119894

sum

119896=1

120573119894119895120588119896119903119894(119909) (20)


120588119896= (119865Ψ

119895119897) =

119899

sum

119896=1

119860119897119896119906119896(119909119895) = 119892119897(119909119895) (21)


119865119898

(119909) =

119898

sum

119894=1

119894

sum

119896=1

120573119894119896120588119896119903119894(119909) = (119891

1119898 1198912119898

119891119899119898

)119879

(22)

and 119891119894119898


for every 119894 = 1 2 119899


minus 1198652



Note that sum119899

119894=1119891119894119898

minus 1198911198942

1198821

2

= 119865119898

minus 1198652

119864rarr 0 Com-


1003816100381610038161003816119891119894119898 minus 119891119894

1003816100381610038161003816 =100381610038161003816100381610038161003816(119891119894119898

(119910) minus 119891119894(119910) 119877

119909(119910))1198821

2

100381610038161003816100381610038161003816

le1003817100381710038171003817119891119894119898 minus 119891

119894

100381710038171003817100381711988212

sdot1003817100381710038171003817119877119909 (119910)

100381710038171003817100381711988212

=1003817100381710038171003817119891119894119898 minus 119891

119894

100381710038171003817100381711988212

radic119877119909(119909)

le radic21003817100381710038171003817119891119894119898 minus 119891

119894

100381710038171003817100381711988212


(23)

It shows that 119891119894119898




119894isin 1198822

2in (1)


2



119873

119894=1119891119894119873


119894 and 119890(119891




2


1198911(119904) = 119892

1(119904) + int

119904

0

(119904 minus 119905)31198911(119905) 119889119905 + int

119904

0

(119904 minus 119905)21198912(119905) 119889119905

1198912(119904) = 119892

2(119904) + int

119904

0

(119904 minus 119905)41198911(119905) 119889119905

+ int

119904

0

(119904 minus 119905)31198912(119905) 119889119905

(24)

where 1198921(119904) and 119892


is 1198911(119904) = 1 + 119904

2 1198912(119904) = 1 + 119904 minus 119904



1198911(119904) = 119892

1(119904) + int

119904

0

(sin (119904 minus 119905) minus 1) 1198911(119905) 119889119905

+ int

119904

0

(1 minus 119905 cos 119904) 1198912(119905) 119889119905

1198912(119904) = 119892

2(119904) + int

119904

0

(1198911(119905)) 119889119905 + int

119904

0

(119904 minus 119905) 1198912(119905) 119889119905

(25)

where 1198921(119904) and 119892


is 1198911(119904) = cos 119904 119891





Nodes 119909119894

Errors 119890(1198911) [11] Errors 119890(119891

1100) Errors 119890(119891

2) [11] Errors 119890(119891

2100) [11]

00 0 158309119864 minus 10 0 398245119864 minus 10












Nodes 119909119894

Errors 119890(1198911) [11] Errors 119890(119891

1100) Errors 119890(119891

2) [11] Errors 119890(119891

2100) [11]

00 0 693348119864 minus 11 0 360316119864 minus 11











5 Conclusion


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119860111199061+ 119860121199062+ sdot sdot sdot + 119860

1119899119906119899= 0

119860211199061+ 119860221199062+ sdot sdot sdot + 119860

2119899119906119899= 0

11986011989911199061+ 11986011989921199062+ sdot sdot sdot + 119860

119899119899119906119899= 0

(16)


1199061 1199062 119906

119899



1119899Ψ21Ψ22 Ψ

2119899 Ψ

1198941 Ψ1198942

Ψ119894119899 denoted by 119903

119894infin

119894=1 that is 119903

1= Ψ11 1199032

= Ψ12

119903119899

= Ψ1119899 119903119899+1

= Ψ21 119903119899+2

= Ψ22 119903

119899+119899= Ψ2119899


= Ψ119894119895 119894 = 1 2 3 119895 = 1





119903119894=

119894

sum

119896=1

120573119894119896119903119896 119894 = 1 2 (17)


119865 (119909) =

infin

sum

119894=1

119894

sum

119896=1

120573119894119896120588119896119903119894(119909) (18)

where 120588119896= (119865(119909) 119903

119896)119864 if 119903119896= Ψ119895119897 then 120588

119896= 119892119897(119909119895)


119894(119909)infin

119894=1in 119864

119865 (119909) =

infin

sum

119894=1

(119865 119903119894)119864119903119894(119909) =

infin

sum

119894=1

119894

sum

119896=1

120573119894119896(119865 119903119896)119864119903119894(119909) (19)

if 120588119896= (119865 119903

119896)119864 then

119865 (119909) =

infin

sum

119894=1

119894

sum

119896=1

120573119894119895120588119896119903119894(119909) (20)


120588119896= (119865Ψ

119895119897) =

119899

sum

119896=1

119860119897119896119906119896(119909119895) = 119892119897(119909119895) (21)


119865119898

(119909) =

119898

sum

119894=1

119894

sum

119896=1

120573119894119896120588119896119903119894(119909) = (119891

1119898 1198912119898

119891119899119898

)119879

(22)

and 119891119894119898


for every 119894 = 1 2 119899


minus 1198652



Note that sum119899

119894=1119891119894119898

minus 1198911198942

1198821

2

= 119865119898

minus 1198652

119864rarr 0 Com-


1003816100381610038161003816119891119894119898 minus 119891119894

1003816100381610038161003816 =100381610038161003816100381610038161003816(119891119894119898

(119910) minus 119891119894(119910) 119877

119909(119910))1198821

2

100381610038161003816100381610038161003816

le1003817100381710038171003817119891119894119898 minus 119891

119894

100381710038171003817100381711988212

sdot1003817100381710038171003817119877119909 (119910)

100381710038171003817100381711988212

=1003817100381710038171003817119891119894119898 minus 119891

119894

100381710038171003817100381711988212

radic119877119909(119909)

le radic21003817100381710038171003817119891119894119898 minus 119891

119894

100381710038171003817100381711988212


(23)

It shows that 119891119894119898




119894isin 1198822

2in (1)


2



119873

119894=1119891119894119873


119894 and 119890(119891




2


1198911(119904) = 119892

1(119904) + int

119904

0

(119904 minus 119905)31198911(119905) 119889119905 + int

119904

0

(119904 minus 119905)21198912(119905) 119889119905

1198912(119904) = 119892

2(119904) + int

119904

0

(119904 minus 119905)41198911(119905) 119889119905

+ int

119904

0

(119904 minus 119905)31198912(119905) 119889119905

(24)

where 1198921(119904) and 119892


is 1198911(119904) = 1 + 119904

2 1198912(119904) = 1 + 119904 minus 119904



1198911(119904) = 119892

1(119904) + int

119904

0

(sin (119904 minus 119905) minus 1) 1198911(119905) 119889119905

+ int

119904

0

(1 minus 119905 cos 119904) 1198912(119905) 119889119905

1198912(119904) = 119892

2(119904) + int

119904

0

(1198911(119905)) 119889119905 + int

119904

0

(119904 minus 119905) 1198912(119905) 119889119905

(25)

where 1198921(119904) and 119892


is 1198911(119904) = cos 119904 119891





Nodes 119909119894

Errors 119890(1198911) [11] Errors 119890(119891

1100) Errors 119890(119891

2) [11] Errors 119890(119891

2100) [11]

00 0 158309119864 minus 10 0 398245119864 minus 10












Nodes 119909119894

Errors 119890(1198911) [11] Errors 119890(119891

1100) Errors 119890(119891

2) [11] Errors 119890(119891

2100) [11]

00 0 693348119864 minus 11 0 360316119864 minus 11











5 Conclusion


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Nodes 119909119894

Errors 119890(1198911) [11] Errors 119890(119891

1100) Errors 119890(119891

2) [11] Errors 119890(119891

2100) [11]

00 0 158309119864 minus 10 0 398245119864 minus 10












Nodes 119909119894

Errors 119890(1198911) [11] Errors 119890(119891

1100) Errors 119890(119891

2) [11] Errors 119890(119891

2100) [11]

00 0 693348119864 minus 11 0 360316119864 minus 11











5 Conclusion


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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