application of reproducing kernel hilbert space method for...
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Research ArticleApplication of Reproducing Kernel Hilbert Space Method forSolving a Class of Nonlinear Integral Equations
Sedigheh Farzaneh Javan1 Saeid Abbasbandy1 andM Ali Fariborzi Araghi2
1Department of Mathematics Science and Research Branch Islamic Azad University Tehran 14515 Iran2Department of Mathematics Central Tehran Branch Islamic Azad University Tehran Iran
Correspondence should be addressed to M Ali Fariborzi Araghi fariborziaraghigmailcom
Received 11 November 2016 Revised 7 February 2017 Accepted 8 February 2017 Published 16 March 2017
Academic Editor Haipeng Peng
Copyright copy 2017 Sedigheh Farzaneh Javan et alThis is an open access article distributed under theCreativeCommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited
A new approach based on the Reproducing Kernel Hilbert Space Method is proposed to approximate the solution of the second-kind nonlinear integral equations In this case the Gram-Schmidt process is substituted by another process so that a satisfactoryresult is obtained In this method the solution is expressed in the form of a series Furthermore the convergence of the proposedtechnique is proved In order to illustrate the effectiveness and efficiency of the method four sample integral equations arising inelectromagnetics are solved via the given algorithm
1 Introduction
Electromagnetics is the phenomenon associated with electricand magnetic fields and their interactions which is generallyone of the most important sciences Exterior calculus is givenin [1 2] inside some textbooks Away to teach electromagnet-ics can be approached via the use of differential forms whichis given in [3] According to electromagnetic field problemsfrommany years ago some solutions via linear and nonlinearintegral equations (NIE) have been given which can be usefulin the field In those methods like block-pulse functions(BPFs) Galerkin and collocation the most important waysare basic functions and appropriate projection Based on theReproducing Kernel Hilbert Space method an approach hasbeen found to solve some electromagnetic issues
Nonlinear integral equations are encountered in differentfields of science and numerous applications as elasticity plas-ticity heat andmass transfer oscillation theory fluid dynam-ics filtration theory electrostatics electrodynamics biome-chanics game theory control queuing theory electricalengineering economics and medicine among others Thereare different types ofNIE usually which cannot beworked outexplicitly so it should be approached approximately
Therefore many researchers studied and focused ondifferent numerical techniques which can work out these
integral equations For instance in [4 5] the authors pre-sented the homotopy analysis method to solve the secondkind of nonlinear Fredholm and Volterra integral equationsThe linear multistep techniques were applied in [6] toobtain the numerical solution of a singular nonlinear Volterraintegral equation In [7] an asymptotic technique to approachnumerically the nonlinear Abel-Volterra integral equationwas applied
Reproducing Kernel Hilbert Space (RKHS) was intro-duced byMinggen et al [8 9] and it was developed in differ-ent areas including approximation theory statistics machinelearning theory group representation theory and variousareas of complex analysis Reproducing Kernel Hilbert SpaceMethod (RKHSM) is a kernel based approximation methodwhich was applied for solving nonlinear boundary valueproblems [7ndash12] generalized singular nonlinear Lane-Emdentype equations [13] integrodifferential equations [14ndash16]integrodifferential fractional equations [17] Bratus Problem[18] and so forth
Consider the following nonlinear integral equation
119906 (119909) + int119887 (or119909)119886
119870 (119909 119905)119873 (119906 (119905)) 119889119905 = 119891 (119909) 119886 le 119909 119905 le 119887 (1)
HindawiMathematical Problems in EngineeringVolume 2017 Article ID 7498136 10 pageshttpsdoiorg10115520177498136
2 Mathematical Problems in Engineering
where 119886 119887 are real constants 119906 isin 11988212 [119886 119887] is an unknownfunction which can be determined 119891 isin 11988212 [119886 119887] is a con-tinuous function on [119886 119887] 119870(119909 119905) is a continuous functionon [119886 119887] times [119886 119887] 119873(V(119909)) is a continuous term in 11988212 [119886 119887]as V(119909) isin 11988212 [119886 119887] 119886 le 119909 le 119887 and11988212 [119886 119887] is ReproducingKernel Space Equation (1) has a continuous solution on [119886 119887][19] The existence and uniqueness conditions of the solutionfor (1) were discussed in [19ndash23]We assume that the solutionof (1) is unique
Over several decades numerical methods in electromag-netic problems have been one of the most important subjectsof extensive researches [1ndash4] On the other hand manyproblems in electromagnetics can be modeled by integralequations mentioned in [24ndash26] for example electric fieldintegral equation (EFIE) andmagnetic field integral equation(MFIE) In recent years several numerical methods forsolving linear and nonlinear integral equations have beenpresented Applicable equations of electromagnetics havebeen implied in the presented paper
In previousworks like [13ndash15] theGram-Schmidt orthog-onalization process has been considered to implementRKHSM Since this process is unstable numerically and itmaytake a lot of time to run the algorithm here we put away thisprocess and act with another way Our approach combinesthe methods mentioned in [13ndash17] More specifically on thecontrary to [13ndash15] without use of the orthogonalizationprocess the RKSHM is applied successfully to solve thenonlinear problem (1)
The structure of this paper would be described as followsIn Section 2 the basic definitions assumptions and prelimi-naries of RKHS are describedThemain idea and convergenceof the proposed scheme are discussed in Section 3 Section 4contains the numerical experiments Finally Section 5 isdedicated to a brief conclusion
2 Preliminaries
In this section some basic definitions and important prop-erties of Reproducing Kernel Hilbert Spaces (RKHS) arementioned [8 9 27ndash29]
Definition 1 AHilbert Space119867 is an inner product space thatis complete and separable with respect to the norm definedby the inner product Completeness of the space 119867 holdsprovided that every Cauchy sequence of points in119867 that hasa limit that is also in119867 and separable of119867 admits a countableorthonormal basis of it
Definition 2 For an abstract setX letH be a Hilbert Spaceof real or complex-valued functions on setX We sayH is aReproducing Kernel Hilbert Space if there exist a linear andbounded evaluation functional 119877119909 overH or equivalently
119877119909 119891 997891997888rarr 119891 (119909) forall119891 isin 119867 (2)
Riesz RepresentationTheorem implies that for all 119909 inXthere exists a unique function119870119909 ofH with the reproducingproperty
119891 (119909) = 119877119909 (119891) = ⟨119891 (119910) 119877119909 (119910)⟩H forall119891 isin H (3)
for each 119910 isin H where ⟨sdot sdot⟩H represent the inner product ofthe Hilbert SpaceH
Definition 3 The space1198821198982 [119886 119887] is interpreted as
1198821198982 [119886 119887] = 119906 (119909) | 119906 [119886 119887] 997888rarr 119877 119906(119898minus1)isin 119860119862 [119886 119887] 119906(119898) isin 1198712 [119886 119887] (4)
The inner product and the norm in1198821198982 [119886 119887] are of forms
⟨119906 (119909) V (119909)⟩1198821198982
= 119898minus1sum119894=0
119906(119894) (119886) V(119894) (119886)
+ int119887119886119906(119898) (120585) V(119898) (120585) 119889120585
119906 V isin 1198821198982 [119886 119887] 119906 = radic⟨119906 (119909) V (119909)⟩119882119898
2
119906 isin 1198821198982 [119886 119887]
(5)
Lemma 4 (see [9 29]) Functional space 1198821198982 [119886 119887] is innerspace
Theorem 5 (see [9 29]) Functional space 1198821198982 [119886 119887] is aHilbert Space
Theorem 6 (see [9 29]) Functional space1198821198982 [119886 119887] is Repro-ducing Kernel Hilbert Space
Now it is taken away that expression form of the Repro-ducing Kernel function 119877119909(119905) isin 1198821198982 [119886 119887]
Based on essay it is easy to prove that 119877119909(119905) is the answerof the following generalized differential equation [9 29]
(minus1)119898 1205972119898119877119909 (119905)1205971199052119898 = 120575 (119905 minus 119909) 120597119894119877119910 (119886)120597119905119894 minus (minus1)119898minus119894minus1 1205972119898minus119894minus1119877119910 (119886)1205971199052119898minus119894minus1 = 0
1205972119898minus119894minus1119877119910 (119887)1205971199052119898minus119894minus1 = 0119894 = 0 1 119898 minus 1
(6)
where 120575 is Diracrsquos delta function While 119909 = 119905 119877119909(119905) isthe answer of the following constant linear homogeneousdifferential equation with 2119898 order
(minus1)119898 1205972119898119877119909 (119905)1205971199052119898 = 0 (7)
Mathematical Problems in Engineering 3
with the boundary conditions
120597119894119877 minus 119910 (119886)120597119905119894 minus (minus1)119898minus119894minus1 1205972119898minus119894minus1119877119910 (119886)1205971199052119898minus119894minus1 = 01205972119898minus119894minus1119877119910 (119887)1205971199052119898minus119894minus1 = 0
119894 = 0 1 119898 minus 1(8)
Equation (7) is characteristic 1205822119898 = 0 Then the general solu-tion of Equation (7) is
119877119909 (119905) =
2119898sum119894=1
119888119894 (119909) 119905119894minus1 119905 le 1199092119898sum119894=1
119889119894 (119909) 119905119894minus1 119905 gt 119909 (9)
where coefficients 119888119894(119909) and 119889119894(119909) 119894 = 1 2 2119898 could becalculated by solving the following linear equations
120597119894119877119909 (119909 + 0)120597119905119894 = 120597119894119877119909 (119909 minus 0)120597119905119894 119894 = 0 1 2119898 minus 2(1205972119898minus1119877119909 (119909 + 0)1205971199052119898minus1 minus 1205972119898minus1119877119909 (119909 minus 0)1205971199052119898minus1 ) = (minus1)119898 120597119894119877119910 (119886)120597119905119894 minus (minus1)119898minus119894minus1 1205972119898minus119894minus1119877119910 (119886)1205971199052119898minus119894minus1 = 01205972119898minus119894minus1119877119910 (119887)1205971199052119898minus119894minus1 = 0 119894 = 0 1 119898 minus 1
(10)
Subsequently the representation of the Reproducing Kernelof11988212 [119886 119887] is provided by
119877119909 (119905) = 1 minus 119886 + 119905 119905 le 1199091 minus 119886 + 119909 119905 gt 119909 (11)
3 Main Idea and Theoretical Discussion
The uniqueness conditions for nonlinear problems exist in[21ndash23] The unique solution of (1) is assumed in this paperThe solution of (1) is given in11988212 [119886 119887] space We consider (1)as
L119906 (119909) = 119906 (119909) (12)
where L119906(119909) = 119891(119909) minus int119887 (or119909)119886
119870(119909 119905)119873(119906(119905))119889119905 It is obviousthatL is the bounded linear operator of11988212 to11988212 Put120593119894(119909) =119877119909119894(119909) and 120595119894(119909) = Llowast120593119894(119909) where Llowast is the adjoint operatorof L In fact for 119906 V isin 11988212 [119886 119887] we have L(119906(119909) + V(119909)) =119906(119909) + V(119909) and L119906(119909)1198821
2[119886119887] = 119906(119909)1198821
2[119886119887]
The orthonormal system of 120595119894(119909)infin119894=1 from the space11988212 [119886 119887] can be derived from Gram-Schmidt orthogonalprocess of 120595119894(119909)infin119894=1
120595119894 (119909) = 119894sum119896=1
120573119894119896120595119896 (119909) 120573119894119894 gt 0 119894 = 1 2 (13)
Definition 7 In a topological space (X 120591) a subset 119860 ofX iscalled dense inX if 119860 = cl119860 = X
Theorem 8 If 119909119894infin119894=1 is dense on [119886 119887] then 120595119894(119909)infin119894=1 is thecomplete function system of the space 11988212 [119886 119887] and 120595119894(119909) =119871 119905119877119909(119905)|119905=119909119894 where the subscript t in the operator L indicatesthat the operator L applies to the function of 119905Proof We have
120595119894 (119909) = ⟨(Llowast120593119894) (119905) 119877119909 (119905)⟩11988212[119886119887]
= ⟨120593119894 (119905) 119871 119905119877119909 (119905)⟩11988212[119886119887] = 119871 119905119877119909 (119905)1003816100381610038161003816119905=119909119894
(14)
Clearly 120595119894(119909) isin 11988212 [119886 119887] For each fixed 119906(119909) isin 11988212 [119886 119887] let⟨119906(119909) 120595119894(119909)⟩11988212[119886119887] = 0 119894 = 1 2 which means
⟨119906 (119909) (Llowast120593119894) (119909)⟩11988212[119886119887] = ⟨L119906 (sdot) 120593119894 (sdot)⟩1198821
2[119886119887]
= (L119906) (119909119894) = 0 (15)
Assume that 119909119894infin119894=1 is dense on [119886 119887] and so (L119906)(119909) = 0 Itfollows that119906 equiv 0 from the existence ofLminus1 Now the theoremis proved
Theorem 9 If 119909119894infin119894=1 is dense on [119886 119887] and the solution of (12)is unique then the solution of (12) is
119906 (119909) = infinsum119894=1
119894sum119896=1
120573119894119896119906 (119909119896) 120595119894 (119909) (16)
Proof Using (13) we have
119906 (119909) = infinsum119894=1
⟨119906 (119909) 120595119894 (119909)⟩11988212[119886119887]
120595119894 (119909)
= infinsum119894=1
⟨119906 (119909) 119894sum119896=1
120573119894119896120595119896 (119909)⟩11988212[119886119887]
120595119894 (119909)
= infinsum119894=1
119894sum119896=1
120573119894119896 ⟨119906 (119909) 120595119896 (119909)⟩11988212[119886119887] 120595119894 (119909)
= infinsum119894=1
119894sum119896=1
120573119894119896 ⟨119906 (119909) Llowast120593119896 (119909)⟩11988212[119886119887] 120595119894 (119909)
= infinsum119894=1
119894sum119896=1
120573119894119896 ⟨L119906 (119909) 120593119896 (119909)⟩11988212[119886119887] 120595119894 (119909)
= infinsum119894=1
119894sum119896=1
120573119894119896L119906 (119909119896) 120595119894 (119909)
= infinsum119894=1
119894sum119896=1
120573119894119896119906 (119909119896) 120595119894 (119909)
(17)
On the other hand 119906(119909) isin 11988212 [119886 119887] and 119906(119909) = suminfin119894=0 119886119894120595119894(119909)119886119894 = ⟨119906(119909) 120595119894(119909)⟩ are the Fourier series expansion about
4 Mathematical Problems in Engineering
normal orthogonal system 120595119894(119909)infin119894=1 and 11988212 [119886 119887] is theHilbert Space Thus the series suminfin119894=0 119886119894120595119894(119909) is convergent inthe sense of sdot 1198821
2
and the proof would be complete
Now the approximate solution 119906119899(119909) can be obtained bythe 119899-term intercept of the exact solution 119906(119909) and
119906119899 (119909) = 119899sum119894=1
119894sum119896=1
120573119894119896119906 (119909119896) 120595119894 (119909) (18)
Theorem 10 If 119906(119909) isin 11988212 [119886 119887] then119872 gt 0 exists such that|119906(119909)| le 119872119906(119909)11988212[119886119887]
Proof We have 119906(119909) = ⟨119906(119905) 119877119909(119905)⟩11988212[119886119887] for any 119909 119905isin [119886 119887] We know 119877119909(119905)1198821
2[119886119887] le 119872 Thus |119906(119909)| = |⟨119906(119905)119877119909(119905)⟩1198821
2[119886119887]| le 119877119909(119905)1198821
2[119886119887]119906(119905)1198821
2[119886119887] le 119872119906(119905)1198821
2[119886119887]
Theorem 11 The approximate solution 119906119899(119909) is uniformlyconvergent
Proof Assuming 119909 isin [119886 119887] by Theorems 9 and 10 it can beproved that
lim119899rarrinfin
1003816100381610038161003816119906119899 (119909) minus 119906 (119909)1003816100381610038161003816= lim119899rarrinfin
100381610038161003816100381610038161003816⟨119906119899 (119909) minus 119906 (119909) 119877119909 (119909)⟩11988221 [119886119887]100381610038161003816100381610038161003816le 119872 lim119899rarrinfin
1003817100381710038171003817119906119899 (119909) minus 119906 (119909)100381710038171003817100381711988221[119886119887] = 0
(19)
In the sequel a new iterative method to achieve thesolution of (12) is presented If
119860 119894 = 119894sum119896=1
120573119894119896119906 (119909119896) (20)
then (16) can be written as
119906 (119909) = infinsum119894=1
119860 119894120595119894 (119909) (21)
Now suppose for some 119909119895 119906(119909119895) is known There is noproblem ifwe assume 119895 = 1Weput1199060(1199091) = 119906(1199091) anddefinethe 119899-term approximation to 119906(119909) by
119906119899 (119909) = 119899sum119894=1
119861119894120595119894 (119909) (22)
where
119861119894 = 119894sum119896=1
120573119894119896119906119896minus1 (119909119896) (23)
In the following it would be proven that the approximatesolution 119906119899(119909) in the iterative (22) is convergent to the exactsolution of (12) uniformly
Theorem 12 Suppose that 11990611989911988221[119886119887] is bounded in (22) If119909119894infin119894=1 is dense on [119886 119887] then 119899-term approximate solution119906119899(119909) in the iterative (22) converges to the exact solution 119906(119909)
of (12) and 119906(119909) = lim119899rarrinfinsum119899119894=1 119861119873119894 120595119894(119909) whereas 119861119894 is givenby (23)
Proof First of all the convergence of 119906119899(119909) from (22) wouldbe proven We infer
119906119899+1 (119909) = 119906119899 (119909) + 119861119899+1120595119899+1 (119909) (24)
Subsequence 120595119894(119909)infin119894=1 is orthogonal and it yields that
1003817100381710038171003817119906119899+11003817100381710038171003817211988221[119886119887] = 1003817100381710038171003817119906119899100381710038171003817100381721198822
1[119886119887] + 1198612119899+1 =
119899sum119894=1
1198612119894 (25)
It is obvious that the sequence 11990611989911988221[119886119887] is monotonically
increasing Because 11990611989911988221[119886119887] is bounded and 1199061198991198822
1[119886119887] is
convergent then suminfin119894=1 1198612119894 is bounded and this implies that119861119894infin119894=1 isin 1198972If119898 gt 119899 then
1003817100381710038171003817119906119898 minus 1199061198991003817100381710038171003817211988221[119886119887] =
1003817100381710038171003817100381710038171003817100381710038171003817119899+1sum119894=119898
(119906119894 minus 119906119894minus1)10038171003817100381710038171003817100381710038171003817100381710038172
11988221[119886119887]
= 119899+1sum119894=119898
1003817100381710038171003817119906119894 minus 119906119894minus11003817100381710038171003817211988221[119886119887]
(26)
So 119906119894 minus 119906119894minus1211988221[119886119887] = 1198612119894 Consequently 119906119898 minus 11990611989921198822
1[119886119887] =sum119899+1119894=119898 1198612119894 rarr 0 as 119899 rarr infin To prove the completeness of11988212 [119886 119887] it requires where isin 11988212 [119886 119887] that 119906119899 rarr as119899 rarr infin Now we can prove is the solution of (12)
If we take limit from (22) we will have (119909) =sum119899119894=1 119861119894120595119894(119909) so L(119909) = suminfin119894=1 119861119894L120595119894(119909) Let 119909119897 isin 119909119894infin119894=1 andthen
L (119909119897) = infinsum119894=1
119861119894L120595119894 (119909119897)
= infinsum119894=1
119861119894 ⟨L120595119894 (119909) 120593119897 (119909)⟩11988212[119886119887]
= infinsum119894=1
119861119894 ⟨120595119894 (119909) Llowast120593119897 (119909)⟩11988212[119886119887]
= infinsum119894=1
119861119894 ⟨120595119894 (119909) 120595119897 (119909)⟩11988212[119886119887]
= infinsum119894=1
119861119894⟨120595119894 (119909) 119894sum119897=1
119861119894119897120595119897 (119909)⟩11988212[119886119887]
(27)
From (23) and (29) it is concluded that L(119909119897) = (119909119897)119909119894infin119894=1 is dense on [119886 119887] For each 119909 isin [119886 119887] 119909119899119894infin119894=1subsequence exists that 119909119899119894 rarr 119909 as 119894 rarr infin Hence when119894 rarr infin we have L119906(119909) = 119906(119909) which indicates that is thesolution of (12)
Mathematical Problems in Engineering 5
The mentioned scheme above is an efficient method ofsolving nonlinear equations [31ndash33] However in implement-ing this algorithm on a computer 120595119894(119909)infin119894=1 is not quiteorthogonal due to rounding errors In other words Gram-Schmidt process is numerically unstable and the computa-tional cost of the algorithm is high Therefore the followingprocess is suggested similar for linear problems in [20 34]This is the subject of the next theorem where the followingnotations are used
120573119899 =[[[[[[[
12057311 0 sdot sdot sdot 012057321 12057322 d 0 d
1205731198991 1205731198992 sdot sdot sdot 120573119899119899
]]]]]]]
120595119899 =[[[[[[[
12059511 12059512 sdot sdot sdot 120595111989912059521 12059522 d 1205952119899 d
1205951198991 1205951198992 sdot sdot sdot 120595119899119899
]]]]]]]
u =[[[[[[[
1199060 (1199091)1199061 (1199092)119906119899minus1 (119909119899)
]]]]]]]
B =[[[[[[[
11986111198612119861119899
]]]]]]]
Λ =[[[[[[[
Λ 1Λ 2Λ 119899
]]]]]]]
119899 = 1 2
(28)
Theorem 13 (let 120574119894119895 = [120595minus1]119894119895) The approximate solutionobtained from (22) is found as follows
119906119899 (119909) = 119899sum119894=1
Λ 119894120595119894 (119909) (29)
where
Λ 119894 = 119894sum119896=1
120574119894119896119906119896minus1 (119909119896) (30)
Proof Suppose that 119906119899(119909) = sum119899119894=1 Λ 119894120595119894(119909) = sum119899119894=1 119861119894120595119894(119909)Since 120595119894(119909) = sum119894119896=1 120573119894119896120595119896(119909)
119899sum119894=1
Λ 119894120595119894 (119909) = 119899sum119894=1
119861119894120595119894 (119909) = 119899sum119894=1
119861119894 119894sum119896=1
120573119894119896120595119896 (119909)
= 119899sum119896=1
119899sum119894=119896
119861119894120573119894119896120595119896 (119909) (31)
120595119894(119909)infin119894=1 and Λ 119896 = sum119899119894=119896 119861119894120573119894119896 (119896 = 1 2 119899) are linearindependence and therefore
Λ = 120573TB (32)
Equation (12) implies L119906119899(119909) = 119906119899(119909) For 119894 = 1 2 119899 wehave
⟨L119906119899 (119909) 120595119894⟩ = ⟨119906119899 (119909) 120595119894⟩ 997904rArr119899sum119895=1
119861119895 ⟨L120595119895 120595119894⟩ = ⟨119906119899 (119909) 120595119894⟩ (33)
Both sides of (33) provide
119899sum119895=1
119861119895 ⟨L120595119894 120595119894⟩ = 119899sum119895=1
119861119895 119894sum119896=1
120573119894119896119895sum119897=1
120573119895119897 ⟨L120595119897 120595119896⟩
= 119899sum119895=1
119861119895 119894sum119896=1
119895sum119897=1
120573119894119896 ⟨L120595119897 120595119896⟩ 120573119879119897119895= 119899sum119895=1
119861119895 (120573120595120573T)119894119895
(34)
⟨119906119899 (119909) 120595119894⟩ = 119861119894 = 119894sum119896=1
120573119894119896119906119896minus1 (119909119896) (using Theorem 12)
(35)
From (32) and (35) the following equation can be reached
120573120595120573TB = 120573u (36)
Equation (32) implies 120573120595Λ = 120573u119899 So120595Λ = u119899 (37)
which proves the theorem
Algorithm 14 The following steps exist for approximatingthe solution without applying Gram-Schmidt orthogonalprocess
Step 1 Fix 119886 le 119909 and 119905 le 119887If 119905 le 119909 set 119877119909(119905) = 1 minus 119886 + 119905Else set 119877119909(119905) = 1 minus 119886 + 119909Step 2 For 119894 = 1 2 119898 set 119909119894 = (119894 minus 1)(119898 minus 1)
6 Mathematical Problems in Engineering
Table 1 The absolute errors comparison between the proposed approach and method [30]
119909 Exact solution Approximate solution Method in [30] Absolute error Absolute error in method [30]01 1105170918 1105154993 1114627560 0000015925 000945664202 1221402757 1221389903 1226758840 0000012855 000535608303 1349858806 1349845490 1345458179 0000013317 000440062704 1491824696 1491803913 1480202470 0000020784 001162222605 1648721268 1648680858 1671769819 0000040412 002304855106 1822118797 1822039541 1838854903 0000079259 001673610607 2013752703 2013608464 2022118086 0000144243 000836538308 2225540923 2225304288 2223139077 0000236640 000240184609 2459603104 2459253211 2443684898 0000349899 0015918206
Table 2 The absolute errors comparison between the proposed approach and method [26]
119909 Exact solution Approximate solution Method in [26] Absolute error Absolute error in method [26]01 0099833 0098524 0101388 0001308 00015502 0198669 0199766 0194073 0001097 000459603 0295520 0294857 0299995 0000662 000447504 0389418 0389658 0387978 0000240 000144005 0479426 0479585 0486266 0000160 000684006 0564642 0563993 0565930 0000648 000128807 0644218 0645477 0640624 0001260 000359408 0717356 0715373 0720611 0001982 000325509 0783327 0785653 0782351 0002326 0000976
Set 120595119894(119909) = 119871 119905119877119909(119905)|119905=119909119894 Step 3 Set 1199060(1199091) = 119906(1199091)Step 4 For 119894 = 1 2 119898 set 120574119894119895 = [120595minus1]119894119895Step 5 119899 = 1Step 6 Set Λ 119899 = sum119899119896=1 120574119899119896119906119896minus1(119909119896)Step 7 Set 119906119899(119909) = sum119899119894=1 Λ 119894120595119894(119909)Step 8 If 119899 lt 119898 then set 119899 = 119899 + 1 and go to step 6Else stop
Algorithm 15 The following steps exist for approximating thesolution by applying Gram-Schmidt orthogonal process
Step 1 Fix 119886 le 119909 and 119905 le 119887If 119905 le 119909 set 119877119909(119905) = 1 minus 119886 + 119905Else set 119877119909(119905) = 1 minus 119886 + 119909Step 2 For 119894 = 1 2 119898 set 119909119894 = (119894 minus 1)(119898 minus 1)Set 120595119894(119909) = 119871 119905119877119909(119905)|119905=119909119894 Step 3 For 119894 = 1 2 119898 and 119895 =1 2 119898 if 119894 = 119895 then set 120573119894119895 =(minus1radic1205951198942 minus sum119894minus1119896=1⟨120595119894(119905) 120595119896(119905)⟩2) sum119894minus1119896=1⟨120595119894(119905) and120595119896(119905)⟩120573119896119895Else 120573119894119895 = 1radic1205951198942 minus sum119894minus1119896=1⟨120595119894(119905) 120595119896(119905)⟩2Else 12057311 = 11205951
Step 4 For 119894 = 1 2 119898 set 120595119894(119909) = sum119894119896=1 120573119894119896120595119896(119909)Step 5 Set 1199060(1199091) = 119906(1199091)Step 6 Set 119899 = 1Step 7 Set 119861119899 = sum119899119896=1 120573119899119896119906119896minus1(119909119896)Step 8 Set 119906119899(119909) = sum119899119894=1 119861119894120595119894(119909)Step 9 If 119899 lt 119898 then set 119899 = 119899 + 1 and go to step 7Else stop
4 Numerical Experiments
In this part four numerical examples are solved for potencyand utility of the present method All computations areperformed by MAPLE package Results which are takenby this method show a proper agreement with the exactsolution A comprehensive applicability of this method isgiven the stability and consistence of the presented methodThe reliability of the method and increasing of the accuracycause this method to be more applicable
Example 1 For first applicable instance we offer nonlinearFredholm integral equations [26 30]
119906 (119909) minus int101199091199101199063 (119910) 119889119910 = 119890119909 minus (1 + 21198903) 119909
9 0 le 119909 le 1
(38)
Mathematical Problems in Engineering 7
1
Error of approximate solution
0 01 02 03 04 05 06 07 08 09
times10minus4Approximate solution and the exact solution
ExactApproximate
0 02 03 04 05 06 07 08 0901 1x
1
12
14
16
18
2
22
24
26
28
minus2minus15
minus1minus05
005
115
2
Abso
lute
erro
r
Figure 1 The absolute errors comparison between the proposed approach and method [30]
Error of approximate solution
10 01 02 03 04 05 06 07 08 09
times10minus3Approximate solution and the exact solution
10 01 02 03 04 05 06 07 08 09
ExactApproximate
0
01
02
03
04
05
06
07
08
09
minus25minus2
minus15minus1
minus050
051
152
Abso
lute
erro
r
Figure 2 The absolute errors comparison between the proposed approach and method [26]
The exact solution of this equation is 119906(119909) = 119890119909 According to(38) we can assume an initial approximation 1199060(0) = 119906(0) =1 The numerical results are given in Table 1 by taking 119909119894 =(119894 minus 1)(119899 minus 1) 119894 = 1 2 119899 and 119899 = 128 In Table 1 acomparison between the absolute errors of our method andthe Haar wavelet method [30] is given Figure 1 shows theapproximate solution and its errors
Example 2 For second applicable example an electromag-netic problem is solved via the presented method It is simu-lated to nonlinear Volterra integral equations model [26 30]
119906 (119909) minus 12 int119909
01199062 (119910) 119889119910 = sin (119909) + 18 sin (2119909) minus 14119909
0 le 119909 le 1(39)
The exact solution of this equation is 119906(119909) = sin(119909)According to (39) we can assume an initial approximation1199060(0) = 119906(0) = 0 Numerical results are given in Table 2 bytaking119909119894 = (119894minus1)(119899minus1) 119894 = 1 2 119899 and 119899 = 128 In Table 2a comparison between the absolute errors of the proposedmethod and the BPFs method [26] is given Figure 2 shows
Table 3 The execution time (seconds) comparison between Algo-rithms 14 and 15
119899 Algorithm 14 (Sec) Algorithm 15 (Sec)4 347 2878 952 48216 5975 3177
the approximate solution and its errors In Table 3 a compar-ison execution time between Algorithms 14 and 15 is given
Example 3 An electromagnetic problem is solved via ourmethod for another applicable example It is simulated tononlinear Fredholm integral equations model [26 30]
119906 (119909) minus 12 int1
01199101199062 (119910) 119889119910 = 1199092 minus 112 0 le 119909 le 1 (40)
The exact solution of this equation is 119906(119909) = 1199092 Accordingto (40) the initial approximation 1199060(0) = 119906(0) = 0 is chosenNumerical results are given in Table 4 by taking 119909119894 = (119894 minus1)(119899minus1) 119894 = 1 2 119899 and 119899 = 128 In Table 4 a comparison
8 Mathematical Problems in Engineering
10
20
30
40
50
60
Error of approximate solutiontimes10minus5Approximate solution and the exact solution
0 01 02 03 04 05 06 07 08 09
ExactApproximate
0 02 03 04 05 06 07 08 0901 1x
0
01
02
03
04
05
06
07
08
minus8minus6minus4minus2
02468
Abso
lute
erro
r
Figure 3 The absolute errors comparison between the proposed approach and method [26]
Table 4 The absolute errors comparison between the proposed approach and method [26]
119909 Exact solution Approximate solution Method in [26] Absolute error Absolute error in method [26]01 0010000 0010762 0010308 0000762 000030802 0040000 0040655 0038140 0000655 000186003 0090000 0090444 0092828 0000444 000282804 0160000 016023 0158746 0000230 000125405 0250000 0250001 0257867 0000001 000786706 0360000 0359606 0361871 0000394 000187107 0490000 0489448 0483453 0000552 000654708 0640000 0639391 0647515 0000609 000751509 0810000 0811638 0807183 0001638 0002817
Table 5 The execution time (seconds) comparison between Algo-rithms 14 and 15
119899 Algorithm 14 (Sec) Algorithm 15 (Sec)4 31 2598 846 50916 342 2889
between the absolute errors of the proposed method and theBPFs method [26] is given Figure 3 shows the approximatesolution and its error In Table 5 a comparison execution timebetween Algorithms 14 and 15 is given
Example 4 A nonlinear Fredholm integral problem [26 30]is solved via our method for this applicable example
119906 (119909) + int10119890119909minus21199101199063 (119910) 119889119910 = 119890119909+1 0 le 119909 lt 1 (41)
The exact solution of this equation is 119906(119909) = 119890119909 According to(41) we consider an initial approximation as 1199060(0) = 119906(0) = 1Numerical results are given in Table 6 by taking 119909119894 = (119894 minus1)(119899minus1) 119894 = 1 2 119899 and 119899 = 128 In Table 6 a comparison
between the absolute errors of the proposed method andthe Haar wavelet method [30] is given Figure 4 shows theapproximate solution and its error
5 Conclusion
According to this essay supplementary of iterative Reproduc-ing Kernel Hilbert SpaceMethod was introduced and appliedto acquire the approximate solution of some nonlinear inte-gral equation In this method unlike other similar methodsorthogonal process is not used However the time is increas-ing and accuracy is also increasing The main point whichis mentioned in this paper is that Algorithm 14 has higherexecution time in comparison with Algorithm 15 but theapproximate solution in Algorithm 14 is more accurate thanAlgorithm 15 Current uniform convergencemethod is statedand proved The obtained numerical results confirm that itis a good candidate for solving the nonlinear integral equa-tion
Conflicts of Interest
The authors declare no conflicts of interest
Mathematical Problems in Engineering 9
Table 6 The absolute errors comparison between the proposed approach and method [30]
119909 Exact solution Approximate solution Method in [30] Absolute error Absolute error in method [30]01 1105170918 1105495796 1111099473 0000324878 000592855502 1221402757 1220931139 1220296615 0000471618 000110614203 1349858806 1349502887 1339817084 0000355919 001004172204 1491824696 1491740269 1471965091 0000084427 001985960505 1648721268 1648649846 1457127691 0000071422 019159357706 1822118797 1822243106 1831888927 0000124309 000977013007 2013752703 2013650232 2011936556 0000102471 000181614708 2225540923 2224684503 2209678381 0000856420 001586254209 2459603104 2458626573 2426854676 0000976531 0032748428
Error of approximate solution
10 01 02 03 04 05 06 07 08 09
times10minus5
Approximate solution and the exact solution
10 01 02 03 04 05 06 07 08 09
ExactApproximate
1
12
14
16
18
2
22
24
26
28
minus10
minus8
minus6
minus4
minus2
0
4
2
Abso
lute
erro
r
x
Figure 4 The absolute errors comparison between the proposed approach and method [30]
Acknowledgments
The authors would like to thank Professor Haipeng Peng forinstructive comments and recommendations to improve thequality of this work and also the Islamic Azad UniversityScience and Research Branch Tehran for supporting thisproject
References
[1] K Meetz and W Engi Electromagnetische Felder SpringerBerlin Germany 1979
[2] W Thirring Lehrbuch der Mathematischen Physik vol 2Springer Vienna Austria 1978
[3] K F Warnick R H Selfridge and D V Arnold ldquoTeachingelectromagnetic field theory using differential formsrdquo IEEETransactions on Education vol 40 no 1 pp 53ndash68 1997
[4] S Abbasbandy ldquoHomotopy analysis method for heat radiationequationsrdquo International Communications in Heat and MassTransfer vol 34 no 3 pp 380ndash387 2007
[5] H Vosughi E Shivanian and S Abbasbandy ldquoA new analyticaltechnique to solve Volterrarsquos integral equationsrdquo MathematicalMethods in the Applied Sciences vol 34 no 10 pp 1243ndash12532011
[6] J-P Kauthen ldquoA survey of singularly perturbed Volterra equa-tionsrdquo Applied Numerical Mathematics An IMACS Journal vol24 no 2-3 pp 95ndash114 1997
[7] A A Kilbas and M Saigo ldquoOn solution of nonlinear Abel-Volterra integral equationrdquo Journal of Mathematical Analysisand Applications vol 229 no 1 pp 41ndash60 1999
[8] C Minggen and D Zhongxing ldquoOn the best operator ofinterpolationrdquo Mathematica Numerica Sinica vol 8 no 2 pp209ndash216 1986
[9] C Minggen and L Yingzhen Nonlinear Numerical Analysis inthe Reproducing Kernel Space Nova Science New York NYUSA 2009
[10] S Abbasbandy B Azarnavid and M S Alhuthali ldquoA shootingreproducing kernel Hilbert spacemethod formultiple solutionsof nonlinear boundary value problemsrdquo Journal of Computa-tional and Applied Mathematics vol 279 pp 293ndash305 2015
[11] N Shawagfeh O Abu Arqub and S Momani ldquoAnalytical solu-tion of nonlinear second-order periodic boundary value prob-lem using reproducing kernel methodrdquo Journal of Computa-tional Analysis andApplications vol 16 no 4 pp 750ndash762 2014
[12] FGeng andMCui ldquoSolving a nonlinear systemof second orderboundary value problemsrdquo Journal of Mathematical Analysisand Applications vol 327 no 2 pp 1167ndash1181 2007
[13] B Azarnavid F Parvaneh and S Abbasbandy ldquoPicard-repro-ducing kernel Hilbert space method for solving generalizedsingular nonlinear Lane-Emden type equationsrdquoMathematicalModelling and Analysis vol 20 no 6 pp 754ndash767 2015
[14] 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
10 Mathematical Problems in Engineering
[15] L Yang and M Cui ldquoNew algorithm for a class of nonlinearintegro-differential equations in the reproducing kernel spacerdquoApplied Mathematics and Computation vol 174 no 2 pp 942ndash960 2006
[16] 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
[17] S Bushnaq B Maayah S Momani and A Alsaedi ldquoA repro-ducing kernel Hilbert space method for solving systems offractional integrodifferential equationsrdquo Abstract and AppliedAnalysis vol 2014 Article ID 103016 6 pages 2014
[18] M Inc A Akgul and F Geng ldquoReproducing kernel Hilbertspace method for solving Bratursquos problemrdquo Bulletin of theMalaysian Mathematical Sciences Society vol 38 no 1 pp 271ndash287 2015
[19] A-M Wazwaz A First Course in Integral Equations WorldScientific 1997
[20] S Javadi E Babolian and E Moradi ldquoNew implementation ofreproducing kernel Hilbert space method for solving a class offunctional integral equationsrdquo Communications in NumericalAnalysis vol 2014 Article ID cna-00205 7 pages 2014
[21] W Jiang and Z Chen ldquoSolving a system of linear Volterraintegral equations using the new reproducing kernel methodrdquoApplied Mathematics and Computation vol 219 no 20 pp10225ndash10230 2013
[22] A-M Wazwaz Linear and Nonlinear Integral Equations Meth-ods and Applications Springer Berlin Germany 2011
[23] I L El-Kalla ldquoConvergence of the Adomian method appliedto a class of nonlinear integral equationsrdquo Applied MathematicsLetters An International Journal of Rapid Publication vol 21 no4 pp 372ndash376 2008
[24] G Gumah KMoaddy M Al-Smadi and I Hashim ldquoSolutionsto uncertain Volterra integral equations by fitted reproducingkernel Hilbert space methodrdquo Journal of Function Spaces vol2016 Article ID 2920463 11 pages 2016
[25] V Sizikov and D Sidorov ldquoGeneralized quadrature for solv-ing singular integral equations of Abel type in applicationto infrared tomographyrdquo Applied Numerical Mathematics AnIMACS Journal vol 106 pp 69ndash78 2016
[26] E Babolian Z Masouri and S Hatamzadeh-Varmazyar ldquoNewdirect method to solve nonlinear Volterra-Fredholm integraland integro-differential equations using operational matrixwith block-pulse functionsrdquo in Progress in ElectromagneticsResearch B vol 8 pp 59ndash76 EMW Publishing CambridgeMass USA 2008
[27] D Alpay Ed Reproducing kernel spaces and applications vol143 of Operator Theory Advances and Applications BirkhauserBasel 2003
[28] S Saitoh D Alpay J A Ball and T Ohsawa ReproducingKernels andTheir Applications vol 11 Springer Science amp Busi-ness Media Berlin Germany 2013
[29] S Saitoh Integral Transforms Reproducing Kernels and TheirApplications vol 369 of Pitman Research Notes in MathematicsSeries Longman Harlow UK 1997
[30] E Babolian andA Shahsavaran ldquoNumerical solution of nonlin-ear Fredholm integral equations of the second kind using Haarwaveletsrdquo Journal of Computational and Applied Mathematicsvol 225 no 1 pp 87ndash95 2009
[31] R Ketabchi R Mokhtari and E Babolian ldquoSome error esti-mates for solving Volterra integral equations by using the
reproducing kernel methodrdquo Journal of Computational andApplied Mathematics vol 273 pp 245ndash250 2015
[32] A Alvandi M Paripour and Z Roshani ldquoReproducing kernelmethod for solving a class of Fredholm integro-differentialequationsrdquo inProceedings of the 46thAnnual IranianMathemat-ics Conference (AIMC 46) p 505 Yazd University 2015
[33] I Komashynska andMAl-Smadi ldquoIterative reproducing kernelmethod for solving second-order integrodifferential equationsof fredholm typerdquo Journal of Applied Mathematics vol 2014Article ID 459509 11 pages 2014
[34] E Babolian S Javadi and E Moradi ldquoError analysis of repro-ducing kernel Hilbert space method for solving functional inte-gral equationsrdquo Journal of Computational and Applied Mathe-matics vol 300 pp 300ndash311 2016
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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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
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
where 119886 119887 are real constants 119906 isin 11988212 [119886 119887] is an unknownfunction which can be determined 119891 isin 11988212 [119886 119887] is a con-tinuous function on [119886 119887] 119870(119909 119905) is a continuous functionon [119886 119887] times [119886 119887] 119873(V(119909)) is a continuous term in 11988212 [119886 119887]as V(119909) isin 11988212 [119886 119887] 119886 le 119909 le 119887 and11988212 [119886 119887] is ReproducingKernel Space Equation (1) has a continuous solution on [119886 119887][19] The existence and uniqueness conditions of the solutionfor (1) were discussed in [19ndash23]We assume that the solutionof (1) is unique
Over several decades numerical methods in electromag-netic problems have been one of the most important subjectsof extensive researches [1ndash4] On the other hand manyproblems in electromagnetics can be modeled by integralequations mentioned in [24ndash26] for example electric fieldintegral equation (EFIE) andmagnetic field integral equation(MFIE) In recent years several numerical methods forsolving linear and nonlinear integral equations have beenpresented Applicable equations of electromagnetics havebeen implied in the presented paper
In previousworks like [13ndash15] theGram-Schmidt orthog-onalization process has been considered to implementRKHSM Since this process is unstable numerically and itmaytake a lot of time to run the algorithm here we put away thisprocess and act with another way Our approach combinesthe methods mentioned in [13ndash17] More specifically on thecontrary to [13ndash15] without use of the orthogonalizationprocess the RKSHM is applied successfully to solve thenonlinear problem (1)
The structure of this paper would be described as followsIn Section 2 the basic definitions assumptions and prelimi-naries of RKHS are describedThemain idea and convergenceof the proposed scheme are discussed in Section 3 Section 4contains the numerical experiments Finally Section 5 isdedicated to a brief conclusion
2 Preliminaries
In this section some basic definitions and important prop-erties of Reproducing Kernel Hilbert Spaces (RKHS) arementioned [8 9 27ndash29]
Definition 1 AHilbert Space119867 is an inner product space thatis complete and separable with respect to the norm definedby the inner product Completeness of the space 119867 holdsprovided that every Cauchy sequence of points in119867 that hasa limit that is also in119867 and separable of119867 admits a countableorthonormal basis of it
Definition 2 For an abstract setX letH be a Hilbert Spaceof real or complex-valued functions on setX We sayH is aReproducing Kernel Hilbert Space if there exist a linear andbounded evaluation functional 119877119909 overH or equivalently
119877119909 119891 997891997888rarr 119891 (119909) forall119891 isin 119867 (2)
Riesz RepresentationTheorem implies that for all 119909 inXthere exists a unique function119870119909 ofH with the reproducingproperty
119891 (119909) = 119877119909 (119891) = ⟨119891 (119910) 119877119909 (119910)⟩H forall119891 isin H (3)
for each 119910 isin H where ⟨sdot sdot⟩H represent the inner product ofthe Hilbert SpaceH
Definition 3 The space1198821198982 [119886 119887] is interpreted as
1198821198982 [119886 119887] = 119906 (119909) | 119906 [119886 119887] 997888rarr 119877 119906(119898minus1)isin 119860119862 [119886 119887] 119906(119898) isin 1198712 [119886 119887] (4)
The inner product and the norm in1198821198982 [119886 119887] are of forms
⟨119906 (119909) V (119909)⟩1198821198982
= 119898minus1sum119894=0
119906(119894) (119886) V(119894) (119886)
+ int119887119886119906(119898) (120585) V(119898) (120585) 119889120585
119906 V isin 1198821198982 [119886 119887] 119906 = radic⟨119906 (119909) V (119909)⟩119882119898
2
119906 isin 1198821198982 [119886 119887]
(5)
Lemma 4 (see [9 29]) Functional space 1198821198982 [119886 119887] is innerspace
Theorem 5 (see [9 29]) Functional space 1198821198982 [119886 119887] is aHilbert Space
Theorem 6 (see [9 29]) Functional space1198821198982 [119886 119887] is Repro-ducing Kernel Hilbert Space
Now it is taken away that expression form of the Repro-ducing Kernel function 119877119909(119905) isin 1198821198982 [119886 119887]
Based on essay it is easy to prove that 119877119909(119905) is the answerof the following generalized differential equation [9 29]
(minus1)119898 1205972119898119877119909 (119905)1205971199052119898 = 120575 (119905 minus 119909) 120597119894119877119910 (119886)120597119905119894 minus (minus1)119898minus119894minus1 1205972119898minus119894minus1119877119910 (119886)1205971199052119898minus119894minus1 = 0
1205972119898minus119894minus1119877119910 (119887)1205971199052119898minus119894minus1 = 0119894 = 0 1 119898 minus 1
(6)
where 120575 is Diracrsquos delta function While 119909 = 119905 119877119909(119905) isthe answer of the following constant linear homogeneousdifferential equation with 2119898 order
(minus1)119898 1205972119898119877119909 (119905)1205971199052119898 = 0 (7)
Mathematical Problems in Engineering 3
with the boundary conditions
120597119894119877 minus 119910 (119886)120597119905119894 minus (minus1)119898minus119894minus1 1205972119898minus119894minus1119877119910 (119886)1205971199052119898minus119894minus1 = 01205972119898minus119894minus1119877119910 (119887)1205971199052119898minus119894minus1 = 0
119894 = 0 1 119898 minus 1(8)
Equation (7) is characteristic 1205822119898 = 0 Then the general solu-tion of Equation (7) is
119877119909 (119905) =
2119898sum119894=1
119888119894 (119909) 119905119894minus1 119905 le 1199092119898sum119894=1
119889119894 (119909) 119905119894minus1 119905 gt 119909 (9)
where coefficients 119888119894(119909) and 119889119894(119909) 119894 = 1 2 2119898 could becalculated by solving the following linear equations
120597119894119877119909 (119909 + 0)120597119905119894 = 120597119894119877119909 (119909 minus 0)120597119905119894 119894 = 0 1 2119898 minus 2(1205972119898minus1119877119909 (119909 + 0)1205971199052119898minus1 minus 1205972119898minus1119877119909 (119909 minus 0)1205971199052119898minus1 ) = (minus1)119898 120597119894119877119910 (119886)120597119905119894 minus (minus1)119898minus119894minus1 1205972119898minus119894minus1119877119910 (119886)1205971199052119898minus119894minus1 = 01205972119898minus119894minus1119877119910 (119887)1205971199052119898minus119894minus1 = 0 119894 = 0 1 119898 minus 1
(10)
Subsequently the representation of the Reproducing Kernelof11988212 [119886 119887] is provided by
119877119909 (119905) = 1 minus 119886 + 119905 119905 le 1199091 minus 119886 + 119909 119905 gt 119909 (11)
3 Main Idea and Theoretical Discussion
The uniqueness conditions for nonlinear problems exist in[21ndash23] The unique solution of (1) is assumed in this paperThe solution of (1) is given in11988212 [119886 119887] space We consider (1)as
L119906 (119909) = 119906 (119909) (12)
where L119906(119909) = 119891(119909) minus int119887 (or119909)119886
119870(119909 119905)119873(119906(119905))119889119905 It is obviousthatL is the bounded linear operator of11988212 to11988212 Put120593119894(119909) =119877119909119894(119909) and 120595119894(119909) = Llowast120593119894(119909) where Llowast is the adjoint operatorof L In fact for 119906 V isin 11988212 [119886 119887] we have L(119906(119909) + V(119909)) =119906(119909) + V(119909) and L119906(119909)1198821
2[119886119887] = 119906(119909)1198821
2[119886119887]
The orthonormal system of 120595119894(119909)infin119894=1 from the space11988212 [119886 119887] can be derived from Gram-Schmidt orthogonalprocess of 120595119894(119909)infin119894=1
120595119894 (119909) = 119894sum119896=1
120573119894119896120595119896 (119909) 120573119894119894 gt 0 119894 = 1 2 (13)
Definition 7 In a topological space (X 120591) a subset 119860 ofX iscalled dense inX if 119860 = cl119860 = X
Theorem 8 If 119909119894infin119894=1 is dense on [119886 119887] then 120595119894(119909)infin119894=1 is thecomplete function system of the space 11988212 [119886 119887] and 120595119894(119909) =119871 119905119877119909(119905)|119905=119909119894 where the subscript t in the operator L indicatesthat the operator L applies to the function of 119905Proof We have
120595119894 (119909) = ⟨(Llowast120593119894) (119905) 119877119909 (119905)⟩11988212[119886119887]
= ⟨120593119894 (119905) 119871 119905119877119909 (119905)⟩11988212[119886119887] = 119871 119905119877119909 (119905)1003816100381610038161003816119905=119909119894
(14)
Clearly 120595119894(119909) isin 11988212 [119886 119887] For each fixed 119906(119909) isin 11988212 [119886 119887] let⟨119906(119909) 120595119894(119909)⟩11988212[119886119887] = 0 119894 = 1 2 which means
⟨119906 (119909) (Llowast120593119894) (119909)⟩11988212[119886119887] = ⟨L119906 (sdot) 120593119894 (sdot)⟩1198821
2[119886119887]
= (L119906) (119909119894) = 0 (15)
Assume that 119909119894infin119894=1 is dense on [119886 119887] and so (L119906)(119909) = 0 Itfollows that119906 equiv 0 from the existence ofLminus1 Now the theoremis proved
Theorem 9 If 119909119894infin119894=1 is dense on [119886 119887] and the solution of (12)is unique then the solution of (12) is
119906 (119909) = infinsum119894=1
119894sum119896=1
120573119894119896119906 (119909119896) 120595119894 (119909) (16)
Proof Using (13) we have
119906 (119909) = infinsum119894=1
⟨119906 (119909) 120595119894 (119909)⟩11988212[119886119887]
120595119894 (119909)
= infinsum119894=1
⟨119906 (119909) 119894sum119896=1
120573119894119896120595119896 (119909)⟩11988212[119886119887]
120595119894 (119909)
= infinsum119894=1
119894sum119896=1
120573119894119896 ⟨119906 (119909) 120595119896 (119909)⟩11988212[119886119887] 120595119894 (119909)
= infinsum119894=1
119894sum119896=1
120573119894119896 ⟨119906 (119909) Llowast120593119896 (119909)⟩11988212[119886119887] 120595119894 (119909)
= infinsum119894=1
119894sum119896=1
120573119894119896 ⟨L119906 (119909) 120593119896 (119909)⟩11988212[119886119887] 120595119894 (119909)
= infinsum119894=1
119894sum119896=1
120573119894119896L119906 (119909119896) 120595119894 (119909)
= infinsum119894=1
119894sum119896=1
120573119894119896119906 (119909119896) 120595119894 (119909)
(17)
On the other hand 119906(119909) isin 11988212 [119886 119887] and 119906(119909) = suminfin119894=0 119886119894120595119894(119909)119886119894 = ⟨119906(119909) 120595119894(119909)⟩ are the Fourier series expansion about
4 Mathematical Problems in Engineering
normal orthogonal system 120595119894(119909)infin119894=1 and 11988212 [119886 119887] is theHilbert Space Thus the series suminfin119894=0 119886119894120595119894(119909) is convergent inthe sense of sdot 1198821
2
and the proof would be complete
Now the approximate solution 119906119899(119909) can be obtained bythe 119899-term intercept of the exact solution 119906(119909) and
119906119899 (119909) = 119899sum119894=1
119894sum119896=1
120573119894119896119906 (119909119896) 120595119894 (119909) (18)
Theorem 10 If 119906(119909) isin 11988212 [119886 119887] then119872 gt 0 exists such that|119906(119909)| le 119872119906(119909)11988212[119886119887]
Proof We have 119906(119909) = ⟨119906(119905) 119877119909(119905)⟩11988212[119886119887] for any 119909 119905isin [119886 119887] We know 119877119909(119905)1198821
2[119886119887] le 119872 Thus |119906(119909)| = |⟨119906(119905)119877119909(119905)⟩1198821
2[119886119887]| le 119877119909(119905)1198821
2[119886119887]119906(119905)1198821
2[119886119887] le 119872119906(119905)1198821
2[119886119887]
Theorem 11 The approximate solution 119906119899(119909) is uniformlyconvergent
Proof Assuming 119909 isin [119886 119887] by Theorems 9 and 10 it can beproved that
lim119899rarrinfin
1003816100381610038161003816119906119899 (119909) minus 119906 (119909)1003816100381610038161003816= lim119899rarrinfin
100381610038161003816100381610038161003816⟨119906119899 (119909) minus 119906 (119909) 119877119909 (119909)⟩11988221 [119886119887]100381610038161003816100381610038161003816le 119872 lim119899rarrinfin
1003817100381710038171003817119906119899 (119909) minus 119906 (119909)100381710038171003817100381711988221[119886119887] = 0
(19)
In the sequel a new iterative method to achieve thesolution of (12) is presented If
119860 119894 = 119894sum119896=1
120573119894119896119906 (119909119896) (20)
then (16) can be written as
119906 (119909) = infinsum119894=1
119860 119894120595119894 (119909) (21)
Now suppose for some 119909119895 119906(119909119895) is known There is noproblem ifwe assume 119895 = 1Weput1199060(1199091) = 119906(1199091) anddefinethe 119899-term approximation to 119906(119909) by
119906119899 (119909) = 119899sum119894=1
119861119894120595119894 (119909) (22)
where
119861119894 = 119894sum119896=1
120573119894119896119906119896minus1 (119909119896) (23)
In the following it would be proven that the approximatesolution 119906119899(119909) in the iterative (22) is convergent to the exactsolution of (12) uniformly
Theorem 12 Suppose that 11990611989911988221[119886119887] is bounded in (22) If119909119894infin119894=1 is dense on [119886 119887] then 119899-term approximate solution119906119899(119909) in the iterative (22) converges to the exact solution 119906(119909)
of (12) and 119906(119909) = lim119899rarrinfinsum119899119894=1 119861119873119894 120595119894(119909) whereas 119861119894 is givenby (23)
Proof First of all the convergence of 119906119899(119909) from (22) wouldbe proven We infer
119906119899+1 (119909) = 119906119899 (119909) + 119861119899+1120595119899+1 (119909) (24)
Subsequence 120595119894(119909)infin119894=1 is orthogonal and it yields that
1003817100381710038171003817119906119899+11003817100381710038171003817211988221[119886119887] = 1003817100381710038171003817119906119899100381710038171003817100381721198822
1[119886119887] + 1198612119899+1 =
119899sum119894=1
1198612119894 (25)
It is obvious that the sequence 11990611989911988221[119886119887] is monotonically
increasing Because 11990611989911988221[119886119887] is bounded and 1199061198991198822
1[119886119887] is
convergent then suminfin119894=1 1198612119894 is bounded and this implies that119861119894infin119894=1 isin 1198972If119898 gt 119899 then
1003817100381710038171003817119906119898 minus 1199061198991003817100381710038171003817211988221[119886119887] =
1003817100381710038171003817100381710038171003817100381710038171003817119899+1sum119894=119898
(119906119894 minus 119906119894minus1)10038171003817100381710038171003817100381710038171003817100381710038172
11988221[119886119887]
= 119899+1sum119894=119898
1003817100381710038171003817119906119894 minus 119906119894minus11003817100381710038171003817211988221[119886119887]
(26)
So 119906119894 minus 119906119894minus1211988221[119886119887] = 1198612119894 Consequently 119906119898 minus 11990611989921198822
1[119886119887] =sum119899+1119894=119898 1198612119894 rarr 0 as 119899 rarr infin To prove the completeness of11988212 [119886 119887] it requires where isin 11988212 [119886 119887] that 119906119899 rarr as119899 rarr infin Now we can prove is the solution of (12)
If we take limit from (22) we will have (119909) =sum119899119894=1 119861119894120595119894(119909) so L(119909) = suminfin119894=1 119861119894L120595119894(119909) Let 119909119897 isin 119909119894infin119894=1 andthen
L (119909119897) = infinsum119894=1
119861119894L120595119894 (119909119897)
= infinsum119894=1
119861119894 ⟨L120595119894 (119909) 120593119897 (119909)⟩11988212[119886119887]
= infinsum119894=1
119861119894 ⟨120595119894 (119909) Llowast120593119897 (119909)⟩11988212[119886119887]
= infinsum119894=1
119861119894 ⟨120595119894 (119909) 120595119897 (119909)⟩11988212[119886119887]
= infinsum119894=1
119861119894⟨120595119894 (119909) 119894sum119897=1
119861119894119897120595119897 (119909)⟩11988212[119886119887]
(27)
From (23) and (29) it is concluded that L(119909119897) = (119909119897)119909119894infin119894=1 is dense on [119886 119887] For each 119909 isin [119886 119887] 119909119899119894infin119894=1subsequence exists that 119909119899119894 rarr 119909 as 119894 rarr infin Hence when119894 rarr infin we have L119906(119909) = 119906(119909) which indicates that is thesolution of (12)
Mathematical Problems in Engineering 5
The mentioned scheme above is an efficient method ofsolving nonlinear equations [31ndash33] However in implement-ing this algorithm on a computer 120595119894(119909)infin119894=1 is not quiteorthogonal due to rounding errors In other words Gram-Schmidt process is numerically unstable and the computa-tional cost of the algorithm is high Therefore the followingprocess is suggested similar for linear problems in [20 34]This is the subject of the next theorem where the followingnotations are used
120573119899 =[[[[[[[
12057311 0 sdot sdot sdot 012057321 12057322 d 0 d
1205731198991 1205731198992 sdot sdot sdot 120573119899119899
]]]]]]]
120595119899 =[[[[[[[
12059511 12059512 sdot sdot sdot 120595111989912059521 12059522 d 1205952119899 d
1205951198991 1205951198992 sdot sdot sdot 120595119899119899
]]]]]]]
u =[[[[[[[
1199060 (1199091)1199061 (1199092)119906119899minus1 (119909119899)
]]]]]]]
B =[[[[[[[
11986111198612119861119899
]]]]]]]
Λ =[[[[[[[
Λ 1Λ 2Λ 119899
]]]]]]]
119899 = 1 2
(28)
Theorem 13 (let 120574119894119895 = [120595minus1]119894119895) The approximate solutionobtained from (22) is found as follows
119906119899 (119909) = 119899sum119894=1
Λ 119894120595119894 (119909) (29)
where
Λ 119894 = 119894sum119896=1
120574119894119896119906119896minus1 (119909119896) (30)
Proof Suppose that 119906119899(119909) = sum119899119894=1 Λ 119894120595119894(119909) = sum119899119894=1 119861119894120595119894(119909)Since 120595119894(119909) = sum119894119896=1 120573119894119896120595119896(119909)
119899sum119894=1
Λ 119894120595119894 (119909) = 119899sum119894=1
119861119894120595119894 (119909) = 119899sum119894=1
119861119894 119894sum119896=1
120573119894119896120595119896 (119909)
= 119899sum119896=1
119899sum119894=119896
119861119894120573119894119896120595119896 (119909) (31)
120595119894(119909)infin119894=1 and Λ 119896 = sum119899119894=119896 119861119894120573119894119896 (119896 = 1 2 119899) are linearindependence and therefore
Λ = 120573TB (32)
Equation (12) implies L119906119899(119909) = 119906119899(119909) For 119894 = 1 2 119899 wehave
⟨L119906119899 (119909) 120595119894⟩ = ⟨119906119899 (119909) 120595119894⟩ 997904rArr119899sum119895=1
119861119895 ⟨L120595119895 120595119894⟩ = ⟨119906119899 (119909) 120595119894⟩ (33)
Both sides of (33) provide
119899sum119895=1
119861119895 ⟨L120595119894 120595119894⟩ = 119899sum119895=1
119861119895 119894sum119896=1
120573119894119896119895sum119897=1
120573119895119897 ⟨L120595119897 120595119896⟩
= 119899sum119895=1
119861119895 119894sum119896=1
119895sum119897=1
120573119894119896 ⟨L120595119897 120595119896⟩ 120573119879119897119895= 119899sum119895=1
119861119895 (120573120595120573T)119894119895
(34)
⟨119906119899 (119909) 120595119894⟩ = 119861119894 = 119894sum119896=1
120573119894119896119906119896minus1 (119909119896) (using Theorem 12)
(35)
From (32) and (35) the following equation can be reached
120573120595120573TB = 120573u (36)
Equation (32) implies 120573120595Λ = 120573u119899 So120595Λ = u119899 (37)
which proves the theorem
Algorithm 14 The following steps exist for approximatingthe solution without applying Gram-Schmidt orthogonalprocess
Step 1 Fix 119886 le 119909 and 119905 le 119887If 119905 le 119909 set 119877119909(119905) = 1 minus 119886 + 119905Else set 119877119909(119905) = 1 minus 119886 + 119909Step 2 For 119894 = 1 2 119898 set 119909119894 = (119894 minus 1)(119898 minus 1)
6 Mathematical Problems in Engineering
Table 1 The absolute errors comparison between the proposed approach and method [30]
119909 Exact solution Approximate solution Method in [30] Absolute error Absolute error in method [30]01 1105170918 1105154993 1114627560 0000015925 000945664202 1221402757 1221389903 1226758840 0000012855 000535608303 1349858806 1349845490 1345458179 0000013317 000440062704 1491824696 1491803913 1480202470 0000020784 001162222605 1648721268 1648680858 1671769819 0000040412 002304855106 1822118797 1822039541 1838854903 0000079259 001673610607 2013752703 2013608464 2022118086 0000144243 000836538308 2225540923 2225304288 2223139077 0000236640 000240184609 2459603104 2459253211 2443684898 0000349899 0015918206
Table 2 The absolute errors comparison between the proposed approach and method [26]
119909 Exact solution Approximate solution Method in [26] Absolute error Absolute error in method [26]01 0099833 0098524 0101388 0001308 00015502 0198669 0199766 0194073 0001097 000459603 0295520 0294857 0299995 0000662 000447504 0389418 0389658 0387978 0000240 000144005 0479426 0479585 0486266 0000160 000684006 0564642 0563993 0565930 0000648 000128807 0644218 0645477 0640624 0001260 000359408 0717356 0715373 0720611 0001982 000325509 0783327 0785653 0782351 0002326 0000976
Set 120595119894(119909) = 119871 119905119877119909(119905)|119905=119909119894 Step 3 Set 1199060(1199091) = 119906(1199091)Step 4 For 119894 = 1 2 119898 set 120574119894119895 = [120595minus1]119894119895Step 5 119899 = 1Step 6 Set Λ 119899 = sum119899119896=1 120574119899119896119906119896minus1(119909119896)Step 7 Set 119906119899(119909) = sum119899119894=1 Λ 119894120595119894(119909)Step 8 If 119899 lt 119898 then set 119899 = 119899 + 1 and go to step 6Else stop
Algorithm 15 The following steps exist for approximating thesolution by applying Gram-Schmidt orthogonal process
Step 1 Fix 119886 le 119909 and 119905 le 119887If 119905 le 119909 set 119877119909(119905) = 1 minus 119886 + 119905Else set 119877119909(119905) = 1 minus 119886 + 119909Step 2 For 119894 = 1 2 119898 set 119909119894 = (119894 minus 1)(119898 minus 1)Set 120595119894(119909) = 119871 119905119877119909(119905)|119905=119909119894 Step 3 For 119894 = 1 2 119898 and 119895 =1 2 119898 if 119894 = 119895 then set 120573119894119895 =(minus1radic1205951198942 minus sum119894minus1119896=1⟨120595119894(119905) 120595119896(119905)⟩2) sum119894minus1119896=1⟨120595119894(119905) and120595119896(119905)⟩120573119896119895Else 120573119894119895 = 1radic1205951198942 minus sum119894minus1119896=1⟨120595119894(119905) 120595119896(119905)⟩2Else 12057311 = 11205951
Step 4 For 119894 = 1 2 119898 set 120595119894(119909) = sum119894119896=1 120573119894119896120595119896(119909)Step 5 Set 1199060(1199091) = 119906(1199091)Step 6 Set 119899 = 1Step 7 Set 119861119899 = sum119899119896=1 120573119899119896119906119896minus1(119909119896)Step 8 Set 119906119899(119909) = sum119899119894=1 119861119894120595119894(119909)Step 9 If 119899 lt 119898 then set 119899 = 119899 + 1 and go to step 7Else stop
4 Numerical Experiments
In this part four numerical examples are solved for potencyand utility of the present method All computations areperformed by MAPLE package Results which are takenby this method show a proper agreement with the exactsolution A comprehensive applicability of this method isgiven the stability and consistence of the presented methodThe reliability of the method and increasing of the accuracycause this method to be more applicable
Example 1 For first applicable instance we offer nonlinearFredholm integral equations [26 30]
119906 (119909) minus int101199091199101199063 (119910) 119889119910 = 119890119909 minus (1 + 21198903) 119909
9 0 le 119909 le 1
(38)
Mathematical Problems in Engineering 7
1
Error of approximate solution
0 01 02 03 04 05 06 07 08 09
times10minus4Approximate solution and the exact solution
ExactApproximate
0 02 03 04 05 06 07 08 0901 1x
1
12
14
16
18
2
22
24
26
28
minus2minus15
minus1minus05
005
115
2
Abso
lute
erro
r
Figure 1 The absolute errors comparison between the proposed approach and method [30]
Error of approximate solution
10 01 02 03 04 05 06 07 08 09
times10minus3Approximate solution and the exact solution
10 01 02 03 04 05 06 07 08 09
ExactApproximate
0
01
02
03
04
05
06
07
08
09
minus25minus2
minus15minus1
minus050
051
152
Abso
lute
erro
r
Figure 2 The absolute errors comparison between the proposed approach and method [26]
The exact solution of this equation is 119906(119909) = 119890119909 According to(38) we can assume an initial approximation 1199060(0) = 119906(0) =1 The numerical results are given in Table 1 by taking 119909119894 =(119894 minus 1)(119899 minus 1) 119894 = 1 2 119899 and 119899 = 128 In Table 1 acomparison between the absolute errors of our method andthe Haar wavelet method [30] is given Figure 1 shows theapproximate solution and its errors
Example 2 For second applicable example an electromag-netic problem is solved via the presented method It is simu-lated to nonlinear Volterra integral equations model [26 30]
119906 (119909) minus 12 int119909
01199062 (119910) 119889119910 = sin (119909) + 18 sin (2119909) minus 14119909
0 le 119909 le 1(39)
The exact solution of this equation is 119906(119909) = sin(119909)According to (39) we can assume an initial approximation1199060(0) = 119906(0) = 0 Numerical results are given in Table 2 bytaking119909119894 = (119894minus1)(119899minus1) 119894 = 1 2 119899 and 119899 = 128 In Table 2a comparison between the absolute errors of the proposedmethod and the BPFs method [26] is given Figure 2 shows
Table 3 The execution time (seconds) comparison between Algo-rithms 14 and 15
119899 Algorithm 14 (Sec) Algorithm 15 (Sec)4 347 2878 952 48216 5975 3177
the approximate solution and its errors In Table 3 a compar-ison execution time between Algorithms 14 and 15 is given
Example 3 An electromagnetic problem is solved via ourmethod for another applicable example It is simulated tononlinear Fredholm integral equations model [26 30]
119906 (119909) minus 12 int1
01199101199062 (119910) 119889119910 = 1199092 minus 112 0 le 119909 le 1 (40)
The exact solution of this equation is 119906(119909) = 1199092 Accordingto (40) the initial approximation 1199060(0) = 119906(0) = 0 is chosenNumerical results are given in Table 4 by taking 119909119894 = (119894 minus1)(119899minus1) 119894 = 1 2 119899 and 119899 = 128 In Table 4 a comparison
8 Mathematical Problems in Engineering
10
20
30
40
50
60
Error of approximate solutiontimes10minus5Approximate solution and the exact solution
0 01 02 03 04 05 06 07 08 09
ExactApproximate
0 02 03 04 05 06 07 08 0901 1x
0
01
02
03
04
05
06
07
08
minus8minus6minus4minus2
02468
Abso
lute
erro
r
Figure 3 The absolute errors comparison between the proposed approach and method [26]
Table 4 The absolute errors comparison between the proposed approach and method [26]
119909 Exact solution Approximate solution Method in [26] Absolute error Absolute error in method [26]01 0010000 0010762 0010308 0000762 000030802 0040000 0040655 0038140 0000655 000186003 0090000 0090444 0092828 0000444 000282804 0160000 016023 0158746 0000230 000125405 0250000 0250001 0257867 0000001 000786706 0360000 0359606 0361871 0000394 000187107 0490000 0489448 0483453 0000552 000654708 0640000 0639391 0647515 0000609 000751509 0810000 0811638 0807183 0001638 0002817
Table 5 The execution time (seconds) comparison between Algo-rithms 14 and 15
119899 Algorithm 14 (Sec) Algorithm 15 (Sec)4 31 2598 846 50916 342 2889
between the absolute errors of the proposed method and theBPFs method [26] is given Figure 3 shows the approximatesolution and its error In Table 5 a comparison execution timebetween Algorithms 14 and 15 is given
Example 4 A nonlinear Fredholm integral problem [26 30]is solved via our method for this applicable example
119906 (119909) + int10119890119909minus21199101199063 (119910) 119889119910 = 119890119909+1 0 le 119909 lt 1 (41)
The exact solution of this equation is 119906(119909) = 119890119909 According to(41) we consider an initial approximation as 1199060(0) = 119906(0) = 1Numerical results are given in Table 6 by taking 119909119894 = (119894 minus1)(119899minus1) 119894 = 1 2 119899 and 119899 = 128 In Table 6 a comparison
between the absolute errors of the proposed method andthe Haar wavelet method [30] is given Figure 4 shows theapproximate solution and its error
5 Conclusion
According to this essay supplementary of iterative Reproduc-ing Kernel Hilbert SpaceMethod was introduced and appliedto acquire the approximate solution of some nonlinear inte-gral equation In this method unlike other similar methodsorthogonal process is not used However the time is increas-ing and accuracy is also increasing The main point whichis mentioned in this paper is that Algorithm 14 has higherexecution time in comparison with Algorithm 15 but theapproximate solution in Algorithm 14 is more accurate thanAlgorithm 15 Current uniform convergencemethod is statedand proved The obtained numerical results confirm that itis a good candidate for solving the nonlinear integral equa-tion
Conflicts of Interest
The authors declare no conflicts of interest
Mathematical Problems in Engineering 9
Table 6 The absolute errors comparison between the proposed approach and method [30]
119909 Exact solution Approximate solution Method in [30] Absolute error Absolute error in method [30]01 1105170918 1105495796 1111099473 0000324878 000592855502 1221402757 1220931139 1220296615 0000471618 000110614203 1349858806 1349502887 1339817084 0000355919 001004172204 1491824696 1491740269 1471965091 0000084427 001985960505 1648721268 1648649846 1457127691 0000071422 019159357706 1822118797 1822243106 1831888927 0000124309 000977013007 2013752703 2013650232 2011936556 0000102471 000181614708 2225540923 2224684503 2209678381 0000856420 001586254209 2459603104 2458626573 2426854676 0000976531 0032748428
Error of approximate solution
10 01 02 03 04 05 06 07 08 09
times10minus5
Approximate solution and the exact solution
10 01 02 03 04 05 06 07 08 09
ExactApproximate
1
12
14
16
18
2
22
24
26
28
minus10
minus8
minus6
minus4
minus2
0
4
2
Abso
lute
erro
r
x
Figure 4 The absolute errors comparison between the proposed approach and method [30]
Acknowledgments
The authors would like to thank Professor Haipeng Peng forinstructive comments and recommendations to improve thequality of this work and also the Islamic Azad UniversityScience and Research Branch Tehran for supporting thisproject
References
[1] K Meetz and W Engi Electromagnetische Felder SpringerBerlin Germany 1979
[2] W Thirring Lehrbuch der Mathematischen Physik vol 2Springer Vienna Austria 1978
[3] K F Warnick R H Selfridge and D V Arnold ldquoTeachingelectromagnetic field theory using differential formsrdquo IEEETransactions on Education vol 40 no 1 pp 53ndash68 1997
[4] S Abbasbandy ldquoHomotopy analysis method for heat radiationequationsrdquo International Communications in Heat and MassTransfer vol 34 no 3 pp 380ndash387 2007
[5] H Vosughi E Shivanian and S Abbasbandy ldquoA new analyticaltechnique to solve Volterrarsquos integral equationsrdquo MathematicalMethods in the Applied Sciences vol 34 no 10 pp 1243ndash12532011
[6] J-P Kauthen ldquoA survey of singularly perturbed Volterra equa-tionsrdquo Applied Numerical Mathematics An IMACS Journal vol24 no 2-3 pp 95ndash114 1997
[7] A A Kilbas and M Saigo ldquoOn solution of nonlinear Abel-Volterra integral equationrdquo Journal of Mathematical Analysisand Applications vol 229 no 1 pp 41ndash60 1999
[8] C Minggen and D Zhongxing ldquoOn the best operator ofinterpolationrdquo Mathematica Numerica Sinica vol 8 no 2 pp209ndash216 1986
[9] C Minggen and L Yingzhen Nonlinear Numerical Analysis inthe Reproducing Kernel Space Nova Science New York NYUSA 2009
[10] S Abbasbandy B Azarnavid and M S Alhuthali ldquoA shootingreproducing kernel Hilbert spacemethod formultiple solutionsof nonlinear boundary value problemsrdquo Journal of Computa-tional and Applied Mathematics vol 279 pp 293ndash305 2015
[11] N Shawagfeh O Abu Arqub and S Momani ldquoAnalytical solu-tion of nonlinear second-order periodic boundary value prob-lem using reproducing kernel methodrdquo Journal of Computa-tional Analysis andApplications vol 16 no 4 pp 750ndash762 2014
[12] FGeng andMCui ldquoSolving a nonlinear systemof second orderboundary value problemsrdquo Journal of Mathematical Analysisand Applications vol 327 no 2 pp 1167ndash1181 2007
[13] B Azarnavid F Parvaneh and S Abbasbandy ldquoPicard-repro-ducing kernel Hilbert space method for solving generalizedsingular nonlinear Lane-Emden type equationsrdquoMathematicalModelling and Analysis vol 20 no 6 pp 754ndash767 2015
[14] 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
10 Mathematical Problems in Engineering
[15] L Yang and M Cui ldquoNew algorithm for a class of nonlinearintegro-differential equations in the reproducing kernel spacerdquoApplied Mathematics and Computation vol 174 no 2 pp 942ndash960 2006
[16] 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
[17] S Bushnaq B Maayah S Momani and A Alsaedi ldquoA repro-ducing kernel Hilbert space method for solving systems offractional integrodifferential equationsrdquo Abstract and AppliedAnalysis vol 2014 Article ID 103016 6 pages 2014
[18] M Inc A Akgul and F Geng ldquoReproducing kernel Hilbertspace method for solving Bratursquos problemrdquo Bulletin of theMalaysian Mathematical Sciences Society vol 38 no 1 pp 271ndash287 2015
[19] A-M Wazwaz A First Course in Integral Equations WorldScientific 1997
[20] S Javadi E Babolian and E Moradi ldquoNew implementation ofreproducing kernel Hilbert space method for solving a class offunctional integral equationsrdquo Communications in NumericalAnalysis vol 2014 Article ID cna-00205 7 pages 2014
[21] W Jiang and Z Chen ldquoSolving a system of linear Volterraintegral equations using the new reproducing kernel methodrdquoApplied Mathematics and Computation vol 219 no 20 pp10225ndash10230 2013
[22] A-M Wazwaz Linear and Nonlinear Integral Equations Meth-ods and Applications Springer Berlin Germany 2011
[23] I L El-Kalla ldquoConvergence of the Adomian method appliedto a class of nonlinear integral equationsrdquo Applied MathematicsLetters An International Journal of Rapid Publication vol 21 no4 pp 372ndash376 2008
[24] G Gumah KMoaddy M Al-Smadi and I Hashim ldquoSolutionsto uncertain Volterra integral equations by fitted reproducingkernel Hilbert space methodrdquo Journal of Function Spaces vol2016 Article ID 2920463 11 pages 2016
[25] V Sizikov and D Sidorov ldquoGeneralized quadrature for solv-ing singular integral equations of Abel type in applicationto infrared tomographyrdquo Applied Numerical Mathematics AnIMACS Journal vol 106 pp 69ndash78 2016
[26] E Babolian Z Masouri and S Hatamzadeh-Varmazyar ldquoNewdirect method to solve nonlinear Volterra-Fredholm integraland integro-differential equations using operational matrixwith block-pulse functionsrdquo in Progress in ElectromagneticsResearch B vol 8 pp 59ndash76 EMW Publishing CambridgeMass USA 2008
[27] D Alpay Ed Reproducing kernel spaces and applications vol143 of Operator Theory Advances and Applications BirkhauserBasel 2003
[28] S Saitoh D Alpay J A Ball and T Ohsawa ReproducingKernels andTheir Applications vol 11 Springer Science amp Busi-ness Media Berlin Germany 2013
[29] S Saitoh Integral Transforms Reproducing Kernels and TheirApplications vol 369 of Pitman Research Notes in MathematicsSeries Longman Harlow UK 1997
[30] E Babolian andA Shahsavaran ldquoNumerical solution of nonlin-ear Fredholm integral equations of the second kind using Haarwaveletsrdquo Journal of Computational and Applied Mathematicsvol 225 no 1 pp 87ndash95 2009
[31] R Ketabchi R Mokhtari and E Babolian ldquoSome error esti-mates for solving Volterra integral equations by using the
reproducing kernel methodrdquo Journal of Computational andApplied Mathematics vol 273 pp 245ndash250 2015
[32] A Alvandi M Paripour and Z Roshani ldquoReproducing kernelmethod for solving a class of Fredholm integro-differentialequationsrdquo inProceedings of the 46thAnnual IranianMathemat-ics Conference (AIMC 46) p 505 Yazd University 2015
[33] I Komashynska andMAl-Smadi ldquoIterative reproducing kernelmethod for solving second-order integrodifferential equationsof fredholm typerdquo Journal of Applied Mathematics vol 2014Article ID 459509 11 pages 2014
[34] E Babolian S Javadi and E Moradi ldquoError analysis of repro-ducing kernel Hilbert space method for solving functional inte-gral equationsrdquo Journal of Computational and Applied Mathe-matics vol 300 pp 300ndash311 2016
Submit your manuscripts athttpswwwhindawicom
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
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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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
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
with the boundary conditions
120597119894119877 minus 119910 (119886)120597119905119894 minus (minus1)119898minus119894minus1 1205972119898minus119894minus1119877119910 (119886)1205971199052119898minus119894minus1 = 01205972119898minus119894minus1119877119910 (119887)1205971199052119898minus119894minus1 = 0
119894 = 0 1 119898 minus 1(8)
Equation (7) is characteristic 1205822119898 = 0 Then the general solu-tion of Equation (7) is
119877119909 (119905) =
2119898sum119894=1
119888119894 (119909) 119905119894minus1 119905 le 1199092119898sum119894=1
119889119894 (119909) 119905119894minus1 119905 gt 119909 (9)
where coefficients 119888119894(119909) and 119889119894(119909) 119894 = 1 2 2119898 could becalculated by solving the following linear equations
120597119894119877119909 (119909 + 0)120597119905119894 = 120597119894119877119909 (119909 minus 0)120597119905119894 119894 = 0 1 2119898 minus 2(1205972119898minus1119877119909 (119909 + 0)1205971199052119898minus1 minus 1205972119898minus1119877119909 (119909 minus 0)1205971199052119898minus1 ) = (minus1)119898 120597119894119877119910 (119886)120597119905119894 minus (minus1)119898minus119894minus1 1205972119898minus119894minus1119877119910 (119886)1205971199052119898minus119894minus1 = 01205972119898minus119894minus1119877119910 (119887)1205971199052119898minus119894minus1 = 0 119894 = 0 1 119898 minus 1
(10)
Subsequently the representation of the Reproducing Kernelof11988212 [119886 119887] is provided by
119877119909 (119905) = 1 minus 119886 + 119905 119905 le 1199091 minus 119886 + 119909 119905 gt 119909 (11)
3 Main Idea and Theoretical Discussion
The uniqueness conditions for nonlinear problems exist in[21ndash23] The unique solution of (1) is assumed in this paperThe solution of (1) is given in11988212 [119886 119887] space We consider (1)as
L119906 (119909) = 119906 (119909) (12)
where L119906(119909) = 119891(119909) minus int119887 (or119909)119886
119870(119909 119905)119873(119906(119905))119889119905 It is obviousthatL is the bounded linear operator of11988212 to11988212 Put120593119894(119909) =119877119909119894(119909) and 120595119894(119909) = Llowast120593119894(119909) where Llowast is the adjoint operatorof L In fact for 119906 V isin 11988212 [119886 119887] we have L(119906(119909) + V(119909)) =119906(119909) + V(119909) and L119906(119909)1198821
2[119886119887] = 119906(119909)1198821
2[119886119887]
The orthonormal system of 120595119894(119909)infin119894=1 from the space11988212 [119886 119887] can be derived from Gram-Schmidt orthogonalprocess of 120595119894(119909)infin119894=1
120595119894 (119909) = 119894sum119896=1
120573119894119896120595119896 (119909) 120573119894119894 gt 0 119894 = 1 2 (13)
Definition 7 In a topological space (X 120591) a subset 119860 ofX iscalled dense inX if 119860 = cl119860 = X
Theorem 8 If 119909119894infin119894=1 is dense on [119886 119887] then 120595119894(119909)infin119894=1 is thecomplete function system of the space 11988212 [119886 119887] and 120595119894(119909) =119871 119905119877119909(119905)|119905=119909119894 where the subscript t in the operator L indicatesthat the operator L applies to the function of 119905Proof We have
120595119894 (119909) = ⟨(Llowast120593119894) (119905) 119877119909 (119905)⟩11988212[119886119887]
= ⟨120593119894 (119905) 119871 119905119877119909 (119905)⟩11988212[119886119887] = 119871 119905119877119909 (119905)1003816100381610038161003816119905=119909119894
(14)
Clearly 120595119894(119909) isin 11988212 [119886 119887] For each fixed 119906(119909) isin 11988212 [119886 119887] let⟨119906(119909) 120595119894(119909)⟩11988212[119886119887] = 0 119894 = 1 2 which means
⟨119906 (119909) (Llowast120593119894) (119909)⟩11988212[119886119887] = ⟨L119906 (sdot) 120593119894 (sdot)⟩1198821
2[119886119887]
= (L119906) (119909119894) = 0 (15)
Assume that 119909119894infin119894=1 is dense on [119886 119887] and so (L119906)(119909) = 0 Itfollows that119906 equiv 0 from the existence ofLminus1 Now the theoremis proved
Theorem 9 If 119909119894infin119894=1 is dense on [119886 119887] and the solution of (12)is unique then the solution of (12) is
119906 (119909) = infinsum119894=1
119894sum119896=1
120573119894119896119906 (119909119896) 120595119894 (119909) (16)
Proof Using (13) we have
119906 (119909) = infinsum119894=1
⟨119906 (119909) 120595119894 (119909)⟩11988212[119886119887]
120595119894 (119909)
= infinsum119894=1
⟨119906 (119909) 119894sum119896=1
120573119894119896120595119896 (119909)⟩11988212[119886119887]
120595119894 (119909)
= infinsum119894=1
119894sum119896=1
120573119894119896 ⟨119906 (119909) 120595119896 (119909)⟩11988212[119886119887] 120595119894 (119909)
= infinsum119894=1
119894sum119896=1
120573119894119896 ⟨119906 (119909) Llowast120593119896 (119909)⟩11988212[119886119887] 120595119894 (119909)
= infinsum119894=1
119894sum119896=1
120573119894119896 ⟨L119906 (119909) 120593119896 (119909)⟩11988212[119886119887] 120595119894 (119909)
= infinsum119894=1
119894sum119896=1
120573119894119896L119906 (119909119896) 120595119894 (119909)
= infinsum119894=1
119894sum119896=1
120573119894119896119906 (119909119896) 120595119894 (119909)
(17)
On the other hand 119906(119909) isin 11988212 [119886 119887] and 119906(119909) = suminfin119894=0 119886119894120595119894(119909)119886119894 = ⟨119906(119909) 120595119894(119909)⟩ are the Fourier series expansion about
4 Mathematical Problems in Engineering
normal orthogonal system 120595119894(119909)infin119894=1 and 11988212 [119886 119887] is theHilbert Space Thus the series suminfin119894=0 119886119894120595119894(119909) is convergent inthe sense of sdot 1198821
2
and the proof would be complete
Now the approximate solution 119906119899(119909) can be obtained bythe 119899-term intercept of the exact solution 119906(119909) and
119906119899 (119909) = 119899sum119894=1
119894sum119896=1
120573119894119896119906 (119909119896) 120595119894 (119909) (18)
Theorem 10 If 119906(119909) isin 11988212 [119886 119887] then119872 gt 0 exists such that|119906(119909)| le 119872119906(119909)11988212[119886119887]
Proof We have 119906(119909) = ⟨119906(119905) 119877119909(119905)⟩11988212[119886119887] for any 119909 119905isin [119886 119887] We know 119877119909(119905)1198821
2[119886119887] le 119872 Thus |119906(119909)| = |⟨119906(119905)119877119909(119905)⟩1198821
2[119886119887]| le 119877119909(119905)1198821
2[119886119887]119906(119905)1198821
2[119886119887] le 119872119906(119905)1198821
2[119886119887]
Theorem 11 The approximate solution 119906119899(119909) is uniformlyconvergent
Proof Assuming 119909 isin [119886 119887] by Theorems 9 and 10 it can beproved that
lim119899rarrinfin
1003816100381610038161003816119906119899 (119909) minus 119906 (119909)1003816100381610038161003816= lim119899rarrinfin
100381610038161003816100381610038161003816⟨119906119899 (119909) minus 119906 (119909) 119877119909 (119909)⟩11988221 [119886119887]100381610038161003816100381610038161003816le 119872 lim119899rarrinfin
1003817100381710038171003817119906119899 (119909) minus 119906 (119909)100381710038171003817100381711988221[119886119887] = 0
(19)
In the sequel a new iterative method to achieve thesolution of (12) is presented If
119860 119894 = 119894sum119896=1
120573119894119896119906 (119909119896) (20)
then (16) can be written as
119906 (119909) = infinsum119894=1
119860 119894120595119894 (119909) (21)
Now suppose for some 119909119895 119906(119909119895) is known There is noproblem ifwe assume 119895 = 1Weput1199060(1199091) = 119906(1199091) anddefinethe 119899-term approximation to 119906(119909) by
119906119899 (119909) = 119899sum119894=1
119861119894120595119894 (119909) (22)
where
119861119894 = 119894sum119896=1
120573119894119896119906119896minus1 (119909119896) (23)
In the following it would be proven that the approximatesolution 119906119899(119909) in the iterative (22) is convergent to the exactsolution of (12) uniformly
Theorem 12 Suppose that 11990611989911988221[119886119887] is bounded in (22) If119909119894infin119894=1 is dense on [119886 119887] then 119899-term approximate solution119906119899(119909) in the iterative (22) converges to the exact solution 119906(119909)
of (12) and 119906(119909) = lim119899rarrinfinsum119899119894=1 119861119873119894 120595119894(119909) whereas 119861119894 is givenby (23)
Proof First of all the convergence of 119906119899(119909) from (22) wouldbe proven We infer
119906119899+1 (119909) = 119906119899 (119909) + 119861119899+1120595119899+1 (119909) (24)
Subsequence 120595119894(119909)infin119894=1 is orthogonal and it yields that
1003817100381710038171003817119906119899+11003817100381710038171003817211988221[119886119887] = 1003817100381710038171003817119906119899100381710038171003817100381721198822
1[119886119887] + 1198612119899+1 =
119899sum119894=1
1198612119894 (25)
It is obvious that the sequence 11990611989911988221[119886119887] is monotonically
increasing Because 11990611989911988221[119886119887] is bounded and 1199061198991198822
1[119886119887] is
convergent then suminfin119894=1 1198612119894 is bounded and this implies that119861119894infin119894=1 isin 1198972If119898 gt 119899 then
1003817100381710038171003817119906119898 minus 1199061198991003817100381710038171003817211988221[119886119887] =
1003817100381710038171003817100381710038171003817100381710038171003817119899+1sum119894=119898
(119906119894 minus 119906119894minus1)10038171003817100381710038171003817100381710038171003817100381710038172
11988221[119886119887]
= 119899+1sum119894=119898
1003817100381710038171003817119906119894 minus 119906119894minus11003817100381710038171003817211988221[119886119887]
(26)
So 119906119894 minus 119906119894minus1211988221[119886119887] = 1198612119894 Consequently 119906119898 minus 11990611989921198822
1[119886119887] =sum119899+1119894=119898 1198612119894 rarr 0 as 119899 rarr infin To prove the completeness of11988212 [119886 119887] it requires where isin 11988212 [119886 119887] that 119906119899 rarr as119899 rarr infin Now we can prove is the solution of (12)
If we take limit from (22) we will have (119909) =sum119899119894=1 119861119894120595119894(119909) so L(119909) = suminfin119894=1 119861119894L120595119894(119909) Let 119909119897 isin 119909119894infin119894=1 andthen
L (119909119897) = infinsum119894=1
119861119894L120595119894 (119909119897)
= infinsum119894=1
119861119894 ⟨L120595119894 (119909) 120593119897 (119909)⟩11988212[119886119887]
= infinsum119894=1
119861119894 ⟨120595119894 (119909) Llowast120593119897 (119909)⟩11988212[119886119887]
= infinsum119894=1
119861119894 ⟨120595119894 (119909) 120595119897 (119909)⟩11988212[119886119887]
= infinsum119894=1
119861119894⟨120595119894 (119909) 119894sum119897=1
119861119894119897120595119897 (119909)⟩11988212[119886119887]
(27)
From (23) and (29) it is concluded that L(119909119897) = (119909119897)119909119894infin119894=1 is dense on [119886 119887] For each 119909 isin [119886 119887] 119909119899119894infin119894=1subsequence exists that 119909119899119894 rarr 119909 as 119894 rarr infin Hence when119894 rarr infin we have L119906(119909) = 119906(119909) which indicates that is thesolution of (12)
Mathematical Problems in Engineering 5
The mentioned scheme above is an efficient method ofsolving nonlinear equations [31ndash33] However in implement-ing this algorithm on a computer 120595119894(119909)infin119894=1 is not quiteorthogonal due to rounding errors In other words Gram-Schmidt process is numerically unstable and the computa-tional cost of the algorithm is high Therefore the followingprocess is suggested similar for linear problems in [20 34]This is the subject of the next theorem where the followingnotations are used
120573119899 =[[[[[[[
12057311 0 sdot sdot sdot 012057321 12057322 d 0 d
1205731198991 1205731198992 sdot sdot sdot 120573119899119899
]]]]]]]
120595119899 =[[[[[[[
12059511 12059512 sdot sdot sdot 120595111989912059521 12059522 d 1205952119899 d
1205951198991 1205951198992 sdot sdot sdot 120595119899119899
]]]]]]]
u =[[[[[[[
1199060 (1199091)1199061 (1199092)119906119899minus1 (119909119899)
]]]]]]]
B =[[[[[[[
11986111198612119861119899
]]]]]]]
Λ =[[[[[[[
Λ 1Λ 2Λ 119899
]]]]]]]
119899 = 1 2
(28)
Theorem 13 (let 120574119894119895 = [120595minus1]119894119895) The approximate solutionobtained from (22) is found as follows
119906119899 (119909) = 119899sum119894=1
Λ 119894120595119894 (119909) (29)
where
Λ 119894 = 119894sum119896=1
120574119894119896119906119896minus1 (119909119896) (30)
Proof Suppose that 119906119899(119909) = sum119899119894=1 Λ 119894120595119894(119909) = sum119899119894=1 119861119894120595119894(119909)Since 120595119894(119909) = sum119894119896=1 120573119894119896120595119896(119909)
119899sum119894=1
Λ 119894120595119894 (119909) = 119899sum119894=1
119861119894120595119894 (119909) = 119899sum119894=1
119861119894 119894sum119896=1
120573119894119896120595119896 (119909)
= 119899sum119896=1
119899sum119894=119896
119861119894120573119894119896120595119896 (119909) (31)
120595119894(119909)infin119894=1 and Λ 119896 = sum119899119894=119896 119861119894120573119894119896 (119896 = 1 2 119899) are linearindependence and therefore
Λ = 120573TB (32)
Equation (12) implies L119906119899(119909) = 119906119899(119909) For 119894 = 1 2 119899 wehave
⟨L119906119899 (119909) 120595119894⟩ = ⟨119906119899 (119909) 120595119894⟩ 997904rArr119899sum119895=1
119861119895 ⟨L120595119895 120595119894⟩ = ⟨119906119899 (119909) 120595119894⟩ (33)
Both sides of (33) provide
119899sum119895=1
119861119895 ⟨L120595119894 120595119894⟩ = 119899sum119895=1
119861119895 119894sum119896=1
120573119894119896119895sum119897=1
120573119895119897 ⟨L120595119897 120595119896⟩
= 119899sum119895=1
119861119895 119894sum119896=1
119895sum119897=1
120573119894119896 ⟨L120595119897 120595119896⟩ 120573119879119897119895= 119899sum119895=1
119861119895 (120573120595120573T)119894119895
(34)
⟨119906119899 (119909) 120595119894⟩ = 119861119894 = 119894sum119896=1
120573119894119896119906119896minus1 (119909119896) (using Theorem 12)
(35)
From (32) and (35) the following equation can be reached
120573120595120573TB = 120573u (36)
Equation (32) implies 120573120595Λ = 120573u119899 So120595Λ = u119899 (37)
which proves the theorem
Algorithm 14 The following steps exist for approximatingthe solution without applying Gram-Schmidt orthogonalprocess
Step 1 Fix 119886 le 119909 and 119905 le 119887If 119905 le 119909 set 119877119909(119905) = 1 minus 119886 + 119905Else set 119877119909(119905) = 1 minus 119886 + 119909Step 2 For 119894 = 1 2 119898 set 119909119894 = (119894 minus 1)(119898 minus 1)
6 Mathematical Problems in Engineering
Table 1 The absolute errors comparison between the proposed approach and method [30]
119909 Exact solution Approximate solution Method in [30] Absolute error Absolute error in method [30]01 1105170918 1105154993 1114627560 0000015925 000945664202 1221402757 1221389903 1226758840 0000012855 000535608303 1349858806 1349845490 1345458179 0000013317 000440062704 1491824696 1491803913 1480202470 0000020784 001162222605 1648721268 1648680858 1671769819 0000040412 002304855106 1822118797 1822039541 1838854903 0000079259 001673610607 2013752703 2013608464 2022118086 0000144243 000836538308 2225540923 2225304288 2223139077 0000236640 000240184609 2459603104 2459253211 2443684898 0000349899 0015918206
Table 2 The absolute errors comparison between the proposed approach and method [26]
119909 Exact solution Approximate solution Method in [26] Absolute error Absolute error in method [26]01 0099833 0098524 0101388 0001308 00015502 0198669 0199766 0194073 0001097 000459603 0295520 0294857 0299995 0000662 000447504 0389418 0389658 0387978 0000240 000144005 0479426 0479585 0486266 0000160 000684006 0564642 0563993 0565930 0000648 000128807 0644218 0645477 0640624 0001260 000359408 0717356 0715373 0720611 0001982 000325509 0783327 0785653 0782351 0002326 0000976
Set 120595119894(119909) = 119871 119905119877119909(119905)|119905=119909119894 Step 3 Set 1199060(1199091) = 119906(1199091)Step 4 For 119894 = 1 2 119898 set 120574119894119895 = [120595minus1]119894119895Step 5 119899 = 1Step 6 Set Λ 119899 = sum119899119896=1 120574119899119896119906119896minus1(119909119896)Step 7 Set 119906119899(119909) = sum119899119894=1 Λ 119894120595119894(119909)Step 8 If 119899 lt 119898 then set 119899 = 119899 + 1 and go to step 6Else stop
Algorithm 15 The following steps exist for approximating thesolution by applying Gram-Schmidt orthogonal process
Step 1 Fix 119886 le 119909 and 119905 le 119887If 119905 le 119909 set 119877119909(119905) = 1 minus 119886 + 119905Else set 119877119909(119905) = 1 minus 119886 + 119909Step 2 For 119894 = 1 2 119898 set 119909119894 = (119894 minus 1)(119898 minus 1)Set 120595119894(119909) = 119871 119905119877119909(119905)|119905=119909119894 Step 3 For 119894 = 1 2 119898 and 119895 =1 2 119898 if 119894 = 119895 then set 120573119894119895 =(minus1radic1205951198942 minus sum119894minus1119896=1⟨120595119894(119905) 120595119896(119905)⟩2) sum119894minus1119896=1⟨120595119894(119905) and120595119896(119905)⟩120573119896119895Else 120573119894119895 = 1radic1205951198942 minus sum119894minus1119896=1⟨120595119894(119905) 120595119896(119905)⟩2Else 12057311 = 11205951
Step 4 For 119894 = 1 2 119898 set 120595119894(119909) = sum119894119896=1 120573119894119896120595119896(119909)Step 5 Set 1199060(1199091) = 119906(1199091)Step 6 Set 119899 = 1Step 7 Set 119861119899 = sum119899119896=1 120573119899119896119906119896minus1(119909119896)Step 8 Set 119906119899(119909) = sum119899119894=1 119861119894120595119894(119909)Step 9 If 119899 lt 119898 then set 119899 = 119899 + 1 and go to step 7Else stop
4 Numerical Experiments
In this part four numerical examples are solved for potencyand utility of the present method All computations areperformed by MAPLE package Results which are takenby this method show a proper agreement with the exactsolution A comprehensive applicability of this method isgiven the stability and consistence of the presented methodThe reliability of the method and increasing of the accuracycause this method to be more applicable
Example 1 For first applicable instance we offer nonlinearFredholm integral equations [26 30]
119906 (119909) minus int101199091199101199063 (119910) 119889119910 = 119890119909 minus (1 + 21198903) 119909
9 0 le 119909 le 1
(38)
Mathematical Problems in Engineering 7
1
Error of approximate solution
0 01 02 03 04 05 06 07 08 09
times10minus4Approximate solution and the exact solution
ExactApproximate
0 02 03 04 05 06 07 08 0901 1x
1
12
14
16
18
2
22
24
26
28
minus2minus15
minus1minus05
005
115
2
Abso
lute
erro
r
Figure 1 The absolute errors comparison between the proposed approach and method [30]
Error of approximate solution
10 01 02 03 04 05 06 07 08 09
times10minus3Approximate solution and the exact solution
10 01 02 03 04 05 06 07 08 09
ExactApproximate
0
01
02
03
04
05
06
07
08
09
minus25minus2
minus15minus1
minus050
051
152
Abso
lute
erro
r
Figure 2 The absolute errors comparison between the proposed approach and method [26]
The exact solution of this equation is 119906(119909) = 119890119909 According to(38) we can assume an initial approximation 1199060(0) = 119906(0) =1 The numerical results are given in Table 1 by taking 119909119894 =(119894 minus 1)(119899 minus 1) 119894 = 1 2 119899 and 119899 = 128 In Table 1 acomparison between the absolute errors of our method andthe Haar wavelet method [30] is given Figure 1 shows theapproximate solution and its errors
Example 2 For second applicable example an electromag-netic problem is solved via the presented method It is simu-lated to nonlinear Volterra integral equations model [26 30]
119906 (119909) minus 12 int119909
01199062 (119910) 119889119910 = sin (119909) + 18 sin (2119909) minus 14119909
0 le 119909 le 1(39)
The exact solution of this equation is 119906(119909) = sin(119909)According to (39) we can assume an initial approximation1199060(0) = 119906(0) = 0 Numerical results are given in Table 2 bytaking119909119894 = (119894minus1)(119899minus1) 119894 = 1 2 119899 and 119899 = 128 In Table 2a comparison between the absolute errors of the proposedmethod and the BPFs method [26] is given Figure 2 shows
Table 3 The execution time (seconds) comparison between Algo-rithms 14 and 15
119899 Algorithm 14 (Sec) Algorithm 15 (Sec)4 347 2878 952 48216 5975 3177
the approximate solution and its errors In Table 3 a compar-ison execution time between Algorithms 14 and 15 is given
Example 3 An electromagnetic problem is solved via ourmethod for another applicable example It is simulated tononlinear Fredholm integral equations model [26 30]
119906 (119909) minus 12 int1
01199101199062 (119910) 119889119910 = 1199092 minus 112 0 le 119909 le 1 (40)
The exact solution of this equation is 119906(119909) = 1199092 Accordingto (40) the initial approximation 1199060(0) = 119906(0) = 0 is chosenNumerical results are given in Table 4 by taking 119909119894 = (119894 minus1)(119899minus1) 119894 = 1 2 119899 and 119899 = 128 In Table 4 a comparison
8 Mathematical Problems in Engineering
10
20
30
40
50
60
Error of approximate solutiontimes10minus5Approximate solution and the exact solution
0 01 02 03 04 05 06 07 08 09
ExactApproximate
0 02 03 04 05 06 07 08 0901 1x
0
01
02
03
04
05
06
07
08
minus8minus6minus4minus2
02468
Abso
lute
erro
r
Figure 3 The absolute errors comparison between the proposed approach and method [26]
Table 4 The absolute errors comparison between the proposed approach and method [26]
119909 Exact solution Approximate solution Method in [26] Absolute error Absolute error in method [26]01 0010000 0010762 0010308 0000762 000030802 0040000 0040655 0038140 0000655 000186003 0090000 0090444 0092828 0000444 000282804 0160000 016023 0158746 0000230 000125405 0250000 0250001 0257867 0000001 000786706 0360000 0359606 0361871 0000394 000187107 0490000 0489448 0483453 0000552 000654708 0640000 0639391 0647515 0000609 000751509 0810000 0811638 0807183 0001638 0002817
Table 5 The execution time (seconds) comparison between Algo-rithms 14 and 15
119899 Algorithm 14 (Sec) Algorithm 15 (Sec)4 31 2598 846 50916 342 2889
between the absolute errors of the proposed method and theBPFs method [26] is given Figure 3 shows the approximatesolution and its error In Table 5 a comparison execution timebetween Algorithms 14 and 15 is given
Example 4 A nonlinear Fredholm integral problem [26 30]is solved via our method for this applicable example
119906 (119909) + int10119890119909minus21199101199063 (119910) 119889119910 = 119890119909+1 0 le 119909 lt 1 (41)
The exact solution of this equation is 119906(119909) = 119890119909 According to(41) we consider an initial approximation as 1199060(0) = 119906(0) = 1Numerical results are given in Table 6 by taking 119909119894 = (119894 minus1)(119899minus1) 119894 = 1 2 119899 and 119899 = 128 In Table 6 a comparison
between the absolute errors of the proposed method andthe Haar wavelet method [30] is given Figure 4 shows theapproximate solution and its error
5 Conclusion
According to this essay supplementary of iterative Reproduc-ing Kernel Hilbert SpaceMethod was introduced and appliedto acquire the approximate solution of some nonlinear inte-gral equation In this method unlike other similar methodsorthogonal process is not used However the time is increas-ing and accuracy is also increasing The main point whichis mentioned in this paper is that Algorithm 14 has higherexecution time in comparison with Algorithm 15 but theapproximate solution in Algorithm 14 is more accurate thanAlgorithm 15 Current uniform convergencemethod is statedand proved The obtained numerical results confirm that itis a good candidate for solving the nonlinear integral equa-tion
Conflicts of Interest
The authors declare no conflicts of interest
Mathematical Problems in Engineering 9
Table 6 The absolute errors comparison between the proposed approach and method [30]
119909 Exact solution Approximate solution Method in [30] Absolute error Absolute error in method [30]01 1105170918 1105495796 1111099473 0000324878 000592855502 1221402757 1220931139 1220296615 0000471618 000110614203 1349858806 1349502887 1339817084 0000355919 001004172204 1491824696 1491740269 1471965091 0000084427 001985960505 1648721268 1648649846 1457127691 0000071422 019159357706 1822118797 1822243106 1831888927 0000124309 000977013007 2013752703 2013650232 2011936556 0000102471 000181614708 2225540923 2224684503 2209678381 0000856420 001586254209 2459603104 2458626573 2426854676 0000976531 0032748428
Error of approximate solution
10 01 02 03 04 05 06 07 08 09
times10minus5
Approximate solution and the exact solution
10 01 02 03 04 05 06 07 08 09
ExactApproximate
1
12
14
16
18
2
22
24
26
28
minus10
minus8
minus6
minus4
minus2
0
4
2
Abso
lute
erro
r
x
Figure 4 The absolute errors comparison between the proposed approach and method [30]
Acknowledgments
The authors would like to thank Professor Haipeng Peng forinstructive comments and recommendations to improve thequality of this work and also the Islamic Azad UniversityScience and Research Branch Tehran for supporting thisproject
References
[1] K Meetz and W Engi Electromagnetische Felder SpringerBerlin Germany 1979
[2] W Thirring Lehrbuch der Mathematischen Physik vol 2Springer Vienna Austria 1978
[3] K F Warnick R H Selfridge and D V Arnold ldquoTeachingelectromagnetic field theory using differential formsrdquo IEEETransactions on Education vol 40 no 1 pp 53ndash68 1997
[4] S Abbasbandy ldquoHomotopy analysis method for heat radiationequationsrdquo International Communications in Heat and MassTransfer vol 34 no 3 pp 380ndash387 2007
[5] H Vosughi E Shivanian and S Abbasbandy ldquoA new analyticaltechnique to solve Volterrarsquos integral equationsrdquo MathematicalMethods in the Applied Sciences vol 34 no 10 pp 1243ndash12532011
[6] J-P Kauthen ldquoA survey of singularly perturbed Volterra equa-tionsrdquo Applied Numerical Mathematics An IMACS Journal vol24 no 2-3 pp 95ndash114 1997
[7] A A Kilbas and M Saigo ldquoOn solution of nonlinear Abel-Volterra integral equationrdquo Journal of Mathematical Analysisand Applications vol 229 no 1 pp 41ndash60 1999
[8] C Minggen and D Zhongxing ldquoOn the best operator ofinterpolationrdquo Mathematica Numerica Sinica vol 8 no 2 pp209ndash216 1986
[9] C Minggen and L Yingzhen Nonlinear Numerical Analysis inthe Reproducing Kernel Space Nova Science New York NYUSA 2009
[10] S Abbasbandy B Azarnavid and M S Alhuthali ldquoA shootingreproducing kernel Hilbert spacemethod formultiple solutionsof nonlinear boundary value problemsrdquo Journal of Computa-tional and Applied Mathematics vol 279 pp 293ndash305 2015
[11] N Shawagfeh O Abu Arqub and S Momani ldquoAnalytical solu-tion of nonlinear second-order periodic boundary value prob-lem using reproducing kernel methodrdquo Journal of Computa-tional Analysis andApplications vol 16 no 4 pp 750ndash762 2014
[12] FGeng andMCui ldquoSolving a nonlinear systemof second orderboundary value problemsrdquo Journal of Mathematical Analysisand Applications vol 327 no 2 pp 1167ndash1181 2007
[13] B Azarnavid F Parvaneh and S Abbasbandy ldquoPicard-repro-ducing kernel Hilbert space method for solving generalizedsingular nonlinear Lane-Emden type equationsrdquoMathematicalModelling and Analysis vol 20 no 6 pp 754ndash767 2015
[14] 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
10 Mathematical Problems in Engineering
[15] L Yang and M Cui ldquoNew algorithm for a class of nonlinearintegro-differential equations in the reproducing kernel spacerdquoApplied Mathematics and Computation vol 174 no 2 pp 942ndash960 2006
[16] 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
[17] S Bushnaq B Maayah S Momani and A Alsaedi ldquoA repro-ducing kernel Hilbert space method for solving systems offractional integrodifferential equationsrdquo Abstract and AppliedAnalysis vol 2014 Article ID 103016 6 pages 2014
[18] M Inc A Akgul and F Geng ldquoReproducing kernel Hilbertspace method for solving Bratursquos problemrdquo Bulletin of theMalaysian Mathematical Sciences Society vol 38 no 1 pp 271ndash287 2015
[19] A-M Wazwaz A First Course in Integral Equations WorldScientific 1997
[20] S Javadi E Babolian and E Moradi ldquoNew implementation ofreproducing kernel Hilbert space method for solving a class offunctional integral equationsrdquo Communications in NumericalAnalysis vol 2014 Article ID cna-00205 7 pages 2014
[21] W Jiang and Z Chen ldquoSolving a system of linear Volterraintegral equations using the new reproducing kernel methodrdquoApplied Mathematics and Computation vol 219 no 20 pp10225ndash10230 2013
[22] A-M Wazwaz Linear and Nonlinear Integral Equations Meth-ods and Applications Springer Berlin Germany 2011
[23] I L El-Kalla ldquoConvergence of the Adomian method appliedto a class of nonlinear integral equationsrdquo Applied MathematicsLetters An International Journal of Rapid Publication vol 21 no4 pp 372ndash376 2008
[24] G Gumah KMoaddy M Al-Smadi and I Hashim ldquoSolutionsto uncertain Volterra integral equations by fitted reproducingkernel Hilbert space methodrdquo Journal of Function Spaces vol2016 Article ID 2920463 11 pages 2016
[25] V Sizikov and D Sidorov ldquoGeneralized quadrature for solv-ing singular integral equations of Abel type in applicationto infrared tomographyrdquo Applied Numerical Mathematics AnIMACS Journal vol 106 pp 69ndash78 2016
[26] E Babolian Z Masouri and S Hatamzadeh-Varmazyar ldquoNewdirect method to solve nonlinear Volterra-Fredholm integraland integro-differential equations using operational matrixwith block-pulse functionsrdquo in Progress in ElectromagneticsResearch B vol 8 pp 59ndash76 EMW Publishing CambridgeMass USA 2008
[27] D Alpay Ed Reproducing kernel spaces and applications vol143 of Operator Theory Advances and Applications BirkhauserBasel 2003
[28] S Saitoh D Alpay J A Ball and T Ohsawa ReproducingKernels andTheir Applications vol 11 Springer Science amp Busi-ness Media Berlin Germany 2013
[29] S Saitoh Integral Transforms Reproducing Kernels and TheirApplications vol 369 of Pitman Research Notes in MathematicsSeries Longman Harlow UK 1997
[30] E Babolian andA Shahsavaran ldquoNumerical solution of nonlin-ear Fredholm integral equations of the second kind using Haarwaveletsrdquo Journal of Computational and Applied Mathematicsvol 225 no 1 pp 87ndash95 2009
[31] R Ketabchi R Mokhtari and E Babolian ldquoSome error esti-mates for solving Volterra integral equations by using the
reproducing kernel methodrdquo Journal of Computational andApplied Mathematics vol 273 pp 245ndash250 2015
[32] A Alvandi M Paripour and Z Roshani ldquoReproducing kernelmethod for solving a class of Fredholm integro-differentialequationsrdquo inProceedings of the 46thAnnual IranianMathemat-ics Conference (AIMC 46) p 505 Yazd University 2015
[33] I Komashynska andMAl-Smadi ldquoIterative reproducing kernelmethod for solving second-order integrodifferential equationsof fredholm typerdquo Journal of Applied Mathematics vol 2014Article ID 459509 11 pages 2014
[34] E Babolian S Javadi and E Moradi ldquoError analysis of repro-ducing kernel Hilbert space method for solving functional inte-gral equationsrdquo Journal of Computational and Applied Mathe-matics vol 300 pp 300ndash311 2016
Submit your manuscripts athttpswwwhindawicom
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
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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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
4 Mathematical Problems in Engineering
normal orthogonal system 120595119894(119909)infin119894=1 and 11988212 [119886 119887] is theHilbert Space Thus the series suminfin119894=0 119886119894120595119894(119909) is convergent inthe sense of sdot 1198821
2
and the proof would be complete
Now the approximate solution 119906119899(119909) can be obtained bythe 119899-term intercept of the exact solution 119906(119909) and
119906119899 (119909) = 119899sum119894=1
119894sum119896=1
120573119894119896119906 (119909119896) 120595119894 (119909) (18)
Theorem 10 If 119906(119909) isin 11988212 [119886 119887] then119872 gt 0 exists such that|119906(119909)| le 119872119906(119909)11988212[119886119887]
Proof We have 119906(119909) = ⟨119906(119905) 119877119909(119905)⟩11988212[119886119887] for any 119909 119905isin [119886 119887] We know 119877119909(119905)1198821
2[119886119887] le 119872 Thus |119906(119909)| = |⟨119906(119905)119877119909(119905)⟩1198821
2[119886119887]| le 119877119909(119905)1198821
2[119886119887]119906(119905)1198821
2[119886119887] le 119872119906(119905)1198821
2[119886119887]
Theorem 11 The approximate solution 119906119899(119909) is uniformlyconvergent
Proof Assuming 119909 isin [119886 119887] by Theorems 9 and 10 it can beproved that
lim119899rarrinfin
1003816100381610038161003816119906119899 (119909) minus 119906 (119909)1003816100381610038161003816= lim119899rarrinfin
100381610038161003816100381610038161003816⟨119906119899 (119909) minus 119906 (119909) 119877119909 (119909)⟩11988221 [119886119887]100381610038161003816100381610038161003816le 119872 lim119899rarrinfin
1003817100381710038171003817119906119899 (119909) minus 119906 (119909)100381710038171003817100381711988221[119886119887] = 0
(19)
In the sequel a new iterative method to achieve thesolution of (12) is presented If
119860 119894 = 119894sum119896=1
120573119894119896119906 (119909119896) (20)
then (16) can be written as
119906 (119909) = infinsum119894=1
119860 119894120595119894 (119909) (21)
Now suppose for some 119909119895 119906(119909119895) is known There is noproblem ifwe assume 119895 = 1Weput1199060(1199091) = 119906(1199091) anddefinethe 119899-term approximation to 119906(119909) by
119906119899 (119909) = 119899sum119894=1
119861119894120595119894 (119909) (22)
where
119861119894 = 119894sum119896=1
120573119894119896119906119896minus1 (119909119896) (23)
In the following it would be proven that the approximatesolution 119906119899(119909) in the iterative (22) is convergent to the exactsolution of (12) uniformly
Theorem 12 Suppose that 11990611989911988221[119886119887] is bounded in (22) If119909119894infin119894=1 is dense on [119886 119887] then 119899-term approximate solution119906119899(119909) in the iterative (22) converges to the exact solution 119906(119909)
of (12) and 119906(119909) = lim119899rarrinfinsum119899119894=1 119861119873119894 120595119894(119909) whereas 119861119894 is givenby (23)
Proof First of all the convergence of 119906119899(119909) from (22) wouldbe proven We infer
119906119899+1 (119909) = 119906119899 (119909) + 119861119899+1120595119899+1 (119909) (24)
Subsequence 120595119894(119909)infin119894=1 is orthogonal and it yields that
1003817100381710038171003817119906119899+11003817100381710038171003817211988221[119886119887] = 1003817100381710038171003817119906119899100381710038171003817100381721198822
1[119886119887] + 1198612119899+1 =
119899sum119894=1
1198612119894 (25)
It is obvious that the sequence 11990611989911988221[119886119887] is monotonically
increasing Because 11990611989911988221[119886119887] is bounded and 1199061198991198822
1[119886119887] is
convergent then suminfin119894=1 1198612119894 is bounded and this implies that119861119894infin119894=1 isin 1198972If119898 gt 119899 then
1003817100381710038171003817119906119898 minus 1199061198991003817100381710038171003817211988221[119886119887] =
1003817100381710038171003817100381710038171003817100381710038171003817119899+1sum119894=119898
(119906119894 minus 119906119894minus1)10038171003817100381710038171003817100381710038171003817100381710038172
11988221[119886119887]
= 119899+1sum119894=119898
1003817100381710038171003817119906119894 minus 119906119894minus11003817100381710038171003817211988221[119886119887]
(26)
So 119906119894 minus 119906119894minus1211988221[119886119887] = 1198612119894 Consequently 119906119898 minus 11990611989921198822
1[119886119887] =sum119899+1119894=119898 1198612119894 rarr 0 as 119899 rarr infin To prove the completeness of11988212 [119886 119887] it requires where isin 11988212 [119886 119887] that 119906119899 rarr as119899 rarr infin Now we can prove is the solution of (12)
If we take limit from (22) we will have (119909) =sum119899119894=1 119861119894120595119894(119909) so L(119909) = suminfin119894=1 119861119894L120595119894(119909) Let 119909119897 isin 119909119894infin119894=1 andthen
L (119909119897) = infinsum119894=1
119861119894L120595119894 (119909119897)
= infinsum119894=1
119861119894 ⟨L120595119894 (119909) 120593119897 (119909)⟩11988212[119886119887]
= infinsum119894=1
119861119894 ⟨120595119894 (119909) Llowast120593119897 (119909)⟩11988212[119886119887]
= infinsum119894=1
119861119894 ⟨120595119894 (119909) 120595119897 (119909)⟩11988212[119886119887]
= infinsum119894=1
119861119894⟨120595119894 (119909) 119894sum119897=1
119861119894119897120595119897 (119909)⟩11988212[119886119887]
(27)
From (23) and (29) it is concluded that L(119909119897) = (119909119897)119909119894infin119894=1 is dense on [119886 119887] For each 119909 isin [119886 119887] 119909119899119894infin119894=1subsequence exists that 119909119899119894 rarr 119909 as 119894 rarr infin Hence when119894 rarr infin we have L119906(119909) = 119906(119909) which indicates that is thesolution of (12)
Mathematical Problems in Engineering 5
The mentioned scheme above is an efficient method ofsolving nonlinear equations [31ndash33] However in implement-ing this algorithm on a computer 120595119894(119909)infin119894=1 is not quiteorthogonal due to rounding errors In other words Gram-Schmidt process is numerically unstable and the computa-tional cost of the algorithm is high Therefore the followingprocess is suggested similar for linear problems in [20 34]This is the subject of the next theorem where the followingnotations are used
120573119899 =[[[[[[[
12057311 0 sdot sdot sdot 012057321 12057322 d 0 d
1205731198991 1205731198992 sdot sdot sdot 120573119899119899
]]]]]]]
120595119899 =[[[[[[[
12059511 12059512 sdot sdot sdot 120595111989912059521 12059522 d 1205952119899 d
1205951198991 1205951198992 sdot sdot sdot 120595119899119899
]]]]]]]
u =[[[[[[[
1199060 (1199091)1199061 (1199092)119906119899minus1 (119909119899)
]]]]]]]
B =[[[[[[[
11986111198612119861119899
]]]]]]]
Λ =[[[[[[[
Λ 1Λ 2Λ 119899
]]]]]]]
119899 = 1 2
(28)
Theorem 13 (let 120574119894119895 = [120595minus1]119894119895) The approximate solutionobtained from (22) is found as follows
119906119899 (119909) = 119899sum119894=1
Λ 119894120595119894 (119909) (29)
where
Λ 119894 = 119894sum119896=1
120574119894119896119906119896minus1 (119909119896) (30)
Proof Suppose that 119906119899(119909) = sum119899119894=1 Λ 119894120595119894(119909) = sum119899119894=1 119861119894120595119894(119909)Since 120595119894(119909) = sum119894119896=1 120573119894119896120595119896(119909)
119899sum119894=1
Λ 119894120595119894 (119909) = 119899sum119894=1
119861119894120595119894 (119909) = 119899sum119894=1
119861119894 119894sum119896=1
120573119894119896120595119896 (119909)
= 119899sum119896=1
119899sum119894=119896
119861119894120573119894119896120595119896 (119909) (31)
120595119894(119909)infin119894=1 and Λ 119896 = sum119899119894=119896 119861119894120573119894119896 (119896 = 1 2 119899) are linearindependence and therefore
Λ = 120573TB (32)
Equation (12) implies L119906119899(119909) = 119906119899(119909) For 119894 = 1 2 119899 wehave
⟨L119906119899 (119909) 120595119894⟩ = ⟨119906119899 (119909) 120595119894⟩ 997904rArr119899sum119895=1
119861119895 ⟨L120595119895 120595119894⟩ = ⟨119906119899 (119909) 120595119894⟩ (33)
Both sides of (33) provide
119899sum119895=1
119861119895 ⟨L120595119894 120595119894⟩ = 119899sum119895=1
119861119895 119894sum119896=1
120573119894119896119895sum119897=1
120573119895119897 ⟨L120595119897 120595119896⟩
= 119899sum119895=1
119861119895 119894sum119896=1
119895sum119897=1
120573119894119896 ⟨L120595119897 120595119896⟩ 120573119879119897119895= 119899sum119895=1
119861119895 (120573120595120573T)119894119895
(34)
⟨119906119899 (119909) 120595119894⟩ = 119861119894 = 119894sum119896=1
120573119894119896119906119896minus1 (119909119896) (using Theorem 12)
(35)
From (32) and (35) the following equation can be reached
120573120595120573TB = 120573u (36)
Equation (32) implies 120573120595Λ = 120573u119899 So120595Λ = u119899 (37)
which proves the theorem
Algorithm 14 The following steps exist for approximatingthe solution without applying Gram-Schmidt orthogonalprocess
Step 1 Fix 119886 le 119909 and 119905 le 119887If 119905 le 119909 set 119877119909(119905) = 1 minus 119886 + 119905Else set 119877119909(119905) = 1 minus 119886 + 119909Step 2 For 119894 = 1 2 119898 set 119909119894 = (119894 minus 1)(119898 minus 1)
6 Mathematical Problems in Engineering
Table 1 The absolute errors comparison between the proposed approach and method [30]
119909 Exact solution Approximate solution Method in [30] Absolute error Absolute error in method [30]01 1105170918 1105154993 1114627560 0000015925 000945664202 1221402757 1221389903 1226758840 0000012855 000535608303 1349858806 1349845490 1345458179 0000013317 000440062704 1491824696 1491803913 1480202470 0000020784 001162222605 1648721268 1648680858 1671769819 0000040412 002304855106 1822118797 1822039541 1838854903 0000079259 001673610607 2013752703 2013608464 2022118086 0000144243 000836538308 2225540923 2225304288 2223139077 0000236640 000240184609 2459603104 2459253211 2443684898 0000349899 0015918206
Table 2 The absolute errors comparison between the proposed approach and method [26]
119909 Exact solution Approximate solution Method in [26] Absolute error Absolute error in method [26]01 0099833 0098524 0101388 0001308 00015502 0198669 0199766 0194073 0001097 000459603 0295520 0294857 0299995 0000662 000447504 0389418 0389658 0387978 0000240 000144005 0479426 0479585 0486266 0000160 000684006 0564642 0563993 0565930 0000648 000128807 0644218 0645477 0640624 0001260 000359408 0717356 0715373 0720611 0001982 000325509 0783327 0785653 0782351 0002326 0000976
Set 120595119894(119909) = 119871 119905119877119909(119905)|119905=119909119894 Step 3 Set 1199060(1199091) = 119906(1199091)Step 4 For 119894 = 1 2 119898 set 120574119894119895 = [120595minus1]119894119895Step 5 119899 = 1Step 6 Set Λ 119899 = sum119899119896=1 120574119899119896119906119896minus1(119909119896)Step 7 Set 119906119899(119909) = sum119899119894=1 Λ 119894120595119894(119909)Step 8 If 119899 lt 119898 then set 119899 = 119899 + 1 and go to step 6Else stop
Algorithm 15 The following steps exist for approximating thesolution by applying Gram-Schmidt orthogonal process
Step 1 Fix 119886 le 119909 and 119905 le 119887If 119905 le 119909 set 119877119909(119905) = 1 minus 119886 + 119905Else set 119877119909(119905) = 1 minus 119886 + 119909Step 2 For 119894 = 1 2 119898 set 119909119894 = (119894 minus 1)(119898 minus 1)Set 120595119894(119909) = 119871 119905119877119909(119905)|119905=119909119894 Step 3 For 119894 = 1 2 119898 and 119895 =1 2 119898 if 119894 = 119895 then set 120573119894119895 =(minus1radic1205951198942 minus sum119894minus1119896=1⟨120595119894(119905) 120595119896(119905)⟩2) sum119894minus1119896=1⟨120595119894(119905) and120595119896(119905)⟩120573119896119895Else 120573119894119895 = 1radic1205951198942 minus sum119894minus1119896=1⟨120595119894(119905) 120595119896(119905)⟩2Else 12057311 = 11205951
Step 4 For 119894 = 1 2 119898 set 120595119894(119909) = sum119894119896=1 120573119894119896120595119896(119909)Step 5 Set 1199060(1199091) = 119906(1199091)Step 6 Set 119899 = 1Step 7 Set 119861119899 = sum119899119896=1 120573119899119896119906119896minus1(119909119896)Step 8 Set 119906119899(119909) = sum119899119894=1 119861119894120595119894(119909)Step 9 If 119899 lt 119898 then set 119899 = 119899 + 1 and go to step 7Else stop
4 Numerical Experiments
In this part four numerical examples are solved for potencyand utility of the present method All computations areperformed by MAPLE package Results which are takenby this method show a proper agreement with the exactsolution A comprehensive applicability of this method isgiven the stability and consistence of the presented methodThe reliability of the method and increasing of the accuracycause this method to be more applicable
Example 1 For first applicable instance we offer nonlinearFredholm integral equations [26 30]
119906 (119909) minus int101199091199101199063 (119910) 119889119910 = 119890119909 minus (1 + 21198903) 119909
9 0 le 119909 le 1
(38)
Mathematical Problems in Engineering 7
1
Error of approximate solution
0 01 02 03 04 05 06 07 08 09
times10minus4Approximate solution and the exact solution
ExactApproximate
0 02 03 04 05 06 07 08 0901 1x
1
12
14
16
18
2
22
24
26
28
minus2minus15
minus1minus05
005
115
2
Abso
lute
erro
r
Figure 1 The absolute errors comparison between the proposed approach and method [30]
Error of approximate solution
10 01 02 03 04 05 06 07 08 09
times10minus3Approximate solution and the exact solution
10 01 02 03 04 05 06 07 08 09
ExactApproximate
0
01
02
03
04
05
06
07
08
09
minus25minus2
minus15minus1
minus050
051
152
Abso
lute
erro
r
Figure 2 The absolute errors comparison between the proposed approach and method [26]
The exact solution of this equation is 119906(119909) = 119890119909 According to(38) we can assume an initial approximation 1199060(0) = 119906(0) =1 The numerical results are given in Table 1 by taking 119909119894 =(119894 minus 1)(119899 minus 1) 119894 = 1 2 119899 and 119899 = 128 In Table 1 acomparison between the absolute errors of our method andthe Haar wavelet method [30] is given Figure 1 shows theapproximate solution and its errors
Example 2 For second applicable example an electromag-netic problem is solved via the presented method It is simu-lated to nonlinear Volterra integral equations model [26 30]
119906 (119909) minus 12 int119909
01199062 (119910) 119889119910 = sin (119909) + 18 sin (2119909) minus 14119909
0 le 119909 le 1(39)
The exact solution of this equation is 119906(119909) = sin(119909)According to (39) we can assume an initial approximation1199060(0) = 119906(0) = 0 Numerical results are given in Table 2 bytaking119909119894 = (119894minus1)(119899minus1) 119894 = 1 2 119899 and 119899 = 128 In Table 2a comparison between the absolute errors of the proposedmethod and the BPFs method [26] is given Figure 2 shows
Table 3 The execution time (seconds) comparison between Algo-rithms 14 and 15
119899 Algorithm 14 (Sec) Algorithm 15 (Sec)4 347 2878 952 48216 5975 3177
the approximate solution and its errors In Table 3 a compar-ison execution time between Algorithms 14 and 15 is given
Example 3 An electromagnetic problem is solved via ourmethod for another applicable example It is simulated tononlinear Fredholm integral equations model [26 30]
119906 (119909) minus 12 int1
01199101199062 (119910) 119889119910 = 1199092 minus 112 0 le 119909 le 1 (40)
The exact solution of this equation is 119906(119909) = 1199092 Accordingto (40) the initial approximation 1199060(0) = 119906(0) = 0 is chosenNumerical results are given in Table 4 by taking 119909119894 = (119894 minus1)(119899minus1) 119894 = 1 2 119899 and 119899 = 128 In Table 4 a comparison
8 Mathematical Problems in Engineering
10
20
30
40
50
60
Error of approximate solutiontimes10minus5Approximate solution and the exact solution
0 01 02 03 04 05 06 07 08 09
ExactApproximate
0 02 03 04 05 06 07 08 0901 1x
0
01
02
03
04
05
06
07
08
minus8minus6minus4minus2
02468
Abso
lute
erro
r
Figure 3 The absolute errors comparison between the proposed approach and method [26]
Table 4 The absolute errors comparison between the proposed approach and method [26]
119909 Exact solution Approximate solution Method in [26] Absolute error Absolute error in method [26]01 0010000 0010762 0010308 0000762 000030802 0040000 0040655 0038140 0000655 000186003 0090000 0090444 0092828 0000444 000282804 0160000 016023 0158746 0000230 000125405 0250000 0250001 0257867 0000001 000786706 0360000 0359606 0361871 0000394 000187107 0490000 0489448 0483453 0000552 000654708 0640000 0639391 0647515 0000609 000751509 0810000 0811638 0807183 0001638 0002817
Table 5 The execution time (seconds) comparison between Algo-rithms 14 and 15
119899 Algorithm 14 (Sec) Algorithm 15 (Sec)4 31 2598 846 50916 342 2889
between the absolute errors of the proposed method and theBPFs method [26] is given Figure 3 shows the approximatesolution and its error In Table 5 a comparison execution timebetween Algorithms 14 and 15 is given
Example 4 A nonlinear Fredholm integral problem [26 30]is solved via our method for this applicable example
119906 (119909) + int10119890119909minus21199101199063 (119910) 119889119910 = 119890119909+1 0 le 119909 lt 1 (41)
The exact solution of this equation is 119906(119909) = 119890119909 According to(41) we consider an initial approximation as 1199060(0) = 119906(0) = 1Numerical results are given in Table 6 by taking 119909119894 = (119894 minus1)(119899minus1) 119894 = 1 2 119899 and 119899 = 128 In Table 6 a comparison
between the absolute errors of the proposed method andthe Haar wavelet method [30] is given Figure 4 shows theapproximate solution and its error
5 Conclusion
According to this essay supplementary of iterative Reproduc-ing Kernel Hilbert SpaceMethod was introduced and appliedto acquire the approximate solution of some nonlinear inte-gral equation In this method unlike other similar methodsorthogonal process is not used However the time is increas-ing and accuracy is also increasing The main point whichis mentioned in this paper is that Algorithm 14 has higherexecution time in comparison with Algorithm 15 but theapproximate solution in Algorithm 14 is more accurate thanAlgorithm 15 Current uniform convergencemethod is statedand proved The obtained numerical results confirm that itis a good candidate for solving the nonlinear integral equa-tion
Conflicts of Interest
The authors declare no conflicts of interest
Mathematical Problems in Engineering 9
Table 6 The absolute errors comparison between the proposed approach and method [30]
119909 Exact solution Approximate solution Method in [30] Absolute error Absolute error in method [30]01 1105170918 1105495796 1111099473 0000324878 000592855502 1221402757 1220931139 1220296615 0000471618 000110614203 1349858806 1349502887 1339817084 0000355919 001004172204 1491824696 1491740269 1471965091 0000084427 001985960505 1648721268 1648649846 1457127691 0000071422 019159357706 1822118797 1822243106 1831888927 0000124309 000977013007 2013752703 2013650232 2011936556 0000102471 000181614708 2225540923 2224684503 2209678381 0000856420 001586254209 2459603104 2458626573 2426854676 0000976531 0032748428
Error of approximate solution
10 01 02 03 04 05 06 07 08 09
times10minus5
Approximate solution and the exact solution
10 01 02 03 04 05 06 07 08 09
ExactApproximate
1
12
14
16
18
2
22
24
26
28
minus10
minus8
minus6
minus4
minus2
0
4
2
Abso
lute
erro
r
x
Figure 4 The absolute errors comparison between the proposed approach and method [30]
Acknowledgments
The authors would like to thank Professor Haipeng Peng forinstructive comments and recommendations to improve thequality of this work and also the Islamic Azad UniversityScience and Research Branch Tehran for supporting thisproject
References
[1] K Meetz and W Engi Electromagnetische Felder SpringerBerlin Germany 1979
[2] W Thirring Lehrbuch der Mathematischen Physik vol 2Springer Vienna Austria 1978
[3] K F Warnick R H Selfridge and D V Arnold ldquoTeachingelectromagnetic field theory using differential formsrdquo IEEETransactions on Education vol 40 no 1 pp 53ndash68 1997
[4] S Abbasbandy ldquoHomotopy analysis method for heat radiationequationsrdquo International Communications in Heat and MassTransfer vol 34 no 3 pp 380ndash387 2007
[5] H Vosughi E Shivanian and S Abbasbandy ldquoA new analyticaltechnique to solve Volterrarsquos integral equationsrdquo MathematicalMethods in the Applied Sciences vol 34 no 10 pp 1243ndash12532011
[6] J-P Kauthen ldquoA survey of singularly perturbed Volterra equa-tionsrdquo Applied Numerical Mathematics An IMACS Journal vol24 no 2-3 pp 95ndash114 1997
[7] A A Kilbas and M Saigo ldquoOn solution of nonlinear Abel-Volterra integral equationrdquo Journal of Mathematical Analysisand Applications vol 229 no 1 pp 41ndash60 1999
[8] C Minggen and D Zhongxing ldquoOn the best operator ofinterpolationrdquo Mathematica Numerica Sinica vol 8 no 2 pp209ndash216 1986
[9] C Minggen and L Yingzhen Nonlinear Numerical Analysis inthe Reproducing Kernel Space Nova Science New York NYUSA 2009
[10] S Abbasbandy B Azarnavid and M S Alhuthali ldquoA shootingreproducing kernel Hilbert spacemethod formultiple solutionsof nonlinear boundary value problemsrdquo Journal of Computa-tional and Applied Mathematics vol 279 pp 293ndash305 2015
[11] N Shawagfeh O Abu Arqub and S Momani ldquoAnalytical solu-tion of nonlinear second-order periodic boundary value prob-lem using reproducing kernel methodrdquo Journal of Computa-tional Analysis andApplications vol 16 no 4 pp 750ndash762 2014
[12] FGeng andMCui ldquoSolving a nonlinear systemof second orderboundary value problemsrdquo Journal of Mathematical Analysisand Applications vol 327 no 2 pp 1167ndash1181 2007
[13] B Azarnavid F Parvaneh and S Abbasbandy ldquoPicard-repro-ducing kernel Hilbert space method for solving generalizedsingular nonlinear Lane-Emden type equationsrdquoMathematicalModelling and Analysis vol 20 no 6 pp 754ndash767 2015
[14] 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
10 Mathematical Problems in Engineering
[15] L Yang and M Cui ldquoNew algorithm for a class of nonlinearintegro-differential equations in the reproducing kernel spacerdquoApplied Mathematics and Computation vol 174 no 2 pp 942ndash960 2006
[16] 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
[17] S Bushnaq B Maayah S Momani and A Alsaedi ldquoA repro-ducing kernel Hilbert space method for solving systems offractional integrodifferential equationsrdquo Abstract and AppliedAnalysis vol 2014 Article ID 103016 6 pages 2014
[18] M Inc A Akgul and F Geng ldquoReproducing kernel Hilbertspace method for solving Bratursquos problemrdquo Bulletin of theMalaysian Mathematical Sciences Society vol 38 no 1 pp 271ndash287 2015
[19] A-M Wazwaz A First Course in Integral Equations WorldScientific 1997
[20] S Javadi E Babolian and E Moradi ldquoNew implementation ofreproducing kernel Hilbert space method for solving a class offunctional integral equationsrdquo Communications in NumericalAnalysis vol 2014 Article ID cna-00205 7 pages 2014
[21] W Jiang and Z Chen ldquoSolving a system of linear Volterraintegral equations using the new reproducing kernel methodrdquoApplied Mathematics and Computation vol 219 no 20 pp10225ndash10230 2013
[22] A-M Wazwaz Linear and Nonlinear Integral Equations Meth-ods and Applications Springer Berlin Germany 2011
[23] I L El-Kalla ldquoConvergence of the Adomian method appliedto a class of nonlinear integral equationsrdquo Applied MathematicsLetters An International Journal of Rapid Publication vol 21 no4 pp 372ndash376 2008
[24] G Gumah KMoaddy M Al-Smadi and I Hashim ldquoSolutionsto uncertain Volterra integral equations by fitted reproducingkernel Hilbert space methodrdquo Journal of Function Spaces vol2016 Article ID 2920463 11 pages 2016
[25] V Sizikov and D Sidorov ldquoGeneralized quadrature for solv-ing singular integral equations of Abel type in applicationto infrared tomographyrdquo Applied Numerical Mathematics AnIMACS Journal vol 106 pp 69ndash78 2016
[26] E Babolian Z Masouri and S Hatamzadeh-Varmazyar ldquoNewdirect method to solve nonlinear Volterra-Fredholm integraland integro-differential equations using operational matrixwith block-pulse functionsrdquo in Progress in ElectromagneticsResearch B vol 8 pp 59ndash76 EMW Publishing CambridgeMass USA 2008
[27] D Alpay Ed Reproducing kernel spaces and applications vol143 of Operator Theory Advances and Applications BirkhauserBasel 2003
[28] S Saitoh D Alpay J A Ball and T Ohsawa ReproducingKernels andTheir Applications vol 11 Springer Science amp Busi-ness Media Berlin Germany 2013
[29] S Saitoh Integral Transforms Reproducing Kernels and TheirApplications vol 369 of Pitman Research Notes in MathematicsSeries Longman Harlow UK 1997
[30] E Babolian andA Shahsavaran ldquoNumerical solution of nonlin-ear Fredholm integral equations of the second kind using Haarwaveletsrdquo Journal of Computational and Applied Mathematicsvol 225 no 1 pp 87ndash95 2009
[31] R Ketabchi R Mokhtari and E Babolian ldquoSome error esti-mates for solving Volterra integral equations by using the
reproducing kernel methodrdquo Journal of Computational andApplied Mathematics vol 273 pp 245ndash250 2015
[32] A Alvandi M Paripour and Z Roshani ldquoReproducing kernelmethod for solving a class of Fredholm integro-differentialequationsrdquo inProceedings of the 46thAnnual IranianMathemat-ics Conference (AIMC 46) p 505 Yazd University 2015
[33] I Komashynska andMAl-Smadi ldquoIterative reproducing kernelmethod for solving second-order integrodifferential equationsof fredholm typerdquo Journal of Applied Mathematics vol 2014Article ID 459509 11 pages 2014
[34] E Babolian S Javadi and E Moradi ldquoError analysis of repro-ducing kernel Hilbert space method for solving functional inte-gral equationsrdquo Journal of Computational and Applied Mathe-matics vol 300 pp 300ndash311 2016
Submit your manuscripts athttpswwwhindawicom
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
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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
Mathematical Problems in Engineering 5
The mentioned scheme above is an efficient method ofsolving nonlinear equations [31ndash33] However in implement-ing this algorithm on a computer 120595119894(119909)infin119894=1 is not quiteorthogonal due to rounding errors In other words Gram-Schmidt process is numerically unstable and the computa-tional cost of the algorithm is high Therefore the followingprocess is suggested similar for linear problems in [20 34]This is the subject of the next theorem where the followingnotations are used
120573119899 =[[[[[[[
12057311 0 sdot sdot sdot 012057321 12057322 d 0 d
1205731198991 1205731198992 sdot sdot sdot 120573119899119899
]]]]]]]
120595119899 =[[[[[[[
12059511 12059512 sdot sdot sdot 120595111989912059521 12059522 d 1205952119899 d
1205951198991 1205951198992 sdot sdot sdot 120595119899119899
]]]]]]]
u =[[[[[[[
1199060 (1199091)1199061 (1199092)119906119899minus1 (119909119899)
]]]]]]]
B =[[[[[[[
11986111198612119861119899
]]]]]]]
Λ =[[[[[[[
Λ 1Λ 2Λ 119899
]]]]]]]
119899 = 1 2
(28)
Theorem 13 (let 120574119894119895 = [120595minus1]119894119895) The approximate solutionobtained from (22) is found as follows
119906119899 (119909) = 119899sum119894=1
Λ 119894120595119894 (119909) (29)
where
Λ 119894 = 119894sum119896=1
120574119894119896119906119896minus1 (119909119896) (30)
Proof Suppose that 119906119899(119909) = sum119899119894=1 Λ 119894120595119894(119909) = sum119899119894=1 119861119894120595119894(119909)Since 120595119894(119909) = sum119894119896=1 120573119894119896120595119896(119909)
119899sum119894=1
Λ 119894120595119894 (119909) = 119899sum119894=1
119861119894120595119894 (119909) = 119899sum119894=1
119861119894 119894sum119896=1
120573119894119896120595119896 (119909)
= 119899sum119896=1
119899sum119894=119896
119861119894120573119894119896120595119896 (119909) (31)
120595119894(119909)infin119894=1 and Λ 119896 = sum119899119894=119896 119861119894120573119894119896 (119896 = 1 2 119899) are linearindependence and therefore
Λ = 120573TB (32)
Equation (12) implies L119906119899(119909) = 119906119899(119909) For 119894 = 1 2 119899 wehave
⟨L119906119899 (119909) 120595119894⟩ = ⟨119906119899 (119909) 120595119894⟩ 997904rArr119899sum119895=1
119861119895 ⟨L120595119895 120595119894⟩ = ⟨119906119899 (119909) 120595119894⟩ (33)
Both sides of (33) provide
119899sum119895=1
119861119895 ⟨L120595119894 120595119894⟩ = 119899sum119895=1
119861119895 119894sum119896=1
120573119894119896119895sum119897=1
120573119895119897 ⟨L120595119897 120595119896⟩
= 119899sum119895=1
119861119895 119894sum119896=1
119895sum119897=1
120573119894119896 ⟨L120595119897 120595119896⟩ 120573119879119897119895= 119899sum119895=1
119861119895 (120573120595120573T)119894119895
(34)
⟨119906119899 (119909) 120595119894⟩ = 119861119894 = 119894sum119896=1
120573119894119896119906119896minus1 (119909119896) (using Theorem 12)
(35)
From (32) and (35) the following equation can be reached
120573120595120573TB = 120573u (36)
Equation (32) implies 120573120595Λ = 120573u119899 So120595Λ = u119899 (37)
which proves the theorem
Algorithm 14 The following steps exist for approximatingthe solution without applying Gram-Schmidt orthogonalprocess
Step 1 Fix 119886 le 119909 and 119905 le 119887If 119905 le 119909 set 119877119909(119905) = 1 minus 119886 + 119905Else set 119877119909(119905) = 1 minus 119886 + 119909Step 2 For 119894 = 1 2 119898 set 119909119894 = (119894 minus 1)(119898 minus 1)
6 Mathematical Problems in Engineering
Table 1 The absolute errors comparison between the proposed approach and method [30]
119909 Exact solution Approximate solution Method in [30] Absolute error Absolute error in method [30]01 1105170918 1105154993 1114627560 0000015925 000945664202 1221402757 1221389903 1226758840 0000012855 000535608303 1349858806 1349845490 1345458179 0000013317 000440062704 1491824696 1491803913 1480202470 0000020784 001162222605 1648721268 1648680858 1671769819 0000040412 002304855106 1822118797 1822039541 1838854903 0000079259 001673610607 2013752703 2013608464 2022118086 0000144243 000836538308 2225540923 2225304288 2223139077 0000236640 000240184609 2459603104 2459253211 2443684898 0000349899 0015918206
Table 2 The absolute errors comparison between the proposed approach and method [26]
119909 Exact solution Approximate solution Method in [26] Absolute error Absolute error in method [26]01 0099833 0098524 0101388 0001308 00015502 0198669 0199766 0194073 0001097 000459603 0295520 0294857 0299995 0000662 000447504 0389418 0389658 0387978 0000240 000144005 0479426 0479585 0486266 0000160 000684006 0564642 0563993 0565930 0000648 000128807 0644218 0645477 0640624 0001260 000359408 0717356 0715373 0720611 0001982 000325509 0783327 0785653 0782351 0002326 0000976
Set 120595119894(119909) = 119871 119905119877119909(119905)|119905=119909119894 Step 3 Set 1199060(1199091) = 119906(1199091)Step 4 For 119894 = 1 2 119898 set 120574119894119895 = [120595minus1]119894119895Step 5 119899 = 1Step 6 Set Λ 119899 = sum119899119896=1 120574119899119896119906119896minus1(119909119896)Step 7 Set 119906119899(119909) = sum119899119894=1 Λ 119894120595119894(119909)Step 8 If 119899 lt 119898 then set 119899 = 119899 + 1 and go to step 6Else stop
Algorithm 15 The following steps exist for approximating thesolution by applying Gram-Schmidt orthogonal process
Step 1 Fix 119886 le 119909 and 119905 le 119887If 119905 le 119909 set 119877119909(119905) = 1 minus 119886 + 119905Else set 119877119909(119905) = 1 minus 119886 + 119909Step 2 For 119894 = 1 2 119898 set 119909119894 = (119894 minus 1)(119898 minus 1)Set 120595119894(119909) = 119871 119905119877119909(119905)|119905=119909119894 Step 3 For 119894 = 1 2 119898 and 119895 =1 2 119898 if 119894 = 119895 then set 120573119894119895 =(minus1radic1205951198942 minus sum119894minus1119896=1⟨120595119894(119905) 120595119896(119905)⟩2) sum119894minus1119896=1⟨120595119894(119905) and120595119896(119905)⟩120573119896119895Else 120573119894119895 = 1radic1205951198942 minus sum119894minus1119896=1⟨120595119894(119905) 120595119896(119905)⟩2Else 12057311 = 11205951
Step 4 For 119894 = 1 2 119898 set 120595119894(119909) = sum119894119896=1 120573119894119896120595119896(119909)Step 5 Set 1199060(1199091) = 119906(1199091)Step 6 Set 119899 = 1Step 7 Set 119861119899 = sum119899119896=1 120573119899119896119906119896minus1(119909119896)Step 8 Set 119906119899(119909) = sum119899119894=1 119861119894120595119894(119909)Step 9 If 119899 lt 119898 then set 119899 = 119899 + 1 and go to step 7Else stop
4 Numerical Experiments
In this part four numerical examples are solved for potencyand utility of the present method All computations areperformed by MAPLE package Results which are takenby this method show a proper agreement with the exactsolution A comprehensive applicability of this method isgiven the stability and consistence of the presented methodThe reliability of the method and increasing of the accuracycause this method to be more applicable
Example 1 For first applicable instance we offer nonlinearFredholm integral equations [26 30]
119906 (119909) minus int101199091199101199063 (119910) 119889119910 = 119890119909 minus (1 + 21198903) 119909
9 0 le 119909 le 1
(38)
Mathematical Problems in Engineering 7
1
Error of approximate solution
0 01 02 03 04 05 06 07 08 09
times10minus4Approximate solution and the exact solution
ExactApproximate
0 02 03 04 05 06 07 08 0901 1x
1
12
14
16
18
2
22
24
26
28
minus2minus15
minus1minus05
005
115
2
Abso
lute
erro
r
Figure 1 The absolute errors comparison between the proposed approach and method [30]
Error of approximate solution
10 01 02 03 04 05 06 07 08 09
times10minus3Approximate solution and the exact solution
10 01 02 03 04 05 06 07 08 09
ExactApproximate
0
01
02
03
04
05
06
07
08
09
minus25minus2
minus15minus1
minus050
051
152
Abso
lute
erro
r
Figure 2 The absolute errors comparison between the proposed approach and method [26]
The exact solution of this equation is 119906(119909) = 119890119909 According to(38) we can assume an initial approximation 1199060(0) = 119906(0) =1 The numerical results are given in Table 1 by taking 119909119894 =(119894 minus 1)(119899 minus 1) 119894 = 1 2 119899 and 119899 = 128 In Table 1 acomparison between the absolute errors of our method andthe Haar wavelet method [30] is given Figure 1 shows theapproximate solution and its errors
Example 2 For second applicable example an electromag-netic problem is solved via the presented method It is simu-lated to nonlinear Volterra integral equations model [26 30]
119906 (119909) minus 12 int119909
01199062 (119910) 119889119910 = sin (119909) + 18 sin (2119909) minus 14119909
0 le 119909 le 1(39)
The exact solution of this equation is 119906(119909) = sin(119909)According to (39) we can assume an initial approximation1199060(0) = 119906(0) = 0 Numerical results are given in Table 2 bytaking119909119894 = (119894minus1)(119899minus1) 119894 = 1 2 119899 and 119899 = 128 In Table 2a comparison between the absolute errors of the proposedmethod and the BPFs method [26] is given Figure 2 shows
Table 3 The execution time (seconds) comparison between Algo-rithms 14 and 15
119899 Algorithm 14 (Sec) Algorithm 15 (Sec)4 347 2878 952 48216 5975 3177
the approximate solution and its errors In Table 3 a compar-ison execution time between Algorithms 14 and 15 is given
Example 3 An electromagnetic problem is solved via ourmethod for another applicable example It is simulated tononlinear Fredholm integral equations model [26 30]
119906 (119909) minus 12 int1
01199101199062 (119910) 119889119910 = 1199092 minus 112 0 le 119909 le 1 (40)
The exact solution of this equation is 119906(119909) = 1199092 Accordingto (40) the initial approximation 1199060(0) = 119906(0) = 0 is chosenNumerical results are given in Table 4 by taking 119909119894 = (119894 minus1)(119899minus1) 119894 = 1 2 119899 and 119899 = 128 In Table 4 a comparison
8 Mathematical Problems in Engineering
10
20
30
40
50
60
Error of approximate solutiontimes10minus5Approximate solution and the exact solution
0 01 02 03 04 05 06 07 08 09
ExactApproximate
0 02 03 04 05 06 07 08 0901 1x
0
01
02
03
04
05
06
07
08
minus8minus6minus4minus2
02468
Abso
lute
erro
r
Figure 3 The absolute errors comparison between the proposed approach and method [26]
Table 4 The absolute errors comparison between the proposed approach and method [26]
119909 Exact solution Approximate solution Method in [26] Absolute error Absolute error in method [26]01 0010000 0010762 0010308 0000762 000030802 0040000 0040655 0038140 0000655 000186003 0090000 0090444 0092828 0000444 000282804 0160000 016023 0158746 0000230 000125405 0250000 0250001 0257867 0000001 000786706 0360000 0359606 0361871 0000394 000187107 0490000 0489448 0483453 0000552 000654708 0640000 0639391 0647515 0000609 000751509 0810000 0811638 0807183 0001638 0002817
Table 5 The execution time (seconds) comparison between Algo-rithms 14 and 15
119899 Algorithm 14 (Sec) Algorithm 15 (Sec)4 31 2598 846 50916 342 2889
between the absolute errors of the proposed method and theBPFs method [26] is given Figure 3 shows the approximatesolution and its error In Table 5 a comparison execution timebetween Algorithms 14 and 15 is given
Example 4 A nonlinear Fredholm integral problem [26 30]is solved via our method for this applicable example
119906 (119909) + int10119890119909minus21199101199063 (119910) 119889119910 = 119890119909+1 0 le 119909 lt 1 (41)
The exact solution of this equation is 119906(119909) = 119890119909 According to(41) we consider an initial approximation as 1199060(0) = 119906(0) = 1Numerical results are given in Table 6 by taking 119909119894 = (119894 minus1)(119899minus1) 119894 = 1 2 119899 and 119899 = 128 In Table 6 a comparison
between the absolute errors of the proposed method andthe Haar wavelet method [30] is given Figure 4 shows theapproximate solution and its error
5 Conclusion
According to this essay supplementary of iterative Reproduc-ing Kernel Hilbert SpaceMethod was introduced and appliedto acquire the approximate solution of some nonlinear inte-gral equation In this method unlike other similar methodsorthogonal process is not used However the time is increas-ing and accuracy is also increasing The main point whichis mentioned in this paper is that Algorithm 14 has higherexecution time in comparison with Algorithm 15 but theapproximate solution in Algorithm 14 is more accurate thanAlgorithm 15 Current uniform convergencemethod is statedand proved The obtained numerical results confirm that itis a good candidate for solving the nonlinear integral equa-tion
Conflicts of Interest
The authors declare no conflicts of interest
Mathematical Problems in Engineering 9
Table 6 The absolute errors comparison between the proposed approach and method [30]
119909 Exact solution Approximate solution Method in [30] Absolute error Absolute error in method [30]01 1105170918 1105495796 1111099473 0000324878 000592855502 1221402757 1220931139 1220296615 0000471618 000110614203 1349858806 1349502887 1339817084 0000355919 001004172204 1491824696 1491740269 1471965091 0000084427 001985960505 1648721268 1648649846 1457127691 0000071422 019159357706 1822118797 1822243106 1831888927 0000124309 000977013007 2013752703 2013650232 2011936556 0000102471 000181614708 2225540923 2224684503 2209678381 0000856420 001586254209 2459603104 2458626573 2426854676 0000976531 0032748428
Error of approximate solution
10 01 02 03 04 05 06 07 08 09
times10minus5
Approximate solution and the exact solution
10 01 02 03 04 05 06 07 08 09
ExactApproximate
1
12
14
16
18
2
22
24
26
28
minus10
minus8
minus6
minus4
minus2
0
4
2
Abso
lute
erro
r
x
Figure 4 The absolute errors comparison between the proposed approach and method [30]
Acknowledgments
The authors would like to thank Professor Haipeng Peng forinstructive comments and recommendations to improve thequality of this work and also the Islamic Azad UniversityScience and Research Branch Tehran for supporting thisproject
References
[1] K Meetz and W Engi Electromagnetische Felder SpringerBerlin Germany 1979
[2] W Thirring Lehrbuch der Mathematischen Physik vol 2Springer Vienna Austria 1978
[3] K F Warnick R H Selfridge and D V Arnold ldquoTeachingelectromagnetic field theory using differential formsrdquo IEEETransactions on Education vol 40 no 1 pp 53ndash68 1997
[4] S Abbasbandy ldquoHomotopy analysis method for heat radiationequationsrdquo International Communications in Heat and MassTransfer vol 34 no 3 pp 380ndash387 2007
[5] H Vosughi E Shivanian and S Abbasbandy ldquoA new analyticaltechnique to solve Volterrarsquos integral equationsrdquo MathematicalMethods in the Applied Sciences vol 34 no 10 pp 1243ndash12532011
[6] J-P Kauthen ldquoA survey of singularly perturbed Volterra equa-tionsrdquo Applied Numerical Mathematics An IMACS Journal vol24 no 2-3 pp 95ndash114 1997
[7] A A Kilbas and M Saigo ldquoOn solution of nonlinear Abel-Volterra integral equationrdquo Journal of Mathematical Analysisand Applications vol 229 no 1 pp 41ndash60 1999
[8] C Minggen and D Zhongxing ldquoOn the best operator ofinterpolationrdquo Mathematica Numerica Sinica vol 8 no 2 pp209ndash216 1986
[9] C Minggen and L Yingzhen Nonlinear Numerical Analysis inthe Reproducing Kernel Space Nova Science New York NYUSA 2009
[10] S Abbasbandy B Azarnavid and M S Alhuthali ldquoA shootingreproducing kernel Hilbert spacemethod formultiple solutionsof nonlinear boundary value problemsrdquo Journal of Computa-tional and Applied Mathematics vol 279 pp 293ndash305 2015
[11] N Shawagfeh O Abu Arqub and S Momani ldquoAnalytical solu-tion of nonlinear second-order periodic boundary value prob-lem using reproducing kernel methodrdquo Journal of Computa-tional Analysis andApplications vol 16 no 4 pp 750ndash762 2014
[12] FGeng andMCui ldquoSolving a nonlinear systemof second orderboundary value problemsrdquo Journal of Mathematical Analysisand Applications vol 327 no 2 pp 1167ndash1181 2007
[13] B Azarnavid F Parvaneh and S Abbasbandy ldquoPicard-repro-ducing kernel Hilbert space method for solving generalizedsingular nonlinear Lane-Emden type equationsrdquoMathematicalModelling and Analysis vol 20 no 6 pp 754ndash767 2015
[14] 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
10 Mathematical Problems in Engineering
[15] L Yang and M Cui ldquoNew algorithm for a class of nonlinearintegro-differential equations in the reproducing kernel spacerdquoApplied Mathematics and Computation vol 174 no 2 pp 942ndash960 2006
[16] 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
[17] S Bushnaq B Maayah S Momani and A Alsaedi ldquoA repro-ducing kernel Hilbert space method for solving systems offractional integrodifferential equationsrdquo Abstract and AppliedAnalysis vol 2014 Article ID 103016 6 pages 2014
[18] M Inc A Akgul and F Geng ldquoReproducing kernel Hilbertspace method for solving Bratursquos problemrdquo Bulletin of theMalaysian Mathematical Sciences Society vol 38 no 1 pp 271ndash287 2015
[19] A-M Wazwaz A First Course in Integral Equations WorldScientific 1997
[20] S Javadi E Babolian and E Moradi ldquoNew implementation ofreproducing kernel Hilbert space method for solving a class offunctional integral equationsrdquo Communications in NumericalAnalysis vol 2014 Article ID cna-00205 7 pages 2014
[21] W Jiang and Z Chen ldquoSolving a system of linear Volterraintegral equations using the new reproducing kernel methodrdquoApplied Mathematics and Computation vol 219 no 20 pp10225ndash10230 2013
[22] A-M Wazwaz Linear and Nonlinear Integral Equations Meth-ods and Applications Springer Berlin Germany 2011
[23] I L El-Kalla ldquoConvergence of the Adomian method appliedto a class of nonlinear integral equationsrdquo Applied MathematicsLetters An International Journal of Rapid Publication vol 21 no4 pp 372ndash376 2008
[24] G Gumah KMoaddy M Al-Smadi and I Hashim ldquoSolutionsto uncertain Volterra integral equations by fitted reproducingkernel Hilbert space methodrdquo Journal of Function Spaces vol2016 Article ID 2920463 11 pages 2016
[25] V Sizikov and D Sidorov ldquoGeneralized quadrature for solv-ing singular integral equations of Abel type in applicationto infrared tomographyrdquo Applied Numerical Mathematics AnIMACS Journal vol 106 pp 69ndash78 2016
[26] E Babolian Z Masouri and S Hatamzadeh-Varmazyar ldquoNewdirect method to solve nonlinear Volterra-Fredholm integraland integro-differential equations using operational matrixwith block-pulse functionsrdquo in Progress in ElectromagneticsResearch B vol 8 pp 59ndash76 EMW Publishing CambridgeMass USA 2008
[27] D Alpay Ed Reproducing kernel spaces and applications vol143 of Operator Theory Advances and Applications BirkhauserBasel 2003
[28] S Saitoh D Alpay J A Ball and T Ohsawa ReproducingKernels andTheir Applications vol 11 Springer Science amp Busi-ness Media Berlin Germany 2013
[29] S Saitoh Integral Transforms Reproducing Kernels and TheirApplications vol 369 of Pitman Research Notes in MathematicsSeries Longman Harlow UK 1997
[30] E Babolian andA Shahsavaran ldquoNumerical solution of nonlin-ear Fredholm integral equations of the second kind using Haarwaveletsrdquo Journal of Computational and Applied Mathematicsvol 225 no 1 pp 87ndash95 2009
[31] R Ketabchi R Mokhtari and E Babolian ldquoSome error esti-mates for solving Volterra integral equations by using the
reproducing kernel methodrdquo Journal of Computational andApplied Mathematics vol 273 pp 245ndash250 2015
[32] A Alvandi M Paripour and Z Roshani ldquoReproducing kernelmethod for solving a class of Fredholm integro-differentialequationsrdquo inProceedings of the 46thAnnual IranianMathemat-ics Conference (AIMC 46) p 505 Yazd University 2015
[33] I Komashynska andMAl-Smadi ldquoIterative reproducing kernelmethod for solving second-order integrodifferential equationsof fredholm typerdquo Journal of Applied Mathematics vol 2014Article ID 459509 11 pages 2014
[34] E Babolian S Javadi and E Moradi ldquoError analysis of repro-ducing kernel Hilbert space method for solving functional inte-gral equationsrdquo Journal of Computational and Applied Mathe-matics vol 300 pp 300ndash311 2016
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
Table 1 The absolute errors comparison between the proposed approach and method [30]
119909 Exact solution Approximate solution Method in [30] Absolute error Absolute error in method [30]01 1105170918 1105154993 1114627560 0000015925 000945664202 1221402757 1221389903 1226758840 0000012855 000535608303 1349858806 1349845490 1345458179 0000013317 000440062704 1491824696 1491803913 1480202470 0000020784 001162222605 1648721268 1648680858 1671769819 0000040412 002304855106 1822118797 1822039541 1838854903 0000079259 001673610607 2013752703 2013608464 2022118086 0000144243 000836538308 2225540923 2225304288 2223139077 0000236640 000240184609 2459603104 2459253211 2443684898 0000349899 0015918206
Table 2 The absolute errors comparison between the proposed approach and method [26]
119909 Exact solution Approximate solution Method in [26] Absolute error Absolute error in method [26]01 0099833 0098524 0101388 0001308 00015502 0198669 0199766 0194073 0001097 000459603 0295520 0294857 0299995 0000662 000447504 0389418 0389658 0387978 0000240 000144005 0479426 0479585 0486266 0000160 000684006 0564642 0563993 0565930 0000648 000128807 0644218 0645477 0640624 0001260 000359408 0717356 0715373 0720611 0001982 000325509 0783327 0785653 0782351 0002326 0000976
Set 120595119894(119909) = 119871 119905119877119909(119905)|119905=119909119894 Step 3 Set 1199060(1199091) = 119906(1199091)Step 4 For 119894 = 1 2 119898 set 120574119894119895 = [120595minus1]119894119895Step 5 119899 = 1Step 6 Set Λ 119899 = sum119899119896=1 120574119899119896119906119896minus1(119909119896)Step 7 Set 119906119899(119909) = sum119899119894=1 Λ 119894120595119894(119909)Step 8 If 119899 lt 119898 then set 119899 = 119899 + 1 and go to step 6Else stop
Algorithm 15 The following steps exist for approximating thesolution by applying Gram-Schmidt orthogonal process
Step 1 Fix 119886 le 119909 and 119905 le 119887If 119905 le 119909 set 119877119909(119905) = 1 minus 119886 + 119905Else set 119877119909(119905) = 1 minus 119886 + 119909Step 2 For 119894 = 1 2 119898 set 119909119894 = (119894 minus 1)(119898 minus 1)Set 120595119894(119909) = 119871 119905119877119909(119905)|119905=119909119894 Step 3 For 119894 = 1 2 119898 and 119895 =1 2 119898 if 119894 = 119895 then set 120573119894119895 =(minus1radic1205951198942 minus sum119894minus1119896=1⟨120595119894(119905) 120595119896(119905)⟩2) sum119894minus1119896=1⟨120595119894(119905) and120595119896(119905)⟩120573119896119895Else 120573119894119895 = 1radic1205951198942 minus sum119894minus1119896=1⟨120595119894(119905) 120595119896(119905)⟩2Else 12057311 = 11205951
Step 4 For 119894 = 1 2 119898 set 120595119894(119909) = sum119894119896=1 120573119894119896120595119896(119909)Step 5 Set 1199060(1199091) = 119906(1199091)Step 6 Set 119899 = 1Step 7 Set 119861119899 = sum119899119896=1 120573119899119896119906119896minus1(119909119896)Step 8 Set 119906119899(119909) = sum119899119894=1 119861119894120595119894(119909)Step 9 If 119899 lt 119898 then set 119899 = 119899 + 1 and go to step 7Else stop
4 Numerical Experiments
In this part four numerical examples are solved for potencyand utility of the present method All computations areperformed by MAPLE package Results which are takenby this method show a proper agreement with the exactsolution A comprehensive applicability of this method isgiven the stability and consistence of the presented methodThe reliability of the method and increasing of the accuracycause this method to be more applicable
Example 1 For first applicable instance we offer nonlinearFredholm integral equations [26 30]
119906 (119909) minus int101199091199101199063 (119910) 119889119910 = 119890119909 minus (1 + 21198903) 119909
9 0 le 119909 le 1
(38)
Mathematical Problems in Engineering 7
1
Error of approximate solution
0 01 02 03 04 05 06 07 08 09
times10minus4Approximate solution and the exact solution
ExactApproximate
0 02 03 04 05 06 07 08 0901 1x
1
12
14
16
18
2
22
24
26
28
minus2minus15
minus1minus05
005
115
2
Abso
lute
erro
r
Figure 1 The absolute errors comparison between the proposed approach and method [30]
Error of approximate solution
10 01 02 03 04 05 06 07 08 09
times10minus3Approximate solution and the exact solution
10 01 02 03 04 05 06 07 08 09
ExactApproximate
0
01
02
03
04
05
06
07
08
09
minus25minus2
minus15minus1
minus050
051
152
Abso
lute
erro
r
Figure 2 The absolute errors comparison between the proposed approach and method [26]
The exact solution of this equation is 119906(119909) = 119890119909 According to(38) we can assume an initial approximation 1199060(0) = 119906(0) =1 The numerical results are given in Table 1 by taking 119909119894 =(119894 minus 1)(119899 minus 1) 119894 = 1 2 119899 and 119899 = 128 In Table 1 acomparison between the absolute errors of our method andthe Haar wavelet method [30] is given Figure 1 shows theapproximate solution and its errors
Example 2 For second applicable example an electromag-netic problem is solved via the presented method It is simu-lated to nonlinear Volterra integral equations model [26 30]
119906 (119909) minus 12 int119909
01199062 (119910) 119889119910 = sin (119909) + 18 sin (2119909) minus 14119909
0 le 119909 le 1(39)
The exact solution of this equation is 119906(119909) = sin(119909)According to (39) we can assume an initial approximation1199060(0) = 119906(0) = 0 Numerical results are given in Table 2 bytaking119909119894 = (119894minus1)(119899minus1) 119894 = 1 2 119899 and 119899 = 128 In Table 2a comparison between the absolute errors of the proposedmethod and the BPFs method [26] is given Figure 2 shows
Table 3 The execution time (seconds) comparison between Algo-rithms 14 and 15
119899 Algorithm 14 (Sec) Algorithm 15 (Sec)4 347 2878 952 48216 5975 3177
the approximate solution and its errors In Table 3 a compar-ison execution time between Algorithms 14 and 15 is given
Example 3 An electromagnetic problem is solved via ourmethod for another applicable example It is simulated tononlinear Fredholm integral equations model [26 30]
119906 (119909) minus 12 int1
01199101199062 (119910) 119889119910 = 1199092 minus 112 0 le 119909 le 1 (40)
The exact solution of this equation is 119906(119909) = 1199092 Accordingto (40) the initial approximation 1199060(0) = 119906(0) = 0 is chosenNumerical results are given in Table 4 by taking 119909119894 = (119894 minus1)(119899minus1) 119894 = 1 2 119899 and 119899 = 128 In Table 4 a comparison
8 Mathematical Problems in Engineering
10
20
30
40
50
60
Error of approximate solutiontimes10minus5Approximate solution and the exact solution
0 01 02 03 04 05 06 07 08 09
ExactApproximate
0 02 03 04 05 06 07 08 0901 1x
0
01
02
03
04
05
06
07
08
minus8minus6minus4minus2
02468
Abso
lute
erro
r
Figure 3 The absolute errors comparison between the proposed approach and method [26]
Table 4 The absolute errors comparison between the proposed approach and method [26]
119909 Exact solution Approximate solution Method in [26] Absolute error Absolute error in method [26]01 0010000 0010762 0010308 0000762 000030802 0040000 0040655 0038140 0000655 000186003 0090000 0090444 0092828 0000444 000282804 0160000 016023 0158746 0000230 000125405 0250000 0250001 0257867 0000001 000786706 0360000 0359606 0361871 0000394 000187107 0490000 0489448 0483453 0000552 000654708 0640000 0639391 0647515 0000609 000751509 0810000 0811638 0807183 0001638 0002817
Table 5 The execution time (seconds) comparison between Algo-rithms 14 and 15
119899 Algorithm 14 (Sec) Algorithm 15 (Sec)4 31 2598 846 50916 342 2889
between the absolute errors of the proposed method and theBPFs method [26] is given Figure 3 shows the approximatesolution and its error In Table 5 a comparison execution timebetween Algorithms 14 and 15 is given
Example 4 A nonlinear Fredholm integral problem [26 30]is solved via our method for this applicable example
119906 (119909) + int10119890119909minus21199101199063 (119910) 119889119910 = 119890119909+1 0 le 119909 lt 1 (41)
The exact solution of this equation is 119906(119909) = 119890119909 According to(41) we consider an initial approximation as 1199060(0) = 119906(0) = 1Numerical results are given in Table 6 by taking 119909119894 = (119894 minus1)(119899minus1) 119894 = 1 2 119899 and 119899 = 128 In Table 6 a comparison
between the absolute errors of the proposed method andthe Haar wavelet method [30] is given Figure 4 shows theapproximate solution and its error
5 Conclusion
According to this essay supplementary of iterative Reproduc-ing Kernel Hilbert SpaceMethod was introduced and appliedto acquire the approximate solution of some nonlinear inte-gral equation In this method unlike other similar methodsorthogonal process is not used However the time is increas-ing and accuracy is also increasing The main point whichis mentioned in this paper is that Algorithm 14 has higherexecution time in comparison with Algorithm 15 but theapproximate solution in Algorithm 14 is more accurate thanAlgorithm 15 Current uniform convergencemethod is statedand proved The obtained numerical results confirm that itis a good candidate for solving the nonlinear integral equa-tion
Conflicts of Interest
The authors declare no conflicts of interest
Mathematical Problems in Engineering 9
Table 6 The absolute errors comparison between the proposed approach and method [30]
119909 Exact solution Approximate solution Method in [30] Absolute error Absolute error in method [30]01 1105170918 1105495796 1111099473 0000324878 000592855502 1221402757 1220931139 1220296615 0000471618 000110614203 1349858806 1349502887 1339817084 0000355919 001004172204 1491824696 1491740269 1471965091 0000084427 001985960505 1648721268 1648649846 1457127691 0000071422 019159357706 1822118797 1822243106 1831888927 0000124309 000977013007 2013752703 2013650232 2011936556 0000102471 000181614708 2225540923 2224684503 2209678381 0000856420 001586254209 2459603104 2458626573 2426854676 0000976531 0032748428
Error of approximate solution
10 01 02 03 04 05 06 07 08 09
times10minus5
Approximate solution and the exact solution
10 01 02 03 04 05 06 07 08 09
ExactApproximate
1
12
14
16
18
2
22
24
26
28
minus10
minus8
minus6
minus4
minus2
0
4
2
Abso
lute
erro
r
x
Figure 4 The absolute errors comparison between the proposed approach and method [30]
Acknowledgments
The authors would like to thank Professor Haipeng Peng forinstructive comments and recommendations to improve thequality of this work and also the Islamic Azad UniversityScience and Research Branch Tehran for supporting thisproject
References
[1] K Meetz and W Engi Electromagnetische Felder SpringerBerlin Germany 1979
[2] W Thirring Lehrbuch der Mathematischen Physik vol 2Springer Vienna Austria 1978
[3] K F Warnick R H Selfridge and D V Arnold ldquoTeachingelectromagnetic field theory using differential formsrdquo IEEETransactions on Education vol 40 no 1 pp 53ndash68 1997
[4] S Abbasbandy ldquoHomotopy analysis method for heat radiationequationsrdquo International Communications in Heat and MassTransfer vol 34 no 3 pp 380ndash387 2007
[5] H Vosughi E Shivanian and S Abbasbandy ldquoA new analyticaltechnique to solve Volterrarsquos integral equationsrdquo MathematicalMethods in the Applied Sciences vol 34 no 10 pp 1243ndash12532011
[6] J-P Kauthen ldquoA survey of singularly perturbed Volterra equa-tionsrdquo Applied Numerical Mathematics An IMACS Journal vol24 no 2-3 pp 95ndash114 1997
[7] A A Kilbas and M Saigo ldquoOn solution of nonlinear Abel-Volterra integral equationrdquo Journal of Mathematical Analysisand Applications vol 229 no 1 pp 41ndash60 1999
[8] C Minggen and D Zhongxing ldquoOn the best operator ofinterpolationrdquo Mathematica Numerica Sinica vol 8 no 2 pp209ndash216 1986
[9] C Minggen and L Yingzhen Nonlinear Numerical Analysis inthe Reproducing Kernel Space Nova Science New York NYUSA 2009
[10] S Abbasbandy B Azarnavid and M S Alhuthali ldquoA shootingreproducing kernel Hilbert spacemethod formultiple solutionsof nonlinear boundary value problemsrdquo Journal of Computa-tional and Applied Mathematics vol 279 pp 293ndash305 2015
[11] N Shawagfeh O Abu Arqub and S Momani ldquoAnalytical solu-tion of nonlinear second-order periodic boundary value prob-lem using reproducing kernel methodrdquo Journal of Computa-tional Analysis andApplications vol 16 no 4 pp 750ndash762 2014
[12] FGeng andMCui ldquoSolving a nonlinear systemof second orderboundary value problemsrdquo Journal of Mathematical Analysisand Applications vol 327 no 2 pp 1167ndash1181 2007
[13] B Azarnavid F Parvaneh and S Abbasbandy ldquoPicard-repro-ducing kernel Hilbert space method for solving generalizedsingular nonlinear Lane-Emden type equationsrdquoMathematicalModelling and Analysis vol 20 no 6 pp 754ndash767 2015
[14] 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
10 Mathematical Problems in Engineering
[15] L Yang and M Cui ldquoNew algorithm for a class of nonlinearintegro-differential equations in the reproducing kernel spacerdquoApplied Mathematics and Computation vol 174 no 2 pp 942ndash960 2006
[16] 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
[17] S Bushnaq B Maayah S Momani and A Alsaedi ldquoA repro-ducing kernel Hilbert space method for solving systems offractional integrodifferential equationsrdquo Abstract and AppliedAnalysis vol 2014 Article ID 103016 6 pages 2014
[18] M Inc A Akgul and F Geng ldquoReproducing kernel Hilbertspace method for solving Bratursquos problemrdquo Bulletin of theMalaysian Mathematical Sciences Society vol 38 no 1 pp 271ndash287 2015
[19] A-M Wazwaz A First Course in Integral Equations WorldScientific 1997
[20] S Javadi E Babolian and E Moradi ldquoNew implementation ofreproducing kernel Hilbert space method for solving a class offunctional integral equationsrdquo Communications in NumericalAnalysis vol 2014 Article ID cna-00205 7 pages 2014
[21] W Jiang and Z Chen ldquoSolving a system of linear Volterraintegral equations using the new reproducing kernel methodrdquoApplied Mathematics and Computation vol 219 no 20 pp10225ndash10230 2013
[22] A-M Wazwaz Linear and Nonlinear Integral Equations Meth-ods and Applications Springer Berlin Germany 2011
[23] I L El-Kalla ldquoConvergence of the Adomian method appliedto a class of nonlinear integral equationsrdquo Applied MathematicsLetters An International Journal of Rapid Publication vol 21 no4 pp 372ndash376 2008
[24] G Gumah KMoaddy M Al-Smadi and I Hashim ldquoSolutionsto uncertain Volterra integral equations by fitted reproducingkernel Hilbert space methodrdquo Journal of Function Spaces vol2016 Article ID 2920463 11 pages 2016
[25] V Sizikov and D Sidorov ldquoGeneralized quadrature for solv-ing singular integral equations of Abel type in applicationto infrared tomographyrdquo Applied Numerical Mathematics AnIMACS Journal vol 106 pp 69ndash78 2016
[26] E Babolian Z Masouri and S Hatamzadeh-Varmazyar ldquoNewdirect method to solve nonlinear Volterra-Fredholm integraland integro-differential equations using operational matrixwith block-pulse functionsrdquo in Progress in ElectromagneticsResearch B vol 8 pp 59ndash76 EMW Publishing CambridgeMass USA 2008
[27] D Alpay Ed Reproducing kernel spaces and applications vol143 of Operator Theory Advances and Applications BirkhauserBasel 2003
[28] S Saitoh D Alpay J A Ball and T Ohsawa ReproducingKernels andTheir Applications vol 11 Springer Science amp Busi-ness Media Berlin Germany 2013
[29] S Saitoh Integral Transforms Reproducing Kernels and TheirApplications vol 369 of Pitman Research Notes in MathematicsSeries Longman Harlow UK 1997
[30] E Babolian andA Shahsavaran ldquoNumerical solution of nonlin-ear Fredholm integral equations of the second kind using Haarwaveletsrdquo Journal of Computational and Applied Mathematicsvol 225 no 1 pp 87ndash95 2009
[31] R Ketabchi R Mokhtari and E Babolian ldquoSome error esti-mates for solving Volterra integral equations by using the
reproducing kernel methodrdquo Journal of Computational andApplied Mathematics vol 273 pp 245ndash250 2015
[32] A Alvandi M Paripour and Z Roshani ldquoReproducing kernelmethod for solving a class of Fredholm integro-differentialequationsrdquo inProceedings of the 46thAnnual IranianMathemat-ics Conference (AIMC 46) p 505 Yazd University 2015
[33] I Komashynska andMAl-Smadi ldquoIterative reproducing kernelmethod for solving second-order integrodifferential equationsof fredholm typerdquo Journal of Applied Mathematics vol 2014Article ID 459509 11 pages 2014
[34] E Babolian S Javadi and E Moradi ldquoError analysis of repro-ducing kernel Hilbert space method for solving functional inte-gral equationsrdquo Journal of Computational and Applied Mathe-matics vol 300 pp 300ndash311 2016
Submit your manuscripts athttpswwwhindawicom
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
Mathematical Problems in Engineering 7
1
Error of approximate solution
0 01 02 03 04 05 06 07 08 09
times10minus4Approximate solution and the exact solution
ExactApproximate
0 02 03 04 05 06 07 08 0901 1x
1
12
14
16
18
2
22
24
26
28
minus2minus15
minus1minus05
005
115
2
Abso
lute
erro
r
Figure 1 The absolute errors comparison between the proposed approach and method [30]
Error of approximate solution
10 01 02 03 04 05 06 07 08 09
times10minus3Approximate solution and the exact solution
10 01 02 03 04 05 06 07 08 09
ExactApproximate
0
01
02
03
04
05
06
07
08
09
minus25minus2
minus15minus1
minus050
051
152
Abso
lute
erro
r
Figure 2 The absolute errors comparison between the proposed approach and method [26]
The exact solution of this equation is 119906(119909) = 119890119909 According to(38) we can assume an initial approximation 1199060(0) = 119906(0) =1 The numerical results are given in Table 1 by taking 119909119894 =(119894 minus 1)(119899 minus 1) 119894 = 1 2 119899 and 119899 = 128 In Table 1 acomparison between the absolute errors of our method andthe Haar wavelet method [30] is given Figure 1 shows theapproximate solution and its errors
Example 2 For second applicable example an electromag-netic problem is solved via the presented method It is simu-lated to nonlinear Volterra integral equations model [26 30]
119906 (119909) minus 12 int119909
01199062 (119910) 119889119910 = sin (119909) + 18 sin (2119909) minus 14119909
0 le 119909 le 1(39)
The exact solution of this equation is 119906(119909) = sin(119909)According to (39) we can assume an initial approximation1199060(0) = 119906(0) = 0 Numerical results are given in Table 2 bytaking119909119894 = (119894minus1)(119899minus1) 119894 = 1 2 119899 and 119899 = 128 In Table 2a comparison between the absolute errors of the proposedmethod and the BPFs method [26] is given Figure 2 shows
Table 3 The execution time (seconds) comparison between Algo-rithms 14 and 15
119899 Algorithm 14 (Sec) Algorithm 15 (Sec)4 347 2878 952 48216 5975 3177
the approximate solution and its errors In Table 3 a compar-ison execution time between Algorithms 14 and 15 is given
Example 3 An electromagnetic problem is solved via ourmethod for another applicable example It is simulated tononlinear Fredholm integral equations model [26 30]
119906 (119909) minus 12 int1
01199101199062 (119910) 119889119910 = 1199092 minus 112 0 le 119909 le 1 (40)
The exact solution of this equation is 119906(119909) = 1199092 Accordingto (40) the initial approximation 1199060(0) = 119906(0) = 0 is chosenNumerical results are given in Table 4 by taking 119909119894 = (119894 minus1)(119899minus1) 119894 = 1 2 119899 and 119899 = 128 In Table 4 a comparison
8 Mathematical Problems in Engineering
10
20
30
40
50
60
Error of approximate solutiontimes10minus5Approximate solution and the exact solution
0 01 02 03 04 05 06 07 08 09
ExactApproximate
0 02 03 04 05 06 07 08 0901 1x
0
01
02
03
04
05
06
07
08
minus8minus6minus4minus2
02468
Abso
lute
erro
r
Figure 3 The absolute errors comparison between the proposed approach and method [26]
Table 4 The absolute errors comparison between the proposed approach and method [26]
119909 Exact solution Approximate solution Method in [26] Absolute error Absolute error in method [26]01 0010000 0010762 0010308 0000762 000030802 0040000 0040655 0038140 0000655 000186003 0090000 0090444 0092828 0000444 000282804 0160000 016023 0158746 0000230 000125405 0250000 0250001 0257867 0000001 000786706 0360000 0359606 0361871 0000394 000187107 0490000 0489448 0483453 0000552 000654708 0640000 0639391 0647515 0000609 000751509 0810000 0811638 0807183 0001638 0002817
Table 5 The execution time (seconds) comparison between Algo-rithms 14 and 15
119899 Algorithm 14 (Sec) Algorithm 15 (Sec)4 31 2598 846 50916 342 2889
between the absolute errors of the proposed method and theBPFs method [26] is given Figure 3 shows the approximatesolution and its error In Table 5 a comparison execution timebetween Algorithms 14 and 15 is given
Example 4 A nonlinear Fredholm integral problem [26 30]is solved via our method for this applicable example
119906 (119909) + int10119890119909minus21199101199063 (119910) 119889119910 = 119890119909+1 0 le 119909 lt 1 (41)
The exact solution of this equation is 119906(119909) = 119890119909 According to(41) we consider an initial approximation as 1199060(0) = 119906(0) = 1Numerical results are given in Table 6 by taking 119909119894 = (119894 minus1)(119899minus1) 119894 = 1 2 119899 and 119899 = 128 In Table 6 a comparison
between the absolute errors of the proposed method andthe Haar wavelet method [30] is given Figure 4 shows theapproximate solution and its error
5 Conclusion
According to this essay supplementary of iterative Reproduc-ing Kernel Hilbert SpaceMethod was introduced and appliedto acquire the approximate solution of some nonlinear inte-gral equation In this method unlike other similar methodsorthogonal process is not used However the time is increas-ing and accuracy is also increasing The main point whichis mentioned in this paper is that Algorithm 14 has higherexecution time in comparison with Algorithm 15 but theapproximate solution in Algorithm 14 is more accurate thanAlgorithm 15 Current uniform convergencemethod is statedand proved The obtained numerical results confirm that itis a good candidate for solving the nonlinear integral equa-tion
Conflicts of Interest
The authors declare no conflicts of interest
Mathematical Problems in Engineering 9
Table 6 The absolute errors comparison between the proposed approach and method [30]
119909 Exact solution Approximate solution Method in [30] Absolute error Absolute error in method [30]01 1105170918 1105495796 1111099473 0000324878 000592855502 1221402757 1220931139 1220296615 0000471618 000110614203 1349858806 1349502887 1339817084 0000355919 001004172204 1491824696 1491740269 1471965091 0000084427 001985960505 1648721268 1648649846 1457127691 0000071422 019159357706 1822118797 1822243106 1831888927 0000124309 000977013007 2013752703 2013650232 2011936556 0000102471 000181614708 2225540923 2224684503 2209678381 0000856420 001586254209 2459603104 2458626573 2426854676 0000976531 0032748428
Error of approximate solution
10 01 02 03 04 05 06 07 08 09
times10minus5
Approximate solution and the exact solution
10 01 02 03 04 05 06 07 08 09
ExactApproximate
1
12
14
16
18
2
22
24
26
28
minus10
minus8
minus6
minus4
minus2
0
4
2
Abso
lute
erro
r
x
Figure 4 The absolute errors comparison between the proposed approach and method [30]
Acknowledgments
The authors would like to thank Professor Haipeng Peng forinstructive comments and recommendations to improve thequality of this work and also the Islamic Azad UniversityScience and Research Branch Tehran for supporting thisproject
References
[1] K Meetz and W Engi Electromagnetische Felder SpringerBerlin Germany 1979
[2] W Thirring Lehrbuch der Mathematischen Physik vol 2Springer Vienna Austria 1978
[3] K F Warnick R H Selfridge and D V Arnold ldquoTeachingelectromagnetic field theory using differential formsrdquo IEEETransactions on Education vol 40 no 1 pp 53ndash68 1997
[4] S Abbasbandy ldquoHomotopy analysis method for heat radiationequationsrdquo International Communications in Heat and MassTransfer vol 34 no 3 pp 380ndash387 2007
[5] H Vosughi E Shivanian and S Abbasbandy ldquoA new analyticaltechnique to solve Volterrarsquos integral equationsrdquo MathematicalMethods in the Applied Sciences vol 34 no 10 pp 1243ndash12532011
[6] J-P Kauthen ldquoA survey of singularly perturbed Volterra equa-tionsrdquo Applied Numerical Mathematics An IMACS Journal vol24 no 2-3 pp 95ndash114 1997
[7] A A Kilbas and M Saigo ldquoOn solution of nonlinear Abel-Volterra integral equationrdquo Journal of Mathematical Analysisand Applications vol 229 no 1 pp 41ndash60 1999
[8] C Minggen and D Zhongxing ldquoOn the best operator ofinterpolationrdquo Mathematica Numerica Sinica vol 8 no 2 pp209ndash216 1986
[9] C Minggen and L Yingzhen Nonlinear Numerical Analysis inthe Reproducing Kernel Space Nova Science New York NYUSA 2009
[10] S Abbasbandy B Azarnavid and M S Alhuthali ldquoA shootingreproducing kernel Hilbert spacemethod formultiple solutionsof nonlinear boundary value problemsrdquo Journal of Computa-tional and Applied Mathematics vol 279 pp 293ndash305 2015
[11] N Shawagfeh O Abu Arqub and S Momani ldquoAnalytical solu-tion of nonlinear second-order periodic boundary value prob-lem using reproducing kernel methodrdquo Journal of Computa-tional Analysis andApplications vol 16 no 4 pp 750ndash762 2014
[12] FGeng andMCui ldquoSolving a nonlinear systemof second orderboundary value problemsrdquo Journal of Mathematical Analysisand Applications vol 327 no 2 pp 1167ndash1181 2007
[13] B Azarnavid F Parvaneh and S Abbasbandy ldquoPicard-repro-ducing kernel Hilbert space method for solving generalizedsingular nonlinear Lane-Emden type equationsrdquoMathematicalModelling and Analysis vol 20 no 6 pp 754ndash767 2015
[14] 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
10 Mathematical Problems in Engineering
[15] L Yang and M Cui ldquoNew algorithm for a class of nonlinearintegro-differential equations in the reproducing kernel spacerdquoApplied Mathematics and Computation vol 174 no 2 pp 942ndash960 2006
[16] 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
[17] S Bushnaq B Maayah S Momani and A Alsaedi ldquoA repro-ducing kernel Hilbert space method for solving systems offractional integrodifferential equationsrdquo Abstract and AppliedAnalysis vol 2014 Article ID 103016 6 pages 2014
[18] M Inc A Akgul and F Geng ldquoReproducing kernel Hilbertspace method for solving Bratursquos problemrdquo Bulletin of theMalaysian Mathematical Sciences Society vol 38 no 1 pp 271ndash287 2015
[19] A-M Wazwaz A First Course in Integral Equations WorldScientific 1997
[20] S Javadi E Babolian and E Moradi ldquoNew implementation ofreproducing kernel Hilbert space method for solving a class offunctional integral equationsrdquo Communications in NumericalAnalysis vol 2014 Article ID cna-00205 7 pages 2014
[21] W Jiang and Z Chen ldquoSolving a system of linear Volterraintegral equations using the new reproducing kernel methodrdquoApplied Mathematics and Computation vol 219 no 20 pp10225ndash10230 2013
[22] A-M Wazwaz Linear and Nonlinear Integral Equations Meth-ods and Applications Springer Berlin Germany 2011
[23] I L El-Kalla ldquoConvergence of the Adomian method appliedto a class of nonlinear integral equationsrdquo Applied MathematicsLetters An International Journal of Rapid Publication vol 21 no4 pp 372ndash376 2008
[24] G Gumah KMoaddy M Al-Smadi and I Hashim ldquoSolutionsto uncertain Volterra integral equations by fitted reproducingkernel Hilbert space methodrdquo Journal of Function Spaces vol2016 Article ID 2920463 11 pages 2016
[25] V Sizikov and D Sidorov ldquoGeneralized quadrature for solv-ing singular integral equations of Abel type in applicationto infrared tomographyrdquo Applied Numerical Mathematics AnIMACS Journal vol 106 pp 69ndash78 2016
[26] E Babolian Z Masouri and S Hatamzadeh-Varmazyar ldquoNewdirect method to solve nonlinear Volterra-Fredholm integraland integro-differential equations using operational matrixwith block-pulse functionsrdquo in Progress in ElectromagneticsResearch B vol 8 pp 59ndash76 EMW Publishing CambridgeMass USA 2008
[27] D Alpay Ed Reproducing kernel spaces and applications vol143 of Operator Theory Advances and Applications BirkhauserBasel 2003
[28] S Saitoh D Alpay J A Ball and T Ohsawa ReproducingKernels andTheir Applications vol 11 Springer Science amp Busi-ness Media Berlin Germany 2013
[29] S Saitoh Integral Transforms Reproducing Kernels and TheirApplications vol 369 of Pitman Research Notes in MathematicsSeries Longman Harlow UK 1997
[30] E Babolian andA Shahsavaran ldquoNumerical solution of nonlin-ear Fredholm integral equations of the second kind using Haarwaveletsrdquo Journal of Computational and Applied Mathematicsvol 225 no 1 pp 87ndash95 2009
[31] R Ketabchi R Mokhtari and E Babolian ldquoSome error esti-mates for solving Volterra integral equations by using the
reproducing kernel methodrdquo Journal of Computational andApplied Mathematics vol 273 pp 245ndash250 2015
[32] A Alvandi M Paripour and Z Roshani ldquoReproducing kernelmethod for solving a class of Fredholm integro-differentialequationsrdquo inProceedings of the 46thAnnual IranianMathemat-ics Conference (AIMC 46) p 505 Yazd University 2015
[33] I Komashynska andMAl-Smadi ldquoIterative reproducing kernelmethod for solving second-order integrodifferential equationsof fredholm typerdquo Journal of Applied Mathematics vol 2014Article ID 459509 11 pages 2014
[34] E Babolian S Javadi and E Moradi ldquoError analysis of repro-ducing kernel Hilbert space method for solving functional inte-gral equationsrdquo Journal of Computational and Applied Mathe-matics vol 300 pp 300ndash311 2016
Submit your manuscripts athttpswwwhindawicom
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 Mathematical Problems in Engineering
10
20
30
40
50
60
Error of approximate solutiontimes10minus5Approximate solution and the exact solution
0 01 02 03 04 05 06 07 08 09
ExactApproximate
0 02 03 04 05 06 07 08 0901 1x
0
01
02
03
04
05
06
07
08
minus8minus6minus4minus2
02468
Abso
lute
erro
r
Figure 3 The absolute errors comparison between the proposed approach and method [26]
Table 4 The absolute errors comparison between the proposed approach and method [26]
119909 Exact solution Approximate solution Method in [26] Absolute error Absolute error in method [26]01 0010000 0010762 0010308 0000762 000030802 0040000 0040655 0038140 0000655 000186003 0090000 0090444 0092828 0000444 000282804 0160000 016023 0158746 0000230 000125405 0250000 0250001 0257867 0000001 000786706 0360000 0359606 0361871 0000394 000187107 0490000 0489448 0483453 0000552 000654708 0640000 0639391 0647515 0000609 000751509 0810000 0811638 0807183 0001638 0002817
Table 5 The execution time (seconds) comparison between Algo-rithms 14 and 15
119899 Algorithm 14 (Sec) Algorithm 15 (Sec)4 31 2598 846 50916 342 2889
between the absolute errors of the proposed method and theBPFs method [26] is given Figure 3 shows the approximatesolution and its error In Table 5 a comparison execution timebetween Algorithms 14 and 15 is given
Example 4 A nonlinear Fredholm integral problem [26 30]is solved via our method for this applicable example
119906 (119909) + int10119890119909minus21199101199063 (119910) 119889119910 = 119890119909+1 0 le 119909 lt 1 (41)
The exact solution of this equation is 119906(119909) = 119890119909 According to(41) we consider an initial approximation as 1199060(0) = 119906(0) = 1Numerical results are given in Table 6 by taking 119909119894 = (119894 minus1)(119899minus1) 119894 = 1 2 119899 and 119899 = 128 In Table 6 a comparison
between the absolute errors of the proposed method andthe Haar wavelet method [30] is given Figure 4 shows theapproximate solution and its error
5 Conclusion
According to this essay supplementary of iterative Reproduc-ing Kernel Hilbert SpaceMethod was introduced and appliedto acquire the approximate solution of some nonlinear inte-gral equation In this method unlike other similar methodsorthogonal process is not used However the time is increas-ing and accuracy is also increasing The main point whichis mentioned in this paper is that Algorithm 14 has higherexecution time in comparison with Algorithm 15 but theapproximate solution in Algorithm 14 is more accurate thanAlgorithm 15 Current uniform convergencemethod is statedand proved The obtained numerical results confirm that itis a good candidate for solving the nonlinear integral equa-tion
Conflicts of Interest
The authors declare no conflicts of interest
Mathematical Problems in Engineering 9
Table 6 The absolute errors comparison between the proposed approach and method [30]
119909 Exact solution Approximate solution Method in [30] Absolute error Absolute error in method [30]01 1105170918 1105495796 1111099473 0000324878 000592855502 1221402757 1220931139 1220296615 0000471618 000110614203 1349858806 1349502887 1339817084 0000355919 001004172204 1491824696 1491740269 1471965091 0000084427 001985960505 1648721268 1648649846 1457127691 0000071422 019159357706 1822118797 1822243106 1831888927 0000124309 000977013007 2013752703 2013650232 2011936556 0000102471 000181614708 2225540923 2224684503 2209678381 0000856420 001586254209 2459603104 2458626573 2426854676 0000976531 0032748428
Error of approximate solution
10 01 02 03 04 05 06 07 08 09
times10minus5
Approximate solution and the exact solution
10 01 02 03 04 05 06 07 08 09
ExactApproximate
1
12
14
16
18
2
22
24
26
28
minus10
minus8
minus6
minus4
minus2
0
4
2
Abso
lute
erro
r
x
Figure 4 The absolute errors comparison between the proposed approach and method [30]
Acknowledgments
The authors would like to thank Professor Haipeng Peng forinstructive comments and recommendations to improve thequality of this work and also the Islamic Azad UniversityScience and Research Branch Tehran for supporting thisproject
References
[1] K Meetz and W Engi Electromagnetische Felder SpringerBerlin Germany 1979
[2] W Thirring Lehrbuch der Mathematischen Physik vol 2Springer Vienna Austria 1978
[3] K F Warnick R H Selfridge and D V Arnold ldquoTeachingelectromagnetic field theory using differential formsrdquo IEEETransactions on Education vol 40 no 1 pp 53ndash68 1997
[4] S Abbasbandy ldquoHomotopy analysis method for heat radiationequationsrdquo International Communications in Heat and MassTransfer vol 34 no 3 pp 380ndash387 2007
[5] H Vosughi E Shivanian and S Abbasbandy ldquoA new analyticaltechnique to solve Volterrarsquos integral equationsrdquo MathematicalMethods in the Applied Sciences vol 34 no 10 pp 1243ndash12532011
[6] J-P Kauthen ldquoA survey of singularly perturbed Volterra equa-tionsrdquo Applied Numerical Mathematics An IMACS Journal vol24 no 2-3 pp 95ndash114 1997
[7] A A Kilbas and M Saigo ldquoOn solution of nonlinear Abel-Volterra integral equationrdquo Journal of Mathematical Analysisand Applications vol 229 no 1 pp 41ndash60 1999
[8] C Minggen and D Zhongxing ldquoOn the best operator ofinterpolationrdquo Mathematica Numerica Sinica vol 8 no 2 pp209ndash216 1986
[9] C Minggen and L Yingzhen Nonlinear Numerical Analysis inthe Reproducing Kernel Space Nova Science New York NYUSA 2009
[10] S Abbasbandy B Azarnavid and M S Alhuthali ldquoA shootingreproducing kernel Hilbert spacemethod formultiple solutionsof nonlinear boundary value problemsrdquo Journal of Computa-tional and Applied Mathematics vol 279 pp 293ndash305 2015
[11] N Shawagfeh O Abu Arqub and S Momani ldquoAnalytical solu-tion of nonlinear second-order periodic boundary value prob-lem using reproducing kernel methodrdquo Journal of Computa-tional Analysis andApplications vol 16 no 4 pp 750ndash762 2014
[12] FGeng andMCui ldquoSolving a nonlinear systemof second orderboundary value problemsrdquo Journal of Mathematical Analysisand Applications vol 327 no 2 pp 1167ndash1181 2007
[13] B Azarnavid F Parvaneh and S Abbasbandy ldquoPicard-repro-ducing kernel Hilbert space method for solving generalizedsingular nonlinear Lane-Emden type equationsrdquoMathematicalModelling and Analysis vol 20 no 6 pp 754ndash767 2015
[14] 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
10 Mathematical Problems in Engineering
[15] L Yang and M Cui ldquoNew algorithm for a class of nonlinearintegro-differential equations in the reproducing kernel spacerdquoApplied Mathematics and Computation vol 174 no 2 pp 942ndash960 2006
[16] 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
[17] S Bushnaq B Maayah S Momani and A Alsaedi ldquoA repro-ducing kernel Hilbert space method for solving systems offractional integrodifferential equationsrdquo Abstract and AppliedAnalysis vol 2014 Article ID 103016 6 pages 2014
[18] M Inc A Akgul and F Geng ldquoReproducing kernel Hilbertspace method for solving Bratursquos problemrdquo Bulletin of theMalaysian Mathematical Sciences Society vol 38 no 1 pp 271ndash287 2015
[19] A-M Wazwaz A First Course in Integral Equations WorldScientific 1997
[20] S Javadi E Babolian and E Moradi ldquoNew implementation ofreproducing kernel Hilbert space method for solving a class offunctional integral equationsrdquo Communications in NumericalAnalysis vol 2014 Article ID cna-00205 7 pages 2014
[21] W Jiang and Z Chen ldquoSolving a system of linear Volterraintegral equations using the new reproducing kernel methodrdquoApplied Mathematics and Computation vol 219 no 20 pp10225ndash10230 2013
[22] A-M Wazwaz Linear and Nonlinear Integral Equations Meth-ods and Applications Springer Berlin Germany 2011
[23] I L El-Kalla ldquoConvergence of the Adomian method appliedto a class of nonlinear integral equationsrdquo Applied MathematicsLetters An International Journal of Rapid Publication vol 21 no4 pp 372ndash376 2008
[24] G Gumah KMoaddy M Al-Smadi and I Hashim ldquoSolutionsto uncertain Volterra integral equations by fitted reproducingkernel Hilbert space methodrdquo Journal of Function Spaces vol2016 Article ID 2920463 11 pages 2016
[25] V Sizikov and D Sidorov ldquoGeneralized quadrature for solv-ing singular integral equations of Abel type in applicationto infrared tomographyrdquo Applied Numerical Mathematics AnIMACS Journal vol 106 pp 69ndash78 2016
[26] E Babolian Z Masouri and S Hatamzadeh-Varmazyar ldquoNewdirect method to solve nonlinear Volterra-Fredholm integraland integro-differential equations using operational matrixwith block-pulse functionsrdquo in Progress in ElectromagneticsResearch B vol 8 pp 59ndash76 EMW Publishing CambridgeMass USA 2008
[27] D Alpay Ed Reproducing kernel spaces and applications vol143 of Operator Theory Advances and Applications BirkhauserBasel 2003
[28] S Saitoh D Alpay J A Ball and T Ohsawa ReproducingKernels andTheir Applications vol 11 Springer Science amp Busi-ness Media Berlin Germany 2013
[29] S Saitoh Integral Transforms Reproducing Kernels and TheirApplications vol 369 of Pitman Research Notes in MathematicsSeries Longman Harlow UK 1997
[30] E Babolian andA Shahsavaran ldquoNumerical solution of nonlin-ear Fredholm integral equations of the second kind using Haarwaveletsrdquo Journal of Computational and Applied Mathematicsvol 225 no 1 pp 87ndash95 2009
[31] R Ketabchi R Mokhtari and E Babolian ldquoSome error esti-mates for solving Volterra integral equations by using the
reproducing kernel methodrdquo Journal of Computational andApplied Mathematics vol 273 pp 245ndash250 2015
[32] A Alvandi M Paripour and Z Roshani ldquoReproducing kernelmethod for solving a class of Fredholm integro-differentialequationsrdquo inProceedings of the 46thAnnual IranianMathemat-ics Conference (AIMC 46) p 505 Yazd University 2015
[33] I Komashynska andMAl-Smadi ldquoIterative reproducing kernelmethod for solving second-order integrodifferential equationsof fredholm typerdquo Journal of Applied Mathematics vol 2014Article ID 459509 11 pages 2014
[34] E Babolian S Javadi and E Moradi ldquoError analysis of repro-ducing kernel Hilbert space method for solving functional inte-gral equationsrdquo Journal of Computational and Applied Mathe-matics vol 300 pp 300ndash311 2016
Submit your manuscripts athttpswwwhindawicom
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
Mathematical Problems in Engineering 9
Table 6 The absolute errors comparison between the proposed approach and method [30]
119909 Exact solution Approximate solution Method in [30] Absolute error Absolute error in method [30]01 1105170918 1105495796 1111099473 0000324878 000592855502 1221402757 1220931139 1220296615 0000471618 000110614203 1349858806 1349502887 1339817084 0000355919 001004172204 1491824696 1491740269 1471965091 0000084427 001985960505 1648721268 1648649846 1457127691 0000071422 019159357706 1822118797 1822243106 1831888927 0000124309 000977013007 2013752703 2013650232 2011936556 0000102471 000181614708 2225540923 2224684503 2209678381 0000856420 001586254209 2459603104 2458626573 2426854676 0000976531 0032748428
Error of approximate solution
10 01 02 03 04 05 06 07 08 09
times10minus5
Approximate solution and the exact solution
10 01 02 03 04 05 06 07 08 09
ExactApproximate
1
12
14
16
18
2
22
24
26
28
minus10
minus8
minus6
minus4
minus2
0
4
2
Abso
lute
erro
r
x
Figure 4 The absolute errors comparison between the proposed approach and method [30]
Acknowledgments
The authors would like to thank Professor Haipeng Peng forinstructive comments and recommendations to improve thequality of this work and also the Islamic Azad UniversityScience and Research Branch Tehran for supporting thisproject
References
[1] K Meetz and W Engi Electromagnetische Felder SpringerBerlin Germany 1979
[2] W Thirring Lehrbuch der Mathematischen Physik vol 2Springer Vienna Austria 1978
[3] K F Warnick R H Selfridge and D V Arnold ldquoTeachingelectromagnetic field theory using differential formsrdquo IEEETransactions on Education vol 40 no 1 pp 53ndash68 1997
[4] S Abbasbandy ldquoHomotopy analysis method for heat radiationequationsrdquo International Communications in Heat and MassTransfer vol 34 no 3 pp 380ndash387 2007
[5] H Vosughi E Shivanian and S Abbasbandy ldquoA new analyticaltechnique to solve Volterrarsquos integral equationsrdquo MathematicalMethods in the Applied Sciences vol 34 no 10 pp 1243ndash12532011
[6] J-P Kauthen ldquoA survey of singularly perturbed Volterra equa-tionsrdquo Applied Numerical Mathematics An IMACS Journal vol24 no 2-3 pp 95ndash114 1997
[7] A A Kilbas and M Saigo ldquoOn solution of nonlinear Abel-Volterra integral equationrdquo Journal of Mathematical Analysisand Applications vol 229 no 1 pp 41ndash60 1999
[8] C Minggen and D Zhongxing ldquoOn the best operator ofinterpolationrdquo Mathematica Numerica Sinica vol 8 no 2 pp209ndash216 1986
[9] C Minggen and L Yingzhen Nonlinear Numerical Analysis inthe Reproducing Kernel Space Nova Science New York NYUSA 2009
[10] S Abbasbandy B Azarnavid and M S Alhuthali ldquoA shootingreproducing kernel Hilbert spacemethod formultiple solutionsof nonlinear boundary value problemsrdquo Journal of Computa-tional and Applied Mathematics vol 279 pp 293ndash305 2015
[11] N Shawagfeh O Abu Arqub and S Momani ldquoAnalytical solu-tion of nonlinear second-order periodic boundary value prob-lem using reproducing kernel methodrdquo Journal of Computa-tional Analysis andApplications vol 16 no 4 pp 750ndash762 2014
[12] FGeng andMCui ldquoSolving a nonlinear systemof second orderboundary value problemsrdquo Journal of Mathematical Analysisand Applications vol 327 no 2 pp 1167ndash1181 2007
[13] B Azarnavid F Parvaneh and S Abbasbandy ldquoPicard-repro-ducing kernel Hilbert space method for solving generalizedsingular nonlinear Lane-Emden type equationsrdquoMathematicalModelling and Analysis vol 20 no 6 pp 754ndash767 2015
[14] 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
10 Mathematical Problems in Engineering
[15] L Yang and M Cui ldquoNew algorithm for a class of nonlinearintegro-differential equations in the reproducing kernel spacerdquoApplied Mathematics and Computation vol 174 no 2 pp 942ndash960 2006
[16] 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
[17] S Bushnaq B Maayah S Momani and A Alsaedi ldquoA repro-ducing kernel Hilbert space method for solving systems offractional integrodifferential equationsrdquo Abstract and AppliedAnalysis vol 2014 Article ID 103016 6 pages 2014
[18] M Inc A Akgul and F Geng ldquoReproducing kernel Hilbertspace method for solving Bratursquos problemrdquo Bulletin of theMalaysian Mathematical Sciences Society vol 38 no 1 pp 271ndash287 2015
[19] A-M Wazwaz A First Course in Integral Equations WorldScientific 1997
[20] S Javadi E Babolian and E Moradi ldquoNew implementation ofreproducing kernel Hilbert space method for solving a class offunctional integral equationsrdquo Communications in NumericalAnalysis vol 2014 Article ID cna-00205 7 pages 2014
[21] W Jiang and Z Chen ldquoSolving a system of linear Volterraintegral equations using the new reproducing kernel methodrdquoApplied Mathematics and Computation vol 219 no 20 pp10225ndash10230 2013
[22] A-M Wazwaz Linear and Nonlinear Integral Equations Meth-ods and Applications Springer Berlin Germany 2011
[23] I L El-Kalla ldquoConvergence of the Adomian method appliedto a class of nonlinear integral equationsrdquo Applied MathematicsLetters An International Journal of Rapid Publication vol 21 no4 pp 372ndash376 2008
[24] G Gumah KMoaddy M Al-Smadi and I Hashim ldquoSolutionsto uncertain Volterra integral equations by fitted reproducingkernel Hilbert space methodrdquo Journal of Function Spaces vol2016 Article ID 2920463 11 pages 2016
[25] V Sizikov and D Sidorov ldquoGeneralized quadrature for solv-ing singular integral equations of Abel type in applicationto infrared tomographyrdquo Applied Numerical Mathematics AnIMACS Journal vol 106 pp 69ndash78 2016
[26] E Babolian Z Masouri and S Hatamzadeh-Varmazyar ldquoNewdirect method to solve nonlinear Volterra-Fredholm integraland integro-differential equations using operational matrixwith block-pulse functionsrdquo in Progress in ElectromagneticsResearch B vol 8 pp 59ndash76 EMW Publishing CambridgeMass USA 2008
[27] D Alpay Ed Reproducing kernel spaces and applications vol143 of Operator Theory Advances and Applications BirkhauserBasel 2003
[28] S Saitoh D Alpay J A Ball and T Ohsawa ReproducingKernels andTheir Applications vol 11 Springer Science amp Busi-ness Media Berlin Germany 2013
[29] S Saitoh Integral Transforms Reproducing Kernels and TheirApplications vol 369 of Pitman Research Notes in MathematicsSeries Longman Harlow UK 1997
[30] E Babolian andA Shahsavaran ldquoNumerical solution of nonlin-ear Fredholm integral equations of the second kind using Haarwaveletsrdquo Journal of Computational and Applied Mathematicsvol 225 no 1 pp 87ndash95 2009
[31] R Ketabchi R Mokhtari and E Babolian ldquoSome error esti-mates for solving Volterra integral equations by using the
reproducing kernel methodrdquo Journal of Computational andApplied Mathematics vol 273 pp 245ndash250 2015
[32] A Alvandi M Paripour and Z Roshani ldquoReproducing kernelmethod for solving a class of Fredholm integro-differentialequationsrdquo inProceedings of the 46thAnnual IranianMathemat-ics Conference (AIMC 46) p 505 Yazd University 2015
[33] I Komashynska andMAl-Smadi ldquoIterative reproducing kernelmethod for solving second-order integrodifferential equationsof fredholm typerdquo Journal of Applied Mathematics vol 2014Article ID 459509 11 pages 2014
[34] E Babolian S Javadi and E Moradi ldquoError analysis of repro-ducing kernel Hilbert space method for solving functional inte-gral equationsrdquo Journal of Computational and Applied Mathe-matics vol 300 pp 300ndash311 2016
Submit your manuscripts athttpswwwhindawicom
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
10 Mathematical Problems in Engineering
[15] L Yang and M Cui ldquoNew algorithm for a class of nonlinearintegro-differential equations in the reproducing kernel spacerdquoApplied Mathematics and Computation vol 174 no 2 pp 942ndash960 2006
[16] 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
[17] S Bushnaq B Maayah S Momani and A Alsaedi ldquoA repro-ducing kernel Hilbert space method for solving systems offractional integrodifferential equationsrdquo Abstract and AppliedAnalysis vol 2014 Article ID 103016 6 pages 2014
[18] M Inc A Akgul and F Geng ldquoReproducing kernel Hilbertspace method for solving Bratursquos problemrdquo Bulletin of theMalaysian Mathematical Sciences Society vol 38 no 1 pp 271ndash287 2015
[19] A-M Wazwaz A First Course in Integral Equations WorldScientific 1997
[20] S Javadi E Babolian and E Moradi ldquoNew implementation ofreproducing kernel Hilbert space method for solving a class offunctional integral equationsrdquo Communications in NumericalAnalysis vol 2014 Article ID cna-00205 7 pages 2014
[21] W Jiang and Z Chen ldquoSolving a system of linear Volterraintegral equations using the new reproducing kernel methodrdquoApplied Mathematics and Computation vol 219 no 20 pp10225ndash10230 2013
[22] A-M Wazwaz Linear and Nonlinear Integral Equations Meth-ods and Applications Springer Berlin Germany 2011
[23] I L El-Kalla ldquoConvergence of the Adomian method appliedto a class of nonlinear integral equationsrdquo Applied MathematicsLetters An International Journal of Rapid Publication vol 21 no4 pp 372ndash376 2008
[24] G Gumah KMoaddy M Al-Smadi and I Hashim ldquoSolutionsto uncertain Volterra integral equations by fitted reproducingkernel Hilbert space methodrdquo Journal of Function Spaces vol2016 Article ID 2920463 11 pages 2016
[25] V Sizikov and D Sidorov ldquoGeneralized quadrature for solv-ing singular integral equations of Abel type in applicationto infrared tomographyrdquo Applied Numerical Mathematics AnIMACS Journal vol 106 pp 69ndash78 2016
[26] E Babolian Z Masouri and S Hatamzadeh-Varmazyar ldquoNewdirect method to solve nonlinear Volterra-Fredholm integraland integro-differential equations using operational matrixwith block-pulse functionsrdquo in Progress in ElectromagneticsResearch B vol 8 pp 59ndash76 EMW Publishing CambridgeMass USA 2008
[27] D Alpay Ed Reproducing kernel spaces and applications vol143 of Operator Theory Advances and Applications BirkhauserBasel 2003
[28] S Saitoh D Alpay J A Ball and T Ohsawa ReproducingKernels andTheir Applications vol 11 Springer Science amp Busi-ness Media Berlin Germany 2013
[29] S Saitoh Integral Transforms Reproducing Kernels and TheirApplications vol 369 of Pitman Research Notes in MathematicsSeries Longman Harlow UK 1997
[30] E Babolian andA Shahsavaran ldquoNumerical solution of nonlin-ear Fredholm integral equations of the second kind using Haarwaveletsrdquo Journal of Computational and Applied Mathematicsvol 225 no 1 pp 87ndash95 2009
[31] R Ketabchi R Mokhtari and E Babolian ldquoSome error esti-mates for solving Volterra integral equations by using the
reproducing kernel methodrdquo Journal of Computational andApplied Mathematics vol 273 pp 245ndash250 2015
[32] A Alvandi M Paripour and Z Roshani ldquoReproducing kernelmethod for solving a class of Fredholm integro-differentialequationsrdquo inProceedings of the 46thAnnual IranianMathemat-ics Conference (AIMC 46) p 505 Yazd University 2015
[33] I Komashynska andMAl-Smadi ldquoIterative reproducing kernelmethod for solving second-order integrodifferential equationsof fredholm typerdquo Journal of Applied Mathematics vol 2014Article ID 459509 11 pages 2014
[34] E Babolian S Javadi and E Moradi ldquoError analysis of repro-ducing kernel Hilbert space method for solving functional inte-gral equationsrdquo Journal of Computational and Applied Mathe-matics vol 300 pp 300ndash311 2016
Submit your manuscripts athttpswwwhindawicom
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 athttpswwwhindawicom
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