Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volumeย 2012ย (2012), Article IDย 839836, 16 pages
http://dx.doi.org/10.1155/2012/839836
Research Article

Application of Reproducing Kernel Method for Solving Nonlinear Fredholm-Volterra Integrodifferential Equations

1Department of Mathematics, Al-Balqa Applied University, Salt 19117, Jordan
2Department of Mathematics and Information Technology, Tafila Technical University, Tafila 66110, Jordan
3Department of Mathematics, University of Jordan, Amman 11942, Jordan

Received 24 June 2012; Revised 24 July 2012; Accepted 24 July 2012

Academic Editor: Irenaย Lasiecka

Copyright ยฉ 2012 Omar Abu Arqub et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper investigates the numerical solution of nonlinear Fredholm-Volterra integro-differential equations using reproducing kernel Hilbert space method. The solution ๐‘ข(๐‘ฅ) is represented in the form of series in the reproducing kernel space. In the mean time, the n-term approximate solution ๐‘ข๐‘›(๐‘ฅ) is obtained and it is proved to converge to the exact solution ๐‘ข(๐‘ฅ). Furthermore, the proposed method has an advantage that it is possible to pick any point in the interval of integration and as well the approximate solution and its derivative will be applicable. Numerical examples are included to demonstrate the accuracy and applicability of the presented technique. The results reveal that the method is very effective and simple.

1. Introduction

In recent years, there has been a growing interest in the integrodifferential equations (IDEs) which are a combination of differential and Fredholm-Volterra integral equations. IDEs are often involved in the mathematical formulation of physical phenomena. IDEs can be encountered in various fields of science such as physics, biology, and engineering. These kinds of equations can also be found in numerous applications, such as biomechanics, electromagnetic, elasticity, electrodynamics, fluid dynamics, heat and mass transfer, and oscillation theory [1โ€“4].

The purpose of this paper is to extend the application of the reproducing kernel Hilbert space (RKHS) method to solve the nonlinear Fredholm-Volterra IDE which is as follows: ๐‘ข๎…ž๎€œ(๐‘ฅ)+๐‘“(๐‘ฅ)๐‘(๐‘ฅ,๐‘ข(๐‘ฅ))+๐‘๐‘Ž๐‘˜1(๐‘ฅ,๐‘ก)๐บ1๎€œ(๐‘ข(๐‘ก))๐‘‘๐‘ก+๐‘ฅ๐‘Ž๐‘˜2(๐‘ฅ,๐‘ก)๐บ2(๐‘ข(๐‘ก))๐‘‘๐‘ก=๐‘”(๐‘ฅ),๐‘Žโ‰ค๐‘ฅ,๐‘กโ‰ค๐‘,(1.1) subject to the initial condition ๐‘ข(๐‘Ž)=๐›ผ,(1.2) where ๐‘Ž,๐‘,๐›ผ are real finite constants, ๐‘ขโˆˆ๐‘Š22[๐‘Ž,๐‘] is an unknown function to be determined, ๐‘“,๐‘”โˆˆ๐‘Š12[๐‘Ž,๐‘], ๐‘˜1(๐‘ฅ,๐‘ก), ๐‘˜2(๐‘ฅ,๐‘ก) are continuous functions on [๐‘Ž,๐‘]ร—[๐‘Ž,๐‘], ๐บ1(๐‘ค), ๐บ2(๐‘ฆ), ๐‘(๐‘ฅ,๐‘ง) are continuous terms in ๐‘Š12[๐‘Ž,๐‘] as ๐‘ค=๐‘ค(๐‘ฅ), ๐‘ฆ=๐‘ฆ(๐‘ฅ), ๐‘ง=๐‘ง(๐‘ฅ)โˆˆ๐‘Š22[๐‘Ž,๐‘], ๐‘Žโ‰ค๐‘ฅโ‰ค๐‘, โˆ’โˆž<๐‘ค,๐‘ฆ,๐‘ง<โˆž and are depending on the problem discussed, and ๐‘Š12[๐‘Ž,๐‘],๐‘Š22[๐‘Ž,๐‘] are reproducing kernel spaces.

In general, nonlinear Fredholm-Volterra IDEs do not always have solutions which we can obtain using analytical methods. In fact, many of real physical phenomena encountered are almost impossible to solve by this technique. Due to this, some authors have proposed numerical methods to approximate the solutions of nonlinear Fredholm-Volterra IDEs. To mention a few, in [5] the authors have discussed the Taylor polynomial method for solving IDEs (1.1) and (1.2) when ๐‘(๐‘ฅ,๐‘ข(๐‘ฅ))=1, ๐บ1(๐‘ข(๐‘ก))=๐‘ข(๐‘ก), and ๐บ2(๐‘ข(๐‘ก))=[๐‘ข(๐‘ก)]๐‘ž, where ๐‘žโˆˆโ„•. The triangular functions method has been applied to solve the same equations when ๐‘(๐‘ฅ,๐‘ข(๐‘ฅ))=๐‘ข(๐‘ฅ), ๐บ1(๐‘ข(๐‘ก))=[๐‘ข(๐‘ก)]๐‘, and ๐บ2(๐‘ข(๐‘ก))=[๐‘ข(๐‘ก)]๐‘ž, where ๐‘, ๐‘žโˆˆโ„• as described in [6]. Furthermore, the operational matrix with block-pulse functions method is carried out in [7] for the aforementioned IDEs in the case ๐‘(๐‘ฅ,๐‘ข(๐‘ฅ))=1, ๐บ1(๐‘ข(๐‘ก))=[๐‘ข(๐‘ก)]๐‘, and ๐บ2(๐‘ข(๐‘ก))=[๐‘ข(๐‘ก)]๐‘ž, where ๐‘,๐‘žโˆˆโ„•. Recently, the Hybrid Legendre polynomials and Block-Pulse functions approach for solving IDEs (1.1) and (1.2) when ๐‘(๐‘ฅ,๐‘ข(๐‘ฅ))=1, ๐บ1(๐‘ข(๐‘ก))=[๐‘ข(๐‘ก)]๐‘, and ๐บ2(๐‘ข(๐‘ก))=[๐‘ข(๐‘ก)]๐‘ž, where ๐‘,๐‘žโˆˆโ„• are proposed in [8]. The numerical solvability of Fredholm and Volterra IDEs and other related equations can be found in [9โ€“11] and references therein. However, none of previous studies propose a methodical way to solve these equations. Moreover, previous studies require more effort to achieve the results, they are not accurate, and usually they are developed for special types of IDEs (1.1) and (1.2).

Reproducing kernel theory has important application in numerical analysis, differential equations, integral equations, probability and statistics, and so on [12โ€“14]. Recently, using the RKHS method, the authors in [15โ€“29] have discussed singular linear two-point boundary value problems, singular nonlinear two-point periodic boundary value problems, nonlinear system of boundary value problems, initial value problems, singular integral equations, nonlinear partial differential equations, operator equations, and fourth-order IDEs.

The outline of the paper is as follows: several reproducing kernel spaces are described in Section 2. In Section 3, a linear operator, a complete normal orthogonal system, and some essential results are introduced. Also, a method for the existence of solutions for (1.1) and (1.2) based on reproducing kernel space is described. In Section 4, we give an iterative method to solve (1.1) and (1.2) numerically in RKHS. Various numerical examples are presented in Section 5. This paper ends in Section 6 with some concluding remarks.

2. Several Reproducing Kernel Spaces

In this section, several reproducing kernels needed are constructed in order to solve (1.1) and (1.2) using RKHS method. Before the construction, we utilize the reproducing kernel concept. Throughout this paper โ„‚ is the set of complex numbers, ๐ฟ2โˆซ[๐‘Ž,๐‘]={๐‘ขโˆฃ๐‘๐‘Ž๐‘ข2(๐‘ฅ)๐‘‘๐‘ฅ<โˆž}, ๐‘™2โˆ‘={๐ดโˆฃโˆž๐‘–=1(๐ด๐‘–)2<โˆž}, and the superscript (๐‘›) in ๐‘ข(๐‘›)(๐‘ก) denotes the ๐‘›-th derivative of ๐‘ข(๐‘ก).

Definition 2.1 (see [18]). Let ๐ธ be a nonempty abstract set. A function ๐พโˆถ๐ธร—๐ธโ†’โ„‚ is a reproducing kernel of the Hilbert space ๐ป if(1)for each ๐‘กโˆˆ๐ธ, ๐พ(โ‹…,๐‘ก)โˆˆ๐ป,(2)for each ๐‘กโˆˆ๐ธ and ๐œ‘โˆˆ๐ป, โŸจ๐œ‘(โ‹…),๐พ(โ‹…,๐‘ก)โŸฉ=๐œ‘(๐‘ก).

The last condition is called โ€œthe reproducing propertyโ€: the value of the function ๐œ‘ at the point ๐‘ก is reproducing by the inner product of ๐œ‘ with ๐พ(โ‹…,๐‘ก). A Hilbert space which possesses a reproducing kernel is called a RKHS [18].

Next, we first construct the space ๐‘Š22[๐‘Ž,๐‘] in which every function satisfies the initial condition (1.2) and then utilize the space ๐‘Š12[๐‘Ž,๐‘].

Definition 2.2 (see [30]). ๐‘Š22[๐‘Ž,๐‘]={๐‘ขโˆถ๐‘ข,๐‘ข๎…ž are absolutely continuous on [๐‘Ž,๐‘], ๐‘ข,๐‘ข๎…ž,๐‘ข๎…ž๎…žโˆˆ๐ฟ2[๐‘Ž,๐‘], and ๐‘ข(๐‘Ž)=0}. The inner product and the norm in ๐‘Š22[๐‘Ž,๐‘] are defined, respectively, by โŸจ๐‘ข,๐‘ฃโŸฉ๐‘Š22=๐‘ข(๐‘Ž)๐‘ฃ(๐‘Ž)+๐‘ข๎…ž(๐‘Ž)๐‘ฃ๎…ž๎€œ(๐‘Ž)+๐‘๐‘Ž๐‘ข๎…ž๎…ž(๐‘ฆ)๐‘ฃ๎…ž๎…ž(๐‘ฆ)๐‘‘๐‘ฆ(2.1) and โ€–๐‘ขโ€–๐‘Š22=๎”โŸจ๐‘ข,๐‘ขโŸฉ๐‘Š22, where ๐‘ข,๐‘ฃโˆˆ๐‘Š22[๐‘Ž,๐‘].

Definition 2.3 (see [23]). ๐‘Š12[๐‘Ž,๐‘]={๐‘ขโˆถ๐‘ข is absolutely continuous on [๐‘Ž,๐‘] and ๐‘ข,๐‘ข๎…žโˆˆ๐ฟ2[๐‘Ž,๐‘]}. The inner product and the norm in ๐‘Š12[๐‘Ž,๐‘] are defined, respectively, by โŸจ๐‘ข,๐‘ฃโŸฉ๐‘Š12=โˆซ๐‘๐‘Ž๐‘ข(๐‘ก)๐‘ฃ(๐‘ก)+๐‘ข๎…ž(๐‘ก)๐‘ฃ๎…ž(๐‘ก)๐‘‘๐‘ก and โ€–๐‘ขโ€–๐‘Š12=๎”โŸจ๐‘ข,๐‘ขโŸฉ๐‘Š12, where ๐‘ข,๐‘ฃโˆˆ๐‘Š12[๐‘Ž,๐‘].

In [23], the authors have proved that the space๐‘Š12[๐‘Ž,๐‘] is a complete reproducing kernel space and its reproducing kernel function is given by ๐‘‡๐‘ฅ1(๐‘ฆ)=๎€บ๎€ท||||.2sinh(๐‘โˆ’๐‘Ž)cosh(๐‘ฅ+๐‘ฆโˆ’๐‘โˆ’๐‘Ž)+cosh๐‘ฅโˆ’๐‘ฆโˆ’๐‘+๐‘Ž๎€ธ๎€ป(2.2) From the definition of the reproducing kernel spaces ๐‘Š12[๐‘Ž,๐‘] and ๐‘Š22[๐‘Ž,๐‘], we get ๐‘Š12[๐‘Ž,๐‘]โŠƒ๐‘Š22[๐‘Ž,๐‘].

The Hilbert space ๐‘Š22[๐‘Ž,๐‘] is called a reproducing kernel if for each fixed ๐‘ฅโˆˆ[๐‘Ž,๐‘] and any ๐‘ข(๐‘ฆ)โˆˆ๐‘Š22[๐‘Ž,๐‘], there exist ๐พ(๐‘ฅ,๐‘ฆ)โˆˆ๐‘Š22[๐‘Ž,๐‘] (simply ๐พ๐‘ฅ(๐‘ฆ)) and ๐‘ฆโˆˆ[๐‘Ž,๐‘] such that โŸจ๐‘ข(๐‘ฆ),๐พ๐‘ฅ(๐‘ฆ)โŸฉ๐‘Š22=๐‘ข(๐‘ฅ). The next theorem formulates the reproducing kernel function ๐พ๐‘ฅ(๐‘ฆ).

Theorem 2.4. The Hilbert space ๐‘Š22[๐‘Ž,๐‘] is a reproducing kernel and its reproducing kernel function ๐พ๐‘ฅ(๐‘ฆ) can be written as ๐พ๐‘ฅ(โŽงโŽชโŽชโŽจโŽชโŽชโŽฉ๐‘ฆ)=4โˆ‘๐‘–=1๐‘๐‘–(๐‘ฅ)๐‘ฆ๐‘–โˆ’1,๐‘ฆโ‰ค๐‘ฅ,4โˆ‘๐‘–=1๐‘ž๐‘–(๐‘ฅ)๐‘ฆ๐‘–โˆ’1,๐‘ฆ>๐‘ฅ,(2.3) where ๐‘๐‘–(๐‘ฅ) and ๐‘ž๐‘–(๐‘ฅ) are unknown coefficients of ๐พ๐‘ฅ(๐‘ฆ).

Proof. Through several integrations by parts for (2.1), we obtain โŸจ๐‘ข(๐‘ฆ),๐พ๐‘ฅ(๐‘ฆ)โŸฉ๐‘Š22=โˆ‘1๐‘–=0๐‘ข(๐‘–)(๐‘Ž)(๐พ๐‘ฅ(๐‘–)(๐‘Ž)+(โˆ’1)๐‘–๐พ๐‘ฅ(3โˆ’๐‘–)โˆ‘(๐‘Ž))+1๐‘–=0(โˆ’1)1โˆ’๐‘–๐‘ข(๐‘–)(๐‘)๐พ๐‘ฅ(3โˆ’๐‘–)โˆซ(๐‘)+๐‘๐‘Ž๐‘ข(๐‘ฆ)๐พ๐‘ฅ(4)(๐‘ฆ)๐‘‘๐‘ฆ. Since ๐พ๐‘ฅ(๐‘ฆ)โˆˆ๐‘Š22[๐‘Ž,๐‘], it follows that ๐พ๐‘ฅ(๐‘Ž)=0. Also, since ๐‘ขโˆˆ๐‘Š22[๐‘Ž,๐‘], one obtains ๐‘ข(๐‘Ž)=0. Thus, if ๐พ๐‘ฅ(๐‘–)(๐‘)=0, ๐‘–=2,3, and ๐พ๎…ž๐‘ฅ(๐‘Ž)โˆ’๐พ๐‘ฅ๎…ž๎…ž(๐‘Ž)=0, then โŸจ๐‘ข(๐‘ฆ),๐พ๐‘ฅ(๐‘ฆ)โŸฉ๐‘Š22=โˆซ๐‘๐‘Ž๐‘ข(๐‘ฆ)๐พ๐‘ฅ(4)(๐‘ฆ)๐‘‘๐‘ฆ. Now, for each ๐‘ฅโˆˆ[๐‘Ž,๐‘], if ๐พ๐‘ฅ(๐‘ฆ) also satisfies ๐พ๐‘ฅ(4)(๐‘ฆ)=๐›ฟ(๐‘ฅโˆ’๐‘ฆ), where ๐›ฟ is the dirac-delta function, then โŸจ๐‘ข(๐‘ฆ),๐พ๐‘ฅ(๐‘ฆ)โŸฉ๐‘Š22=๐‘ข(๐‘ฅ). Obviously, ๐พ๐‘ฅ(๐‘ฆ) is the reproducing kernel function of the space ๐‘Š22[๐‘Ž,๐‘].
Next, we give the expression of the reproducing kernel function ๐พ๐‘ฅ(๐‘ฆ). The characteristic equation of ๐พ๐‘ฅ(4)(๐‘ฆ)=๐›ฟ(๐‘ฅโˆ’๐‘ฆ) is ๐œ†4=0, and their characteristic values are ๐œ†=0 with 4 multiple roots. So, let the kernel ๐พ๐‘ฅ(๐‘ฆ) be as defined in (2.3).
On the other hand, let ๐พ๐‘ฅ(๐‘ฆ) satisfy ๐พ๐‘ฅ(๐‘š)(๐‘ฅ+0)=๐พ๐‘ฅ(๐‘š)(๐‘ฅโˆ’0), ๐‘š=0,1,2. Integrating ๐พ๐‘ฅ(4)(๐‘ฆ)=๐›ฟ(๐‘ฅโˆ’๐‘ฆ) from ๐‘ฅโˆ’๐œ€ to ๐‘ฅ+๐œ€ with respect to ๐‘ฆ and letting ๐œ€โ†’0, we have the jump degree of ๐พ๐‘ฅ(3)(๐‘ฆ) at ๐‘ฆ=๐‘ฅ given by ๐พ๐‘ฅ(3)(๐‘ฅโˆ’0)โˆ’๐พ๐‘ฅ(3)(๐‘ฅ+0)=โˆ’1. Through the last descriptions the unknown coefficients of (2.3) can be obtained. This completes the proof.

By using Mathematica 7.0 software package, the representation of the reproducing kernel function ๐พ๐‘ฅ(๐‘ฆ) is provided by ๐พ๐‘ฅโŽงโŽชโŽจโŽชโŽฉ1(๐‘ฆ)=6๎€ท(๐‘ฆโˆ’๐‘Ž)2๐‘Ž2โˆ’๐‘ฆ2๎€ธ1+3๐‘ฅ(2+๐‘ฆ)โˆ’๐‘Ž(6+3๐‘ฅ+๐‘ฆ),๐‘ฆโ‰ค๐‘ฅ,6๎€ท(๐‘ฅโˆ’๐‘Ž)2๐‘Ž2โˆ’๐‘ฅ2๎€ธ+3๐‘ฆ(2+๐‘ฅ)โˆ’๐‘Ž(6+๐‘ฅ+3๐‘ฆ),๐‘ฆ>๐‘ฅ.(2.4)

The following corollary summarizes some important properties of the reproducing kernel function ๐พ๐‘ฅ(๐‘ฆ).

Corollary 2.5. The reproducing kernel function ๐พ๐‘ฅ(๐‘ฆ) is symmetric, unique, and ๐พ๐‘ฅ(๐‘ฅ)โ‰ฅ0 for any fixed ๐‘ฅโˆˆ[๐‘Ž,๐‘].

Proof. By the reproducing property, we have ๐พ๐‘ฅ(๐‘ฆ)=โŸจ๐พ๐‘ฅ(๐œ‰),๐พ๐‘ฆ(๐œ‰)โŸฉ=โŸจ๐พ๐‘ฆ(๐œ‰),๐พ๐‘ฅ(๐œ‰)โŸฉ=๐พ๐‘ฆ(๐‘ฅ) for each ๐‘ฅ and ๐‘ฆ. Now, let ๐พ1๐‘ฅ(๐‘ฆ) and ๐พ2๐‘ฅ(๐‘ฆ) be all the reproducing kernels of the space ๐‘Š22[๐‘Ž,๐‘]; then ๐พ1๐‘ฅ(๐‘ฆ)=โŸจ๐พ1๐‘ฅ(๐œ‰),๐พ2๐‘ฆ(๐œ‰)โŸฉ=โŸจ๐พ2๐‘ฆ(๐œ‰),๐พ1๐‘ฅ(๐œ‰)โŸฉ=๐พ2๐‘ฆ(๐‘ฅ)=๐พ2๐‘ฅ(๐‘ฆ). Finally, we note that ๐พ๐‘ฅ(๐‘ฅ)=โŸจ๐พ๐‘ฅ(๐œ‰),๐พ๐‘ฅ(๐œ‰)โŸฉ=โ€–๐พ๐‘ฅ(๐œ‰)โ€–2โ‰ฅ0.

3. Introduction to a Linear Operator and a Normal Orthogonal System in ๐‘Š22[๐‘Ž,๐‘]

In this section, we construct an orthogonal function system of ๐‘Š22[๐‘Ž,๐‘]. Also, representation of the solution of (1.1) and (1.2) is given in the reproducing kernel space ๐‘Š22[๐‘Ž,๐‘].

To do this, we define a differential operator ๐ฟโˆถ๐‘Š22[๐‘Ž,๐‘]โ†’๐‘Š12[๐‘Ž,๐‘] such that ๐ฟ๐‘ข(๐‘ฅ)=๐‘ข๎…ž(๐‘ฅ). After homogenization of the initial condition (1.2), IDEs (1.1) and (1.2) can be converted into the equivalent form as follows: ๐‘ข๐ฟ๐‘ข(๐‘ฅ)=๐น(๐‘ฅ,๐‘ข(๐‘ฅ),๐‘‡๐‘ข(๐‘ฅ),๐‘†๐‘ข(๐‘ฅ)),๐‘Žโ‰ค๐‘ฅโ‰ค๐‘,(๐‘Ž)=0,(3.1) such that ๐น(๐‘ฅ,๐‘ข(๐‘ฅ),๐‘‡๐‘ข(๐‘ฅ),๐‘†๐‘ข(๐‘ฅ))=๐‘”(๐‘ฅ)โˆ’๐‘“(๐‘ฅ)๐‘(๐‘ฅ,๐‘ข(๐‘ฅ))โˆ’๐‘‡๐‘ข(๐‘ฅ)โˆ’๐‘†๐‘ข(๐‘ฅ), โˆซ๐‘‡๐‘ข(๐‘ฅ)=๐‘๐‘Ž๐‘˜1(๐‘ฅ,๐‘ก)๐บ1(๐‘ข(๐‘ก))๐‘‘๐‘ก, and โˆซ๐‘†๐‘ข(๐‘ฅ)=๐‘ฅ๐‘Ž๐‘˜2(๐‘ฅ,๐‘ก)๐บ2(๐‘ข(๐‘ก))๐‘‘๐‘ก, where ๐‘ข(๐‘ฅ)โˆˆ๐‘Š22[๐‘Ž,๐‘] and ๐น(๐‘ฅ,๐‘ค,๐‘ฆ,๐‘ง)โˆˆ๐‘Š12[๐‘Ž,๐‘] for ๐‘ฅโˆˆ[๐‘Ž,๐‘] and ๐‘ค=๐‘ค(๐‘ฅ),๐‘ฆ=๐‘ฆ(๐‘ฅ),๐‘ง=๐‘ง(๐‘ฅ)โˆˆ๐‘Š22[๐‘Ž,๐‘], โˆ’โˆž<๐‘ค,๐‘ฆ,๐‘ง<โˆž. It is easy to show that ๐ฟ is a bounded linear operator from ๐‘Š22[๐‘Ž,๐‘] to ๐‘Š12[๐‘Ž,๐‘].

Now, we construct an orthogonal function system of ๐‘Š22[๐‘Ž,๐‘]. Let ๐œ‘๐‘–(๐‘ฅ)=๐‘‡๐‘ฅ๐‘–(๐‘ฅ) and ๐œ“๐‘–(๐‘ฅ)=๐ฟโˆ—๐œ‘๐‘–(๐‘ฅ), where {๐‘ฅ๐‘–}โˆž๐‘–=1 is dense on [๐‘Ž,๐‘] and ๐ฟโˆ— is the adjoint operator of ๐ฟ. From the properties of the reproducing kernel ๐‘‡๐‘ฅ(๐‘ฆ), we have โŸจ๐‘ข(๐‘ฅ),๐œ‘๐‘–(๐‘ฅ)โŸฉ๐‘Š12=โŸจ๐‘ข(๐‘ฅ),๐‘‡๐‘ฅ๐‘–(๐‘ฅ)โŸฉ๐‘Š12=๐‘ข(๐‘ฅ๐‘–) for every ๐‘ข(๐‘ฅ)โˆˆ๐‘Š12[๐‘Ž,๐‘]. In terms of the properties of ๐พ๐‘ฅ(๐‘ฆ), one obtains โŸจ๐‘ข(๐‘ฅ),๐œ“๐‘–(๐‘ฅ)โŸฉ๐‘Š22=โŸจ๐‘ข(๐‘ฅ),๐ฟโˆ—๐œ‘๐‘–(๐‘ฅ)โŸฉ๐‘Š22=โŸจ๐ฟ๐‘ข(๐‘ฅ),๐œ‘๐‘–(๐‘ฅ)โŸฉ๐‘Š12=๐ฟ๐‘ข(๐‘ฅ๐‘–), ๐‘–=1,2,โ‹ฏ.

It is easy to see that ๐œ“๐‘–(๐‘ฅ)=๐ฟโˆ—๐œ‘๐‘–(๐‘ฅ)=โŸจ๐ฟโˆ—๐œ‘๐‘–(๐‘ฅ),๐พ๐‘ฅ(๐‘ฆ)โŸฉ๐‘Š22=โŸจ๐œ‘๐‘–(๐‘ฅ),๐ฟ๐‘ฆ๐พ๐‘ฅ(๐‘ฆ)โŸฉ๐‘Š12=๐ฟ๐‘ฆ๐พ๐‘ฅ(๐‘ฆ)|๐‘ฆ=๐‘ฅ๐‘–. Thus, ๐œ“๐‘–(๐‘ฅ) can be expressed in the form ๐œ“๐‘–(๐‘ฅ)=๐ฟ๐‘ฆ๐พ๐‘ฅ(๐‘ฆ)|๐‘ฆ=๐‘ฅ๐‘–, where ๐ฟ๐‘ฆ indicates that the operator ๐ฟ applies to the function of ๐‘ฆ.

Theorem 3.1. For (3.1), if {๐‘ฅ๐‘–}โˆž๐‘–=1 is dense on [๐‘Ž,๐‘], then {๐œ“๐‘–(๐‘ฅ)}โˆž๐‘–=1 is the complete function system of the space ๐‘Š22[๐‘Ž,๐‘].

Proof. Clearly, ๐œ“๐‘–(๐‘ฅ)โˆˆ๐‘Š22[๐‘Ž,๐‘]. For each fixed ๐‘ข(๐‘ฅ)โˆˆ๐‘Š22[๐‘Ž,๐‘], let โŸจ๐‘ข(๐‘ฅ),๐œ“๐‘–(๐‘ฅ)โŸฉ๐‘Š22=0, ๐‘–=1,2,โ€ฆ, which means that โŸจ๐‘ข(๐‘ฅ),๐œ“๐‘–(๐‘ฅ)โŸฉ๐‘Š22=โŸจ๐‘ข(๐‘ฅ),๐ฟโˆ—๐œ‘๐‘–(๐‘ฅ)โŸฉ๐‘Š22=โŸจ๐ฟ๐‘ข(๐‘ฅ),๐œ‘๐‘–(๐‘ฅ)โŸฉ๐‘Š12=๐ฟ๐‘ข(๐‘ฅ๐‘–)=0. Note that {๐‘ฅ๐‘–}โˆž๐‘–=1 is dense on [๐‘Ž,๐‘]; therefore, ๐ฟ๐‘ข(๐‘ฅ)=0. It follows that ๐‘ข(๐‘ฅ)=0 from the existence of ๐ฟโˆ’1. So, the proof of the theorem is complete.

The orthonormal function system {๐œ“๐‘–(๐‘ฅ)}โˆž๐‘–=1 of the space ๐‘Š22[๐‘Ž,๐‘] can be derived from Gram-Schmidt orthogonalization process of {๐œ“๐‘–(๐‘ฅ)}โˆž๐‘–=1 as follows: ๐œ“๐‘–(๐‘ฅ)=๐‘–๎“๐‘˜=1๐›ฝ๐‘–๐‘˜๐œ“๐‘˜(๐‘ฅ),(3.2) where ๐›ฝ๐‘–๐‘˜ are orthogonalization coefficients given as ๐›ฝ11=1/โ€–๐œ“1โ€–, ๐›ฝ๐‘–๐‘–=1/๐‘‘๐‘–๐‘˜, and ๐›ฝ๐‘–๐‘—=โˆ’(1/๐‘‘๐‘–๐‘˜)โˆ‘๐‘–โˆ’1๐‘˜=๐‘—๐‘๐‘–๐‘˜๐›ฝ๐‘˜๐‘— for ๐‘—<๐‘– in which ๐‘‘๐‘–๐‘˜=๎”โ€–๐œ“๐‘–โ€–2โˆ’โˆ‘๐‘–โˆ’1๐‘˜=1๐‘2๐‘–๐‘˜, ๐‘๐‘–๐‘˜=โŸจ๐œ“๐‘–,๐œ“๐‘˜โŸฉ๐‘Š22, and {๐œ“๐‘–(๐‘ฅ)}โˆž๐‘–=1 is the orthonormal system in the space ๐‘Š22[๐‘Ž,๐‘].

Theorem 3.2. For each ๐‘ข(๐‘ฅ)โˆˆ๐‘Š22[๐‘Ž,๐‘], the series โˆ‘โˆž๐‘–=1โŸจ๐‘ข(๐‘ฅ),๐œ“๐‘–(๐‘ฅ)โŸฉ๐œ“๐‘–(๐‘ฅ) is convergent in the norm of ๐‘Š22[๐‘Ž,๐‘]. On the other hand, if {๐‘ฅ๐‘–}โˆž๐‘–=1 is dense on [๐‘Ž,๐‘] and ๐‘ข(๐‘ฅ) is the exact solution of (3.1), then ๐‘ข(๐‘ฅ)=โˆž๎“๐‘–๐‘–=1๎“๐‘˜=1๐›ฝ๐‘–๐‘˜๐น๎€ท๐‘ฅ๐‘˜๎€ท๐‘ฅ,๐‘ข๐‘˜๎€ธ๎€ท๐‘ฅ,๐‘‡๐‘ข๐‘˜๎€ธ๎€ท๐‘ฅ,๐‘†๐‘ข๐‘˜๎€ธ๎€ธ๐œ“๐‘–(๐‘ฅ).(3.3)

Proof. Since ๐‘ข(๐‘ฅ)โˆˆ๐‘Š22[๐‘Ž,๐‘], โˆ‘โˆž๐‘–=1โŸจ๐‘ข(๐‘ฅ),๐œ“๐‘–(๐‘ฅ)โŸฉ๐œ“๐‘–(๐‘ฅ) is the Fourier series expansion about normal orthogonal system {๐œ“๐‘–(๐‘ฅ)}โˆž๐‘–=1, and ๐‘Š22[๐‘Ž,๐‘] is the Hilbert space, then the series โˆ‘โˆž๐‘–=1โŸจ๐‘ข(๐‘ฅ),๐œ“๐‘–(๐‘ฅ)โŸฉ๐œ“๐‘–(๐‘ฅ) is convergent in the sense of โ€–โ‹…โ€–๐‘Š22. On the other hand, using (3.2), we have ๐‘ข(๐‘ฅ)=โˆž๎“๐‘–=1๎ซ๐‘ข(๐‘ฅ),๐œ“๐‘–๎ฌ(๐‘ฅ)๐‘Š22๐œ“๐‘–(๐‘ฅ)=โˆž๎“๐‘–=1๎„”๐‘ข(๐‘ฅ),๐‘–๎“๐‘˜=1๐›ฝ๐‘–๐‘˜๐œ“๐‘˜๎„•(๐‘ฅ)๐‘Š22๐œ“๐‘–=(๐‘ฅ)โˆž๎“๐‘–๐‘–=1๎“๐‘˜=1๐›ฝ๐‘–๐‘˜โŸจ๐‘ข(๐‘ฅ),๐œ“๐‘˜(๐‘ฅ)โŸฉ๐‘Š22๐œ“๐‘–(๐‘ฅ)=โˆž๎“๐‘–๐‘–=1๎“๐‘˜=1๐›ฝ๐‘–๐‘˜โŸจ๐‘ข(๐‘ฅ),๐ฟโˆ—๐œ‘๐‘˜(๐‘ฅ)โŸฉ๐‘Š22๐œ“๐‘–=(๐‘ฅ)โˆž๎“๐‘–๐‘–=1๎“๐‘˜=1๐›ฝ๐‘–๐‘˜โŸจ๐ฟ๐‘ข(๐‘ฅ),๐œ‘๐‘˜(๐‘ฅ)โŸฉ๐‘Š12๐œ“๐‘–(๐‘ฅ)=โˆž๎“๐‘–๐‘–=1๎“๐‘˜=1๐›ฝ๐‘–๐‘˜๎€ท๐‘ฅ๐ฟ๐‘ข๐‘˜๎€ธ๐œ“๐‘–(๐‘ฅ).(3.4) But since ๐‘ข(๐‘ฅ) is the exact solution of (3.1), then ๐ฟ๐‘ข(๐‘ฅ)=๐น(๐‘ฅ,๐‘ข(๐‘ฅ),๐‘‡๐‘ข(๐‘ฅ),๐‘†๐‘ข(๐‘ฅ)) and ๐‘ข(๐‘ฅ)=โˆž๎“๐‘–๐‘–=1๎“๐‘˜=1๐›ฝ๐‘–๐‘˜๐น๎€ท๐‘ฅ๐‘˜๎€ท๐‘ฅ,๐‘ข๐‘˜๎€ธ๎€ท๐‘ฅ,๐‘‡๐‘ข๐‘˜๎€ธ๎€ท๐‘ฅ,๐‘†๐‘ข๐‘˜๎€ธ๎€ธ๐œ“๐‘–(๐‘ฅ).(3.5) So, the proof of the theorem is complete.

Note that we denote to the approximate solution of ๐‘ข(๐‘ฅ) by ๐‘ข๐‘›(๐‘ฅ)=๐‘›๎“๐‘–๐‘–=1๎“๐‘˜=1๐›ฝ๐‘–๐‘˜๐น๎€ท๐‘ฅ๐‘˜๎€ท๐‘ฅ,๐‘ข๐‘˜๎€ธ๎€ท๐‘ฅ,๐‘‡๐‘ข๐‘˜๎€ธ๎€ท๐‘ฅ,๐‘†๐‘ข๐‘˜๎€ธ๎€ธ๐œ“๐‘–(๐‘ฅ).(3.6)

Theorem 3.3. If ๐‘ข(๐‘ฅ)โˆˆ๐‘Š22[๐‘Ž,๐‘], then there exists ๐‘€>0 such that ||๐‘ข(๐‘–)(๐‘ฅ)||๐ถโ‰ค๐‘€||๐‘ข(๐‘ฅ)||๐‘Š22, ๐‘–=0,1, where ||๐‘ข(๐‘ฅ)||๐ถ=max๐‘Žโ‰ค๐‘ฅโ‰ค๐‘|๐‘ข(๐‘ฅ)|.

Proof. For any ๐‘ฅ,๐‘ฆโˆˆ[๐‘Ž,๐‘], we have ๐‘ข(๐‘–)(๐‘ฅ)=โŸจ๐‘ข(๐‘ฆ),๐พ๐‘ฅ(๐‘–)(๐‘ฆ)โŸฉ๐‘Š22,๐‘–=0,1. By the expression of ๐พ๐‘ฅ(๐‘ฆ), it follows that โ€–๐พ๐‘ฅ(๐‘–)(๐‘ฆ)โ€–๐‘Š22โ‰ค๐‘€๐‘–, ๐‘–=0,1. Thus, |๐‘ข(๐‘–)(๐‘ฅ)|=|โŸจ๐‘ข(๐‘ฅ),๐พ๐‘ฅ(๐‘–)(๐‘ฅ)โŸฉ๐‘Š22|โ‰คโ€–๐‘ข(๐‘ฅ)โ€–๐‘Š22โ€–๐พ๐‘ฅ(๐‘–)(๐‘ฅ)โ€–๐‘Š22โ‰ค๐‘€๐‘–โ€–๐‘ข(๐‘ฅ)โ€–๐‘Š22, ๐‘–=0,1. Hence, ||๐‘ข(๐‘–)(๐‘ฅ)||๐ถโ‰ค๐‘€||๐‘ข(๐‘ฅ)||๐‘Š22,โ€‰โ€‰๐‘–=0,1, where ๐‘€=max{๐‘€0,๐‘€1}. The proof is complete.

Corollary 3.4. The approximate solution ๐‘ข๐‘›(๐‘ฅ) and its derivative ๐‘ข๎…ž๐‘›(๐‘ฅ) are uniformly convergent.

Proof. By Theorems 3.2 and 3.3, for any ๐‘ฅโˆˆ[๐‘Ž,๐‘], we get ||๐‘ข๐‘›||=|||(๐‘ฅ)โˆ’๐‘ข(๐‘ฅ)โŸจ๐‘ข๐‘›(๐‘ฅ)โˆ’๐‘ข(๐‘ฅ),๐พ๐‘ฅ(๐‘ฅ)โŸฉ๐‘Š22|||โ‰คโ€–โ€–๐พ๐‘ฅโ€–โ€–(๐‘ฅ)๐‘Š22โ€–โ€–๐‘ข๐‘›โ€–โ€–(๐‘ฅ)โˆ’๐‘ข(๐‘ฅ)๐‘Š22โ‰ค๐‘€0โ€–โ€–๐‘ข๐‘›โ€–โ€–(๐‘ฅ)โˆ’๐‘ข(๐‘ฅ)๐‘Š22.(3.7)
On the other hand, ||๐‘ข๎…ž๐‘›(๐‘ฅ)โˆ’๐‘ข๎…ž||=|||๎ซ๐‘ข(๐‘ฅ)๐‘›(๐‘ฅ)โˆ’๐‘ข(๐‘ฅ),๐พ๎…ž๐‘ฅ๎ฌ(๐‘ฅ)๐‘Š22|||โ‰คโ€–โ€–๐พ๎…ž๐‘ฅโ€–โ€–(๐‘ฅ)๐‘Š22โ€–โ€–๐‘ข๐‘›โ€–โ€–(๐‘ฅ)โˆ’๐‘ข(๐‘ฅ)๐‘Š22โ‰ค๐‘€1โ€–โ€–๐‘ข๐‘›โ€–โ€–(๐‘ฅ)โˆ’๐‘ข(๐‘ฅ)๐‘Š22.(3.8)
Hence, |๐‘ข๐‘›(๐‘–)(๐‘ฅ)โˆ’๐‘ข(๐‘–)(๐‘ฅ)|โ‰คโ€–๐‘ข๐‘›(๐‘–)(๐‘ฅ)โˆ’๐‘ข(๐‘–)(๐‘ฅ)โ€–๐ถโ‰ค๐‘€๐‘–โ€–๐‘ข๐‘›(๐‘ฅ)โˆ’๐‘ข(๐‘ฅ)โ€–๐‘Š22, where ๐‘€0 and ๐‘€1 are positive constants. Hence, if โ€–๐‘ข๐‘›(๐‘ฅ)โˆ’๐‘ข(๐‘ฅ)โ€–๐‘Š22โ†’0 as ๐‘›โ†’โˆž, the approximate solutions ๐‘ข๐‘›(๐‘ฅ) and ๐‘ข๎…ž๐‘›(๐‘ฅ) converge uniformly to the exact solution ๐‘ข(๐‘ฅ) and its derivative, respectively.

4. Iterative Method and Convergence Theorem

In this section, an iterative method of obtaining the solution of (3.1) is presented in the reproducing kernel space ๐‘Š22[๐‘Ž,๐‘].

First of all, we will mention the following remark in order to solve (1.1) and (1.2) numerically. If (1.1) is linear, then the exact and approximate solutions can be obtained directly from (3.5) and (3.6), respectively. On the other hand, if (1.1) is nonlinear, then the exact and approximate solutions can be obtained using the following iterative method.

According to (3.5), the representation of the solution of (1.1) can be denoted by ๐‘ข(๐‘ฅ)=โˆž๎“๐‘–=1๐ด๐‘–๐œ“๐‘–(๐‘ฅ),(4.1) where ๐ด๐‘–=โˆ‘๐‘–๐‘˜=1๐›ฝ๐‘–๐‘˜๐น(๐‘ฅ๐‘˜,๐‘ข(๐‘ฅ๐‘˜),๐‘‡๐‘ข(๐‘ฅ๐‘˜),๐‘†๐‘ข(๐‘ฅ๐‘˜)). In fact, ๐ด๐‘–, ๐‘–=1,2,โ€ฆ, in (4.1) are unknown, and we will approximate ๐ด๐‘– using known ๐ต๐‘–. For a numerical computation, we define initial function ๐‘ข0(๐‘ฅ1)=0 and the ๐‘›-term approximation to ๐‘ข(๐‘ฅ) by ๐‘ข๐‘›(๐‘ฅ)=๐‘›๎“๐‘–=1๐ต๐‘–๐œ“๐‘–(๐‘ฅ),(4.2) where the coefficients ๐ต๐‘– are given as ๐ต1=๐›ฝ11๐น๎€ท๐‘ฅ1,๐‘ข0๎€ท๐‘ฅ1๎€ธ,๐‘‡๐‘ข0๎€ท๐‘ฅ1๎€ธ,๐‘†๐‘ข0๎€ท๐‘ฅ1,๐‘ข๎€ธ๎€ธ1(๐‘ฅ)=๐ต1๐œ“1๐ต(๐‘ฅ),2=2๎“๐‘˜=1๐›ฝ2๐‘˜๐น๎€ท๐‘ฅ๐‘˜,๐‘ข๐‘˜โˆ’1๎€ท๐‘ฅ๐‘˜๎€ธ,๐‘‡๐‘ข๐‘˜โˆ’1๎€ท๐‘ฅ๐‘˜๎€ธ,๐‘†๐‘ข๐‘˜โˆ’1๎€ท๐‘ฅ๐‘˜,๐‘ข๎€ธ๎€ธ2(๐‘ฅ)=2๎“๐‘–=1๐ต๐‘–๐œ“๐‘–โ‹ฎ๐‘ข(๐‘ฅ),๐‘›โˆ’1(๐‘ฅ)=๐‘›โˆ’1๎“๐‘–=1๐ต๐‘–๐œ“๐‘–(๐ต๐‘ฅ),๐‘›=๐‘›๎“๐‘˜=1๐›ฝ๐‘›๐‘˜๐น๎€ท๐‘ฅ๐‘˜,๐‘ข๐‘˜โˆ’1๎€ท๐‘ฅ๐‘˜๎€ธ,๐‘‡๐‘ข๐‘˜โˆ’1๎€ท๐‘ฅ๐‘˜๎€ธ,๐‘†๐‘ข๐‘˜โˆ’1๎€ท๐‘ฅ๐‘˜.๎€ธ๎€ธ(4.3)

We mention here the following remark: in the iterative process of (4.2), we can guarantee that the approximation ๐‘ข๐‘›(๐‘ฅ) satisfies the initial condition (1.2).

Now, the approximate solution ๐‘ข๐‘๐‘›(๐‘ฅ) can be obtained by taking finitely many terms in the series representation of ๐‘ข๐‘›(๐‘ฅ) and ๐‘ข๐‘๐‘›(๐‘ฅ)=๐‘๎“๐‘–๐‘–=1๎“๐‘˜=1๐›ฝ๐‘–๐‘˜๐น๎€ท๐‘ฅ๐‘˜,๐‘ข๐‘›โˆ’1๎€ท๐‘ฅ๐‘˜๎€ธ,๐‘‡๐‘ข๐‘›โˆ’1๎€ท๐‘ฅ๐‘˜๎€ธ,๐‘†๐‘ข๐‘›โˆ’1๎€ท๐‘ฅ๐‘˜๎€ธ๎€ธ๐œ“๐‘–(๐‘ฅ).(4.4)

Next, we will prove that ๐‘ข๐‘›(๐‘ฅ) in the iterative formula (4.2) is convergent to the exact solution ๐‘ข(๐‘ฅ) of (1.1).

Lemma 4.1. If โ€–๐‘ข๐‘›(๐‘ฅ)โˆ’๐‘ข(๐‘ฅ)โ€–๐‘Š22โ†’0, ๐‘ฅ๐‘›โ†’๐‘ฆ as ๐‘›โ†’โˆž and ๐น(๐‘ฅ,๐‘ฃ,๐‘ค,๐‘ง) is continuous in [๐‘Ž,๐‘] with respect to ๐‘ฅ,๐‘ฃ,๐‘ค,๐‘ง, for ๐‘ฅโˆˆ[๐‘Ž,๐‘] and ๐‘ฃ,๐‘ค,๐‘งโˆˆ(โˆ’โˆž,โˆž), then as ๐‘›โ†’โˆž, one has ๐น(๐‘ฅ๐‘›,๐‘ข๐‘›โˆ’1(๐‘ฅ๐‘›),๐‘‡๐‘ข๐‘›โˆ’1(๐‘ฅ๐‘›),๐‘†๐‘ข๐‘›โˆ’1(๐‘ฅ๐‘›))โ†’๐น(๐‘ฆ,๐‘ข(๐‘ฆ),๐‘‡๐‘ข(๐‘ฆ),๐‘†๐‘ข(๐‘ฆ)).

Proof. Firstly, we will prove that ๐‘ข๐‘›โˆ’1(๐‘ฅ๐‘›)โ†’๐‘ข(๐‘ฆ) in the sense of โ€–โ‹…โ€–๐‘Š22. Since ||๐‘ข๐‘›โˆ’1๎€ท๐‘ฅ๐‘›๎€ธ||=||๐‘ขโˆ’๐‘ข(๐‘ฆ)๐‘›โˆ’1๎€ท๐‘ฅ๐‘›๎€ธโˆ’๐‘ข๐‘›โˆ’1(๐‘ฆ)+๐‘ข๐‘›โˆ’1||โ‰ค||๐‘ข(๐‘ฆ)โˆ’๐‘ข(๐‘ฆ)๐‘›โˆ’1๎€ท๐‘ฅ๐‘›๎€ธโˆ’๐‘ข๐‘›โˆ’1(||+||๐‘ข๐‘ฆ)๐‘›โˆ’1(||,๐‘ฆ)โˆ’๐‘ข(๐‘ฆ)(4.5) by reproducing kernel property of ๐พ๐‘ฅ(๐‘ฆ), we have ๐‘ข๐‘›โˆ’1(๐‘ฅ๐‘›) = โŸจ๐‘ข๐‘›โˆ’1(๐‘ฅ),๐พ๐‘ฅ๐‘›(๐‘ฅ)โŸฉ and ๐‘ข๐‘›โˆ’1(๐‘ฆ)=โŸจ๐‘ข๐‘›โˆ’1(๐‘ฅ),๐พ๐‘ฆ(๐‘ฅ)โŸฉ. Thus, |๐‘ข๐‘›โˆ’1(๐‘ฅ๐‘›)โˆ’๐‘ข๐‘›โˆ’1(๐‘ฆ)| = |โŸจ๐‘ข๐‘›โˆ’1(๐‘ฅ),๐พ๐‘ฅ๐‘›(๐‘ฅ)โˆ’๐พ๐‘ฆ(๐‘ฅ)โŸฉ๐‘Š22|โ‰คโ€–๐‘ข๐‘›โˆ’1(๐‘ฅ)โ€–๐‘Š22โ€–๐พ๐‘ฅ๐‘›(๐‘ฅ)โˆ’๐พ๐‘ฆ(๐‘ฅ)โ€–๐‘Š22. From the symmetry of ๐พ๐‘ฅ(๐‘ฆ), it follows that โ€–๐พ๐‘ฅ๐‘›(๐‘ฅ)โˆ’๐พ๐‘ฆ(๐‘ฅ)โ€–๐‘Š22โ†’0 as ๐‘›โ†’โˆž. Hence, |๐‘ข๐‘›โˆ’1(๐‘ฅ๐‘›)โˆ’๐‘ข๐‘›โˆ’1(๐‘ฆ)|โ†’0 as soon as ๐‘ฅ๐‘›โ†’๐‘ฆ.
On the other hand, by Corollary 3.4, for any ๐‘ฆโˆˆ[๐‘Ž,๐‘], it holds that |๐‘ข๐‘›โˆ’1(๐‘ฆ)โˆ’๐‘ข(๐‘ฆ)|โ†’0. Therefore, ๐‘ข๐‘›โˆ’1(๐‘ฅ๐‘›)โ†’๐‘ข(๐‘ฆ) in the sense of โ€–โ‹…โ€–๐‘Š22 as ๐‘ฅ๐‘›โ†’๐‘ฆ and ๐‘›โ†’โˆž.
Thus, by means of the continuation of ๐บ1, ๐บ2, and ๐‘, it is obtained that ๐บ1(๐‘ข๐‘›โˆ’1(๐‘ฅ๐‘›))โ†’๐บ1(๐‘ข(๐‘ฆ)), ๐บ2(๐‘ข๐‘›โˆ’1(๐‘ฅ๐‘›))โ†’๐บ2(๐‘ข(๐‘ฆ)), and ๐‘(๐‘ฅ๐‘›,๐‘ข๐‘›โˆ’1(๐‘ฅ๐‘›))โ†’๐‘(๐‘ฆ,๐‘ข(๐‘ฆ)) as ๐‘›โ†’โˆž. This shows that ๐‘‡๐‘ข๐‘›โˆ’1(๐‘ฅ๐‘›)โ†’๐‘‡๐‘ข(๐‘ฆ) and ๐‘†๐‘ข๐‘›โˆ’1(๐‘ฅ๐‘›)โ†’๐‘†๐‘ข(๐‘ฆ) as ๐‘›โ†’โˆž. Hence, the continuity of ๐น gives the result.

Lemma 4.2. {๐‘ข๐‘›}โˆž๐‘›=1 in (4.2) is monotonically increasing in the sense of the norm of ๐‘Š22[๐‘Ž,๐‘].

Proof. By Theorem 3.1, {๐œ“๐‘–}โˆž๐‘–=1 is the complete orthonormal system in the space ๐‘Š22[๐‘Ž,๐‘]. Hence, we have โ€–๐‘ข๐‘›โ€–2๐‘Š22=โŸจ๐‘ข๐‘›(๐‘ฅ),๐‘ข๐‘›(๐‘ฅ)โŸฉ๐‘Š22โˆ‘=โŸจ๐‘›๐‘–=1๐ต๐‘–๐œ“๐‘–โˆ‘(๐‘ฅ),๐‘›๐‘–=1๐ต๐‘–๐œ“๐‘–(๐‘ฅ)โŸฉ๐‘Š22=โˆ‘๐‘›๐‘–=1(๐ต๐‘–)2. Therefore, โ€–๐‘ข๐‘›โ€–๐‘Š22 is monotonically increasing.

Lemma 4.3. One has ๐ฟ๐‘ข๐‘›(๐‘ฅ๐‘—)=๐น(๐‘ฅ๐‘—,๐‘ข๐‘—โˆ’1(๐‘ฅ๐‘—),๐‘‡๐‘ข๐‘—โˆ’1(๐‘ฅ๐‘—),๐‘†๐‘ข๐‘—โˆ’1(๐‘ฅ๐‘—)),๐‘—โ‰ค๐‘›.

Proof. The proof will be obtained by mathematical induction as follows: if ๐‘—โ‰ค๐‘›, then ๐ฟ๐‘ข๐‘›(๐‘ฅ๐‘—) = โˆ‘๐‘›๐‘–=1๐ต๐‘–๐ฟ๐œ“๐‘–(๐‘ฅ๐‘—) = โˆ‘๐‘›๐‘–=1๐ต๐‘–โŸจ๐ฟ๐œ“๐‘–(๐‘ฅ),๐œ‘๐‘—(๐‘ฅ)โŸฉ๐‘Š12 = โˆ‘๐‘›๐‘–=1๐ต๐‘–โŸจ๐œ“๐‘–(๐‘ฅ),๐ฟโˆ—๐‘—๐œ‘(๐‘ฅ)โŸฉ๐‘Š22 = โˆ‘๐‘›๐‘–=1๐ต๐‘–โŸจ๐œ“๐‘–(๐‘ฅ),๐œ“๐‘—(๐‘ฅ)โŸฉ๐‘Š22. Thus, ๐ฟ๐‘ข๐‘›๎€ท๐‘ฅ๐‘—๎€ธ=๐‘›๎“๐‘–=1๐ต๐‘–๎ซ๐œ“๐‘–(๐‘ฅ),๐œ“๐‘—๎ฌ(๐‘ฅ)๐‘Š22.(4.6)
Multiplying both sides of (4.6) by ๐›ฝ๐‘—๐‘™, summing for ๐‘™ from 1 to ๐‘—, and using the orthogonality of {๐œ“๐‘–(๐‘ฅ)}โˆž๐‘–=1 yield that ๐‘—๎“๐‘™=1๐›ฝ๐‘—๐‘™๐ฟ๐‘ข๐‘›๎€ท๐‘ฅ๐‘™๎€ธ=๐‘›๎“๐‘–=1๐ต๐‘–๎„”๐œ“๐‘–(๐‘ฅ),๐‘—๎“๐‘™=1๐›ฝ๐‘—๐‘™๐œ“๐‘™๎„•(๐‘ฅ)๐‘Š22=๐‘›๎“๐‘–=1๐ต๐‘–๎ซ๐œ“๐‘–(๐‘ฅ),๐œ“๐‘—๎ฌ(๐‘ฅ)๐‘Š22=๐ต๐‘—=๐‘—๎“๐‘™=1๐›ฝ๐‘—๐‘™๐น๎€ท๐‘ฅ๐‘™,๐‘ข๐‘™โˆ’1๎€ท๐‘ฅ๐‘™๎€ธ,๐‘‡๐‘ข๐‘™โˆ’1๎€ท๐‘ฅ๐‘™๎€ธ,๐‘†๐‘ข๐‘™โˆ’1๎€ท๐‘ฅ๐‘™.๎€ธ๎€ธ(4.7)
Now, if ๐‘—=1, then ๐ฟ๐‘ข๐‘›(๐‘ฅ1)=๐น(๐‘ฅ1,๐‘ข0(๐‘ฅ1),๐‘‡๐‘ข0(๐‘ฅ1),๐‘†๐‘ข0(๐‘ฅ1)). On the other hand, if ๐‘—=2, then ๐›ฝ21๐ฟ๐‘ข๐‘›(๐‘ฅ1)+๐›ฝ22๐ฟ๐‘ข๐‘›(๐‘ฅ2)=๐›ฝ21๐น(๐‘ฅ1,๐‘ข0(๐‘ฅ1),๐‘‡๐‘ข0(๐‘ฅ1),๐‘†๐‘ข0(๐‘ฅ1))+๐›ฝ22๐น(๐‘ฅ2,๐‘ข1(๐‘ฅ2),๐‘‡๐‘ข1(๐‘ฅ2),๐‘†๐‘ข1(๐‘ฅ2)). Thus, ๐ฟ๐‘ข๐‘›(๐‘ฅ2)=๐น(๐‘ฅ2,๐‘ข1(๐‘ฅ2),๐‘‡๐‘ข1(๐‘ฅ2),๐‘†๐‘ข1(๐‘ฅ2)). It is easy to see that ๐ฟ๐‘ข๐‘›(๐‘ฅ๐‘—)=๐น(๐‘ฅ๐‘—,๐‘ข๐‘—โˆ’1(๐‘ฅ๐‘—),๐‘‡๐‘ข๐‘—โˆ’1(๐‘ฅ๐‘—),๐‘†๐‘ข๐‘—โˆ’1(๐‘ฅ๐‘—)) by using mathematical induction.

Lemma 4.4. One has ๐ฟ๐‘ข๐‘›(๐‘ฅ๐‘—)=๐ฟ๐‘ข(๐‘ฅ๐‘—),๐‘—โ‰ค๐‘›.

Proof. It is clear that on taking limits in (4.2) โˆ‘๐‘ข(๐‘ฅ)=โˆž๐‘–=1๐ต๐‘–๐œ“๐‘–(๐‘ฅ). Therefore, ๐‘ข๐‘›(๐‘ฅ)=๐‘ƒ๐‘›๐‘ข(๐‘ฅ), where ๐‘ƒ๐‘› is an orthogonal projector from ๐‘Š22[๐‘Ž,๐‘] to Span{๐œ“1,๐œ“2,โ€ฆ,๐œ“๐‘›}. Thus, ๐ฟ๐‘ข๐‘›(๐‘ฅ๐‘—) = โŸจ๐ฟ๐‘ข๐‘›(๐‘ฅ),๐œ‘๐‘—(๐‘ฅ)โŸฉ๐‘Š12 = โŸจ๐‘ข๐‘›(๐‘ฅ),๐ฟโˆ—๐‘—๐œ‘(๐‘ฅ)โŸฉ๐‘Š22 = โŸจ๐‘ƒ๐‘›๐‘ข(๐‘ฅ),๐œ“๐‘—(๐‘ฅ)โŸฉ๐‘Š22 = โŸจ๐‘ข(๐‘ฅ),๐‘ƒ๐‘›๐œ“๐‘—(๐‘ฅ)โŸฉ๐‘Š22 = โŸจ๐‘ข(๐‘ฅ),๐œ“๐‘—(๐‘ฅ)โŸฉ๐‘Š22 = โŸจ๐ฟ๐‘ข(๐‘ฅ),๐œ‘๐‘—(๐‘ฅ)โŸฉ๐‘Š12=๐ฟ๐‘ข(๐‘ฅ๐‘—).

Theorem 4.5. Suppose that โ€–๐‘ข๐‘›โ€–๐‘Š22 is bounded in (4.2). If {๐‘ฅ๐‘–}โˆž๐‘–=1 is dense on [๐‘Ž,๐‘], then the ๐‘›-term approximate solution ๐‘ข๐‘›(๐‘ฅ) in the iterative formula (4.2) converges to the exact solution ๐‘ข(๐‘ฅ) of (3.1) in the space ๐‘Š22[๐‘Ž,๐‘] and โˆ‘๐‘ข(๐‘ฅ)=โˆž๐‘–=1๐ต๐‘–๐œ“๐‘–(๐‘ฅ), where ๐ต๐‘– is given by (4.3).

Proof. First of all, we will prove the convergence of ๐‘ข๐‘›(๐‘ฅ). From (4.2), we infer that ๐‘ข๐‘›+1(๐‘ฅ)=๐‘ข๐‘›(๐‘ฅ)+๐ต๐‘›+1๐œ“๐‘›+1(๐‘ฅ). The orthogonality of {๐œ“๐‘–(๐‘ฅ)}โˆž๐‘–=1 yields that ||||๐‘ข๐‘›+1||||2๐‘Š22=||||๐‘ข๐‘›||||2๐‘Š22+๎€ท๐ต๐‘›+1๎€ธ2=||||๐‘ข๐‘›โˆ’1||||2๐‘Š22+๎€ท๐ต๐‘›๎€ธ2+๎€ท๐ต๐‘›+1๎€ธ2||||๐‘ข=โ‹ฏ=0||||2๐‘Š22+๐‘›+1๎“๐‘–=1๎€ท๐ต๐‘–๎€ธ2.(4.8)
From Lemma 4.2, the sequence โ€–๐‘ข๐‘›โ€–๐‘Š22 is monotonically increasing. Due to the condition that โ€–๐‘ข๐‘›โ€–๐‘Š22 is bounded, โ€–๐‘ข๐‘›โ€–๐‘Š22 is convergent as ๐‘›โ†’โˆž. Then, there exists a constant ๐‘ such that โˆ‘โˆž๐‘–=1(๐ต๐‘–)2=๐‘. This implies that ๐ต๐‘–=โˆ‘๐‘–๐‘˜=1๐›ฝ๐‘–๐‘˜๐น(๐‘ฅ๐‘˜,๐‘ข๐‘˜โˆ’1(๐‘ฅ๐‘˜),๐‘‡๐‘ข๐‘˜โˆ’1(๐‘ฅ๐‘˜),๐‘†๐‘ข๐‘˜โˆ’1(๐‘ฅ๐‘˜))โˆˆ๐‘™2,โ€‰โ€‰๐‘–=1,2,โ€ฆ.
If ๐‘š>๐‘›, using (๐‘ข๐‘šโˆ’๐‘ข๐‘šโˆ’1)โŸ‚(๐‘ข๐‘šโˆ’1โˆ’๐‘ข๐‘šโˆ’2)โŸ‚โ‹ฏโŸ‚(๐‘ข๐‘›+1โˆ’๐‘ข๐‘›), then one gets ||||๐‘ข๐‘š(๐‘ฅ)โˆ’๐‘ข๐‘›||||(๐‘ฅ)2๐‘Š22=||||๐‘ข๐‘š(๐‘ฅ)โˆ’๐‘ข๐‘šโˆ’1(๐‘ฅ)+๐‘ข๐‘šโˆ’1(๐‘ฅ)โˆ’โ‹ฏ+๐‘ข๐‘›+1(๐‘ฅ)โˆ’๐‘ข๐‘›||||(๐‘ฅ)2๐‘Š22=||||๐‘ข๐‘š(๐‘ฅ)โˆ’๐‘ข๐‘šโˆ’1||||(๐‘ฅ)2๐‘Š22||||๐‘ข+โ‹ฏ+๐‘›+1(๐‘ฅ)โˆ’๐‘ข๐‘›||||(๐‘ฅ)2๐‘Š22.(4.9) Furthermore, ||๐‘ข๐‘š(๐‘ฅ)โˆ’๐‘ข๐‘šโˆ’1(๐‘ฅ)||2๐‘Š22=(๐ต๐‘š)2. Consequently, ||๐‘ข๐‘š(๐‘ฅ)โˆ’๐‘ข๐‘›(๐‘ฅ)||2๐‘Š22=โˆ‘๐‘š๐‘–=๐‘›+1(๐ต๐‘–)2โ†’0 as ๐‘›,๐‘šโ†’โˆž. Considering the completeness of the space ๐‘Š22[๐‘Ž,๐‘], there exists a ๐‘ข(๐‘ฅ)โˆˆ๐‘Š22[๐‘Ž,๐‘] such that ๐‘ข๐‘›(๐‘ฅ)โ†’๐‘ข(๐‘ฅ) as ๐‘›โ†’โˆž in the sense of โ€–โ‹…โ€–๐‘Š22.
Secondly, we will prove that ๐‘ข(๐‘ฅ) is the solution of (3.1). Since {๐‘ฅ๐‘–}โˆž๐‘–=1 is dense on [๐‘Ž,๐‘], for any ๐‘ฅโˆˆ[๐‘Ž,๐‘], there exists subsequence {๐‘ฅ๐‘›๐‘—}โˆž๐‘—=1 such that ๐‘ฅ๐‘›๐‘—โ†’๐‘ฅ as ๐‘—โ†’โˆž. From Lemmas 4.3 and 4.4, it is easy to see that ๐ฟ๐‘ข(๐‘ฅ๐‘›๐‘—)=๐น(๐‘ฅ๐‘›๐‘—,๐‘ข๐‘›๐‘—โˆ’1(๐‘ฅ๐‘˜),๐‘‡๐‘ข๐‘›๐‘—โˆ’1(๐‘ฅ๐‘˜),๐‘†๐‘ข๐‘›๐‘—โˆ’1(๐‘ฅ๐‘˜)). Hence, letting ๐‘—โ†’โˆž, by Lemma 4.1 and the continuity of ๐น, we have ๐ฟ๐‘ข(๐‘ฅ)=๐น(๐‘ฅ,๐‘ข(๐‘ฅ),๐‘‡๐‘ข(๐‘ฅ),๐‘†๐‘ข(๐‘ฅ)). That is, ๐‘ข(๐‘ฅ) is the solution of (3.1).
Since ๐œ“๐‘–(๐‘ฅ)โˆˆ๐‘Š22[๐‘Ž,๐‘], clearly, ๐‘ข(๐‘ฅ) satisfies the initial condition (1.2). In other words, ๐‘ข(๐‘ฅ) is the solution of (1.1) and (1.2), where โˆ‘๐‘ข(๐‘ฅ)=โˆž๐‘–=1๐ต๐‘–๐œ“๐‘–(๐‘ฅ) and ๐ต๐‘– is given by (4.3). The proof is complete.

Theorem 4.6. Assume that ๐‘ข(๐‘ฅ) is the solution of (3.1) and ๐‘Ÿ๐‘›(๐‘ฅ) is the difference between the approximate solution ๐‘ข๐‘›(๐‘ฅ) and the exact solution ๐‘ข(๐‘ฅ). Then, ๐‘Ÿ๐‘›(๐‘ฅ) is monotonically decreasing in the sense of the norm of ๐‘Š22[๐‘Ž,๐‘].

Proof. It obvious that ||๐‘Ÿ๐‘›(๐‘ฅ)||2๐‘Š22=||๐‘ข(๐‘ฅ)โˆ’๐‘ข๐‘›(๐‘ฅ)||2๐‘Š22โˆ‘=โ€–โˆž๐‘–=๐‘›+1๐ต๐‘–๐œ“๐‘–(๐‘ฅ)โ€–2๐‘Š22=โˆ‘โˆž๐‘–=๐‘›+1(๐ต๐‘–)2 and ||๐‘Ÿ๐‘›โˆ’1(๐‘ฅ)||2๐‘Š22=โˆ‘โˆž๐‘–=๐‘›(๐ต๐‘–)2. Thus, ||๐‘Ÿ๐‘›(๐‘ฅ)||๐‘Š22โ‰ค||๐‘Ÿ๐‘›โˆ’1(๐‘ฅ)||๐‘Š22; consequently, the difference ๐‘Ÿ๐‘›(๐‘ฅ) is monotonically decreasing in the sense of โ€–โ‹…โ€–๐‘Š22. So, the proof of the theorem is complete.

5. Numerical Examples

In this section, some numerical examples are studied to demonstrate the accuracy and applicability of the present method. Results obtained are compared with the exact solution of each example and are found to be in good agreement with each other. In the process of computation, all the symbolic and numerical computations were performed by using Mathematica 7.0 software package.

Example 5.1. Consider the nonlinear Fredholm-Volterra IDE: ๐‘ข๎…ž1(๐‘ฅ)+๐‘ข(๐‘ฅ)โˆ’4๎€œ10๐‘ก๐‘ข31(๐‘ก)๐‘‘๐‘ก+2๎€œ๐‘ฅ0๐‘ฅ๐‘ข2(๐‘ก)๐‘‘๐‘ก=๐‘”(๐‘ฅ),0โ‰ค๐‘ฅ,๐‘กโ‰ค1,๐‘ข(0)=0,(5.1) where ๐‘”(๐‘ฅ)=(1/10)๐‘ฅ6+๐‘ฅ2+2๐‘ฅโˆ’(1/32). The exact solution is ๐‘ข(๐‘ฅ)=๐‘ฅ2.

Using RKHS method, taking ๐‘ฅ๐‘–=(๐‘–โˆ’1)/(๐‘โˆ’1), ๐‘–=1,2,โ€ฆ,๐‘, with the reproducing kernel function ๐พ๐‘ฅ(๐‘ฆ) on [0,1], the approximate solution ๐‘ข๐‘๐‘›(๐‘ฅ) is calculated by (4.4). The numerical results at some selected grid points for ๐‘=26 and ๐‘›=5 are given in Table 1.

tab1
Table 1: Numerical results for Example 5.1.

As we mention, we used the grid nodes mentioned earlier in order to obtain approximate solutions. Moreover, it is possible to pick any point in [๐‘Ž,๐‘] and as well the approximate solution and its derivative will be applicable. Next, the numerical results for Example 5.1 at some selected gird nodes in [0,1] of ๐‘ข๎…ž(๐‘ฅ) are given in Table 2.

tab2
Table 2: Numerical results of ๐‘ข๎…ž(๐‘ฅ) for Example 5.1.

Table 3 shows, a comparison between the absolute errors of our method together with triangular functions method [6], operational matrix with block-pulse functions method [7], and Hybrid Legendre polynomials and block-pulse functions method [8]. As it is evident from the comparison results, it was found that our method in comparison with the mentioned methods is better with a view to accuracy and utilization.

tab3
Table 3: Numerical comparison of absolute error for Example 5.1.

Example 5.2. Consider the nonlinear Fredholm-Volterra IDE: ๐‘ข๎…ž๎€œ(๐‘ฅ)+2๐‘ฅ๐‘ข(๐‘ฅ)โˆ’10๎€œ(๐‘ฅโˆ’๐‘ก)๐‘ข(๐‘ก)๐‘‘๐‘กโˆ’๐‘ฅ0(๐‘ฅ+๐‘ก)๐‘ข3(๐‘ก)๐‘‘๐‘ก=๐‘”(๐‘ฅ),0โ‰ค๐‘ฅ,๐‘กโ‰ค1,๐‘ข(0)=1,(5.2) where ๐‘”(๐‘ฅ)=(โˆ’(2/3)๐‘ฅ+(1/9))๐‘’3๐‘ฅ+(2๐‘ฅ+1)๐‘’๐‘ฅ+((4/3)โˆ’๐‘’)๐‘ฅ+(8/9). The exact solution is ๐‘ข(๐‘ฅ)=๐‘’๐‘ฅ.

Using RKHS method, taking ๐‘ฅ๐‘–=(๐‘–โˆ’1)/(๐‘โˆ’1), ๐‘–=1,2,โ€ฆ,๐‘, with the reproducing kernel function ๐พ๐‘ฅ(๐‘ฆ) on [0,1], the approximate solution ๐‘ข๐‘๐‘›(๐‘ฅ) is calculated by (4.4). The numerical results at some selected grid points for ๐‘=26 and ๐‘›=1 are given in Table 4.

tab4
Table 4: Numerical results for Example 5.2.

The comparison among the RKHS solution besides the solutions of triangular functions [6], operational matrix with block-pulse functions solution [7], and exact solutions are shown in Table 5.

tab5
Table 5: Numerical comparison of approximate solution for Example 5.2.

Example 5.3. Consider the nonlinear Fredholm-Volterra IDE: ๐‘ข๎…ž(๐‘ฅ)+๐‘ข2๎€œ(๐‘ฅ)โˆ’10๎€œ(๐‘ก+1)sinh(๐‘ข(๐‘ก)โˆ’1)๐‘‘๐‘กโˆ’๐‘ฅ0๐‘ฅ๐‘’๐‘ข(๐‘ก)๐‘‘๐‘ก=๐‘”(๐‘ฅ),0โ‰ค๐‘ฅ,๐‘กโ‰ค1,๐‘ข(0)=1,(5.3) where ๐‘”(๐‘ฅ)=(ln(๐‘ฅ+1)+1)2+(๐‘ฅ+1)โˆ’1โˆ’(๐‘’/2)(๐‘ฅ+2)๐‘ฅ2โˆ’(2/3). The exact solution is ๐‘ข(๐‘ฅ)=1+ln(๐‘ฅ+1).

Using RKHS method, taking ๐‘ฅ๐‘–=(๐‘–โˆ’1)/(๐‘โˆ’1), ๐‘–=1,2,โ€ฆ,๐‘, with the reproducing kernel function ๐พ๐‘ฅ(๐‘ฆ) on [0,1], the approximate solution ๐‘ข๐‘๐‘›(๐‘ฅ) is calculated by (4.4). The numerical results at some selected grid points for ๐‘=51 and ๐‘›=1 are given in Table 6.

tab6
Table 6: Numerical results for Example 5.3.

Example 5.4. Consider the nonlinear Fredholm-Volterra IDE: ๐‘ข๎…ž๎€œ(๐‘ฅ)+cos(๐‘ข(๐‘ฅ))โˆ’10๎€œ(๐‘ก+๐‘ฅ)๐‘ข(๐‘ก)๐‘‘๐‘กโˆ’๐‘ฅ0๐‘ก๐‘’๐‘ข(๐‘ก)๐‘‘๐‘ก=๐‘”(๐‘ฅ),0โ‰ค๐‘ฅ,๐‘กโ‰ค1,๐‘ข(0)=1,(5.4) where ๐‘”(๐‘ฅ)=cos(1โˆ’๐‘ฅ)+๐‘’1โˆ’๐‘ฅ(1+๐‘ฅ)โˆ’(1/2)๐‘ฅโˆ’(7/6)โˆ’๐‘’. The exact solution is ๐‘ข(๐‘ฅ)=1โˆ’๐‘ฅ.

Using RKHS method, taking ๐‘ฅ๐‘–=(๐‘–โˆ’1)/(๐‘โˆ’1), ๐‘–=1,2,โ€ฆ,๐‘, with the reproducing kernel function ๐พ๐‘ฅ(๐‘ฆ) on [0,1], the approximate solution ๐‘ข๐‘๐‘›(๐‘ฅ) is calculated by (4.4). The numerical results at some selected grid points for ๐‘=51 and ๐‘›=5 are given in Table 7.

tab7
Table 7: Numerical results for Example 5.4.

Example 5.5. Consider the nonlinear Fredholm-Volterra IDE: ๐‘ข๎…ž(๐‘ฅ)+๐‘ฅ๐‘ข2๎€œ(๐‘ฅ)โˆ’10๎€ท(๐‘ก+๐‘ฅ)1+๐‘ข2๎€ธ๎€œ(๐‘ก)๐‘‘๐‘กโˆ’๐‘ฅ0๐‘ฅcos(๐‘ข(๐‘ก))๐‘‘๐‘ก=๐‘”(๐‘ฅ),0โ‰ค๐‘ฅ,๐‘กโ‰ค1,๐‘ข(0)=0,(5.5) where ๐‘”(๐‘ฅ)=โˆ’๐‘ฅsin(๐‘ฅ)+๐‘ฅ3โˆ’(4/3)๐‘ฅ+(1/4). The exact solution is ๐‘ข(๐‘ฅ)=๐‘ฅ.
Using RKHS method, taking ๐‘ฅ๐‘–=(๐‘–โˆ’1)/(๐‘โˆ’1), ๐‘–=1,2,โ€ฆ,๐‘ with the reproducing kernel function ๐พ๐‘ฅ(๐‘ฆ) on [0,1], the approximate solution ๐‘ข๐‘๐‘›(๐‘ฅ) is calculated by (4.4). The numerical results at some selected grid points for ๐‘=26 and ๐‘›=5 are given in Table 8.

tab8
Table 8: Numerical results for Example 5.5.

6. Conclusion

In this paper, the RKHS method was employed to solve the nonlinear Fredholm-Volterra IDEs (1.1) and (1.2). The solution ๐‘ข(๐‘ฅ) and the approximate solution ๐‘ข๐‘›(๐‘ฅ) are represented in the form of series in the space ๐‘Š22[๐‘Ž,๐‘]. Moreover, the approximate solution and its derivative converge uniformly to the exact solution and its derivative, respectively. Meanwhile, the error of the approximate solution is monotonically decreasing in the sense of the norm of ๐‘Š22[๐‘Ž,๐‘].

References

  1. F. Bloom, โ€œAsymptotic bounds for solutions to a system of damped integro-differential equations of electromagnetic theory,โ€ Journal of Mathematical Analysis and Applications, vol. 73, no. 2, pp. 524โ€“542, 1980. View at Publisher ยท View at Google Scholar
  2. K. Holmåker, โ€œGlobal asymptotic stability for a stationary solution of a system of integro-differential equations describing the formation of liver zones,โ€ SIAM Journal on Mathematical Analysis, vol. 24, no. 1, pp. 116โ€“128, 1993. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  3. L. K. Forbes, S. Crozier, and D. M. Doddrell, โ€œCalculating current densities and fields produced by shielded magnetic resonance imaging probes,โ€ SIAM Journal on Applied Mathematics, vol. 57, no. 2, pp. 401โ€“425, 1997. View at Publisher ยท View at Google Scholar
  4. R. P. Kanwal, Linear Integral Differential Equations Theory and Technique, Academic Press, New York, NY, USA, 1971.
  5. K. Maleknejad and Y. Mahmoudi, โ€œTaylor polynomial solution of high-order nonlinear Volterra-Fredholm integro-differential equations,โ€ Applied Mathematics and Computation, vol. 145, no. 2-3, pp. 641โ€“653, 2003. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  6. E. Babolian, Z. Masouri, and S. Hatamzadeh-Varmazyar, โ€œNumerical solution of nonlinear Volterra-Fredholm integro-differential equations via direct method using triangular functions,โ€ Computers and Mathematics with Applications, vol. 58, no. 2, pp. 239โ€“247, 2009. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  7. E. Babolian, Z. Masouri, and S. Hatamzadeh-Varmazyar, โ€œNew direct method to solve nonlinear Volterra-Fredholm integral and integro-differential equations using operational matrix with block-pulse functions,โ€ Progress In Electromagnetics Research B, vol. 8, pp. 59โ€“76, 2008. View at Google Scholar
  8. K. Maleknejad, B. Basirat, and E. Hashemizadeh, โ€œHybrid Legendre polynomials and block-pulse functions approach for nonlinear Volterra-Fredholm integro-differential equations,โ€ Computers and Mathematics with Applications, vol. 61, no. 9, pp. 2821โ€“2828, 2011. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  9. S. Momani and R. Qaralleh, โ€œAn efficient method for solving systems of fractional integro-differential equations,โ€ Computers and Mathematics with Applications, vol. 52, no. 3-4, pp. 459โ€“470, 2006. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  10. M. Ghasemi, M. Tavassoli Kajani, and E. Babolian, โ€œApplication of He's homotopy perturbation method to nonlinear integro-differential equations,โ€ Applied Mathematics and Computation, vol. 188, no. 1, pp. 538โ€“548, 2007. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  11. A. El-Ajou, O. Abu Arqub, and S. Momani, โ€œHomotopy analysis method for second-order boundary value problems of integro-differential equations,โ€ Discrete Dynamics in Nature and Society. In press.
  12. A. Berlinet and C. Thomas-Agnan, Reproducing Kernel Hilbert Space in Probability and Statistics, Kluwer Academic, Boston, Mass, USA, 2004. View at Publisher ยท View at Google Scholar
  13. M. Cui and Y. Lin, Nonlinear Numercial Analysis in the Reproducing Kernel Space, Nova Science, New York, NY, USA, 2008.
  14. A. Daniel, Reproducing Kernel Spaces and Applications, Springer, New York, NY, USA, 2003.
  15. M. Cui and H. Du, โ€œRepresentation of exact solution for the nonlinear Volterra-Fredholm integral equations,โ€ Applied Mathematics and Computation, vol. 182, no. 2, pp. 1795โ€“1802, 2006. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  16. H. Du and J. Shen, โ€œReproducing kernel method of solving singular integral equation with cosecant kernel,โ€ Journal of Mathematical Analysis and Applications, vol. 348, no. 1, pp. 308โ€“314, 2008. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  17. F. Geng, โ€œA new reproducing kernel Hilbert space method for solving nonlinear fourth-order boundary value problems,โ€ Applied Mathematics and Computation, vol. 213, no. 1, pp. 163โ€“169, 2009. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  18. F. Geng, โ€œSolving singular second order three-point boundary value problems using reproducing kernel Hilbert space method,โ€ Applied Mathematics and Computation, vol. 215, no. 6, pp. 2095โ€“2102, 2009. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  19. F. Geng and M. Cui, โ€œSolving singular nonlinear two-point boundary value problems in the reproducing kernel space,โ€ Journal of the Korean Mathematical Society, vol. 45, no. 3, pp. 631โ€“644, 2008. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  20. F. Geng and M. Cui, โ€œSolving a nonlinear system of second order boundary value problems,โ€ Journal of Mathematical Analysis and Applications, vol. 327, no. 2, pp. 1167โ€“1181, 2007. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  21. F. Geng and M. Cui, โ€œSolving singular nonlinear second-order periodic boundary value problems in the reproducing kernel space,โ€ Applied Mathematics and Computation, vol. 192, no. 2, pp. 389โ€“398, 2007. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  22. J. Li, โ€œA computational method for solving singularly perturbed two-point singular boundary value problem,โ€ International Journal of Mathematical Analysis, vol. 2, no. 21-24, pp. 1089โ€“1096, 2008. View at Google Scholar ยท View at Zentralblatt MATH
  23. C.-l. Li and M.-g. Cui, โ€œThe exact solution for solving a class nonlinear operator equations in the reproducing kernel space,โ€ Applied Mathematics and Computation, vol. 143, no. 2-3, pp. 393โ€“399, 2003. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  24. Y. Li, F. Geng, and M. Cui, โ€œThe analytical solution of a system of nonlinear differential equations,โ€ International Journal of Mathematical Analysis, vol. 1, no. 9-12, pp. 451โ€“462, 2007. View at Google Scholar ยท View at Zentralblatt MATH
  25. Y. Z. Lin, M. G. Cui, and L. H. Yang, โ€œRepresentation of the exact solution for a kind of nonlinear partial differential equation,โ€ Applied Mathematics Letters, vol. 19, no. 8, pp. 808โ€“813, 2006. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  26. X. Lü and M. Cui, โ€œSolving a singular system of two nonlinear ODEs,โ€ Applied Mathematics and Computation, vol. 198, no. 2, pp. 534โ€“543, 2008. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  27. L. Yang and M. Cui, โ€œNew algorithm for a class of nonlinear integro-differential equations in the reproducing kernel space,โ€ Applied Mathematics and Computation, vol. 174, no. 2, pp. 942โ€“960, 2006. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  28. Y. Lin, P. Chung, and M. Cui, โ€œA solution of an infinite system of quadratic equations in reproducing kernel space,โ€ Complex Analysis and Operator Theory, vol. 1, no. 4, pp. 571โ€“579, 2007. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  29. M. Al-Smadi, O. Abu Arqub, and N. Shawagfeh, โ€œApproximate solution of BVPs for 4th-order IDEs by using RKHS method,โ€ Applied Mathematical Sciences, vol. 6, pp. 2453โ€“2464, 2012. View at Google Scholar
  30. Y. Zhou, M. Cui, and Y. Lin, โ€œNumerical algorithm for parabolic problems with non-classical conditions,โ€ Journal of Computational and Applied Mathematics, vol. 230, no. 2, pp. 770โ€“780, 2009. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH