Abstract

We present an efficient iterative method for solving a class of nonlinear second-order Fredholm integrodifferential equations associated with different boundary conditions. A simple algorithm is given to obtain the approximate solutions for this type of equations based on the reproducing kernel space method. The solution obtained by the method takes form of a convergent series with easily computable components. Furthermore, the error of the approximate solution is monotone decreasing with the increasing of nodal points. The reliability and efficiency of the proposed algorithm are demonstrated by some numerical experiments.

1. Introduction

The topic of integrodifferential equations (IDEs) which has attracted growing interest for some time has been recently developed in many applied fields, so a wide variety of problems in the physical sciences and engineering can be reduced to IDEs, in particular in relation to mathematical modeling of biological phenomena [1–3], aeroelasticity phenomena [4], population dynamics [5], neural networks [6], electrocardiology [7], electromagnetic [8], electrodynamics [9], and so on. Thus, it is important to study boundary value problems (BVPs) for especially the nonlinear IDEs, which can be classified into two types: Fredholm and Volterra IDEs, where the upper bound of the integral part of Fredholm type is a fixed number whilst it is a variable for Volterra type [10]. In this point, these types of IDEs arise in the theories of singular integral equations with degenerate symbol and BVPs for mixed type partial differential equations. Therefore, the investigations in this area are of great interest; see [11] and the references therein for an overview of the current state of the art in their numerical methods; also it is well known that it is extremely difficult to analytically solve nonlinear IDEs. Unfortunately, few of these equations can be solved explicitly. Thus, it is required to obtain an efficient approximation method in order to solve these types of IDEs. So far, several numerical methods are currently improved in this regard.

The existence and uniqueness of the solutions for the BVPs of higher-order IDEs have been investigated by Agarwal [12]; but no numerical method was presented. Particularly, the analytical approximate solutions for first-order up to higher-order IDEs have been obtained by the numerical integration techniques such as Runge-Kutta methods [13], Euler-Chebyshev methods [14], Wavelet-Galerkin method [15], and Taylor polynomials method [16] and by semianalytical-numerical techniques such as Adomian decomposition method [17], reproducing kernel Hilbert space (RKHS) method [18–22], homotopy analysis method [23], and variational iteration method [24].

In this paper, we apply the RKHS technique to develop a novel numerical method in the space for solving second-order Fredholm IDEs of the following form: where and are real finite constants, is an unknown function to be determined, and the forcing function can be linear or nonlinear function of . Subject to the typical boundary conditions, where and are already known boundary values.

Consequently here, we assume that IDEs (1) and (2) satisfy the following two assumptions. Firstly, the forcing function and all its partial derivatives are continuous, and is nonpositive for . Secondly, the kernel function satisfies the positive definite property: where is any continuous nonzero function and holds

For a comprehensive introduction about the existence and uniqueness theory of solution of such problems, we refer to [25, 26]. Additionally, we assume that IDEs (1) and (2) have a unique solution under the above two assumptions on the given interval.

In this paper, the attention is given to obtain the approximate solution of second-order Fredholm IDEs with different boundary conditions using the RKHS method. The present method can approximate the solutions and their derivatives at every point of the range of integration; also it has several advantages such that the conditions for determining solutions can be imposed on the reproducing kernel space, the conditions about the nonlinearity of the forcing function are simple and may include , , or any others operator of , and the iterative sequence of approximate solutions converges in to the solution .

This paper is comprised of five sections including the introduction. The next section is devoted to several reproducing kernel spaces and essential theorems. An associated linear operator and solution representation in are obtained in Section 3. Meanwhile, an iterative method is developed for the existence of solutions for IDEs (1) and (2) based on reproducing kernel space. The applications of the proposed numerical scheme are illustrated in Section 4. Conclusions are presented in Section 5.

2. Analysis of Reproducing Kernel Hilbert Space (RKHS)

In functional analysis, the RKHS is a Hilbert space of functions in which pointwise evaluation is a continuous linear functional. Equivalently, they are spaces that can be defined by reproducing kernels. In this section, we utilize the reproducing kernel concept to construct two reproducing kernel Hilbert spaces and to find out their representation of reproducing functions for solving the IDEs (1) and (2) via RKHS technique.

Definition 1 (see [27]). Hilbert spaces of functions on a nonempty abstract set are called a reproducing kernel Hilbert spaces if there exists a reproducing kernel of .

It is worth mentioning that the reproducing kernel of a Hilbert space is unique, and the existence of is due to the Riesz representation theorem, where completely determines the space . Moreover, every sequence of functions which converges strongly to a function in , converges also in the pointwise sense. This convergence is uniform on every subset on on which is bounded. In this occasion, these spaces have wide applications including complex analysis, harmonic analysis, quantum mechanics, statistics, and machine learning. Subsequently, the space is constructed in which every function satisfies the boundary conditions (2) and then utilized the space . For the theoretical background of reproducing kernel Hilbert space theory and its applications, we refer the reader to [27–31].

Definition 2. The inner product space is defined as is an absolutely continuous  function in , , , and is equipped with inner product and norm, respectively, , where and
The Hilbert space is called a reproducing kernel if, for each , there exist , simply , such that for any and .

Theorem 3. The Hilbert space is a complete reproducing kernel and its reproducing kernel function can be written as where the functions and can be determined by , and , in which , , and , .

By using the Mathematica 7.0 software package, the coefficients of the reproducing kernel are given in the appendix, whilst, the proof of completeness and the process of obtaining the coefficients of the reproducing kernel are similar to the proof of Theorem  2.1 in [20].

Remark 4. The reproducing kernel function is symmetric and unique, and for any fixed .

Definition 5. The inner product space is defined as .
On the other hand, the inner product and norm in are defined, respectively, by and , where .
In [29], the authors had proved that is complete reproducing kernel space and its reproducing kernel is

Lemma 6. The space is imbedded to space .

Theorem 7. An arbitrary bounded set of   is a compact set of .

Proof. Let be a bounded set of . Assume that , where is positive constant. From the representation of , we get Since , , is uniformly bounded about and , so we have . Hence, . Now, we need to prove that is a compact set of ; that is, are equicontinuous functions. Again, from the property of , it follows that By the symmetry of and the mean value theorem of differentials, it follows that Thus, if , we can get .

3. The Exact and Approximate Solution

In this section, formulation of a differential linear operator and implementation method are presented in the spaces and . Meanwhile, we construct an associated orthogonal function system based on Gram-Schmidt orthogonalization process in order to obtain the exact and approximate solutions of IDEs (1) and (2). In order to apply the RKHS method, as in [30–33], we firstly define a differential linear operator such that . After homogenization of the boundary conditions of Equation (2), the IDEs (1) and (2) can be transformed into the equivalent operator equation as where , and for , , and . It is clear that is a bounded linear operator and exists.

Lemma 8. Let and , , where is the reproducing kernel of . If is a dense subset on , then is the complete system of the space .

Proof. For all , let , ; then . Thus, by the density of and the continuity of , we have . Thus, is the complete system of . Therefore, is also the complete system of .
The normal orthogonal system of the space of the sequence can be constructed as , , where , , are the coefficients of orthogonalization.

Lemma 9. Let , , where is the adjoint operator of . If is dense in , then , .

Proof. We note that . The subscript by the operator indicates that the operator applies to the function of . Clearly .

Corollary 10. For (11), if is dense in , then is the complete system of .

The normal orthogonal system of the space can be derived from Gram-Schmidt orthogonalization process of the sequence as follows: where the orthogonalization coefficients are given by , , for ,   and , and such that , .

Theorem 11. For all , the series is convergent in the sense of the norm . On the other hand, if is the exact solution of (11), then

Proof. Since can be expanded in the form of Fourier series about normal orthogonal system as and since the space is Hilbert space, so the series is convergent in the norm of . From (12) and (14), it can be written as .
If is the exact solution of (11) and , for , then .
As a result, the approximate solution can be obtained by the -term intercept of the exact solution , and it is given by

Remark 12. If (11) is linear, then the approximate solution to (11) can be obtained directly from (15), while, if (11) is nonlinear, then the exact and approximate solutions can be obtained using the following iterative algorithm.
According to (14), we construct the iterative sequences : where the coefficients of ,  , are given as
In the iteration process of (16), we can guarantee that the approximation solution satisfies the boundary conditions of (2).

Corollary 13. The sequence in (16) is monotone increasing in the sense of the norm .

Proof. Since is the complete orthonormal system in , one can get . Hence, is monotone increasing.
Since is Hilbert space, then and thus is convergent. On the other hand, the solution of IDEs (1) and (2) is considered the fixed point of the following functional under the suitable choice of the initial term :

Theorem 14 (see [28]). Assume that is a Banach space and is a nonlinear mapping and suppose that , for some constants α < 1. Then has a unique fixed point. Furthermore, the sequence , with an arbitrary choice of , converges to the fixed point of .

According to above theorem, for the nonlinear mapping, A sufficient condition for convergence of the present iteration method is strictly contraction of . Furthermore, the sequence (16) converges to the fixed point of which is also the solution of IDEs (1) and (2).

The approximate solution can be obtained by taking finitely many terms in the series representation of , given by

4. Numerical Results and Discussions

To illustrate the accuracy and applicability of the RKHS method, three examples are given in this section. 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 are performed by using Mathematica 7.0 software package.

Example 15 (see [34]). Consider the following Fredholm IDE: subject to the boundary conditions , with the exact solution .

We employ the RKHS method to solve this example. Taking , , the approximate solutions are obtained using (15) and the reproducing kernel function on is given such that where and are given in the appendix.

In Table 1, there is a comparison of the numerical result against the Taylor polynomial solution, used in [35], and Tau-Chebyshev and Legendre method, used in [36], at some selected grid points on . It is worth noting that the RKHS results become very highly accurate only with a few iterations and become very close to the exact solution.

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.

Example 16 (see [37]). Consider the following nonlinear Fredholm IDE: subject to the boundary conditions with the exact solution .

Again, we employ the RKHS method to solve this example, taking , , . The approximate solutions at some selected gird points on are obtained using (16). The results and errors are reported in Table 2.