Abstract and Applied AnalysisVolume 2012 (2012), Article ID 971593, 10 pagesdoi:10.1155/2012/971593

Research Article

Reproducing Kernel Space Method for the Solution of Linear Fredholm Integro-Differential Equations and Analysis of Stability

Xueqin Lv and Yue Gao

School of Mathematical Sciences, Harbin Normal University, Heilongjiang, Harbin 150025, China

Received 2 February 2012; Accepted 7 March 2012

Academic Editor: Ljubisa Kocinac

Copyright © 2012 Xueqin Lv and Yue Gao. 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.

We present a numerical method to solve the linear Fredholm integro-differential equation in reproducing kernel space. A simple algorithm is given to obtain the approximate solutions of the equation. Through the comparison of approximate and true solution, we can find that the method can effectively solve the linear Fredholm integro-differential equation. At the same time the numerical solution of the equation is stable.

1. Introduction

In this paper, we consider the following first-order Fredholm type integro-differential equation: with the initial condition

The equation is discussed by Yusufoğlu in [1], where , , and are sufficiently regular given functions. Integro-differential system is an important tool in solving real-world problems. A wide variety of natural phenomena are modelled by Fredholm type integro-differential equations. The ordinary integro-differential system has been applied to many problems in fluid dynamics, engineering, chemical reactions, and so on.

In the recent years, there are some methods to solve the Fredholm type integro-differential equations [2–8]. At this point, a new method is presented to solve the integro-differential equations. The method is established in reproducing kernel space; the problem of solving the integro-differential problem with a perturbation can be converted into the simple problem of solving the equation. The representation of all the solutions for Fredholm type integro-differential equation is given if it has solutions. The stability is important and references are there in [9, 10]. There are many discussions about the solutions in reproducing kernel space in [11–15]. In this paper, we discuss the Fredholm type integro-differential equation. In the last section, CAS wavelet approximating methods [5], differential transformation methods [6], HPM [1], and reproducing kernel space method are compared, then we can get some effective data. The numerical experiments show that this kind of method is stable in the reproducing kernel space.

2. Two Reproducing Kernel Spaces

2.1. The Reproducing Kernel Space

The reproducing kernel space is defined as follows:

The inner product and norm in are defined respectively by

Then is a complete reproducing kernel space. That is, there exists a function , for each fixed ,  , and for any and , satisfying By using Mathematica, is given by

2.2. The Reproducing Kernel Space

The construction of reproducing kernel space can be found in [14] and its reproducing kernel function is

3. Analysis of the Solution of (1.1)

Let , such that

where , it is easy to know that is a linear bounded operator and (1.1) can be converted into the equivalent form

In order to obtain the representation of all the solutions of (1.1), let , , where is dense in .

From the definition of the reproducing kernel, we have

where is a conjugate operator of . Practise Gram-Schmidt orthonormalization for where are coefficients of Gram-Schmidt orthonormalization and is orthonormal system in .

Theorem 3.1. If (3.2) has solutions, the results are proved by the following formula:

Proof. The results are proved by the following formula:

Now, the approximate solution can be obtained by the -term truncation of (3.5), that is

Theorem 3.2. Assume is the solution of (3.2) and is the error between the approximate solution and the exact solution . Then .

Proof. In the following we prove the sequence is monotone decreasing in the sense of . From (3.5) and (3.7), we have By (3.8), we know that . Then is monotone decreasing in . Since the series is convergent in , we obtain .

4. The Stability of the Approximate Solution

In order to consider the stability of the approximate solution, we add a perturbation in the right-hand side, then (3.2) becomes

Now, we discuss the representation of the solution for (4.1).

4.1. Representation of All the Solutions of (4.1)

In order to study the stability of (4.1), let be a projection operator from to , where is given in (3.4), and satisfies . Moreover

We have

where is a solution of (3.2) in .

Define then we have

Theorem 4.1. If is dense in , then the form (4.6) is the solution of (3.2) in .

Proof. We have since is dense in , then .It is easy to know that the solution of (3.2) is unique in (see [16]).

The following lemma holds.

Lemma 4.2. , where and is a null space of , that is .

Proof. For any , Since is dense in , then . We can obtain . Obviously, .

Then .

The following theorem is obvious.

Theorem 4.3. Let be any dense subset of , if (4.1) has solutions, then its solution can be represented as follows: where .

Assume that is a basis of . Orthonormalzing , we obtain

Hence, is a normal orthogonal basis in .

According to Theorem 4.3, we can obtain the solution of (4.1). Hence, through the Gram-Schmidt process, we have are the complete orthonormal system in , and is a complete orthonormal system in .

4.2. The Stability of the Solution for (3.2) in Reproducing Kernel Space

Let the space be complete. be a restricted operator of in , we have the converse operator exists and is bounded.

Lemma 4.4. If is given by (4.6), then is the minimal norm solution.

Proof. Let be a solution of (3.2). We have where and . The following holds.It is pointed that is the minimal norm solution of (3.2).

Theorem 4.5. If the (4.1) has solutions and let be the minimal norm solution, then where is a truncation of . Hence, is stable in .

Proof. Let and , where is a perturbation and in .On the one hand, since , , it follows thatOn the other hand, , from the form (4.6), we havewhere and then From the continuity of and in , we have

5. Numerical Experiments

To illustrate the effectiveness of the above method, we give an example as follows:

Example 5.1. Consider the following first-order Fredholm type integro-differential equation (see [1]): The analytic solution of this equation is .From Tables 1, 2, and 3, we can see that the absolute error, relative error and the root-mean-square are small.

Table 1: Comparison of absolute errors of approximate solutions, for example, obtained by reproducing kernel space methods, CAS wavelet approximating [5], differential transformation [6] methods and HPM [1].

Node
Method in [5]
Method in [6]
Method in [1]
Reproducing kernel methods

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Table 2: Relative error of approximate solutions with .

Node
Approximate solution
True solution
Relative error

0.1
0.110575
0.110517

0.2
0.244350
0.244281

0.3
0.405040
0.404958

0.4
0.596828
0.596730

0.5
0.824476
0.824361

0.6
1.093410
1.093270

0.7
1.409790
1.409630

0.8
1.780620
1.780430

0.9
2.213850
2.213640

1
2.718520
2.718280

Table 3: The root-mean-square, for example, with .

6. Conclusions

From the previous numerical results, we can see that the error is quite small and the numerical solution is stable when the right-hand side with a small perturbation. It illustrates that the method is given in the paper is valid.

Acknowledgments

The research was supported by the National Natural Science Foundation of China (61071181), the Excellent Middle-Young Scientific Research Award Foundation of Heilongjiang Province (QC2010036), academic foundation for youth of Harbin Normal University (10KXQ-05), and science and technology development project of Harbin Normal University (08XYG-13).

References

E. Yusufoğlu, “Improved homotopy perturbation method for solving Fredholm type integro-differential equations,” Chaos, Solitons and Fractals, vol. 41, pp. 28–37, 2009.

M. T. Kajani, M. Ghasemi, and E. Babolian, “Comparison between the homotopy perturbation method and the sine-cosine wavelet method for solving linear integro-differential equations,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 1162–1168, 2007.

J. Rashidinia and M. Zarebnia, “The numerical solution of integro-differential equation by means of the sinc method,” Applied Mathematics and Computation, vol. 188, no. 2, pp. 1124–1130, 2007.

A. Golbabai and S. Seifollahi, “Radial basis function networks in the numerical solution of linear integro-differential equations,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 427–432, 2007.

H. Danfu and S. Xufeng, “Numerical solution of integro-differential equations by using CAS wavelet operational matrix of integration,” Applied Mathematics and Computation, vol. 194, no. 2, pp. 460–466, 2007.

P. Darania and A. Ebadian, “A method for the numerical solution of the integro-differential equations,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 657–668, 2007.

J. Zhang, H. Han, and X. Pan, “Legendre wavelets method for solving system of linear Fredholm integro-differential equations,” Science Technology and Engineering, no. 24, pp. 7283–7287, 2009.

Z. Shi and Z. Deng, “Numbrical solution of Fredholm integro-differential equations by using Legendre wavelets,” Journal of Mathematical Study, vol. 24, pp. 10–13, 2009.

B. Yıldız, H. Yetişkin, and A. Sever, “A stability estimate on the regularized solution of the backward heat equation,” Applied Mathematics and Computation, vol. 135, no. 2-3, pp. 561–567, 2003.

Y. Yildiz, “Central difference schemes in time and error estimate on a non-standard inverse heat conduction problem,” Applied Mathematics and Computation, vol. 157, no. 1, pp. 77–91, 2004.

C. C. Tisdell, “Existence of solutions to first-order periodic boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 323, no. 2, pp. 1325–1332, 2006.

E. M. E. Elbarbary and M. El-Kady, “Chebyshev finite difference approximation for the boundary value problems,” Applied Mathematics and Computation, vol. 139, no. 2-3, pp. 513–523, 2003.

Z. Chen and Y. Lin, “The exact solution of a linear integral equation with weakly singular kernel,” Journal of Mathematical Analysis and Applications, vol. 344, no. 2, pp. 726–734, 2008.

M. Gui and Y. Lin, Nonlinear Numercial Analysis in the Reproducing Kernel Space, Nova Science, New York, NY, USA, 2008.

C. Li and M. Cui, “The exact solution for solving a class nonlinear operator equations in the reproducing kernel space,” Applied Mathematics and Computation, vol. 143, no. 2-3, pp. 393–399, 2003.

H. Du and M. Cui, “Approximate solution of the Fredholm integral equation of the first kind in a reproducing kernel Hilbert space,” Applied Mathematics Letters, vol. 21, no. 6, pp. 617–623, 2008.