- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Abstract and Applied Analysis

Volume 2013 (2013), Article ID 196308, 5 pages

http://dx.doi.org/10.1155/2013/196308

## Solving a System of Linear Volterra Integral Equations Using the Modified Reproducing Kernel Method

College of Science, Harbin Engineering University, Heilongjiang 150001, China

Received 16 May 2013; Accepted 30 September 2013

Academic Editor: Rodrigo Lopez Pouso

Copyright © 2013 Li-Hong Yang 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

A numerical technique based on reproducing kernel methods for the exact
solution of linear Volterra integral equations system of the second kind is
given. The traditional reproducing kernel method requests that operator a satisfied linear operator equation , is bounded and its image space is the reproducing kernel space . It limits its application. Now, we modify the reproducing kernel method such that it can be more widely
applicable. The *n*-term approximation solution obtained by the modified
method is of high accuracy. The numerical example compared with other
methods shows that the modified method is more efficient.

#### 1. Introduction

The purpose of this paper is to solve a system of linear Volterra integral equations where

In (1), the functions and are given, and is the solution to be determined. We assume that (1) has a unique solution. Volterra integral equation arises in many physical applications, for example, potential theory and Dirichlet problems, electrostatics, mathematical problems of radiative equilibrium, the particle transport problems of astrophysics and reactor theory, and radiative heat transfer problems [1–5]. Several valid methods for solving Volterra integral equation have been developed in recent years, including power series method [6], Adomain’s decomposition method [7], homotopy perturbation method [8, 9], block by block method [10], and expansion method [11].

Since the reproducing kernel space , which is a special Hilbert space, is constructed in 1986 [12], the reproducing kernel theory has been applied successfully to many linear and nonlinear problems, such as differential equation, population model, and many other equations appearing in physics and engineering [12–21]. The traditional reproducing kernel method is limited, because it requires that the image space of operator in linear operator equation is and operator must be bounded. In order to enlarge its application range, the MRKM removes the boundedness of and weakens its image space to . Subsequently, we apply the MRKM to obtain the series expression of the exact solution for (1). The *n*-term approximation solution is provided by truncating the series. The final numerical comparisons between our method and other methods show the efficiency of the proposed method. It is worth to mention that the MRKM can be generalized to solve other system of linear equations.

#### 2. Preliminaries

##### 2.1. The Reproducing Kernel Space

The reproducing kernel space consists of all absolute continuous real-valued functions, which defined on the closed interval , and the first derivative functions belong to .

The inner product and the norm are equipped with

Theorem 1. *is a reproducing kernel space with reproducing kernel [22]
**
that is, for every and , it follows that
*

##### 2.2. The Reproducing Kernel Space

The reproducing kernel space consists of all real-valued functions in which the first derivative functions are absolute continuous on the closed interval and the second derivative functions belong to .

The inner product and the norm are equipped with

Theorem 2. *is a reproducing kernel space with reproducing kernel [22]
**
that is, for every and , it follows that
*

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

##### 2.3. Hilbert Space

Hilbert space is defined by The inner product and the norm are given by

It is easy to prove that is a Hilbert space.

#### 3. The Exact Solution of (1)

##### 3.1. Identical Transformation of (1)

Consider the th equation of (1):

Define operator , where . Then, (1) can be turned into where .

##### 3.2. The Exact solution of (1)

Let be a dense subset of interval , and define for every ; the subscript of means that the operator acts on the function of . It is easy to prove that .

Theorem 3. * is complete in .*

*Proof. *Take such that for every .

From this fact, it holds that

for every . The dense assumes that

Since (16) has a unique solution, it follows that . This completes the proof.

We arrange , ,, , , , , , , , ,, ,, denoted by ; that is, . In a general way, . The orthogonal basis in from Gram-Schmidt orthogonalization of is as follows:

Theorem 4. *The exact solution of (1) can be expressed by
**
where ; if , then .*

*Proof. *Assume that is the exact solution of (1). can be expanded to Fourier series in terms of normal orthogonal basis in :
if , then
When , it holds that

Corollary 5. *The approximate solution of (1) is
**
and converges uniformly to on as for every . *

*Proof. *Obviously, holds as ; that is, is the approximate solution of (1).

Note that . Combining with the expression of , we have
It shows that converges uniformly to on as for every . So the proof is complete.

*Remark 6. *If and in (1), then it is reasonable to regard the unknown functions as the elements of .

#### 4. Numerical Examples

Taking nodes , is the approximate solutions of , and denotes the absolute errors of . According to Remark 6, we solve the following two examples appearing in [11] in .

*Example 7. *Consider the following system of Volterra integral equations of the second kind [11]:
where and are chosen such that the exact solution is . The numerical results obtained by using the present method are compared with [11] in Table 1.

*Example 8. *Consider the following system of linear Volterra integral equations of the second kind [11]:
where and are chosen such that the exact solution is . The numerical results obtained by using the present method are compared with [11] in Table 2.

#### 5. Conclusion

In this paper, we modify the traditional reproducing kernel method to enlarge its application range. The new method named MRKM is applied successfully to solve a system of linear Volterra integral equations. The numerical results show that our method is effective. It is worth to be pointed out that the MRKM is still suitable for solving other systems of linear equations.

#### Acknowledgments

The research was supported by the Fundamental Research Funds for the Central Universities.

#### References

- Y. Ren, B. Zhang, and H. Qiao, “A simple Taylor-series expansion method for a class of second kind integral equations,”
*Journal of Computational and Applied Mathematics*, vol. 110, no. 1, pp. 15–24, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Maleknejad and Y. Mahmoudi, “Numerical solution of linear Fredholm integral equation by using hybrid Taylor and block-pulse functions,”
*Applied Mathematics and Computation*, vol. 149, no. 3, pp. 799–806, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Yalçınbaş and M. Sezer, “The approximate solution of high-order linear Volterra-Fredholm integro-differential equations in terms of Taylor polynomials,”
*Applied Mathematics and Computation*, vol. 112, no. 2-3, pp. 291–308, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Yalçinbaş, “Taylor polynomial solutions of nonlinear Volterra-Fredholm integral equations,”
*Applied Mathematics and Computation*, vol. 127, no. 2-3, pp. 195–206, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. Deeba, S. A. Khuri, and S. Xie, “An algorithm for solving a nonlinear integro-differential equation,”
*Applied Mathematics and Computation*, vol. 115, no. 2-3, pp. 123–131, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Tahmasbi and O. S. Fard, “Numerical solution of linear Volterra integral equations system of the second kind,”
*Applied Mathematics and Computation*, vol. 201, no. 1-2, pp. 547–552, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. Babolian, J. Biazar, and A. R. Vahidi, “On the decomposition method for system of linear equations and system of linear Volterra integral equations,”
*Applied Mathematics and Computation*, vol. 147, no. 1, pp. 19–27, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Biazar and H. Ghazvini, “He's homotopy perturbation method for solving systems of Volterra integral equations of the second kind,”
*Chaos, Solitons & Fractals*, vol. 39, no. 2, pp. 770–777, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. Yusufoğlu (Agadjanov), “A homotopy perturbation algorithm to solve a system of Fredholm-Volterra type integral equations,”
*Mathematical and Computer Modelling*, vol. 47, no. 11-12, pp. 1099–1107, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Katani and S. Shahmorad, “Block by block method for the systems of nonlinear Volterra integral equations,”
*Applied Mathematical Modelling*, vol. 34, no. 2, pp. 400–406, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Rabbani, K. Maleknejad, and N. Aghazadeh, “Numerical computational solution of the Volterra integral equations system of the second kind by using an expansion method,”
*Applied Mathematics and Computation*, vol. 187, no. 2, pp. 1143–1146, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. G. Cui and Z. X. Deng, “On the best operator of interpolation in ${W}_{2}^{1}[a,b]$,”
*Mathematica Numerica Sinica*, vol. 2, pp. 209–216, 1986. - M. Cui and F. Geng, “A computational method for solving one-dimensional variable-coefficient Burgers equation,”
*Applied Mathematics and Computation*, vol. 188, no. 2, pp. 1389–1401, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Cui and F. Geng, “A computational method for solving third-order singularly perturbed boundary-value problems,”
*Applied Mathematics and Computation*, vol. 198, no. 2, pp. 896–903, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Cui and Z. Chen, “The exact solution of nonlinear age-structured population model,”
*Nonlinear Analysis. Real World Applications*, vol. 8, no. 4, pp. 1096–1112, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Z. Geng and M. G. Cui, “Solving singular nonlinear two-point boundary value problems in the reproducing kernel space,”
*Journal of the Korean Mathematical Society*, vol. 45, no. 3, pp. 631–644, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. G. Cui and Y. B. Yan, “The representation of the solution of a kind operator
equation $Au=f$,”
*Journal of Chinese Universities*, vol. 3, pp. 82–86, 1995. - O. Abu Arqub, M. Al-Smadi, and S. Momani, “Application of reproducing kernel method for solving nonlinear Fredholm-Volterra integrodifferential equations,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 839836, 16 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - O. A. Arqub, M. Al-Smadi, and N. Shawagfeh, “Solving Fredholm integro-differential equations using reproducing kernel Hilbert space method,”
*Applied Mathematics and Computation*, vol. 219, no. 17, pp. 8938–8948, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - M. Al-Smadi, O. Abu Arqub, and S. Momani, “A computational method for two-point boundary value problems of fourth-order mixed integrodifferential equations,”
*Mathematical Problems in Engineering*, vol. 2013, Article ID 832074, 10 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - M. Al-Smadi, O. Abu Arqub, and N. Shawagfeh, “Approximate solution of BVPs for 4th-order IDEs by using RKHS method,”
*Applied Mathematical Sciences*, vol. 6, no. 49–52, pp. 2453–2464, 2012. View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Cui and Y. Lin,
*Nonlinear Numerical Analysis in the Reproducing Kernel Space*, Nova Science Publishers, New York, NY, USA, 2009. View at MathSciNet