Journal of Mathematics

Volume 2017, Article ID 8213932, 6 pages

https://doi.org/10.1155/2017/8213932

## Approximate Solution of Perturbed Volterra-Fredholm Integrodifferential Equations by Chebyshev-Galerkin Method

^{1}Department of Statistics and Mathematical Sciences, Kwara State University, PMB 1530, Malete, Ilorin, Kwara State, Nigeria^{2}Department of Mathematics, Islamic Azad University, Darab Branch, Darab, Iran

Correspondence should be addressed to K. Issa; moc.oohay@meezakassi

Received 4 October 2016; Accepted 22 December 2016; Published 12 January 2017

Academic Editor: Liwei Zhang

Copyright © 2017 K. Issa and F. Salehi. 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

In this work, we obtain the approximate solution for the integrodifferential equations by adding perturbation terms to the right hand side of integrodifferential equation and then solve the resulting equation using Chebyshev-Galerkin method. Details of the method are presented and some numerical results along with absolute errors are given to clarify the method. Where necessary, we made comparison with the results obtained previously in the literature. The results obtained reveal the accuracy of the method presented in this study.

#### 1. Introduction

Integrodifferential equations (IDEs) arise in many branches of mathematical sciences such as financial mathematics, mathematical modeling, and control theory, which are usually difficult to solve analytically, so numerical approach is required. Different methods have been used to obtain the solution of both linear and nonlinear IDEs such as Galerkin method [1–4], homotopy perturbation [5, 6], Tau method [7–9], spline collocation method [10–12], Taylor collocation [13–17], finite element approach [18–23], Legendre polynomials [24], variational iteration method [25], Haar wavelets [26], Krein’s method [27], and Bessel collocation method [28], to mention a few.

The present work is motivated by the desire to obtain numerical solutions to initial value problems for integrodifferential equations via perturbed Chebyshev-Galerkin method. This paper is organized as follows. In Section 2, Chebyshev polynomial is discussed. In Section 3, preliminary steps towards application of the perturbed Chebyshev-Galerkin method are introduced. In Section 4, some numerical results are provided to demonstrate the efficiency and accuracy of using perturbed Chebyshev-Galerkin method and compared with those of [1, 29] and lastly, Section 5 is the conclusion.

#### 2. Chebyshev Polynomial

Chebyshev polynomials are widely used in applications in mathematics, mathematical physics, engineering, and computer science. Chebyshev polynomial is an orthogonal polynomial which satisfy the recurrence relationIn recent years, a lot of attention has been devoted to the study of Chebyshev methods to investigate various scientific models. Using these methods made it possible to solve differential equations of different forms [1, 7, 30–32], irrespective of the order of the differential equations.

In certain applications we need expressions for the products like , which comes easily from

#### 3. Perturbed Chebyshev-Galerkin Method

We will consider the numerical solution of a class of linear Volterra-Fredholm integrodifferential initial value problems of the formwhere is the unknown function, , , and , , are known functions, is the order of (3), and , , are real numbers. Unless otherwise stated, will always be the independent variable of the functions which appear throughout this paper and will be defined in a finite interval . Moreover suppose that be the approximate solution of degree to , so we writewhere are determined by adding perturbation termsto the right hand side of (3); we obtain

Applying Chebyshev-Galerkin approach discussed in [1], that is, multiplying both sides of (7) by , , , and then integrating the resulting equation over interval , we obtainFrom (8), we havewhere is a matrix of , and are column matrices of , and the other equations are derived from the initial conditions (4); that is,Substitute the values of obtained from (9) and (10) in (5) to obtain the approximate solution of degree .

#### 4. Numerical Examples

*Example 1 (see [1, 33]). *Consider the Fredholm integrodifferential equationsubject to initial conditionThe absolute errors are tabulated in Table 1 at different . Table 5 exhibits a comparison between the errors obtained by using the perturbed Chebyshev-Galerkin and using Galerkin method [1]. Figure 1 shows the absolute errors at different and Figure 1(b) presents the perturbed Chebyshev-Galerkin method and exact solutions.