Numerical solution of the system of Fredholm integro-differential equations by the Tau method

Abstract

The Tau method, produces approximate polynomial solution of differential, integral and integro-differential equations (see [E.l,Ortiz, The Tau method, SIAM J. Numer. Anal. 6 (3) (1969) 480–492; E.l. Ortiz, H. Samara, An operational approach to the Tau method for the numerical solution of non-linear differential equations, Computing 27 (1981) 15–25; S.M. Hosseini, S. Shahmorad, A matrix formulation of the Tau for Fredholm and Volterra linear integro-differential equations, The Korean J. Comput. Appl. Math. 9 (2) (2002) 497–507; S.M. Hosseini, S. Shahmorad, Numerical solution of a class of integro-differential equations by the Tau method with an error estimation, Appl. Math. Comput. 136 (2003) 559–570]). In this paper, we extend the Tau method for the numerical solution of integro-differential equations systems (IDES). We also give a brief description of the structure of the Tau program by the Maple software.

An efficient error estimation of the numerical solution of the method is also introduced. Some examples are given to clarify the efficiency and high accuracy of the method.

Introduction

In recent years the operational technique of the Tau method has developed to cover the numerical solution of differential, integral and integro-differential equations [6], [7], [2], [3], [4]. Liu and Pan [5] presented extension of the operational approach to the Tau method for the numerical solution of mixed-order systems of linear ordinary differential equations with polynomial or rational polynomial coefficients, together with initial or boundary conditions. In this paper, we consider Ortiz and Samara’s operational approach to the Tau method for the numerical solution of the system of Fredholm integro-differential equations

$$\sum _{j=1}^m[D_{\mathit{ij}}{u}_j(x)+\lambda \underset{a}{\overset{b}{\int }}{K}_{\mathit{ij}}(x\text{,}t)u_j(t)\mathrm{d}t]=f_i(x)\text{,}x\in [a\text{,}b]\text{,}i=1\text{,}\dots \text{,}m$$

with the supplementary conditions

$$\sum _{j=1}^m(l_{\mathit{rj}}\text{,}{u}_j)={\mathrm{d}}_r\text{,}r=1\text{,}2\text{,}\dots \text{,}\omega \text{,}$$

where

$$\begin{array}{cc}\hfill & D_{\mathit{ij}}=\sum _{r=0}^{{\mathit{nd}}_{\mathit{ij}}}{P}_{\mathit{ijr}}(x)\frac{{\mathrm{d}}^r}{\mathrm{d}{x}^r}=\sum _{r=0}^{{\mathit{nd}}_{\mathit{ij}}}\sum _{s=0}^{{\beta }_{\mathit{ijr}}}{p}_{\mathit{ijrs}}{x}^s\frac{{\mathrm{d}}^r}{\mathrm{d}{x}^r}\text{,}\hfill \\ \hfill & {\mathit{nd}}_{\mathit{ij}}⩽{\mathit{nd}}_{\mathit{ii}}\text{,}j\ne i\text{,}i=1\text{,}2\text{,}\dots \text{,}m\text{,}\omega =\sum _{i=1}^m{\mathit{nd}}_{\mathit{ii}}\text{,}\hfill \end{array}$$

f_i(x)∈C[a, b] and l_rj is a linear point evaluation functionals acting on u_j(x).

In this paper Chebyshev basis is applied for the numerical solution of system (1), that leads to remarkable computational accuracy and simplicity.

The paper is organized as follows. In Section 2, we review the operational approach to the Tau Method. Section 3 is devoted to apply the Tau method with arbitrary basis to cover solving of IDES (1). An efficient Tau error estimator is introduced in Section 4. Section 5 explains structure of the Tau program in Maple codes and some numerical results are provided to illustrate the efficiency of applying Chebyshev bases in numerical results. Closing section includes some useful remarks and conclusion.