Numerical solution of second kind Fredholm integral equations system by using a Taylor-series expansion method

Abstract

In this paper, we present a Taylor-series expansion method for a second kind Fredholm integral equations system with smooth or weakly singular kernels. This method reduce the system of integral equations to a linear system of ordinary differential equation. After constructing boundary conditions this system reduce to a system of equations that can be solved easily with any of the usual methods. Finally, we show the efficiency of the method using some numerical examples.

Introduction

In this paper, we use a modified Taylor-series expansion method for solving Fredholm integral equations systems of second kind. This method first presented in [4] for solving Fredholm integral equations of second kind and then in [3] for solving Volterra integral equations of second kind.

Consider the second kind Fredholm integral equations system of the form

$$\mathbf{F}(s)=\mathbf{G}(s)+{\int }_0^1\mathbf{K}(s\text{,}t)\mathbf{F}(t)\mathrm{d}t\text{,}0⩽s⩽1\text{,}$$

where

$$\begin{array}{cc}\hfill & \mathbf{F}(s)=[f_1(s)\text{,}{f}_2(s)\text{,}\dots \text{,}{f}_n(s){]}^{\mathrm{T}}\text{,}\hfill \\ \hfill & \mathbf{G}(s)=[g_1(s)\text{,}{g}_2(s)\text{,}\dots \text{,}{g}_n(s){]}^{\mathrm{T}}\text{,}\hfill \\ \hfill & \mathbf{K}(s\text{,}t)=[k_{\mathit{ij}}(s\text{,}t)]\text{,}i\text{,}j=1\text{,}2\text{,}\dots \text{,}n.\hfill \end{array}$$

In Eq. (1) the functions K and G are given, and F is the solution to be determined [1], [2]. We assume that (1) has a unique solution.

Consider the ith equation of (1):

$$f_i(s)=g_i(s)+{\int }_0^1\sum _{j=1}^n{k}_{\mathit{ij}}(s\text{,}t)f_j(t)\mathrm{d}t\text{,}i=1\text{,}2\text{,}\dots \text{,}n\text{.}$$

A Taylor-series expansion can be made for the solution f_j(t) in the integral Eq. (2):

$$f_j(t)=f_j(s)+f_j^{\prime }(s)(t-s)+\cdots +\frac{1}{m!}{f}_j^{(m)}(s)(t-s{)}^m+E(t)\text{,}$$

where E(t) denotes the error between f_j(t) and its Taylor-series expansion (3).

$$E(t)=\frac{1}{(m+1)!}{f}_j^{(m+1)}(s)(t-s{)}^{(m+1)}+\cdots$$

If we use the first m term of Taylor-series expansion (3) (as an approximate for f_j(t) in (2)) and neglige the ${\int }_0^1{\sum }_{j=1}^n{k}_{\mathit{ij}}(s\text{,}t)E(t)\mathrm{d}t$, then substituting (3) for f_j(t) in the integral in Eq. (2), we have

$$f_i(s)\simeq g_i(s)+{\int }_0^1\sum _{j=1}^n{k}_{\mathit{ij}}(s\text{,}t)\sum _{r=0}^m\frac{1}{r!}(t-s{)}^r{f}_j^{(r)}(s)\mathrm{d}t\text{,}i=1\text{,}2\text{,}\dots \text{,}n\text{,}$$

$$f_i(s)\simeq g_i(s)+\sum _{j=1}^n\sum _{r=0}^m\frac{1}{r!}{f}_j^{(r)}(s){\int }_0^1{k}_{\mathit{ij}}(s\text{,}t)(t-s{)}^r\mathrm{d}t\text{,}i=1\text{,}2\text{,}\dots \text{,}n\text{,}$$

$$f_i(s)-\sum _{j=1}^n\sum _{r=0}^m\frac{1}{r!}{f}_j^{(r)}(s)\left[{\int }_0^1{k}_{\mathit{ij}}(s\text{,}t)(t-s{)}^r\mathrm{d}t\right]\simeq g_i(s)\text{,}i=1\text{,}2\text{,}\dots \text{,}n\text{,}$$

if the integrals in Eq. (6) can be solved analytically, then the bracketed quantities are functions of s alone. So Eqs. (6) becomes a linear system of ordinary differential equations that can be solved. However, this requires the manufacture of an appropriate number of boundary conditions. Now we present a method to manufacturing boundary conditions in easy way.