Numerical computational solution of the Volterra integral equations system of the second kind by using an expansion method
Introduction
In this paper, we use a modified Taylor-series expansion method for solving Volterra integral equations system of the second kind. This method was first presented in 2 for solving Fredholm integral equations of second kind and then in [3], [4] for solving Volterra integral equations system and Fredholm integral equations system of second kind.
Consider the second kind Volterra integral equations system of the form:where
In Eq. (1) the functions K and G are given, and F is the solution to be determined [1].
We assume that (1) has a unique solution.
Consider the ith equation of (1):
A Taylor-series expansion can be made for the solution fj(t) in the integral Eq. (2):where E(t) denotes the error between fj(t) and its Taylor-series expansion (3).
If we use the first m terms of the Taylor-series expansion (3) and neglige the term in Eq. (2), then by substituting (3) for fj(t) in the integral in Eq. (2), we have:If the integrals in Eq. (6) can be solved analytically, then the bracketed quantities are functions of s alone. So Eq. (6) becomes a linear system of ordinary differential equations that can be solved. However, this requires the construction of an appropriate number of boundary conditions.
Section snippets
Boundary conditions
In order to construct boundary conditions, we first differentiate both sides of Eq. (2) to get the interval 0 < s < 1 and i = 1, 2, … , n:where . Substitute fj(s) for fj(t) in the integrals in Eqs. (7), (8) to obtain for 0 < s < 1 and i = 1, 2, … , n:
Numerical examples
Example 1 For the first example, consider the following Volterra system of integral equations:g1(s) and g2(s) are chosen such that the exact solution is f1(s) = s2 + 1, f2(s) = 1 − s3 + s. Numerical results are given in Table 1. Example 2 For the second example, consider the following Volterra system of integral equations:g1(s) and g2(s) are
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