STANDARD COLLOCATION AND LEAST SQUARES METHODS FOR SOLVING LINEAR VOLTERRA-FREDHOLM INTEGRO-DIFFERENTIAL EQUATION USING CHEBYSHEV POLYNOMIAL AS THE BASIS FUNCTION

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1 Mathematics, Physical Science, University of Ilorin

2 Mathematics Programme,Physical Sciences, College of Pure and Applied sciences. Kwara State

3 Mathematics, Physical Sciences, University of Ilorin

Abstract

This research employs the standard collocation and least squares methods to numerically solve Linear Volterra-Fredholm integro-differential equations, utilizing Chebyshev polynomials of the first kind as basis functions. Both methods start by assuming an approximate solution represented by Chebyshev polynomials, which are then substituted into the integro-differential equation. The coefficients of these polynomials are collected and simplified accordingly. In the standard collocation method, the resulting equations are evaluated at equally spaced interior points within the domain, ensuring that the solution satisfies the original equation at these discrete points. Conversely, the least squares method involves minimizing the residual error over the entire domain using the least squares approximation, thereby fitting the approximate solution as closely as possible to the exact solution in a global sense. Both approaches lead to the formation of an algebraic linear system of equations. This system is solved using Gaussian elimination to determine the unknown constants. These constants are then substituted back into the assumed Chebyshev polynomial solution to derive the final approximate solution. Numerical examples are provided to illustrate the accuracy of both methods, with results indicating that the accuracy improves as the degree of the polynomial approximant increases, as evidenced by the presented tables and graphs.

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