Collocation Method for the Direct Solution of Volterra and Fredholm Integro-differential Equations

Abstract

Integro-differential equations play a pivotal role in modeling a wide range of physical, biological, and engineering systems. These equations combine differential operators with integral terms, posing unique challenges for numerical solutions. This paper presents a comprehensive examination of the collocation method as a powerful and versatile technique for directly solving Volterra and Fredholm integro-differential equations. The collocation method is a numerical approach that involves discretizing the differential and integral components of the equation at specific collocation points. By appropriately choosing these points, the method transforms the integro-differential equation into a system of algebraic equations, which can be efficiently solved using standard numerical techniques. This paper explores the mathematical foundations of the collocation method and its applicability to various classes of integro-differential equations.