An Investigation for a Second Order Volterra-Fredholm Integro-differential Equation With Two Algebraic Weakly Singular Kernels

Abstract

This research focuses on the analytical and numerical analysis of a nonlinear second-order Volterra-Fredholm integro-differential equation with two algebraic weakly singular kernels. We rigorously establish the existence and uniqueness of the solution using Krasnoselskii's fixed-point theorem, which elegantly addresses the nonlinear structure of the equation. To approximate the solution, we employ the Nystr\"{o}m method combined with the product integration technique, specifically designed to overcome the challenges posed by the weak singularities.

We conduct extensive numerical experiments to demonstrate the performance and the accuracy of our proposed approach. The results not only validate our theoretical findings but also underscore the method's effectiveness in solving similar classes of integro-differential equations. This study advances our understanding of numerical methods for singular and nonlinear equations, offering valuable insights into their potential applications.