Finite Difference Method for Solving Nonlinear Fredholm Integro-Differential Equations

Abstract

This thesis presents a novel numerical approach for solving nonlinear Fredholm integro-differential equations involving both the unknown function and its derivative. The proposed method combines the backward finite difference technique with the Nyström method to significantly reduce the size of the resulting nonlinear system, compared to the classical approach introduced by Bounaya et al. [1]. This reduction enhances computational efficiency, especially for large integration intervals where traditional methods become impractical. To guarantee convergence and stability, we construct an appropriate norm on R N+1 . Numerical experiments confirm the effectiveness of the proposed scheme, demonstrating improved accuracy and reduced execution time. This work contributes to the development of reliable numerical tools for solving nonlinear integro-differential equations, with potential applications in various fields of science and engineering.