Stability And Convergence Analysis Of Numerical Schemes For Nonlinear Volterra Integro-Differential Equations

Abstract

This article provides a complete review of the stability and convergence aspects of numerical methods that are used in the process of solving nonlinear Volterra integro-differential equations (VIDEs). These equations, which are distinguished by the presence of differential and integral elements, are used in a variety of scientific and technical applications. Some examples of these applications include population dynamics, viscoelastic materials, and financial modeling. There are difficulties associated with numerical computing as a result of the intrinsic complexity of VIDEs, which is caused by their nonlinearity and memory-dependent components. In this paper, we investigate a number of numerical schemes, including implicit, explicit, and hybrid approaches, with a particular emphasis on the theoretical foundations of these schemes and their performance in practice. Strategies such as the von Neumann method and energy methods are used in order to carry out a comprehensive investigation of the stability of the system. Following that, we conduct an analysis of the convergence rates of the schemes, which is backed by error estimates and instances that illustrate the points. The findings that we have obtained provide valuable insights into the selection of numerical techniques that are suitable for solving nonlinear VIDEs. These approaches guarantee both stability and accuracy over a wide variety of applications.