A Novel Third Order Numerical Method for Solving Volterra Integro-Differential Equations

TL;DR

This paper introduces a new third-order numerical method for nonlinear Volterra integro-differential equations, combining implicit trapezium rule discretization with DJM, demonstrating improved efficiency and stability over existing methods.

Contribution

The paper presents a novel third-order numerical scheme for solving nonlinear Volterra integro-differential equations, including theoretical analysis and comparative error assessment.

Findings

The method achieves higher accuracy than existing approaches.

It demonstrates stability and convergence in various test cases.

The approach is more efficient in computational experiments.

Abstract

In this paper we introduce a numerical method for solving nonlinear Volterra integro-differential equations. In the first step, we apply implicit trapezium rule to discretize the integral in given equation. Further, the Daftardar-Gejji and Jafari technique (DJM) is used to find the unknown term on the right side. We derive existence-uniqueness theorem for such equations by using Lipschitz condition. We further present the error, convergence, stability and bifurcation analysis of the proposed method. We solve various types of equations using this method and compare the error with other numerical methods. It is observed that our method is more efficient than other numerical methods.