Stability analysis of an implicit and explicit numerical method for Volterra integro-differential equations with kernel K(x,y(t),t)

TL;DR

This paper develops and analyzes implicit and explicit numerical methods for Volterra integro-differential equations with general kernels, focusing on their stability properties through numerical stability regions.

Contribution

It introduces extended implicit and explicit algorithms for general kernels and provides a stability analysis with derived stability regions.

Findings

Implicit method has an unbounded stability region.

Explicit method's stability region is bounded and near the origin.

Numerical calculations confirm the stability analysis.

Abstract

We present implicit and explicit versions of a numerical algorithm for solving a Volterra integro-differential equation. These algorithms are an extension of our previous work, and cater for a kernel of general form. We use an appropriate test equation to study the stability of both algorithms, numerically deriving stability regions. The region for the implicit method appears to be unbounded, while the explicit has a bounded region close to the origin. We perform a few calculations to demonstrate our results.