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

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. We prove that the implicit method is unconditionally stable in the third quadrant. We determine the stability region in the third quadrant for the explicit method numerically. The explicit method has a bounded region close to the origin. We perform several calculations to demonstrate our results.