Trigonometric-Interpolation Based Approach for Second-Order Volterra Integro-Differential Equations

TL;DR

This paper introduces a trigonometric interpolation method for solving second-order Volterra integro-differential equations, achieving high accuracy and flexibility similar to previous methods for Fredholm equations.

Contribution

It extends the trigonometric interpolation approach to second-order Volterra equations, demonstrating its effectiveness and generalizability for various boundary conditions.

Findings

High accuracy with moderate grid points

Effective handling of kernel singularities

Good performance across different boundary conditions

Abstract

The trigonometric interpolation has been recently applied to solve a second-order Fredholm integro-differentiable equation (FIDE). It achieves high accuracy with a moderate size of grid points and effectively addresses singularities of kernel functions. In addition, it work well with general boundary conditions and the framework can be generalized to work for FIDEs with a high-order ODE component. In this paper, we apply the same idea to develop an algorithm for the solution of a second-order Volterra integro-differentiable equation (VIDE) with the same advantages as in the study of FIDE. Numerical experiments with various boundary conditions are conducted with decent performances as expected.