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

TL;DR

This work extends a trigonometric interpolation framework to second-order Volterra integro-differentiable equations (VIDEs) by reformulating the problem as a linear algebraic system that leverages a $2$-D sine expansion of the kernel and a cutoff-based handling of nonperiodicity. The method treats the ODE and integral components coherently, yielding representations that express the integral term and derivatives as linear functions of the unknown coefficient vectors, suitable for both continuous and integrable kernels, including those with singularities. Numerical experiments across various boundary conditions and kernel types demonstrate decent to high accuracy with moderate grid sizes, with Dirichlet boundaries often providing the best performance and convergence improving with higher discretization parameter $q$. The approach offers a practical, high-accuracy tool for VIDEs and can be extended to IDEs with higher-order ODE components, aiding applications in physics and engineering where such equations arise.

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.