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

TL;DR

This paper extends a trigonometric interpolation algorithm to two dimensions and applies it to efficiently solve second order Fredholm integro-differential equations with high accuracy and robustness to singularities.

Contribution

The paper develops a 2D trigonometric interpolation method and demonstrates its effectiveness for solving second order Fredholm integro-differential equations with various boundary conditions.

Findings

High accuracy with moderate grid points

Effective handling of singular kernel functions

Robust performance across different boundary conditions

Abstract

A trigonometric interpolation algorithm for non-periodic functions has been recently proposed and applied to study general ordinary differential equation (ODE). This paper enhances the algorithm to approximate functions in $2$-dim space. Performance of the enhanced algorithm is expected to be similar as in $1$-dim case and achieve accuracy aligned with the smoothness of the target function, which is confirmed by numerical examples. As an application, the $2$-dim trigonometric interpolation method is used to develop an algorithm for the solution of a second order Fredholm integro-differential equation (FIDE). There are several advantages of the algorithm. First of all, it converges quickly and high accuracy can be achieved with a moderate size of grid points; Secondly, it can effectively address singularities of kernel functions and work well with general boundary conditions. Finally, it can be enhanced to copy with other IDE such as Volterra IDE or IDE with high order ODE component. The tests conducted in this paper include various boundary conditions with both continuous kernels and integrable ones with singularity. Decent performance is observed across all covered scenarios with a moderate size of grid points.