Discrete approximations of differential equations via trigonometric interpolation

TL;DR

This paper introduces a method for approximating solutions to linear differential equations using trigonometric interpolation, resulting in high accuracy and rapid convergence in numerical tests.

Contribution

It develops a novel approach projecting solution spaces onto trigonometric polynomial spaces and analyzes the rank of associated matrix representations.

Findings

High accuracy in numerical solutions

Fast convergence observed in tests

Effective for boundary and eigenvalue problems

Abstract

To approximate solutions of a linear differential equation, we project, via trigonometric interpolation, its solution space onto a finite-dimensional space of trigonometric polynomials and construct a matrix representation of the differential operator associated with the equation. We compute the ranks of the matrix representations of a certain class of linear differential operators. Our numerical tests show high accuracy and fast convergence of the method applied to several boundary and eigenvalue problems.