An ultraspherical spectral method for linear Fredholm and Volterra integro-differential equations of convolution type

TL;DR

This paper introduces an ultraspherical spectral method that efficiently solves linear Fredholm and Volterra integro-differential equations with convolution kernels, achieving spectral accuracy and linear computational complexity.

Contribution

It combines Legendre-based spectral techniques with convolution formulas to create a new, highly accurate method for convolution-type integro-differential equations.

Findings

Spectral accuracy for smooth kernels and coefficients.

Linear complexity linear system solutions.

Effective for convolution-type integro-differential equations.

Abstract

The Legendre-based ultraspherical spectral method for ordinary differential equations is combined with a formula for the convolution of two Legendre series to produce a new technique for solving linear Fredholm and Volterra integro-differential equations with convolution-type kernels. When the kernel and coefficient functions are sufficiently smooth then the method is spectrally-accurate and the resulting almost-banded linear systems can be solved with linear complexity.