A fast and spectrally convergent algorithm for rational-order fractional integral and differential equations

TL;DR

This paper introduces a fast, spectrally convergent algorithm for solving linear variable-coefficient rational-order fractional integral and differential equations, leveraging ultraspherical polynomial bases for efficiency.

Contribution

It develops a novel spectral method using ultraspherical and Jacobi polynomial bases to efficiently solve fractional equations with variable coefficients.

Findings

Algorithm is linear in degrees of freedom.

Demonstrates geometric convergence for smooth problems.

Operators are banded or nearly banded, enabling fast computation.

Abstract

A fast algorithm (linear in the degrees of freedom) for the solution of linear variable-coefficient rational-order fractional integral and differential equations is described. The approach is related to the ultraspherical method for ordinary differential equations [Olver & Townsend 2013], and involves constructing two different bases, one for the domain of the operator and one for the range of the operator. The bases are constructed from direct sums of suitably weighted ultraspherical or Jacobi polynomial expansions, for which explicit representations of fractional integrals and derivatives are known, and are carefully chosen so that the resulting operators are banded or almost-banded. Geometric convergence is demonstrated for numerous model problems when the variable coefficients and right-hand side are sufficiently smooth.