On a spectral solver for highly oscillatory and non-smooth solutions of a class of linear fractional differential systems

TL;DR

This paper introduces a spectral Galerkin method using M"{u}ntz-Jacobi functions for efficiently solving linear fractional differential systems with highly oscillatory and non-smooth solutions, ensuring high accuracy and stability.

Contribution

It develops a recurrence-based spectral solver tailored for challenging fractional systems with oscillatory and non-smooth behavior, improving computational efficiency and stability.

Findings

Achieves exponential accuracy in the $L^2$-norm.

Effectively captures highly oscillatory solutions.

Demonstrates stability at high approximation degrees.

Abstract

This study discusses a class of linear systems of fractional differential equations with non-constant coefficients, with a particular focus on problems exhibiting highly oscillatory and non-smooth behavior. We first establish the regularity properties of the solutions under specific conditions on the input data. A spectral Galerkin method based on M\"{u}ntz-Jacobi functions is developed that efficiently handle the non-smooth and highly oscillatory solutions. A key advantage of the proposed approach is the ability to compute the approximate solution via recurrence relations, avoiding the need to solve complex algebraic systems. Moreover, the method remains stable even at higher approximation degrees, effectively capturing highly oscillatory solutions with high accuracy. The well-known exponential accuracy is established in the $L^2$-norm, and some numerical examples are provided to demonstrate both the validity of the theoretical analysis and the efficiency of the proposed algorithm.