A temporal multiscale method and its analysis for a system of fractional differential equations

TL;DR

This paper develops and analyzes a multiscale numerical method for efficiently solving a coupled system of fractional differential equations with multiple time scales, demonstrating significant computational savings and accuracy improvements.

Contribution

The paper introduces a novel multiscale approach for fractional differential equations with multiple time scales, including error analysis and validation through numerical experiments.

Findings

Significant reduction in computational time compared to full simulations.

High accuracy of the multiscale method for small time scale separation.

Method's effectiveness increases with larger time scale separation.

Abstract

In this paper, a nonlinear system of fractional ordinary differential equations with multiple scales in time is investigated. We are interested in the effective long-term computation of the solution. The main challenge is how to obtain the solution of the coupled problem at a lower computational cost. We analysize a multiscale method for the nonlinear system where the fast system has a periodic applied force and the slow equation contains fractional derivatives as a simplication of the atherosclerosis with a plaque growth. A local periodic equation is derived to approximate the original system and the error estimates are given. Then a finite difference method is designed to approximate the original and the approximate problems. We construct four examples, including three with exact solutions and one following the original problem setting, to test the accuracy and computational efficiency of the proposed method. It is observed that, the computational time is very much reduced and the multiscale method performs very well in comparison to fully resolved simulation for the case of small time scale separation. The larger the time scale separation is, the more effective the multiscale method is.