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

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.

Averaging principle for fractional stochastic differential equations with Lp convergence

“…The averaging principle, an important method for studying complex differential equation systems, can simplify an existing problem to an approximate one to reduce the complexity of the problem or increase the computational efficiency. Recently, the averaging method has been applied to address the temporal multiscale problem of the atherosclerosis with a plaque growth [1,2]. For the averaging theory of stochastic differential equations (SDEs), Khasminskii [3] first introduced the averaging principle of SDEs driven by Brownian motion.…”