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In this paper, a nonlinear system of fractional ordinary differentialequations with multiple scales in time is investigated. We are interested inthe effective long-term computation of the solution. The main challenge is howto obtain the solution of the coupled problem at a lower computational cost. Weanalysize a multiscale method for the nonlinear system where the fast systemhas a periodic applied force and the slow equation contains fractionalderivatives as a simplication of the atherosclerosis with a plaque growth. Alocal periodic equation is derived to approximate the original system and theerror estimates are given. Then a finite difference method is designed toapproximate the original and the approximate problems. We construct fourexamples, including three with exact solutions and one following the originalproblem setting, to test the accuracy and computational efficiency of theproposed method. It is observed that, the computational time is very muchreduced and the multiscale method performs very well in comparison to fullyresolved simulation for the case of small time scale separation. The larger thetime scale separation is, the more effective the multiscale method is.

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