  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.
