  Fractional derivative relaxation type equations (FREs) including fractional
diffusion equation and fractional relaxation equation, have been widely used to
describe anomalous phenomena in physics. To utilize the characteristics of
fractional dynamic systems, this paper proposes a scale-dependent finite
difference method (S-FDM) in which the non-uniform mesh depends on the time
fractional derivative order of FRE. The purpose is to establish a stable
numerical method with low computation cost for FREs by making a bridge between
the fractional derivative order and space-time discretization steps. The
proposed method is proved to be unconditional stable with (2-{\alpha})-th
convergence rate. Moreover, three examples are carried out to make a comparison
among the uniform difference method, common non-uniform method and S-FDM in
term of accuracy, convergence rate and computational costs. It has been
confirmed that the S-FDM method owns obvious advantages in computational
efficiency compared with uniform mesh method, especially for long-time range
computation (e.g. the CPU time of S-FDM is ~1/400 of uniform mesh method with
better relative error for time T=500 and fractional derivative order
alpha=0.4).
