A scale-dependent finite difference method for time fractional derivative relaxation type equations

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-α)-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 α=0.4)..