Finite Difference/Spectral Methods For Variable-Order Fractional Diffusion Equations

Variable-order fractional differential equations appear in modeling the dynamics of some particular materials,such as viscoelastic deformation process and motion of particles suspended in a viscous fluid with drag force.There is still very limited theoretical and numerical investigations in this field.The aim of this paper is to design and analyze a numerical method for a basic variable-order fractional differential equation,i.e.,the variable-order time fractional diffusion equation.The constructed scheme makes use of the idea proposed by Lv and Xu in[International Journal of Numerical Analysis&Modeling,12(2),p384-400],which is a combination of the finite difference scheme in time and spectral method in space.A theoretical analysis is carried to establish the stability of the overall scheme and derive the optimal error estimate.The main result is of(2-max?(t))order convergence in time and spectral accuracy in space,where?(t)is the fractional order function of the underlying equation.Finally a series of numerical examples is provided to confirm the theoretical claims.