Adaptive Algorithms In The Time Domain For Solving Fractional Diffusion Equations

With the extensive application of the fractional differential equation in hydrodynamics, viscoelasticity,biology,finance and physics,considerable attention has been given to the solutions of fractional order partial differential equations.In general,analytical solutions of fractional order partial differential equations are difficult to get.So it is very important to develop effective numerical methods for fractional orderpartial diffusion equations.In this paper,numerical solutions for two classes fractional differential equation are researched. The first class is the time-fractional diffusion equation with a time-fractional derivativeα(0＜α＜1).By expanding variables with fractional power at discretized time intervals,a time-fractional diffusion equation with initial and boundary conditions is converted into a series of boundary value problems which are solved by FDM.The computing accuracy is maintained via an adaptive procedure for different time step sizes;The second class is the space-fractional diffusion equation with a space-fractional derivativeβ(1＜β＜2).By expanding variables with fractional power at discredited time intervals,a space-fractional diffusion equation with initial and boundary conditions is converted into a series of boundary value problems,and they can be solved with a space difference scheme whose convergent order is super linear in space domain.The computing accuracy is maintained via an adaptive procedure for different time step sizes.Two illustrative examples aregiven to demonstrate the efficiency of the proposed algorithms.