Parallel Computation Methods Of Finite Difference For Time-Fractional Fisher Equations

The time-fractional Fisher equation is a mathematical model containing linear diffusion and nonlinear growth.Due to the profound physical background,rich theoretical connotation and difficult to show its analytical solution,the fast algorithm of numerical solution for time-fractional Fisher equation has important theoretical significance and practical engineering value in many fields,such as physics,biology,chemistry and so on.In this paper,we propose a class of parallel computation methods of finite difference for one and two dimensional time fractional Fisher equations.Firstly,the hybrid alternating segment Crank-Nicolson(HASC-N)parallel finite difference method is constructed for the one-dimensional time-fractional Fisher equation.The main idea is to combine the classical explicit scheme,the classical implicit scheme,and the Crank-Nicolson(C-N)scheme using the alternating segment difference technology.The existence and uniqueness of the numerical solution of the HASC-N difference method,the unconditional stability and convergence of HASC-N difference method are analyzed theoretically.Numerical experiments show that the HASC-N difference method has 2-α order time precision and 2 order space precision under strong regularity,α order time precision and 2 order space precision under weak regularity of fractional derivative discontinuity.Secondly,based on alternate banding technique,the HASC-N difference method for solving one-dimensional time-fractional Fisher equation is extended to solving two-dimensional time-fractional Fisher equation,and the hybrid alternating band C-N(HABC-N)parallel finite difference method is proposed.The existence and uniqueness of the numerical solution of the HABC-N difference method,unconditional stability and convergence of the HABC-N difference method are proved theoretically.Numerical experiments verify the theoretical analysis,and show that the HABC-N difference method has high precision and obvious parallel computing characteristics,which is an efficient method to solve the two-dimensional time-fractional Fisher equation.