Alternating Direction Implicit Difference Method For Two-Dimensional Fractional Order Delay Wave Equations

The fractional order calculus model are widely used in the fields of natural science and engineering.In particular,the time-space fractional delay differential equation can accurately describe some complex problems,such as anomalous sub-diffusion phenomenon,super-anomalous diffusion phenomenon,porous media problems,etc.Therefore,how to propose an efficient and stable difference scheme for the two-dimensional time-space fractional delay wave equation,that is a valuable work.This paper is given an alternating direction implicit difference scheme for the twodimensional time-space fractional delay wave equations.In present scheme,the 1 interpolation is adopted to approximate the time Caputo fractional derivative;the shifted Grünwald-Letnikov?G-L?approximate formula is used to discretize the spatial Riesz fractional derivative;the simple Taylor expansion is applied to approximate the delay nonlinear source term;and alternative directions iterate method is used to solve the resulting discretion equation.Some theoretical analysis are given for the proposed implicit difference scheme.By using mathematical induction,the existence and uniqueness of the solution is proved.Further proved its convergence order is O(?^3-?+h_x²+h_y²).Numerical tests are performed to validate the present alternating direction implicit difference scheme.Here,different time fractional order ,timestep and grid resolutions are considred.Numerical results shows that the present scheme algorithm is convergent both in time and spatial,which is consistent with the theoretical analysis results.