Difference Method For A Class Of Fractional Order Nonlinear Equation With Time Delay

Fractional nonlinear differential equation with time delay can describe many com-plex physical phenomena with memory or delay effect,and have important applications in ecology,materials science and physical science and so on.Therefore,it is of great theoret-ical and practical significance to study the numerical methods of such equations.In this thesis,we mainly studies the difference method of two-dimensional fractional nonlinear time-delay diffusion equation.This thesis includes the following two parts.In the first part,the compact difference method of the time-fractional nonlinear fourth-order diffusion equation with time delay is studied.First,by introducing an in-termediate variable,the original fourth-order equations is transformed into a coupled second-order system.Then,the space derivative term is discretized by the fourth-order compact difference method and the Caputo derivative is discretized by L2-(?) formula.The extrapolation method is used to deal with nonlinear source terms with time delay and the compact difference scheme is constructed to solve the problem.Next,the exis-tence and uniqueness of the numerical solution and the convergence and stability of the compact difference scheme are proved.Finally,numerical examples are given to verify the theoretical analysis results and the effectiveness of the scheme.Then,in the second part,the difference method of the time-space fractional nonlinear diffusion equations with time delay is studied.First,by introducing an intermediate variable,the original equation is transformed into a system of coupled equations.Then,the space Riesz fractional derivative term is discretized by the fractional central difference formula,the second derivative of space is discretized by central difference formula and the time Caputo derivative is discretized by the L2-(?) formula.The extrapolation method is used to deal with nonlinear source terms with time delay and the difference scheme is constructed to solve the problem.Next,the existence and uniqueness of the numerical solution and the convergence and stability of the difference scheme are proved.Finally,a numerical example is given to verify the theoretical analysis results.