Two Kinds Of Predictor-corrector Difference Schemes For Time-fractional Reaction-diffusion Equations

The fractional reaction-diffusion equation is a kind of mathematical models widely used in complex physics,mechanics,biology and engineering science,and the research of its numerical solution has scientific significance and engineering application value.With the deepening of fractional reaction-diffusion equation application,the solutions of this equation have been an urgent research work.The solutions of nonlinear fractional reaction-diffusion equations are difficult to express explicitly.Even if the analytical solutions exist,most of them contain special functions,which are computationally complex and slow to converge,which makes the study of fast numerical algorithms is great important.Firstly,for the time-fractional generalized Huxley equation,this paper constructs a new predictor-corrector difference(P-CD)scheme.The predictor formula adopts the linearized implicit difference scheme and the corrector formula adopts the Crank-Nicolson(C-N)scheme.The nonlinear time-fractional generalized Huxley equation is transformed into two linear difference equations.Theoretical analysis proves that the solution of P-CD scheme exists and is unique.Combining mathematical induction and matrix methods,the unconditional stability and convergence of the P-CD scheme are proved.Numerical analysis and experiments show that P-CD scheme has(2-α)-order accuracy in time and second-order accuracy in space.It is shown that P-CD scheme is efficient for solving time-fractional generalized Huxley equation.Secondly,for solving nonlinear time-fractional reaction-diffusion equation with nonhomogeneous terms,a fast predictor-corrector difference(FP-CD)scheme is constructed based on fast L1 approximation of Caputo fractional derivative.Theoretical analysis proves that FP-CD scheme is convergent and stable unconditionally.Numerical analysis and experiments show that the convergence accuracy of FP-CD scheme is O(τ2-α+h2)under the strong regularity condition and O(τα+h2)under the weak regularity condition.Compared with the classical P-CD scheme based on standard L1 approximation,the FP-CD format is more complex.With the double acceleration of P-CD scheme and fast L1 approximation,the FP-CD scheme improves the computational efficiency without losing the computational accuracy.It is shown that the FP-CD scheme is an efficient method to solve nonlinear time-fractional reaction-diffusion equation.