Numerical Algorithm Of Fractional Diffusion Equation

Fractional diffusion equation is an extension of integer diffusion equation,which is widely used in financial theory,groundwater infiltration,image analysis and stochastic processes.However,the analytical solutions of fractional diffusion equations are generally not easy to obtain,so it is particularly important to design efficient discrete schemes to solve them numerically.In this paper,a higher-order numerical scheme of Caputo fractional derivative is constructed,and the integer Allen-Cahn equation,the time fractional convection diffusion equation and the time fractional phase field equation are solved step by step.The specific contents include the following aspects:Firstly of all,the two-dimensional Allen-Cahn equation is decomposed into nonlinear equation and heat conduction equation by operator splitting method.Among them,the nonlinear equation is solved analytically.The Crank-Nicolson(CN)scheme is used for time discretization of heat conduction equation,and central difference and Padé approximation are used for space discretization,respectively,to obtain two numerical schemes.Numerical examples show the superiority of the scheme and confirm that Padé approximation is superior to central difference and the law of energy decreasing.Secondly,a high-order scheme of two-dimensional time fractional convection diffusion equation is proposed by Padé approximation and the (3-α) order discrete scheme of Caputo fractional derivative which is given by using quadratic interpolation function.Numerical examples verify the effectivity and accuracy of the scheme.Finally,the spectral method and the (3-α) order numerical scheme proposed in chapter third are applied to the numerical discretization of the time fractional Allen-Cahn equation and the time fractional Cahn-Hilliard equation.Numerical examples demonstrate the effectiveness of the scheme.