Numerical Methods For Several Kinds Of Fractional Diffusion Equations

Fractional partial differential equations(FPDEs)play an important role in many fields,such as electromagnetic theory,economics,finance,control theory,environ-mental science,polymer material.FPDEs can provide a deeper and more accurate physical explanation for the complex dynamic behaviors with memory function,self-similarity and heredity,and hence possess incomparable superiority than the tradi-tional PDEs.However,it is extremely difficult to obtain analytical solutions to the FPDEs.Therefore,it is of great theoretical significance and Practical application value to develop the numerical methods for FPDEs.The aim of this paper is to study some difference methods for several kinds of fractional diffusion equations.First,the difference schemes of the one-dimensional(1D)and two-dimensional(2D)nonlinear time-space distributed-order diffusion equations are discussed,where the nonlinear terms satisfy the local Lipschitz conditions.For the 1D equation,the L1 scheme and the fractional central difference scheme are used in the temporal and spatial discretizations,respectively,then the corresponding difference scheme is given.Furthermore,an alternating direction implicit(ADI)difference scheme of the 2D problem is proposed on the basis of the scheme for the 1D problem.We also analyze the well-posedness and error estimates of the both two schemes.The numerical tests are executed to verify the correctness of the results of theoretical analyses.Second,we investigate the difference schemes of the 1D and 2D nonlinear time-space distributed-order diffusion wave equations,where the nonlinear terms satisfy the local Lipschitz conditions.We construct the difference scheme for 1D problem and ADI difference scheme for 2D problem by using the L2 scheme and the fractional central difference scheme in the temporal and spacial discretizations,respectively.The well-posedness of the difference schemes are analyzed.The numerical tests are given to demonstrate the efficiency of the schemes at last.Third,we develop a finite difference/finite element method for the 2D time-space distributed-order reaction-diffusion equation with low regularity solution at the initial time.A fast evaluation of the distributed-order time fractional derivative based on graded time mesh is obtained by replacing the weak singular kernel with an exponential sum.The stability and convergence of the developed semi-discrete variational formulation are discussed,and the convergence of the corresponding fully discrete scheme is investigated.Finally,some numerical tests are given to verify the obtained theoretical results and demonstrate the efficiency of the method.Finally,we develop numerical algorithms for the 1D fractional diffusion equation with fractional Laplacian operator.The fractional Laplacian operator can be charac-terized as a weak singular integral under zero boundary condition,and the fractional diffusion equation can be converted to a convective equation and a singular integral equation.We propose a finite difference scheme to solve the fractional diffusion e-quation using the linear interpolation functions,and it is shown that the scheme is convergent with the accuracy O(h~2)in the maximum norm.We also present an im-proved numerical algorithm to obtain non-smooth solutions to the fractional diffuse equation.Some numerical tests are given to verify the efficiency of the algorithm.