High Order Schemes Of Multi-term Tempered Time Fractional Diffusion Equation

The tempered fractional calculus have been successfully applied for describing the time evolution of the diffusion particles in a non-Markovian system.The related governing equations are a series of partial differential equations with tempered fractional derivatives.One of the challenging of this problem lies in the existence of singular kernel and smooth kernel.The paper uses interpolation technique combined with finite difference and local discontinuous Galerkin finite element method to solve the tempered fractional differential equations and construct high order numerical schemes for partial differential equations which with different physical backgrounds.(1)The difference schemes of one-dimensional and two-dimensional tempered time fractional diffusion equation are constructed.By using the polynomial interpolation technique,in this paper we present three efficient numerical formulas,namely tempered L1 formula,tempered L1-2 formula and tempered L2-1? formula,to approximate the Caputo tempered fractional derivative of order ?(0<?<1).The truncation error of tempered L1 formula is of order 2-?,and the tempered L1-2 formula and L2-1? formula are of order 3-?.As an application,we construct implicit schemes and alternating direction implicit schemes for one dimensional and two dimensional time tempered fractional diffusion equations,respectively.Furthermore,the theorems of stability and convergence of two developed difference schemes with tempered L1 and tempered L2-1? formulas are proved by Fourier analysis method.Finally,according to the results of the numerical examples,the established schemes are compared,and the advantages and disadvantages of the numerical schemes are analyzed.(2)Two difference schemes of the tempered time fractional Burgers equation are established.Firstly,the tempered fractional Burgers equation is derived from the thermodynamic model in porous media,and a priori error estimation is given.Then,using tempered L1 and tempered L2-1? approximation formulas to discrete time derivative,central difference for the space component,and the nonlinear term is linearized.In addition,adopting linearized approach for the nonlinear term to avoid additional and unnecessary iteration.The stability and convergence are analyzed by L2 error estimation.Finally,numerical examples and numerical simulations verified the feasibility of the difference scheme and the rationality of theoretical analysis.(3)Two local discontinuous Galerkin finite element schemes for multi-term tempered time fractional diffusion equation are developed.Two approximated formulas,called tempered L1 and tempered L2-1? formulas are used in time discretization and local discontinuous Galerkin method with k+1 convergence order for the space component.Choosing the alternating numerical fluxes,and the stability and convergence of the difference schemes are studied in detail.Then,without loss of generality,the numerical results of solving the two-term and four-term tempered fractional diffusion equation are given in the numerical examples,which can be used to verify the validity of the proposed schemes.