The Study Of Fast Method For Time Distributed Order Diffusion Equations

In recent years,the study of the time distributed-order diffusion equa-tions has been widely concerned.These equations can be used to describe the complex diffusion processes in which many diffusion indices change over time,such as sub-diffusion of deceleration,hyper-diffusion of acceleration,hyper-diffusion of deceleration,and sub-diffusion of acceleration.At present,the time distributed-order diffusion equations has been widely used in various en-gineering fields such as rheological properties of composite materials,signal control and processing,dielectric induction and diffusion,and stress-strain be-havior of viscoelastic materials.In this paper,we will introduce the fast calculation method of two kinds of finite difference schemes for the time distributed-order diffusion equation and a finite difference scheme for the quasilinear time distributed-order diffusion equation.The model is as follows:where ?1(0)= 0,?2(0)= 0,and w(?)0>,?01 w(?)d?=c0>0,f,?1 and ?2 are given functions.Because of the nonlocal property of the time distributed-order derivative,using the finite difference method to solve the numerical solution of the equa-tion at the current time step,we need to use the function value of all the previous time steps,which leads to a computational complexity of O(MN2J)by assembling the right-hand side,where M is the number of spatial nodes and N is the number of time steps,J is the number of subdivisions of the distributed order derivative.When the mesh is refined,the computational work and the memory requirement will be more and more significant,and the computational time will be longer.By studying the structure of coefficients matrix,we can obtain a matrix formed by block Toeplitz matrix since the co-efficients matrix is transformed through a reasonable way.Combining the fast Fourier transform and the fast Fourier inverse transform,we can get the fast method of this model.This new method can reduce the computation amount to O(MN log N)+ O(NJ)and improve the computational efficiency.Then,we will consider the case of the right-hand side f as the quasilinear term.We set the f is local bounded and satisfies the local Lipschitz condition,then we will propose a finite difference scheme.The convergence order O(?+h2+??2)of this scheme is proved..In this paper,the fast algorithm is discussed for the above model,and the quasilinear condition of the right-hand side is analyzed.This paper is divided into four chapters:Chapter 1:The background of the time distributed-order diffusion equa-tions,the model described as above and domestic and foreign research litera-ture review are introduced.Chapter 2:Combining two finite difference schemes of the time distributed-order diffusion equation,the fast method of these two schemes is deduced,and the effectiveness of the fast method are verified by a numerical example.Chapter 3:The finite difference scheme for the quasilinear time distributed-order diffusion equation is given,the convergence of this scheme is proved,and the theoretical results are verified by a numerical example.Chapter 4:A summary and outlook of this paper are given.