Finite Difference Scheme For Distributed Orcder Partial Differential Equation With Non-uniform Meshes

In recent years,more and more attention has been focused on fractional differential equations,which are generalizations of classical partial differential equations.Such equations have attracted wide attention in the fields of electro-chemical processes,dielectric polarization,colored noise,anomalous diffusion,signal processing,and control optics.These models are increasingly used in fluid flow,finance and other fields.Most fractional differential equations have no analytical solution,so numerical methods are needed for analysis.Since fractional order operators are non-local,the computational com-plexity required to solve fractional-order partial differential equations with finite-difference methods atuniform time steps is very large.In addition,the solutions to these problems usually involve significantly different time scales,which leads to a large numerical error.One way to solve these problems is to use a finite difference algorithm with non-uniform grids.The finite difference algorithm on non-uniform grids has a good development in approximating classical integrals and derivatives.Due to the memory and non-locality of fractional operators,it is difficult to generalize them to fractional-order mod-els.The main research of this paper is the finite difference algorithm for time distributed-order differential equations with non-uniform meshes.The research is mainly divided into three parts:In the first part,the initial boundary value problem of one-dimensional time distributed-order differential equation is studied.The non-uniform time step is used to discretize the time term.Firstly,the distribution order integral is discretized.The Caputo derivative is discretized by L1 scheme,and the com-pact finite difference scheme is obtained.Then,the stability and convergence of the scheme is proved by the energy method.In the second part,the time distributed-order differential equations with the variable coefficient is studied.Since the spatial partial derivative contains the variable coefficient a(x),the integer pointcenter difference cannot effective-ly discretize the spatial partial derivative.We introduce a semi-integer point and a new compact operator,and obtain a compact finite difference scheme for the time distributed-order fractional equations with the variable coefficient.The stability and convergence of the scheme are proved by the energy method.In the third partthe two-dimensional timedistributed order diffusion equation is studied.The dispersion of the integral term of the time term and the distribution order is the same as that of the first part.The finite difference scheme in two-dimensional is obtained by the alternating direction method.The stability and convergence of the scheme are proved by the energy method.In the fourth part,the numerical examples of time distributed-order differ-ential equations are used to illustrate the effectiveness of non-uniform meshes.