Research On Sinc Method For Time Nonlocal Palabolic Equation

In recent years,research on non-local problems has attracted more and more attention from researchers.Non-local models can solve discontinuous problems such as fau,lts and cracks,so that the problems of continuous media,cracks and particles can be reasonably unified in a model framework.At present discrete methods for non-local operators include finite element methods,finite difference methods,and so on.In this thesis,the finite difference method and the Sinc method are used to construct a fully discrete format for the initial boundary value problem of time nonlocal parabolic equations.The specific work is as follows:(1)The Sinc-Galerkin method is used to numerically solve the initial boundary value problem of Burgers equation.Firstly,we transforme the nonlinear Burgers equation into a linear equation using the Hopf-Cole transformation..The derivative of time is discretized in weighted format the spatial derivative is discretized by Sinc-Galerkin method,and the weight function is introduced at the endpoint to process the transformed second type of boundary condition.Finally,the exponential convergence of the Sinc-Galerkin method is verified by numerical examples.By comparing with the exact solution,the numerical format constructed in this paper is highly accurate and can effectively capture physical phenomena such as shock waves.(2)For the initial boundary value problem of one-dimensional nonlocal parabolic equations,the finite difference method is used to discretize the time variables to obtain the semi-discrete format,and the local truncation error is given.Next we use the Sinc-Galerkin method and the Sinc-Collocation method to discretize the spatial differential operators,and two fully discrete schemes for solving the time nonlocal parabolic equations are established:DSG scheme and DSC scheme.Finally,the numerical example is presented to verify the effectiveness of DSG scheme DSC-I scheme and DSC-? scheme,and the influence of parameters on numerical solution in non-local parabolic equations is analyzed.(3)For the two-dimensional time nonlocal problem,the time nonlocal operator is still discretized by the finite difference method,and the spatial derivative is discretized by the two-dimensional Sinc-Galerkin method and the Sinc-Collocation method.Considering the different parameters in the discrete scheme,four fully discrete scheme have been establish:TDSG-?,TDSG-?,TDSC-?,TDSC-?.The numerical example is presented to verify the effectiveness of these four numerical schemes,and the influence of the parameter values on numerical solution is analyzed.The four schemes are compared and analyzed.