A Study Of Numerical Method For Solving Quasilinear Degenerate Parabolic Problem

The degenerate parabolic equations are mathematical models that describe and study the laws of change in physics,chemistry,biology and economics.The quasilinear degenerate parabolic equation has been received extensive research and attention,and it is also of great significance for the study of nonlinear partial d-ifferential equation theory.Because the quasilinear equations with degenerate and singularity can better reflect the physical reality,and the quasilinear degenerate parabolic problem has widely application prospect in the mathematical models of non-Newtonian fluid motion and in image problems.Therefore,the numerical solu-tion for studying the quasilinear degenerate parabolic problem has a rich physical background and practical application prospect.The paper analyzes and studies the solution of quasilinear degenerate parabolic problems from both theoretical and numerical aspects.Firstly,we define the weak solution to the quasilinear degenerate parabolic problem.Secondly,the regulariza-tion equation of the quasilinear degenerate parabolic equation is given,the weak solution of the regularization equation is defined and the existence and uniqueness of the weak solution are proved by the regularization method.Finally,it is proved that the weak solution of the regularization problem converges to the weak solution of the quasilinear degenerate parabolic problem,so that the weak solution of the quasilinear degenerate parabolic problem exists and is unique.In order to solve the quasilinear degenerate parabolic problem,this paper proposes the shifted Legendre reproducing kernel Galerkin method（SLRKGM for short）which includes the following four steps:firstly,the time derivative and the quasilinear degenerate term are discretized by the finite difference method,thereby transforming the quasilinear degenerate parabolic equation into an iterative scheme with respect to time and confirming the stability of the iterative scheme,secondly,a new reproducing kernel space[0,1]is constructed,and then the basis function of[0,1]is constructed using the shifted Legendre polynomial to construct a numer-ical solution,finally,a test function is established based on the reproducing kernel function of the reproducing kernel space(2²[0,1],thereby the Galerkin method is used to determine the unknown coefficients of the numerical solution.Besides,the uniformly convergence and corresponding error estimates of the numerical solution based on the shifted Legendre reproducing kernel Galerkin method are given.The stability of the method with respect to the right-hand side of the equation is also discussed.Numerical examples show the effectiveness of the method.