Theoretical Analysis And Numerical Method Of Radially Symmetric Free Boundary Value Problem

Free boundary problem is a classical problem to describe the spreading of new species or invasive species,and it can express the spreading state and spreading speed of the species on invasive region more exactly.Therefore,it has been an important research focus of mathematical biology.However,due to the uncertainty of the free boundary,it is difficult to solve this problem accurately by mathematic viewpoint.In recent years,domestic and foreign scholars have only done a lot of research on the free boundary problem in one-dimensional space,occasionally involving the analysis in high-dimensional space.This thesis considers a free boundary problem in a radially symmetric setting under the high-dimensional space.The existence and uniqueness of the solution for the free boundary problem are analyed and a finite difference scheme is proposed.As application,the numerical simulation of KPP model with a free boundary is performed.The study contents and corresponding research results are as follows:Firstly,the mathematical model of free boundary problem in radially symmetric setting is established.The free boundary of the model is transformed into a fixed domain by front-fixing approach.Then,the existence and uniqueness of local solution is proved by using the contraction mapping principle.The local solution is extended to global solution by using comparison principle,maximum principle and Poisson formula.Based on Bessel function,the eigenvalue formula of the free boundary problem in radially symmetric setting is derived.Secondly,the finite difference schemes for the free boundary problem is presented.The singularity of the solution at some points is avoided by using the half-point shift.The local truncation errors are calculated by using Taylor series expansion,which proves the consistency of the numerical solution.The positive of the free boundary and the numerical solution are obtained by induction argument.The stability of the numerical solution is obtained by analyzing the parameters of the finite difference schemes.The convergence of the numerical solution is proved by using Lax theorem,and the effectiveness of the difference scheme is verified.Finally,the numerical simulation of KPP model with a free boundary is performed according to the difference scheme.The spreading speed of the extended front and the range of parameters in the Stefan condition are estimated,finally the rationality of the numerical method is verified by comparing with the theoretical results of relevant foreign literatures.