Two Kinds Of Inverse Free Boundary Problems And Numerical Calculation

Recently,the inverse problem of mathematical physics has developed extremely fast,this is driven by the needs of other disciplines and engineering disciplines.Inverse problems have applications in many disciplines,such as geophysics,materials science,pattern recognition,image processing and more.Most of the inverse problems we have studied have relatively complete or adequate research.This paper considers an inverse problem for a diffusive Logistic model with free boundary and a ductal carcinoma in situ(DCIS)model with free boundary.Based on a fixed point argument,we prove the local in time existence and uniqueness of the Logistic model.We obtain a global-in-time uniqueness of the DCIS model.Then based on the optimization method,we present a regularization algorithm to recover coefficient.Then our numerical experiment shows the effectiveness of the proposed numerical method.The structure of this article are as follows.In the first chapter,we first summarize the research fields on the inverse problems in mathematical physics.Then the physical background and research development of the Logistic equation and the DCIS equation are described.Finally the main work of this paper is introduced.In the second chapter,we discuss the inverse source problem of the Logistic e-quation with free boundary.We first transfer our inverse problem to an equivalent problem.Then a local existence and uniqueness of the equivalent problem is obtained by the contraction mapping.In the third chapter,we discuss the inverse problem for a mathematical model with free boundary related to DCIS.We first obtain a global-in-time uniqueness of our inverse problem.Then based on the optimization method,we present a regulariza-tion algorithm to recover the coefficient.Finally,our numerical experiment shows the effectiveness of the proposed numerical method.