Research On Parabolic Inverse Source Problem With Integral Observation Data

Due to the actual needs of many engineering fields,in the past 30 years,inverse problem has become one of the fastest growing fields in applied mathematics.There are many kinds of inverse problems,and they are often ill-posed in the classical sense.The ill-posedness of inverse problems has become the focus and difficulty of research.Among them,the inverse problem of heat conduction equation is an important branch of inverse problem,which has been studied by many mathematicians.In this paper,the source term reconstruction of parabolic equation with variable coefficients is studied by different methods,and then the degenerate parabolic equation is studied.In this paper,different models are studied by using new methods.The conditions and some important conclusions that must be satisfied for the existence,uniqueness,stability of solutions of different models are obtained,and some periodic results are obtained.This paper is mainly divided into the following five chapters:The first chapter is the introduction,which briefly introduces the model and the related research on the inverse problem.In Chapter 2,The problem of source term reconstruction for a class of parabolic equation with variable coefficients is studied,the source term here is only time dependent.Different from the previous work,the additional condition in this paper is about the integration of the spatial variable,this type of additional conditions can help to eliminate the errors caused by random selection,but at the same time,it will lead to many analysis methods unavailable,such as the conjugate theory of parabolic equation.Based on the variational theory,firstly,the variational formula is given and the uniqueness of the solution is proved by the variational formula;Secondly,the time discrete model is given,and based on the variational form of linear discretization,a series of priori estimates are derived and the existence of weak solutions is proved.Chapter 3 is based on the model of Chapter 2 and studies the transformation of the original problem into the optimal control problem based on the optimal theory.Firstly,the regularity proof of the solution of the positive problem is given by using the variational theory.Then,the source term reconstruction problem of parabolic equation with variable coefficients is transformed into the optimal control problem,and the existence,uniqueness and stability of the solution of the optimal control problem are proved.In the fourth chapter,on the basis of the second chapter model,The source terminversion problem for Degenerate Parabolic Equations with a class of integral observational data is studied.This kind of problem occurs widely in the heat conduction phenomena with composite materials.Firstly,the rationality of the initial and boundary conditions of the degenerate model is proved.By introducing the corresponding test function space,the variational formulas are obtained by seeking the appropriate test function,and the uniqueness of the solution is proved based on the variational theory.Secondly,the corresponding time discrete model is given because of the backward Euler method and a series of prior estimators are derived.Finally,some convergence properties and existence proof of the solution are obtained.The fifth chapter is the summary and outlook part,which makes a brief summary and further outlook for the work of this paper.In this paper,the one-dimensional parabolic inverse source problem is theoretically analyzed.It is hoped that this problem will play a good guiding role in the high-dimensional case,and better methods are needed to make the solution of this kind of problem more stable and convergent.