Inverse Problems Of Identifying The Coefficient Of Second Order Parabolic Equation With Integral Source Term

Due to the urgent needs of the development of porous media fluid mechanics,medical research,geological exploration and other fields,there are more and more articles are studying the inverse problems of partial differential equations,especially the inverse problems of parabolic equations.This article reconstructs zero-order coefficients and initial values of second-order parabolic equations,which is different from the common inverse problem of parabolic equation.The source term of the equation contains integral terms,and even terminal observation data with small errors will cause the solution to be completely distorted.In this paper,the optimal control theory is used to establish the control function.The existence,uniqueness and stability of the minimal element of the optimal control problem are proved under appropriate additional conditions.Finally,the Landweber iterative algorithm is used for numerical simulation,and good results are obtained.This article is mainly divided into the following four chapters:The first chapter briefly introduces the current situation of this article and the two important chapters of the thesis.The second chapter studies the problem of inverse radiation coefficient of the second-order parabolic equation with integral source term,and uses an optimal problem to replace the original ill-posed problem.Firstly,the existence of the optimal solution of this optimization problem is proved,and then it derives the necessary condition that the optimal solution satisfies.Secondly,the uniqueness and stability of the optimal solution are proved through several priori estimations of direct problem.The third chapter mainly discusses the inverse problem of reconstructing the initial value of the second-order parabolic equation with integral source term.Similar to the second chapter,the well-posedness of the optimal solution is proved firstly,then the two difference schemes of the direct problem are established by the finite volume method,and their stability is proved.Finally,the Landweber iterative algorithm is used for numerical simulation,and good numerical results are obtained.These numerical results prove that the reconstruction of the initial value are very effective.The fourth chapter is a summary and the prospect of this article.This article mainly discusses the coefficients of inversion problem of second-order parabolic equations with integral source term in one-dimensional situations.The follow-up work can be discussed from a high-dimensional perspective and applied to practical problems to realize the value of research.In addition,we can also try more effective and convenient numerical algorithms to perform numerical calculations on this equation.