Studies On Reconstructing The Heat Source And The Initial Distribution Based On The Expansion Of Eigenfunctions

With the development of science and technology,a lot of works have been done in mathematical and physical inverse problems in the past decades.Inverse problem of heat conduction equations problems arise in many branches of applied science and engineering and technology,purpose is determining the unknown source or unknown source and the initial distribution from some measurable information related to the source.As we all know,this problem is ill-posed since small errors inherently presented in the practical measurement can induce dramatic change errors in reconstructing the solution.This paper mainly studies on reconstructing the heat source and the initial distribution based on the expansion of eigenfunctions in an inverse heat problem.The main results of this paper are as follows:In the first chapter,we introduce the research significance,the dynamic as well as the studied contents of this thesis.In the second chapter,we consider a kind of regularization method about inverse source problem of the higher-dimensional heat equation.Firstly,by the Fourier expansion of the final temperature in the rectangular area,a kind of regularized approximate problem is constructed for obtaining a regularization solution of the source.Then the stability and the convergence for the regularization solution are presented.Furthermore,the converge rate of the regularized solution is analyzed in the prior and a posteriori selection of regularization parameter.Compared with the previous regularization methods,the convergence rate is improved.Finally,we generalize inverse source problem of the regularization method in the general area.Numerical results show that the proposed regularization method is very stable.In the third chapter,we consider the regularization method about simultaneous determination of a source and the initial distribution in an inverse heat conduction problem.First of all,we study the regularization method and research ideas in the general area.For convenience,we introduce the processes of regularization method for an inverse source and the initial distribution problem of one-dimensional heat equation as an example.The stability and the convergence for the regularization solution are presented.Furthermore,the converge rate of the regularized solution is analyzed in the prior and aposteriori selection of regularization parameter.Numerical results show that the proposed regularization method is very stable.