Simultaneous Inversions Of The Source Term And The Initial Distribution In Two Classes Of Parabolic Equation And Theirs Algorithms

A class of physical process of diffusion,conduction and transportation of natural materials is usually described by the parabolic partial differential equation,whose researches of inverse problem play an important role in various scientific and engineering fields.In this thesis,we mainly consider the simultaneous inverse problems of reconstructing the space-dependent source term and the initial distribution for two classes of parabolic equation,namely,the simultaneous inversions of a parabolic equation with elliptic operator and a degenerate parabolic equation and theirs numerical algorithms.In the first chapter,we introduce the research significance of inverse problem of parabolic equation and its research trends of simultaneous recovering the source term and the initial value,as well as the studied content of this thesis.In the second chapter,some kind of preliminary knowledge of the function space,the Fichera's theory to degenerate partial differential equation and inequalities are given.The simultaneous inverse problem of the source term and the initial distribution for a parabolic equation with elliptic operator is researched in the third chapter.By transforming information of the initial value into the source term and obtaining a combined source term,the parabolic equation problem is converted into a parabolic problem with homogeneous initial-boundary conditions.Then the considered inverse problem is formulated into a regularized functional minimization problem.Based on the superposition principle of linear problem,the regularized functional minimization problem is discreted into linear algebraic equations,whose coefficient matrix and the right side are solved by a series of well-posed direct problems by using the finite element method.Therefore,the approximate solutions can be obtained by a non-iterative method,directly.The uniqueness of inverse solutions is proved by the solvability of the corresponding variational problem,and the error estimate as well as the convergence rate of regularized solutions are also provided.Then,the error estimate of approximate regularization solutions is presented in the finite dimensional space.Finally,the results of several numerical examples show that the proposed method is efficient and robust with respect to data noise.The simultaneous inverse problem of a degenerate parabolic equation and its numerical algorithms are studied in the fourth chapter.Respected to the direct problem of the degenerate parabolic equation,the form of weak solution of the direct problem and its regularity are given in a weight Sobolev space.Then,the simultaneous inversion is reduced to a regularized functional minimization problem by the classical Tikhonov regularized method,the uniqueness of the minimizers is proved rigourously.In addition,the error estimate of regularized solutions is given with the help of the theory of best approximation.The numerical algorithm of the inverse problem is constructed by the conjugate gradient method.In order to solve the inverse problem,a Crank-Nicolson difference scheme based on the finite volume method is deduced to compute the direct problem,furthermore the stability of the numerical scheme is obtained.Finally,numerical examples illustrate the proposed method is valid and effective.Some conclusions and future works are contained in the fifth chapter.