Regularization Methods For Several Inverse Problems Related To Heat Conduction In Symmetric Region

Inverse problems related to heat conduction, especially inverse heat conduction problems and their steady-state, i.e., the Cauchy problem for the Laplace equation, backward heat conduction problems, unknown source identification problems have extensive application in engineering field. These problems are classical ill-posed problems. Some profound theories and extremely effective algorithm have been obtained for the problems in one-dimensional space. It is very difficult to deal with the problems in higher dimensional space, just as D.A.Murio said. So far there is hardly theoretic research, especially application of regularization method and accurate estimate of convergence rate for the problems in higher dimensional space except for some results in numerical simulation. In this thesis, we will explore the theory for several inverse problems of heat conduction in higher-dimensional space. Due to the difficulty of the problems, we mainly investigate them in some symmetric regions in two- and three-dimensional space. The problems either have obvious application background or can provide some breakthrough for us in theoretic research.The main contents of this thesis consist of four chapters. In the second chapter we mainly consider two backward heat conduction problems in two symmetric regions. The problems are ill-posed. Under some a-priori smoothness assumptions, we apply a simplified Tikhonov regularization method and a spectral cutoff method to recover the stability of solution. Moreover, by choosing proper regularization parameters we obtain some quite sharp error estimates between the approximate solution and exact solution in interval 0 < t < T and logarithmic type convergence estimates at initial time t = 0.The third chapter deals with higher-dimensional inverse heat conduction problems in some symmetric regions. First, we use Fourier transformation and properties of some special functions to derive the solutions of the corresponding direct problems, and explain the ill-posedness of the problems. Next, we apply three regularization methods to recover the stability of solution. In order to obtain stability analysis, we overcome many difficulties, combining with properties of the special functions, constructing and proving related inequalities. With suitable choices of regularization parameters, quite sharp error estimates between the approximate solution and exact solution are obtained for all the proposed methods. Mean- while, parts of the proposed methods are also added for the stability estimates of heat flux. The results for temperature distribution in spherical region have been accepted by two journals.In the fourth chapter we research two unknown heat source identification problems in two symmetric regions. Combining method of separation of variables with properties of some special functions, we obtain the expression of solutions of the problems, and expatiate the ill-posedness of them. Then, two regularization methods are applied to formulate regularized solutions which are stably convergent to the exact ones. Finally, logarithmic type error estimates between the approximate solution and exact solution are both obtained with suitable choices of regularization parameters. The uniqueness of solution is proven as well.The fifth chapter, we propose a regularization method for solving Cauchy problem of Laplace equation in higher-dimensional space to recover the stability of solution. A logarithmic-Holder type error estimate between the approximate solution and exact solution is obtained by introducing a rather technical inequality and improving a-priori smoothness assumption, and the convergence on an inaccessible boundary is also found.