Studies On Numerical Methods For Several Inverse Problems For Parabolic Equations

In this thesis, we consider several classes of inverse problems for parabolic equations, including simultaneous reconstruction of the heat source and the initial temperature, and determining the boundary coefficient of two-dimensional parabolic equation with the integral condition. There are two kinds of reconstructions of initial temperature in the first class, i.e., the partial and entire initial values. Besides, the boundary conditions are different from each other; one is under Dirichlet conditions, and the other is under Neumann conditions.Firstly, we give the inverse heat conduction problem of simultaneous reconstruction of the heat source and the partial initial temperature. By using an integral transformation of the unknown heat source function, we change the former problem into a homogeneous one, and transform the inverse source problem into an inverse boundary value problem. Hence, we can use the method of fundamental solutions.Secondly, we mainly investigate the problem of reconstructing the heat source and the entire initial value. We give a proof of the uniqueness theorem of our proposed problem by applying a function transformation. Meanwhile, we also change the problem into a backward heat conduction problem and an inverse source problem. For the backward heat conduction problem, the method of fundamental solutions is applied to solve it. In the process of solving the heat source, we use the trick of the numerical differentiation.Finally, we study the inverse boundary coefficient problem of two-dimensional parabolic equation with the integral condition. Because of the homogeneity of the equation, we can use the two-dimensional fundamental solution method.Since the resultant matrices of the method of fundamental solutions are highly ill-conditioned, we utilize the Tikhonov regularization method to solve them. For the choices of the regularization parameters, we use two kinds of a-posteriori schemes, i.e., the generalized cross-validation and L-curve methods. Besides, we choose the regularization parameters byα-priori scheme in the numerical differentiation.For the above problems, we give several typical numerical examples to show the stability and efficiency of our proposed methods. From the numerical results, we can obtain the conclusion that our methods work effectively on these ill-posed problems. Moreover, we also summarize some conclusions about our methods through data analysis.