The Regularization Methods For Three Non-classical Inverse Heat Conduction Problems

This thesis studies three non-classical inverse heat conduction problems, namely, the heat conduction problem with the convection term, the inverse heat conduction problem without the initial data and the spherically symmetric inverse heat conduction problem. These problems are seriously ill-posed. It has important theoretical value and practical significance to apply feasible regularization method for restoring the stability of the solution.The main contents of this thesis consist of the following three chapters. The second chapter studies a nonstandard inverse heat conduction problem. The ill-posedness of the problem is proved. We use a Fourier regularization method and a quasi-reversibility regularization method to formulate regularized solutions which are stably convergent to the exact ones. We obtain some quite sharp error estimates between the approximate solution and exact solution in interval under the suitable choices of regularization parameters. The third chapter studies a inverse heat conduction problem without initial data. We use spectrum truncated regularization method to find the regularized solution, and obtain the logarithmic error estimate between exact solution and regularized solution. The fourth chapter studies a spherically symmetric inverse heat conduction problem. We use the Meyer wavelet regularization method to deal with the ill-posed problem for receiving the regularization solution of the problem, and obtain the optimal order error estimate under the suitable choice of regularization parameter. Finally we demonstrate the effectiveness of the regularization method.