Several Regularization Methods For Backward Heat Conduction Problem

In the field of natural science and engineering, one wishes to reconstruct the initial temperature distributions from a measured temperature at a fixed time t =T >0 in a heat conduction body.This problem is called the backward heat conduction problem.This problem is ill-posed. It is one of the main contents that using appropriate regularization method to restore the stability of the solution for the problem. In this thesis, we will use three regularization methods to study three non-standard backward heat conduction problems.The main contents of this thesis consist of the following three chapters. The first chapter mainly introduces backward heat conduction problem. then introduces the current situation of Fourier regularization method. Tikhonov regularization method and quasi-reversibility regularization method. In the second chapter, we consider a one dimensional backward heat conduction problem with convection term in unbounded region. We provide a modified Fourier regularization method to formulate regularized solution which is stably convergent to the exact ones. Moreover, we obtain some quite sharp error estimate. The third chapter uses a modified Tikhonov regularization method for solving the one dimensional non-standard backward heat conduction problem in bounded area. By constructing inequalities and selecting appropriate regularization parameters, we obtain the error analysis for the regularized solution. In the last chapter, we apply a quasi-reversibility and quasi-final value regularization methods for a backward heat conduction problem with robin boundary value and one dimensional non-standard backward heat conduction problem, respectively. And the stabilities of the solutions for the problems are given. Quite sharp error estimates between the approximate solution and exact solution are obtained for the two proposed methods under the suitable choices of regularization parameters.