| In this work, we study the Cauchy problem for the heat equation, as well as the inverse heat conduction problem, both of which are ill-posed problems in the sense of Hadamard.; The first chapter provides the background material about the previous investigations on the ill-posed Cauchy problem for the heat equation and the inverse heat conduction problem by other mathematicians. The method of Quasi-Reversibility is also introduced.; In the second chapter we apply the method of Quasi-Reversibility to the Cauchy problem for the heat equation and obtain a formal approximate solution. We prove that the convergence of the approximate solution to the presumed exact solution holds under certain conditions. Based on this result, we propose a modified numerical scheme in Chapter 3. It is implemented using the finite difference method, and the computational results for some example problems are satisfactory.; For the inverse heat conduction problem, we first investigate a simple case of positive heat flux at the boundary of the object over a finite time. Some estimates are obtained for that special case. In considering the general case of the inverse heat conduction problem, we smooth the given temperature at the given interior location, then transform the problem into its frequency domain by Fourier transform. An approximate solution is then constructed in the form of an inverse Fourier transform. We can prove that the data of the approximate solution at the given location could be very close to the given temperature, which validates the effectiveness of the application of the method of Quasi-Reversibility to the inverse heat conduction problem. |
