Regularization Methods For Inverse Problems In Heat Conduction

The purpose of this thesis is to study inverse heat conduction problem and backward heatconduction problems. These problems reflect a variety of complex physical phenomenonassociated with heat exchange. They are both severely ill-posed problems，and theill-posedness becomes sharper as the unknown solutions are closer to the boundary point.Therefore, restoring stability of the solution, especially restoring stability of the solutionon the boundary has important theoretical significance and extensive applicationbackground. Regularization is its theoretical basis. So far the results on the problems aremainly devoted to numerical simulation and some research on the error analysis is given.But the optimality analysis work is less, the boundary is even more rare to stabilityanalysis of the results.The second chapter mainly research the IHCP of the Burgers equation which basedon the linear equationu t u x uxx.Firstly we illustrate the ill-posedness of problem andthen uses the modified Tikhonov method and wavelet dual least squares method to dealwith the problem to restore the stability of the solution. By constructing skill inequalityand choosing appropriate regularization parameters, these methods both obtain sharperror estimates (on temperature distribution and heat flux distribution) especially give thestabilities of solution on the boundary x0.The third chapter mainly studies a backward heat conduction problem. By twomethods of Fourier truncation, we get the two corresponding regular solutions. And thenwe provide error estimates of Logarithmic Holder between exact value and regularsolutions got by two methods of Fourier truncation. Especially we give the stabilities ofsolution at the initial moment t0.