Solving Methods For Several Kinds Of Ill-posed Problems Of Mathematical Physics

Driven by practical problems, Inverse Problems has become one of the fastest growing branches in the field of Applied Mathematics. Most of the inverse problems are ill-posed, especially, small changes in the data may cause dramatically large errors in the solution, and how to restore the stability of the solution is the dominant research interest of the ill-posed problems. This thesis mainly investigates the solving methods of some difficult inverse problems of Mathematical Physics involving high dimension, multiple independent variables or variable coefficient.This thesis considers the Cauchy problems for the Laplace equation in two kinds of cylinders. We apply the Fourier method and the modified Tikhonov method to solve the Cauchy problem for the Laplace equation in a semi-infinitely long cylinder and obtain the Holder type error estimate, respectively. We use the truncated regu-larization method combined with a priori and a posteriori parameter choice rule to solve the Cauchy problem for the Laplace equation in a bounded hollow cylinder, and obtain the corresponding explicit error estimate.This thesis adopts the variational method together with the conjugate gradient method to identify an unknown source with multiple independent variables, and fur-ther study the backward heat conduction problem with space-time dependent variable coefficient. The necessary theoretical analysis are given as far as possible. After trans-forming the original problem into an optimization problem, we prove the uniqueness of the minimal element theoretically, and give the expression of Frechet derivative of the minimization functional. The numerical results are satisfactory.We also carry out preliminary research on predictor-corrector method solving some difficult ill-posed problems from the perspective of numerical simulation. The identification problems of robin coefficient and unknown source from part of boundary are considered, respectively. In addition, different types of numerical examples are presented to show the validity of the solving methods.