The Lie-group Shooting Method For Two Classical Ill-posed Problems

In this paper, we mainly study two classical ill-posed problems by combining the lie-group shooting method with the quasi-boundary regularization method, i.e., the inverse heat conduction problem in an annulus domain and the Cauchy problem for the elliptic equation in a general annular domain. Since these two problems are ill-posed, we first use the qusi-boundary regularization method to transform the two ill-posed problems into well-posed problems and then use the idea of the semi-discretization to transform the well-posed problems into two points boundary problem of the ordinary differential equations, and then the nonlinear target alge-bra equations can be obtained by using the structure and property of the lie-group and group preserve scheme. Further, by choosing the suitable weight factor and the iterative scheme we can get the approximate solution of the ill-posed problem. Specifically, for the Cauchy problem of the elliptic equation in a general annular domain, we transform the interior curve edge into the straight on the boundary with our goal being to use the lie-group shooting method to solve it. The advantage of the lie-group method is that it does't need the prior information, the computational cost is very saving and the numerical implement is easy. Finally, the numerical results show that the lie-group shooting method is a effective and stable.