抄録
We show a numerical method for the Cauchy problem of the Laplace equation and the backward heat conduction problem with ill-posedness. The numerical method consists of the multi-precision arithmetic and a high order finite difference method in which sampling points can be arbitrarily located in the domain of the problem. It is our strategy to suppress the influence on the accuracy of the numerical solution from the rounding error and the discretization error because of the instability of the solution. In numerical examples, we can obtain the numerical solution with high accuracy of the ill-posed problems.
