A Novel Method for Solving the Cauchy Problem of Laplace Equation Using the Fictitious Time Integration Method

In this study, a novel method for solving the Cauchy problem of Laplace equation is developed. Through the fictitious time integration method (FTIM), the finding of the root of the resulting linear equations call be transformed into for finding the fixed point of it system of first order ordinary differential equations, ill which a fictitious time variable is introduced. In such a sense, the inverse of ill-posed leading matrix is not necessary for the FTIM. This method uses the residual of each equation to control the evolution of unknowns in the fictitious time, and it is different from the conventional iteration method where an artificial iteration rule is required. Comparing to the Tikhonov's regularization method, the FTIM does not need to seek for the optimal regularization parameter, and it also needs not to seek for the inverse of the leading coefficient matrix in each step. Numerical results are given to show the validity of the current approach and it can be seen that this method can obtain reasonable results with or without noise. It shows a better noise resistance than the Tikhonov's regularization method.