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

Chih-Chang Chi1, Weichung Yeih1,2, Chein-Shan Liu3
Computation and Simulation Center, National Taiwan Ocean University, Keelung 20224, Taiwan, ROC. Email of the corresponding author: d88520002@mail.ntou.edu.tw
Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung 20224,Taiwan, ROC
Department of Civil Engineering, National Taiwan University, Taipei 10671, Taiwan, ROC.

Computer Modeling in Engineering & Sciences 2009, 47(2), 167-190. https://doi.org/10.3970/cmes.2009.047.167

### Abstract

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 can be transformed into for finding the fixed point of a system of first order ordinary differential equations, in 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.

### Keywords

Inverse Cauchy problem, Laplace equation, Boundary element method, Fictitious Time Integration Method, Tikhonov's regularization method

Chi, C., Yeih, W., Liu, C. (2009). A Novel Method for Solving the Cauchy Problem of Laplace Equation Using the Fictitious Time Integration Method. CMES-Computer Modeling in Engineering & Sciences, 47(2), 167–190.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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