Browsing by Author "Guan, Pai-Chen"

Now showing 1 - 2 of 2

Solving Inverse Laplace Equation with Singularity by Weighted Reproducing Kernel Collocation Method
(2017-07-01)
Yang, Judy P.
;
Guan, Pai-Chen
;
Fan, Chia-Ming
;
土木工程學系
;
Department of Civil Engineering
This work introduces the weighted collocation method with reproducing kernel approximation to solve the inverse Laplace equations. As the inverse problems in consideration are equipped with over-specified boundary conditions, the resulting equations yield an overdetermined system. Following our previous work, the weighted collocation method using a least-squares minimization has shown to solve the inverse Cauchy problems efficiently without using techniques such as iteration and regularization. In this work, we further consider solving the inverse problems of Laplace type and introduce the Shepard functions to deal with singularity. Numerical examples are provided to demonstrate the validity of the method.
Weighted Reproducing Kernel Collocation Method and Error Analysis for Inverse Cauchy Problems
(2016-04)
Yang, Judy P.
;
Guan, Pai-Chen
;
Fan, Chia-Ming
;
土木工程學系
;
Department of Civil Engineering
In this work, the weighted reproducing kernel collocation method (weighted RKCM) is introduced to solve the inverse Cauchy problems governed by both homogeneous and inhomogeneous second-order linear partial differential equations. As the inverse Cauchy problem is known for the incomplete boundary conditions, how to numerically obtain an accurate solution to the problem is a challenging task. We first show that the weighted RKCM for solving the inverse Cauchy problems considered is formulated in the least-squares sense. Then, we provide the corresponding error analysis to show how the errors in the domain and on the boundary can be balanced with proper weights. The numerical examples demonstrate that the weighted discrete systems improve the accuracy of solutions and exhibit optimal convergence rates in comparison with those obtained by the traditional direct collocation method. It is shown that neither implementation of regularization nor implementation of iteration is needed to reach the desired accuracy. Further, the locality of reproducing kernel approximation gets rid of the ill-conditioned system.