Solving Inverse Laplace Equation with Singularity by Weighted Reproducing Kernel Collocation Method

2017 ◽  
Vol 09 (05) ◽  
pp. 1750065 ◽  
Author(s):  
Judy P. Yang ◽  
Pai-Chen Guan ◽  
Chia-Ming Fan
Keyword(s):  
Inverse Problems ◽  
Collocation Method ◽  
Reproducing Kernel ◽  
Cauchy Problems ◽  
Overdetermined System ◽  
Numerical Examples ◽  
Kernel Approximation ◽  
Laplace Equations ◽  
Laplace Type ◽  
Least Squares Minimization

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.

Investigation of Multiply Connected Inverse Cauchy Problems by Efficient Weighted Collocation Method

2020 ◽  
Vol 12 (01) ◽  
pp. 2050012 ◽  
Author(s):  
Judy P. Yang ◽  
Qizheng Lin
Keyword(s):  
Inverse Problems ◽  
Boundary Conditions ◽  
Collocation Method ◽  
Reproducing Kernel ◽  
Dirichlet Boundary Conditions ◽  
Benchmark Problems ◽  
Cauchy Problems ◽  
Multiply Connected ◽  
Simply Connected ◽  
Gradient Approach

This work introduces an efficient weighted collocation method to solve inverse Cauchy problems. As it is known that the reproducing kernel approximation takes time to compute the second-order derivatives in the meshfree strong form method, the gradient approach alleviates such a drawback by approximating the first-order derivatives in a similar way to the primary unknown. In view of the overdetermined system derived from inverse Cauchy problems with incomplete boundary conditions, the weighted gradient reproducing kernel collocation method (G-RKCM) is further introduced in the analysis. The convergence of the method is first demonstrated by the simply connected inverse problems, in which the same set of source points and collocation points is adopted. Then, the multiply connected inverse problems are investigated to show that high accuracy of approximation can be reached. The sensitivity and stability of the method is tested through the disturbance added on both Neumann and Dirichlet boundary conditions. From the investigation of four benchmark problems, it is concluded that the weighted gradient reproducing kernel collocation method is more efficient than the reproducing kernel collocation method.

Gradient Enhanced Localized Radial Basis Collocation Method for Inverse Analysis of Cauchy Problems

2020 ◽  
Vol 12 (09) ◽  
pp. 2050107 ◽  
Author(s):  
Judy P. Yang ◽  
Yuan-Chia Chen
Keyword(s):  
Boundary Conditions ◽  
Collocation Method ◽  
Reproducing Kernel ◽  
Shape Function ◽  
Dirichlet Boundary Conditions ◽  
Cauchy Problems ◽  
Second Derivatives ◽  
Radial Basis ◽  
Global Nature ◽  
Direct Problems

This work proposes a gradient enhanced localized radial basis collocation method (GL-RBCM) for solving boundary value problems. In particular, the attention is paid to the solution of inverse Cauchy problems. It is known that the approximation by radial basis functions often leads to ill-conditioned systems due to the global nature. To this end, the reproducing kernel shape function and gradient reproducing kernel shape function are proposed to localize the radial basis function while the gradient approximation is aimed at reducing the computational intensity of carrying out the second derivatives of reproducing kernel shape function. In the proposed weighted collocation method, the weights on Neumann and Dirichlet boundary conditions are determined for both direct problems and inverse problems. From stability analysis, it is shown that the GL-RBCM can maintain high accuracy of approximating the first derivatives even under irregular perturbation added to boundary conditions. By comparing with the localized RBCM, the CPU saving of the GL-RBCM is manifested. The efficacy of the proposed method is therefore demonstrated.

Weighted Reproducing Kernel Collocation Method and Error Analysis for Inverse Cauchy Problems

2016 ◽  
Vol 08 (03) ◽  
pp. 1650030 ◽  
Author(s):  
Judy P. Yang ◽  
Pai-Chen Guan ◽  
Chia-Ming Fan
Keyword(s):  
Error Analysis ◽  
Collocation Method ◽  
Reproducing Kernel ◽  
Convergence Rates ◽  
Discrete Systems ◽  
Accurate Solution ◽  
Cauchy Problems ◽  
Linear Partial Differential Equations ◽  
Direct Collocation ◽  
Optimal Convergence Rates

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.

scholarly journals Solving Some Differential Equations Arising in Electric Engineering Using Hermite Polynomials

2020 ◽  
Vol 12 (4) ◽  
pp. 517-523
Author(s):  
G. Singh ◽  
I. Singh
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Collocation Method ◽  
Hermite Polynomials ◽  
Electric Circuit ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Electric Engineering ◽  
Collocation Points ◽  
Linear Algebraic Equations

In this paper, a collocation method based on Hermite polynomials is presented for the numerical solution of the electric circuit equations arising in many branches of sciences and engineering. By using collocation points and Hermite polynomials, electric circuit equations are transformed into a system of linear algebraic equations with unknown Hermite coefficients. These unknown Hermite coefficients have been computed by solving such algebraic equations. To illustrate the accuracy of the proposed method some numerical examples are presented.

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