Ziqiang Bai¹, Wenzhen Qu^2,*, Guanghua Wu^3,*

CMES-Computer Modeling in Engineering & Sciences, Vol.138, No.3, pp. 2955-2972, 2024, DOI:10.32604/cmes.2023.031474

Abstract In the past decade, notable progress has been achieved in the development of the generalized finite difference method (GFDM). The underlying principle of GFDM involves dividing the domain into multiple sub-domains. Within each sub-domain, explicit formulas for the necessary partial derivatives of the partial differential equations (PDEs) can be obtained through the application of Taylor series expansion and moving-least square approximation methods. Consequently, the method generates a sparse coefficient matrix, exhibiting a banded structure, making it highly advantageous for large-scale engineering computations. In this study, we present the application of the GFDM to numerically solve inverse Cauchy problems in two-… More >

Chih-Wen Chang^*

CMC-Computers, Materials & Continua, Vol.72, No.2, pp. 3231-3245, 2022, DOI:10.32604/cmc.2022.024563

Abstract In this paper, we address 3D inverse Cauchy issues of highly nonlinear elliptic equations in large cuboids by utilizing the new 3D homogenization functions of different orders to adapt all the specified boundary data. We also add the average classification as an approximate solution to the nonlinear operator part, without requiring to cope with nonlinear equations to resolve the weighting coefficients because these constructions are owned many conditions about the true solution. The unknown boundary conditions and the result can be retrieved straightway by coping with a small-scale linear system when the outcome is described by a new 3D homogenization… More >

Chein-Shan Liu¹

CMES-Computer Modeling in Engineering & Sciences, Vol.35, No.2, pp. 91-112, 2008, DOI:10.3970/cmes.2008.035.091

Abstract We consider the inverse Cauchy problems for Laplace equation in simply and doubly connected plane domains by recoverning the unknown boundary value on an inaccessible part of a noncircular contour from overspecified data. A modified Trefftz method is used directly to solve those problems with a simple collocation technique to determine unknown coefficients, which is named a modified collocation Trefftz method (MCTM). Because the condition number is small for the MCTM, we can apply it to numerically solve the inverse Cauchy problems without needing of an extra regularization, as that used in the solutions of direct problems for Laplace equation.… More >

Chein-Shan Liu¹, Chung-Lun Kuo²

CMES-Computer Modeling in Engineering & Sciences, Vol.77, No.1, pp. 57-80, 2011, DOI:10.3970/cmes.2011.077.057

Abstract In this paper, the solutions of inverse Cauchy problems for quasi-linear elliptic equations are resorted to an unusual mixed group-preserving scheme (MGPS). The bottom of a finite rectangle is imposed by overspecified boundary data, and we seek unknown data on the top side. The spring-damping regularization method (SDRM) is introduced by converting the governing equation into a new one, which includes a spring term and a damping term. The SDRM can further stabilize the inverse Cauchy problems, such that we can apply a direct numerical integration method to solve them by using the MGPS. Several numerical examples are examined to… More >

Leevan Ling¹, Tomoya Takeuchi²

CMES-Computer Modeling in Engineering & Sciences, Vol.29, No.1, pp. 45-54, 2008, DOI:10.3970/cmes.2008.029.045

Abstract The method of fundamental solutions is coupled with the boundary control technique to solve the Cauchy problems of the Laplace Equations. The main idea of the proposed method is to solve a sequence of direct problems instead of solving the inverse problem directly. In particular, we use a boundary control technique to obtain an approximation of the missing Dirichlet boundary data; the Tikhonov regularization technique and the L-curve method are employed to achieve such goal stably. Once the boundary data on the whole boundary are known, the numerical solution to the Cauchy problem can be obtained by solving a direct… More >

Chein-Shan Liu^1,2, Chung-Lun Kuo³, Dongjie Liu⁴

CMC-Computers, Materials & Continua, Vol.24, No.2, pp. 105-124, 2011, DOI:10.3970/cmc.2011.024.105

Abstract The inverse Cauchy problems for elliptic equations, such as the Laplace equation, the Poisson equation, the Helmholtz equation and the modified Helmholtz equation, defined in annular domains are investigated. The outer boundary of the annulus is imposed by overspecified boundary data, and we seek unknown data on the inner boundary through the numerical solution by a spring-damping regularization method and its Lie-group shooting method (LGSM). Several numerical examples are examined to show that the LGSM can overcome the ill-posed behavior of inverse Cauchy problem against the disturbance from random noise, and the computational cost is very cheap. More >

Weichung Yeih¹, Chein-Shan Liu², Chung-Lun Kuo³, Satya N. Atluri⁴

CMC-Computers, Materials & Continua, Vol.17, No.3, pp. 275-302, 2010, DOI:10.3970/cmc.2010.017.275

Abstract In this paper, a multiple-source-point boundary-collocation Trefftz method, with characteristic lengths being introduced in the basis functions, is proposed to solve the direct, as well as inverse Cauchy problems of the Laplace equation for a multiply connected domain. When a multiply connected domain with genus p (p>1) is considered, the conventional Trefftz method (T-Trefftz method) will fail since it allows only one source point, but the representation of solution using only one source point is impossible. We propose to relax this constraint by allowing many source points in the formulation. To set up a complete set of basis functions, we… More >