Journal Article10.1080/17415977.2014.933831
Generalized finite difference method for solving two-dimensional inverse Cauchy problems
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TL;DR: In this article, a meshless numerical scheme is adopted for solving two-dimensional inverse Cauchy problems which are governed by second-order linear partial differential equations, which can overcome time-consuming mesh generation and numerical quadrature.
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Abstract: In this paper, a meshless numerical scheme is adopted for solving two-dimensional inverse Cauchy problems which are governed by second-order linear partial differential equations. In Cauchy problems, over-specified boundary conditions are imposed on portions of the boundary while on parts of boundary no boundary conditions are imposed. The application of conventional numerical methods to Cauchy problems yields highly ill-conditioned matrices. Hence, small noise added in the boundary conditions will tremendously enlarge the computational errors. The generalized finite difference method (GFDM), which is a newly developed domain-type meshless method, is adopted to solve in a stable manner the two-dimensional Cauchy problems. The GFDM can overcome time-consuming mesh generation and numerical quadrature. Besides, Cauchy problems can be solved stably and accurately by the GFDM. We present three numerical examples to validate the accuracy and the simplicity of the meshless scheme. In addition, different levels o...
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Citations
Application of the meshless generalized finite difference method to inverse heat source problems
TL;DR: In this article, the generalized finite difference method (GFDM) is applied to the heat source recovery problem in steady-state heat conduction problems, and the authors show that the proposed algorithm is accurate, computationally efficient and numerically stable for numerical solution of inverse heat source problems.
119
Generalized finite difference method for two-dimensional shallow water equations
Po-Wei Li,Chia-Ming Fan +1 more
TL;DR: A new way to determine the shape of star in the GFDM is proposed in this paper to capture the wave transmission and validate the accuracy and the consistency of the proposed meshless numerical scheme.
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A meshless method for solving three-dimensional time fractional diffusion equation with variable-order derivatives
Yan Gu,HongGuang Sun +1 more
TL;DR: In this article, a new framework for solving 3D time fractional diffusion equation with variable-order derivatives is presented, which is truly meshless and can be used to solve problems defined on an arbitrary domain in 3D.
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A generalized finite difference method based on the Peridynamic differential operator for the solution of problems in bounded and unbounded domains
TL;DR: A comprehensive study on the accuracy, convergence, and behavior of the GFDM through a patch test is conducted and it is shown that it generates a well-conditioned stiffness matrix for both structured and unstructured discretization.
82
Localized MFS for the inverse Cauchy problems of two-dimensional Laplace and biharmonic equations
TL;DR: Numerical experiments demonstrate the validity and accuracy of the proposed LMFS for the inverse Cauchy problems of two-dimensional Laplace and biharmonic equations with noisy boundary data.
76
References
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