Direct solution of ill-posed boundary value problems by radial basis function collocation method

2005 ◽  
Vol 64 (1) ◽  
pp. 45-64 ◽  
Author(s):  
A. H.-D. Cheng ◽  
J. J. S. P. Cabral
Keyword(s):  
Radial Basis Function ◽  
Boundary Value Problems ◽  
Collocation Method ◽  
Basis Function ◽  
Boundary Value ◽  
Direct Solution ◽  
Radial Basis ◽  
Ill Posed

scholarly journals The Conical Radial Basis Function for Partial Differential Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
J. Zhang ◽  
F. Z. Wang ◽  
E. R. Hou
Keyword(s):  
Radial Basis Function ◽  
Boundary Value Problems ◽  
Basis Function ◽  
Boundary Value ◽  
Computational Cost ◽  
Basis Functions ◽  
Simple Scheme ◽  
Collocation Points ◽  
Radial Basis ◽  
Radial Basis Function Method

The performance of the parameter-free conical radial basis functions accompanied with the Chebyshev node generation is investigated for the solution of boundary value problems. In contrast to the traditional conical radial basis function method, where the collocation points are placed uniformly or quasi-uniformly in the physical domain of the boundary value problems in question, we consider three different Chebyshev-type schemes to generate the collocation points. This simple scheme improves accuracy of the method with no additional computational cost. Several numerical experiments are given to show the validity of the newly proposed method.

scholarly journals Efficient Solving of Boundary Value Problems Using Radial Basis Function Networks Learned by Trust Region Method

2018 ◽  
Vol 2018 ◽  
pp. 1-4 ◽  
Author(s):  
Mohie Mortadha Alqezweeni ◽  
Vladimir Ivanovich Gorbachenko ◽  
Maxim Valerievich Zhukov ◽  
Mustafa Sadeq Jaafar
Keyword(s):  
Radial Basis Function ◽  
Boundary Value Problems ◽  
Basis Function ◽  
Boundary Value ◽  
Trust Region Method ◽  
Trust Region ◽  
Radial Basis Function Networks ◽  
Structure Selection ◽  
Region Method ◽  
Radial Basis

A method using radial basis function networks (RBFNs) to solve boundary value problems of mathematical physics is presented in this paper. The main advantages of mesh-free methods based on RBFN are explained here. To learn RBFNs, the Trust Region Method (TRM) is proposed, which simplifies the process of network structure selection and reduces time expenses to adjust their parameters. Application of the proposed algorithm is illustrated by solving two-dimensional Poisson equation.

A novel local radial basis function collocation method for multi-dimensional piezoelectric problems

2021 ◽  
pp. 1045389X2110572
Author(s):  
Amir Noorizadegan ◽  
Der Liang Young ◽  
Chuin-Shan Chen
Keyword(s):  
Radial Basis Function ◽  
Collocation Method ◽  
Basis Function ◽  
Shape Parameter ◽  
Piezoelectric Medium ◽  
The Finite Element Method ◽  
Mechanical Field ◽  
Radial Basis ◽  
2D And 3D

The local radial basis function collocation method (LRBFCM), a strong-form formulation of the meshless numerical method, is proposed for solving piezoelectric medium problems. The proposed numerical algorithm is based on the local Kansa method using variable shape parameter. We introduce a novel technique for the determination of shape parameter in the LRBFCM, which leads to greater accuracy, and simplicity. The implemented algorithm is first verified with a 2D Poisson equation. Then, we employed LRBFCM in a numerical simulation for 2D and 3D piezoelectric problems involving mutual coupling of the electric field and elastodynamic equations for mechanical field. The presented meshless method is verified using corresponding results obtained from the finite element method and moving least squares meshless local Petrov–Galerkin method. In particular, the 2D piezoelectric problem is verified with an exact solution.

scholarly journals A Novel Meshfree Approach with a Radial Polynomial for Solving Nonhomogeneous Partial Differential Equations

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 270
Author(s):  
Cheng-Yu Ku ◽  
Jing-En Xiao ◽  
Chih-Yu Liu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Radial Basis Function ◽  
Collocation Method ◽  
Basis Function ◽  
Numerical Solutions ◽  
Polynomial Basis ◽  
Partial Differential ◽  
Minimum Order ◽  
Radial Basis

In this article, a novel radial–based meshfree approach for solving nonhomogeneous partial differential equations is proposed. Stemming from the radial basis function collocation method, the novel meshfree approach is formulated by incorporating the radial polynomial as the basis function. The solution of the nonhomogeneous partial differential equation is therefore approximated by the discretization of the governing equation using the radial polynomial basis function. To avoid the singularity, the minimum order of the radial polynomial basis function must be greater than two for the second order partial differential equations. Since the radial polynomial basis function is a non–singular series function, accurate numerical solutions may be obtained by increasing the terms of the radial polynomial. In addition, the shape parameter in the radial basis function collocation method is no longer required in the proposed method. Several numerical implementations, including homogeneous and nonhomogeneous Laplace and modified Helmholtz equations, are conducted. The results illustrate that the proposed approach may obtain highly accurate solutions with the use of higher order radial polynomial terms. Finally, compared with the radial basis function collocation method, the proposed approach may produce more accurate solutions than the other.

Sign in / Sign up

Export Citation Format

Share Document