scholarly journals Solving high-order partial differential equations with indirect radial basis function networks

2005 ◽  
Vol 63 (11) ◽  
pp. 1636-1654 ◽  
Author(s):  
N. Mai-Duy ◽  
R. I. Tanner
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Radial Basis Function ◽  
Basis Function ◽  
High Order ◽  
Radial Basis Function Networks ◽  
Partial Differential ◽  
Radial Basis

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.

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