Abstract
In this article, a novel infinitely smooth polyharmonic radial basis function (PRBF) collocation method for solving elliptic partial differential equations (PDEs) is presented. The PRBF with natural logarithm is a piecewise smooth function in the conventional radial basis function collocation method for solving governing equations. We converted the piecewise smooth PRBF into an infinitely smooth PRBF using source points collocated outside the domain to ensure that the radial distance was always greater than zero to avoid the singularity of the conventional PRBF. Accordingly, the PRBF and its derivatives in the governing PDEs were always continuous. The seismic wave propagation problem, groundwater flow problem, unsaturated flow problem, and groundwater contamination problem were investigated to reveal the robustness of the proposed PRBF. Comparisons of the conventional PRBF with the proposed method were carried out as well. The results illustrate that the proposed approach could provide more accurate solutions for solving PDEs than the conventional PRBF, even with the optimal order. Furthermore, we also demonstrated that techniques designed to deal with the singularity in the original piecewise smooth PRBF are no longer required.
Highlights
A large number of multidisciplinary problems may require solutions involving complicated mathematical models and numerical techniques [1,2]
The meshfree method has gained the attention of researchers from different scientific fields due to its capability of dealing with partial differential equations (PDEs) with complicated and irregular geometry [3,4,5,6,7]
The radial basis function collocation method (RBFCM) is one meshfree approach for analyzing governing equations, where the unknowns are represented by a function approximation [8,9,10]
Summary
A large number of multidisciplinary problems may require solutions involving complicated mathematical models and numerical techniques [1,2]. Since the distance between the source and the interior point is always greater than zero, the MQ and its derivatives are always smooth and globally infinitely differentiable without using the shape parameter. Motivated by this concept, a novel infinitely smooth polyharmonic radial basis function (PRBF) collocation method for solving elliptic partial differential equations (PDEs) is presented. We propose a novel idea to convert the piecewise smooth PRBF into an infinitely smooth PRBF using source points collocated outside the domain to ensure that the radial distance is always greater than zero, avoiding the singularity of the conventional PRBF. The concept of fictitious sources is elaborated
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