International Journal of Nonlinear Analysis and Applications

An efficient RBF-FD method using polyharmonic splines alongside polynomials for the numerical solution of two-dimensional PDEs held on irregular domains and subject to Dirichlet and Robin boundary conditions

Document Type : Research Paper

Authors

1 Department of Applied Mathematics, Imam Khomeini International University, Qazvin, 34149-16818, Iran

2 Department of Applied Mathematics, Imam Khomeini International University, Qazvin, 34148-96818, Iran

Abstract

In the present paper, the relatively new method of Radial Basis Function-Generated Finite Difference (RBF-FD) is used to solve a class of Partial Differential Equations (PDEs) with Dirichlet and Robin boundary conditions. For this approximation, Polyharmonic Splines (PHS) are used alongside Polynomials. This combination has many benefits. On the other hand, Polyharmonic Splines have no shape parameter and therefore relieve us of the hassle of calculating the optimal shape parameter. As the first problem, a two-dimensional Poisson equation with the Dirichlet boundary condition is investigated in various domains. Then, an elliptic PDE with the Robin boundary condition is solved by the proposed method. The results of numerical studies indicate the excellent efficiency, accuracy and high speed of the method, while for these studies, very fluctuating and special test functions have been used.

Keywords

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International Journal of Nonlinear Analysis and Applications
Volume 15, Issue 4
April 2024
Pages 173-184
  • Receive Date: 23 August 2022
  • Revise Date: 03 March 2023
  • Accept Date: 23 March 2023