A polynomial-augmented RBF collocation method for fourth-order boundary value problems. (1st March 2023)

• Record Type: Journal Article
• Authors: Cao, Dingding; Li, Xinxiang; Zhu, Huiqing
• Is Part Of: Computers & mathematics with applications. Volume 133(2023)
• Page Start: 1
• Page End: 11
• Publication Date: 2023-03-01
• Subjects: Radial basis functions -- Shape parameter -- Fictitious centres -- Fourth-order partial differential equations; Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2022.12.014
• Languages: English
• ISSNs: 0898-1221

Abstract

Abstract: In this paper, we consider a polynomial-augmented radial basis function collocation method (RBFCM) for solving fourth-order multi-dimensional elliptic partial differential equations (PDEs). Instead of choosing the RBF centres inside the problem domain, we use fictitious centres that can be located outside it to deal with the double boundary conditions of the fourth-order PDEs, and to also improve the computational accuracy. Furthermore, additional polynomial constraints applied to the approximation matrix of the RBFCM using fictitious centres, significantly alleviate the sensitivity of the method to the variation of the shape parameter, whereby it makes the RBFCM more practical for solving partial differential equations. We also apply the proposed method to a fourth-order elliptic PDE with random forcing terms (input data of the model). The input data are assumed to depend on a finite number of random variables at the collocation points. Numerical experiments are presented to demonstrate the effectiveness of the proposed method in terms of the selection of the shape parameter. In addition, the numerical results show that the proposed method achieves a higher accuracy than the conventional RBFCM.