Radial boundary elements method, a new approach on using radial basis functions to solve partial differential equations, efficiently

IF 3.5 2区 数学 Q1 MATHEMATICS, APPLIED Applied Mathematics and Computation Pub Date : 2025-01-02 DOI:10.1016/j.amc.2024.129252
Hossein Hosseinzadeh, Zeinab Sedaghatjoo
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Abstract

Conventionally, piecewise polynomials have been used in the boundary element method (BEM) to approximate unknown boundary values. However, since infinitely smooth radial basis functions (RBFs) are more stable and accurate than the polynomials for high dimensional domains, this paper proposes approximating the unknown values using RBFs. This new formulation is called the radial BEM. To calculate the singular boundary integrals in the radial BEM, the authors propose a new distribution of boundary source points that removes the singularity from the integrals. This allows the boundary integrals to be precisely calculated using the standard Gaussian quadrature rule with 16 quadrature nodes. Several numerical examples are presented to evaluate the efficiency of the radial BEM compared to standard BEM and RBF collocation method for solving partial differential equations (PDEs). The analytical and numerical studies demonstrate that the radial BEM is a superior version of BEM that will significantly enhance the application of BEM and RBFs in solving PDEs.
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径向边界元法是利用径向基函数求解偏微分方程的一种新方法
传统的边界元法采用分段多项式来逼近未知的边界值。然而,由于无限光滑径向基函数(rbf)在高维域上比多项式更稳定和精确,本文提出了利用rbf逼近未知值的方法。这个新公式被称为径向边界元。为了计算径向边界元中的奇异边界积分,作者提出了一种新的边界源点分布,消除了积分中的奇异性。这使得边界积分可以使用标准高斯正交规则精确计算16个正交节点。给出了几个数值算例,比较了径向边界元法与标准边界元法和RBF配置法求解偏微分方程的效率。分析和数值研究表明,径向边界元法是边界元法的一种优越版本,将大大提高边界元法和径向边界元法在求解偏微分方程中的应用。
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来源期刊
CiteScore
7.90
自引率
10.00%
发文量
755
审稿时长
36 days
期刊介绍: Applied Mathematics and Computation addresses work at the interface between applied mathematics, numerical computation, and applications of systems – oriented ideas to the physical, biological, social, and behavioral sciences, and emphasizes papers of a computational nature focusing on new algorithms, their analysis and numerical results. In addition to presenting research papers, Applied Mathematics and Computation publishes review articles and single–topics issues.
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