Abstract In this paper stability of numerical solution of Fredholm integral equation of the first kind is studied for radial basis kernels which possess positive Fourier transform. As a result, the equivalence relation between strong and weak forms of partial differential equations (PDEs) is proved for some special radial test functions. Also the stability of boundary elements method (BEM) is proved analytically for Laplace and Helmholtz equations by obtaining Fourier transform of singular fundamental solutions applied in BEM. Analytical result presented in this paper is an extension of stability idea of radial basis functions (RBFs) used to interpolate scattered data described by Wendland in [51] . Similar to the interpolation, it is proved here mathematically that integral operators which have radial kernels with positive Fourier transform are strictly positive definite. Thanks to the stability idea presented in [51] , a positive lower bound for eigenvalues of these integral operators is found here, explicitly.

中文翻译：

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第一类Fredholm积分方程数值解的稳定性研究及其在边界元法中的应用

摘要 本文研究了具有正傅里叶变换的径向基核的第一类Fredholm积分方程数值解的稳定性。结果，证明了偏微分方程（PDE）的强弱形式之间的等价关系对于一些特殊的径向测试函数。此外，通过获得应用于边界元法的奇异基本解的傅立叶变换，分析证明了边界元法（BEM）的稳定性，用于拉普拉斯方程和亥姆霍兹方程。本文中提出的分析结果是对 Wendland 在 [51] 中描述的用于插值散射数据的径向基函数 (RBF) 稳定性思想的扩展。类似于插值，这里在数学上证明了具有正傅立叶变换的径向核的积分算子是严格正定的。由于 [51] 中提出的稳定性思想，这里明确地找到了这些积分运算符的特征值的正下限。