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Abstract
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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.
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