The stability study of numerical solution of Fredholm integral equations of the first kind with emphasis on its application in boundary elements method, By Hossein Hosseinzadeh

Research

Title	The stability study of numerical solution of Fredholm integral equations of the first kind with emphasis on its application in boundary elements method
Type	Article
Keywords	Radial basis functions Partial differential equations Boundary elements method Fredholm integral equations Weak and strong forms Numerical Stability Analysis of deep level transient spectroscopy data Analysis of nuclear magnetic resonance data Analysis of static light scattering data
Journal	APPLIED NUMERICAL MATHEMATICS
DOI	10.1016/j.apnum.2020.07.011
Researchers	Hossein Hosseinzadeh (First researcher) , mehdi dehghan (Second researcher) , zeinab Sedaghatjoo (Third researcher)

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.