A kernel-based technique to solve three-dimensional linear Fredholm integral equations of the second kind over general d
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(2019) 38:181
A kernel-based technique to solve three-dimensional linear Fredholm integral equations of the second kind over general domains Hamid Esmaeili1 · Davoud Moazami2 Received: 30 September 2018 / Revised: 22 August 2019 / Accepted: 24 September 2019 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019
Abstract In this article, we study a kernel-based method to solve three-dimensional linear Fredholm integral equations of the second kind over general domains. The radial kernels are utilized as a basis in the discrete collocation method to reduce the solution of linear integral equations to that of a linear system of algebraic equations. Integrals appeared in the scheme are approximately computed by the Gauss–Legendre and Monte Carlo quadrature rules. The method does not require any background mesh or cell structures, so it is mesh free and accordingly independent of the domain geometry. Thus, for the three-dimensional linear Fredholm integral equation, an irregular domain can be considered. The convergence analysis is also given for the method. Finally, numerical examples are presented to show the efficiency and accuracy of the technique. Keywords Linear integral equation · Three-dimensional Fredholm integral equation · Radial kernels · Meshfree method · General domains · Convergence analysis Mathematics Subject Classification 33E30 · 45A05 · 45B05 · 41A63
1 Introduction Integral equation models have numerous applications in the field of physics, mechanics and engineering. These mathematical models are used in the formulation of electromagnetic waves, scattering problems, continuum mechanics, potential theory, geophysics, electricity,
Communicated by Hui Liang.
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Hamid Esmaeili [email protected] Davoud Moazami [email protected]
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Department of Mathematics, Faculty of Science, Bu-Ali Sina University, Hamedan, Iran
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Department of Mathematics, Faculty of Mathematics Science and Statistics, Malayer University, Malayer, Iran
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kinetic theory of gases, hereditary phenomena in physics and biology, quantum mechanics, radiation, optimization, renewal theory, optimal control systems, communication theory, mathematical economics, population genetics, queueing theory, medicine, etc. (Farengo et al. 1983; Hatamzadeh-Varmazyar and Masouri 2011; Radlow 1964; Manzhirov 1985; Mirkin and Bard 1992; Boersma and Danicki 1993; Bremer et al. 2010; Li and Rong 2002; Jerri 1999; Atkinson 1997). Therefore, the study of these types of equations and methods for solving them is very useful in application. Three-dimensional integral equations are usually difficult to solve analytically, so, it is required to obtain their approximate solutions. There are many different numerical methods to solve integral equations in high dimensions. The projection, iterated projection, and Nystrom methods (Atkinson 1997; Atkinson et al. 1983; Atkinson and Potra 1989; Han and Wang 2002) are the commonly used approaches to solve multidimensional integral e
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