A discrete collocation scheme to solve Fredholm integral equations of the second kind in high dimensions using radial ke
- PDF / 3,646,729 Bytes
- 25 Pages / 439.37 x 666.142 pts Page_size
- 70 Downloads / 165 Views
A discrete collocation scheme to solve Fredholm integral equations of the second kind in high dimensions using radial kernels H. Esmaeili1 · F. Mirzaee2 · D. Moazami2 Received: 16 April 2020 / Accepted: 29 August 2020 © Sociedad Española de Matemática Aplicada 2020
Abstract In this paper, we propose a kernel-based method to solve multidimensional linear Fredholm integral equations of the second kind over general domains. The discrete collocation method in combination with radial kernels interpolation method is utilized to convert these types of equations to a linear system of equations that can be solved numerically by a suitable numerical method. Integrals appeared in the scheme are approximately computed by the Gauss–Legendre and Monte Carlo quadrature rules. The proposed scheme does not require a structured grid, and thus can be used to solve complex geometry problems based on a set of scattered points that can be arbitrarily chosen. Thus, for the multidimensional linear Fredholm integral equation, an irregular region can be considered. The convergence analysis of the approach is studied for the presented method. The accuracy and efficiency of the new technique are illustrated by several numerical examples. Keywords Linear integral equation · Multidimensional Fredholm integral equation · Radial kernels · Discrete collocation method · General domains · Convergence analysis Mathematics Subject Classification 65R20 · 45A05 · 45B05 · 41A63
1 Introduction Various problems in physics, engineering, mechanics, nonhomogeneous elasticity, electrostatics, population and many other fields give rise to the second-kind multidimensional
B
H. Esmaeili [email protected] F. Mirzaee [email protected] D. Moazami [email protected]
1
Department of Mathematics, Faculty of Science, Bu-Ali Sina University, Hamedan, Iran
2
Department of Mathematics, Faculty of Mathematical Sciences and Statistics, Malayer University, Malayer, Iran
123
H. Esmaeili et al.
integral equations [4,8,15,16,21,23]. Since analytical solutions of integral equations are not generally available, numerical methods become a practical way to solve this type of equations. Therefore, it is important to note that the survey of these problems is very useful in applications. There are several numerical techniques for approximating the solution of integral equations in high dimensions. The Nystrom type methods, iterative methods and projection methods [4,5,13] which include: the well-known collocations and Galerkin methods [6,7,14] are the most important approaches to obtain the numerical solution of these equations. These approaches are based on the domain discretization. The discretization involved in all of these methods requires some sort of underlying computational mesh, e.g., a triangulation of the region of interest. Creation of these meshes becomes a rather difficult task in three dimensions, and virtually impossible for higher-dimensional problems. One of the ways to overcome problems related to the meshing are so-called meshless
Data Loading...
