Open Access

ARTICLE

A Novel Meshless Method for Solving the Second Kind of Fredholm Integral Equations

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1 School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Republic of Singapore
2 Corresponding author, Tel.: + 65 6790 4953; Fax: + 65 6792 4062; Email address: lihua@ntu.edu.sg (Hua Li)

Computer Modeling in Engineering & Sciences 2010, 67(1), 55-78. https://doi.org/10.3970/cmes.2010.067.055

Abstract

A novel meshless technique termed the Random Integral Quadrature (RIQ) method is developed in this paper for solving the generalized integral equations. By the RIQ method, the governing equations in the integral form are discretized directly with the field nodes distributed randomly or uniformly, which is achieved by discretizing the integral governing equations with the generalized integral quadrature (GIQ) technique over a set of background virtual nodes, and then interpolating the function values at the virtual nodes over a set of field nodes with Local Kriging method, where the field nodes are distributed either randomly or uniformly. The RIQ method is a meshless technique since it doesn't require any approximation cells, but a set of field nodes distributed either randomly or uniformly in the computational domain. In order to validate the RIQ method, the second kind of Fredholm integral equations in one-, two- and three-dimensional domains are solved via both randomly and uniformly distributed field nodes. Corresponding convergence rate is also studied for each of the case studies. The numerical solutions of all these case studies demonstrate that the RIQ method can achieve highly computational accuracy, even if only a few field nodes are scattered in the domains, and also make good convergence rates.

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