Analytical and Numerical Investigation of the Fredholm Integral Equation for the Heat Radiation Problem

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Journal Title, Volume, Page: 
Applied Mathematics and Computation Volume 175, Issue 1, 1 April 2006, Pages 149–170
Year of Publication: 
2006
Authors: 
Naji Qatanani
Department of Mathematics, College of Science and Technology, Al-Quds University, P.O. Box 20002, Abu Dies, Jerusalem, Palestine
Current Affiliation: 
Department of Mathematics, Faculty of Science, An-Najah National University, Nablus, Palestine
Monika Schulz
University of Stuttgart, Mathematical Institute A, Pfaffenwaldring 57, D-70569 Stuttgart, Germany
Preferred Abstract (Original): 

This article deals with the mathematical and the numerical aspects of the Fredholm integral equation of the second kind arising as a result of the heat energy exchange inside a convex and non-convex enclosure geometries. Some mathematical results concerning the integral operator are presented. The Banach fixed point theorem is used to guarantee the existence and the uniqueness of the solution of the integral equation. For the non-convex geometries the visibility function has to be taken into consideration, then a geometrical algorithm is developed to provide an efficient detection of the shadow zones. For the numerical simulation of the integral equation we use the boundary element method based on the Galerkin discretization scheme. Some iterative methods for the discrete radiosity equation are implemented. Several two- and three-dimensional numerical test cases for convex and non-convex geometries are included. This give concrete hints which iterative scheme might be more useful for such practical applications.

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