An approximate method for evaluating hypersingular integrals |
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| Affiliation: | 1. Department of Mathematics, The Penza State University, 40, Krasnay Str., Penza 440017, Russia Federation;2. Department of Mathematics, Kansas State University, USA;3. Department of Engineering Science and Mechanics, The Pennsylvania State University, 203 B Earth & Engineering Science Building, University Park, PA 16802-1401, USA;1. Department of Mathematics, Nanjing University, Nanjing, 210093, PR China;2. School of Mathematics and Statistics, Xi''an Jiaotong University, Xi''an, 710049, PR China;1. Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Santiago, Chile;2. School of Mathematics and Statistics, The University of New South Wales, Sydney 2052, Australia |
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| Abstract: | This paper describes a new method for direct numerical evaluation of multidimensional hypersingular integrals assigned on smooth curves and surfaces. These integrals arise when the boundary integral equations are used to solve problems of mechanics, electrodynamics, aerodynamics, etc. The hypersingular integrals are considered, in the sense of Hadamard, as finite parts. The main advantage of the proposed method is the numerical computation of the hypersingular integrals by the direct application of the developed cubature formulas, thus requiring little analytical pre-work. The method is not restricted to the type of problem however and may be easily applied to any hypersingular integrals. The convergence of the proposed technique has been proved and error estimates are given. An illustrative example demonstrates the accuracy and efficiency of the method. |
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