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Investigation on the Singularities of Some Singular Integrals

Zai You Yan1, Qiang Zhang1
Department of Aerodynamics, Nanjing University of Aeronautics and Astronautics, Yu Dao Street 29, City of Nan Jing, Jiang Su Province, P. R.China, 210016. Email: jutsjtu@yahoo.com.cn

Computer Modeling in Engineering & Sciences 2011, 75(3&4), 205-222. https://doi.org/10.3970/cmes.2011.075.205

Abstract

In a boundary element method, the treatment of all the possible singular integrals is very important for the correctness and accuracy of the solutions. Generally, the directional derivative of a weakly singular integral is computed by an integral in the sense of Cauchy principal value if the directional derivative of the weakly singular integral kernel is strongly singular or in the sense of Hadamard finite part integral if it is hypersingular. In this paper, we try to discover how the strongly singular and hypersingular integrals are generated and propose an idea to avoid the appearance of such kind of strongly singular and hypersingular integrals. This idea is termed as the 'exact derivation' of the directional derivative of a weakly singular integral. Using some simple examples, we proof that the directional derivative of a weakly singular integral found by this idea can still be a weakly singular integral. That is none strongly or hypersingular integrals are generated in such a process. Therefore, Cauchy principal value and Hadamard finite part integral are not indispensable.

Keywords

weakly singular integral, strongly singular integral, hypersingular integral, Cauchy principal value, Hadamard finite part, improper integral

Cite This Article

Yan, Z. Y., Zhang, Q. (2011). Investigation on the Singularities of Some Singular Integrals. CMES-Computer Modeling in Engineering & Sciences, 75(3&4), 205–222.



This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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