Regularization of divergent integrals: A comparison of the classical and generalized-functions approaches
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Regularization of divergent integrals: A comparison of the classical and generalized-functions approaches V. V. Zozulya
Received: 13 March 2012 / Accepted: 29 August 2013 / Published online: 19 June 2015 © Springer Science+Business Media New York 2015
Abstract This article considers methods of weakly singular and hypersingular integral regularization based on the theory of distributions. For regularization of divergent integrals, the Gauss–Ostrogradskii theorem and the second Green’s theorem in the sense of the theory of distribution have been used. Equations that allow easy calculation of weakly singular, singular, and hypersingular integrals in oneand two-dimensional cases for any sufficiently smooth function have been obtained. These equations are compared with classical methods of regularization. The results of numerical calculation using classical approaches and those based of the theory of generalized functions, along with a comparison for different functions, are presented in tables and graphs of the values of divergent integrals versus the position of the colocation point. Keywords Divergent · Singular · Hypersingular · Regularization · Generalized function · BIE · BEM Mathematics Subject Classification (2010) 35J08 · 35j25 · 45E05 · 65R20
1 Introduction As mentioned in [7], divergent integrals were first considered by Cauchy, in 1826. He called such integrals “extraordinary.” Cauchy also remarked that differentiation and integration with respect to a parameter are permissible with these “extraordinary”
Communicated by: Leslie Greengard V. V. Zozulya () Centro de Investigacion Cientifica de Yucatan A.C., Calle 43, No 130, Colonia: Chuburn´a de Hidalgo, C.P. 97200, M´erida, Yucat´an, M´exico e-mail: [email protected]
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V. V. Zozulya
integrals. The next significant step in the definition and application of divergent integrals was taken by Hadamard, in about 1900. He extended the definition of divergent integrals to the multidimensional case and applied it to the solution of the Cauchy problem for differential equations of hyperbolic type [44]. But it is only recently, in connection with the development of boundary integral equations (BIE) and boundary element methods (BEM), that research in this area has been intensively pursued and applied to the solution of scientific and engineering problems. Over the past few decades, thousands of publications on this topic in both theoretical and applied branches of mathematics have appeared. We cannot, therefore, give here anything approaching a comprehensive literature review, and so we instead consider issues related to the regularization and computation of divergent integrals. In this more reduced area of research, we consider in detail an approach associated with the use of generalized functions and compare it with the classical approach. We shall give references only to the most important works of the past, those works directly related to the topic of our research, and works of a fundamental nature, mostly monographs. Divergent integrals arise in various fie
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