The fourth double-layer potential for a generalized bi-axially symmetric Helmholtz equation | Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika – Tomsk State University Journal of Mathematics and Mechanics. 2017. № 50. DOI: 10.17223/19988621/50/4

The fourth double-layer potential for a generalized bi-axially symmetric Helmholtz equation

Applying a method of complex analysis (based upon analytic functions), R.P. Gilbert in 1969 constructed an integral representation of solutions of the generalized bi-axially symmetric Helmholtz equation. Fundamental solutions of this equation were constructed recently. In fact, when the spectral parameter is zero, fundamental solutions of the generalized bi-axially symmetric Helmholtz equation can be expressed in terms of Appell’s hypergeometric function of two variables of the second kind. All the fundamental solutions of the generalized bi-axially symmetric Helmholtz equation are known, and only for the first one the theory of potential was constructed. In this paper, we aim at constructing a theory of double-layer potentials corresponding to the fourth fundamental solution. Using some properties of Appell’s hypergeometric functions of two variables, we prove limiting theorems and derive integral equations containing double-layer potential densities in the kernel.

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Keywords

обобщенное двуосесимметрическое уравнение Гельмгольца, формула Грина, фундаментальное решение, четвертый потенциал двойного слоя, гипергеометрические функции Аппеля от двух переменных, интегральные уравнения с плотностью потенциала двойного слоя в ядре, generalized bi-axially symmetric Helmholtz equation, Green’s formula, fundamental solution, fourth double-layer potential, Appell’s hypergeometric functions of two variables, integral equations with double-layer potential density

Authors

Name	Organization	E-mail
Ehrgashev Tuhtasin G.	Tashkent Institute of Irrigation and Agricultural Mechanization Engineers	ertuhtasin@mail.ru

Всего: 1

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