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Vestnik Tomskogo Gosudarstvennogo Universiteta. Matematika i Mekhanika, 2017, Number 50, Pages 45–56
DOI: https://doi.org/10.17223/19988621/50/4
(Mi vtgu617)
 

This article is cited in 12 scientific papers (total in 12 papers)

MATHEMATICS

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

T. G. Ehrgashev

Tashkent Institute of Irrigation and Agricultural Mechanization Engineers, Tashkent, Uzbekistan
References:
Abstract: 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.
Keywords: 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.
Received: 12.08.2017
Bibliographic databases:
Document Type: Article
UDC: 517.956.6; 517.44
Language: Russian
Citation: T. G. Ehrgashev, “The fourth double-layer potential for a generalized bi-axially symmetric Helmholtz equation”, Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2017, no. 50, 45–56
Citation in format AMSBIB
\Bibitem{Erg17}
\by T.~G.~Ehrgashev
\paper The fourth double-layer potential for a generalized bi-axially symmetric Helmholtz equation
\jour Vestn. Tomsk. Gos. Univ. Mat. Mekh.
\yr 2017
\issue 50
\pages 45--56
\mathnet{http://mi.mathnet.ru/vtgu617}
\crossref{https://doi.org/10.17223/19988621/50/4}
\elib{https://elibrary.ru/item.asp?id=30778971}
Linking options:
  • https://www.mathnet.ru/eng/vtgu617
  • https://www.mathnet.ru/eng/vtgu/y2017/i50/p45
  • This publication is cited in the following 12 articles:
    1. Tukhtasin Ergashev, Zafarzhon Arzikulov, Mamirzhon Kholmirzaev, “FORMULY RAZLOZhENIYa DLYa GIPERGEOMETRIChESKIKh FUNKTsII DVUKh PEREMENNYKh I IKh PRIMENENIE K TEORII SINGULYaRNYKh ELLIPTIChESKIKh URAVNENII”, VOGUMFT, 2023, no. 2(3), 149  crossref
    2. T. G. Ergashev, “Potentsialy dlya trekhmernogo ellipticheskogo uravneniya s odnim singulyarnym koeffitsientom i ikh primenenie”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 25:2 (2021), 257–285  mathnet  crossref  zmath  elib
    3. T. G. Ergashev, Z. R. Tulakova, “The Dirichlet problem for an elliptic equation with several singular coefficients in an infinite domain”, Russian Math. (Iz. VUZ), 65:7 (2021), 71–80  mathnet  crossref  crossref
    4. T. G. Ergashev, N. D. Komilova, “Zadacha Kholmgrena dlya mnogomernogo ellipticheskogo uravneniya s dvumya singulyarnymi koeffitsientami”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2020, no. 63, 47–59  mathnet  crossref
    5. A. A. Abdullaev, T. G. Ergashev, “Zadacha Puankare–Trikomi dlya uravneniya smeshannogo elliptiko-giperbolicheskogo tipa vtorogo roda”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2020, no. 65, 5–21  mathnet  crossref
    6. T. G. Ergashev, “Potentials for the singular elliptic equations and their application”, Results Appl. Math, 7 (2020), 100126  crossref  zmath  isi  scopus
    7. T. G. Ergashev, “Potentials for three-dimensional singular elliptic equation and their application to the solving a mixed problem”, Lobachevskii J. Math., 41:6, SI (2020), 1067–1077  crossref  mathscinet  zmath  isi  scopus
    8. H. M. Srivastava, A. Hasanov, T. G. Ergashev, “A family of potentials for elliptic equations with one singular coefficient and their applications”, Math. Meth. Appl. Sci., 43:10 (2020), 6181–6199  crossref  mathscinet  zmath  isi  scopus
    9. T. G. Ergashev, “Fundamental solutions of the generalized helmholtz equation with several singular coefficients and confluent hypergeometric functions of many variables”, Lobachevskii J. Math., 41:1, SI (2020), 15–26  crossref  mathscinet  zmath  isi  scopus
    10. T. G. Ergashev, N. M. Safarbaeva, “Zadacha Dirikhle dlya mnogomernogo uravneniya Gelmgoltsa s odnim singulyarnym koeffitsientom”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2019, no. 62, 55–67  mathnet  crossref
    11. V. V. Katrakhov, S. M. Sitnik, “Metod operatorov preobrazovaniya i kraevye zadachi dlya singulyarnykh ellipticheskikh uravnenii”, Singulyarnye differentsialnye uravneniya, SMFN, 64, no. 2, Rossiiskii universitet druzhby narodov, M., 2018, 211–426  mathnet  crossref
    12. Tuhtasin G. Ergashev, “The Dirichlet problem for elliptic equation with several singular coefficients”, e-Journal of Analysis and Applied Mathematics, 2018:1 (2018), 81  crossref
    Citing articles in Google Scholar: Russian citations, English citations
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    Вестник Томского государственного университета. Математика и механика
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