Abstract:
The double-layer potential plays an important role in solving boundary value problems for elliptic equations, and in studying this potential, the properties of the fundamental solutions of the given equation are used. At present, all fundamental solutions to the generalized bi-axially symmetric Helmholtz equation are known but nevertheless, only for the first of them the potential theory was constructed. In this paper we study the double layer potential corresponding to the third fundamental solution. By using properties of Appell hypergeometric functions of two variables, we prove limiting theorems and derive integral equations involving the density of double-layer potentials in their kernels.
Keywords:
generalized bi-axially symmetric Helmholtz equation, Green formula, fundamental solution, third double-layer potential, Appell hypergeometric functions of two variables, integral equations with a density of double-layer potential in their kernel.
\Bibitem{Erg18}
\by T.~G.~Ergashev
\paper Third double-layer potential for a generalized bi-axially symmetric Helmholtz equation
\jour Ufa Math. J.
\yr 2018
\vol 10
\issue 4
\pages 111--121
\mathnet{http://mi.mathnet.ru/eng/ufa453}
\crossref{https://doi.org/10.13108/2018-10-4-111}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85073673237}
Linking options:
https://www.mathnet.ru/eng/ufa453
https://doi.org/10.13108/2018-10-4-111
https://www.mathnet.ru/eng/ufa/v10/i4/p111
This publication is cited in the following 14 articles:
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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
T. G. Ergashev, “Double- and simple-layer potentials for a three-dimensional elliptic equation with a singular coefficient and their applications”, Russian Math. (Iz. VUZ), 65:1 (2021), 72–86
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
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T. G. Ergashev, A. Hasanov, “Holmgren problem for elliptic equation with singular coefficients”, Vestnik KRAUNTs. Fiz.-mat. nauki, 32:3 (2020), 114–126
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
T. G. Ergashev, “Potentials for the singular elliptic equations and their application”, Results Appl. Math, 7 (2020), 100126
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
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
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
Tuhtasin G. Ergashev, “The Dirichlet problem for elliptic equation with several singular coefficients”, e-Journal of Analysis and Applied Mathematics, 2018:1 (2018), 81
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