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
This publication is cited in the following 12 articles:
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, “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
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
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
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
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
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
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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