Geodesic
The fourth double-layer potential for a generalized bi-axially symmetric Helmholtz equation
Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 50 (2017), pp. 45-56 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     title = {The fourth double-layer potential for a generalized bi-axially symmetric {Helmholtz} equation},
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T. G. Ehrgashev. The fourth double-layer potential for a generalized bi-axially symmetric Helmholtz equation. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 50 (2017), pp. 45-56. http://geodesic.mathdoc.fr/item/VTGU_2017_50_a3/

[1] Miranda C., Partial Differential Equations of Elliptic Type, Translated from the Italian edition by Z. C. Motteler, Ergebnisse der Mathematik und ihrer Grenzgebiete, 2, Second Revised Edition, Springer-Verlag, Berlin–Heidelberg–New York, 1970 | MR

[2] Gunter N. M., Potential Theory and Its Applications to Basic Problems of Mathematical Physics, Translated from the Russian edition by J. R. Schulenberger, Frederick Ungar Publishing Company, New York, 1967 | MR | Zbl

[3] Gilbert R. P., Theoretic Methods in Partial Differential Equations. Mathematics in Science and Engineering, A Series of Monographs and Textbooks, 54, Academic Press, New York–London, 1969, 308 pp. | MR

[4] Hasanov A., “Fundamental solutions of generalized bi-axially symmetric Helmholtz equation”, Complex variables and Elliptic Equations, 52 (2007), 673–683 | DOI | MR | Zbl

[5] Appell P., Kampe de Feriet J., Fonctions Hypergeometriques et Hyperspheriques: Polynomes d'Hermite, Gauthier-Villars, Paris, 1926, 440 pp. | Zbl

[6] Erdelyi A., Magnus W., Oberhettinger F., Tricomi F. G., Higher Transcendental Functions, v. I, McGraw-Hill Book Company, New York–Toronto–London, 1953

[7] Srivastava H. M., Karlsson P. W., Multipl. Gaussian Hypergeometric Series, Halsted Press, Ellis Horwood Limited, Chicherster; John Wiley and Sons, New York–Chichester–Brisbane–Toronto, 1985, 386 pp. | MR

[8] Smirnov M. M., Degenerate elliptic and hyperbolic equations, Nauka, M., 1966

[9] Smirnov M. M., Equations of mixed type, Vysshaya Shkola, M., 1985

[10] Srivastava H. M., Hasanov A., Choi J., “Double-Layer Potentials for a Generalized Bi-Axially Symmetric Helmholtz Equation”, Sohag J. Math., 2:1 (2015), 1–10

[11] Burchnall J. L., Chaundy T. W., “Expansions of Appell's double hypergeometric functions”, Quart. J. Math. Oxford Ser., 11 (1940), 249–270 | DOI | MR