A THREE-VARIABLE ANALOGUE OF THE HUMBERT FUNCTION WITH APPLICATIONS TO SOLVING DIRICHLET PROBLEM FOR A SINGULAR HELMHOLTZ EQUATION
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confluent hypergeometric function of three variables; system of partial differential equations; asymptotic formula; three-dimensional Helmholtz equation with three singular coefficients; Dirichlet problem in the first infinite octantAnnotatsiya
In this paper we introduce a new confluent hypergeometric function of three variables, list its elementary properties, and construct a system of partial differential equations that the new function satisfies. We study the behavior of the function and establish an asymptotic formula for a large values of arguments. The results obtained are applied to solving the Dirichlet problem for the three-dimensional Helmholtz equation with three singular coefficients in the first infinite octant. The uniqueness of the solution of the Dirichlet problem in an infinite domain is proved using the extremum principle for elliptic equations. Thanks to the established properties of the confluent hypergeometric function of three variables, the unique solution to the posed boundary value problem is written out in explicit form.
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[1] L.Bers, Mathematical aspects of subsonic and transonic gas dynamics.(AM-206). New York, Dover Publications Inc, 1958.
[2] G. Lohofer, "Theory of an electromagnetically deviated metal sphere. 1: Abcorbed power" , SIAM Journal Applied Mathematics, 49(1989) 567–581.
[3] A.W. Niukkanen, "Generalized hypergeometric series arising in physical and quantum chemical applications" , J.Phys.A:Math.Gen., 16(9) (1983) 1813–1825.
[4] J. Horn, "Über die Convergenz der hypergeometrischen Reihen zweier und dreier Veränderlichen" , Math. Ann., 34(4) (1889) 544–600.
[5] G. Lauricella, "Sulle funzioni ipergeometriche a piu variabili" , Rend. Circ. Mat. Palermo., 7 (1893) 111–158.
[6] S. Saran, "Hypergeometric functions of three variables" , Ganita., 5 (1954) 77–91, Corrigendum. ibid., 7(1956) 65.
[7] G.K. Dhawan, "Hypergeometric functions in three variables" , Proc. Nat. Acad. Sci. India Sect. A., 40 (1970) 43-48.
[8] M.S. Samar, "Some definite integrals" , Vijnana Parishad Anusandhan Patrika., 16 (1973) 7–11.
[9] H. Exton, "Hypergeometric functions of three variables" , J. Indian Acad. Math., 4 (1982) 113–119.
[10] H.M. Srivastava, P.W. Karlsson, Multiple Gaussian Hypergeometric Series. (AM-428). New York, Chichester, Brisbane and Toronto: Halsted Press, 1985.
[11] R.N. Jain, "The confluent hypergeometric functions of three variables" , Proc. Nat.Acad.Sci.India. Sect. A., 36 (1966) 395-408.
[12] H. Exton, "On certain confluent hypergeometric of three variables", Ganita., 21(2) (1970) 79–92.
[13] Z.O. Arzikulov, T.G. Ergashev, "Some systems of PDE associated with the multiple confluent hypergeometric functions and their applications", Lobachevskii Journal of Mathematics., 45(2) (2024) 591 – 603.
[14] A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, (AM-302). New York; Toronto; London: McGraw-Hill Book Company, 1953.
[15] P. Appell. "Sur les séries hypergéométriques de deux variables, et sur des équations différentielles linéaires aux dérivées partielles", C.R. Acad. Sci. Paris., 90 (1880) 296–298.
[16] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional integrals and derivatives. Theory and applications., (AM-976). Amsterdam: Gordon and Breach Science Publishers, 1993.
[17] O.I. Marichev, Handbook of integral transforms of higher transcendental functions, theory and algorithmic tables., (AM-336). Chichester, New York: Ellis Horwood Ltd, 1982.
[18] O.A. Repin, M.E. Lerner, "On the Dirichlet problem for the generalized bioxially symmetric Helmholtz equation in the first quadrant" , Vestnik Samarsk. Gos. Tekh. Universiteta, Ser. fiz.-matem. nauki., 6 (1998) 5–8.
[19] A. Hasanov, T.G. Ergashev, "On potential theory for the generalized bi-axially symmetric elliptic equation in the plane", KazNU Bulletin Mathematics Mechanics Computer Science Series., 1 (2021) 3–24.
[20] N.J. Komilova, "Dirichlet problem for multidimensional elliptic equation with two singular coefficients", Uzbek Mathematical Journal., 65(1) (2021) 98 – 109.
[21] Z.R. Tulakova, "Spatial mixed problems and Neumann problem for the three-dimensional elliptic equation with the two singular coefficients" Uzbek Mathematical Journal., 68(3) (2024) 150 –157.
[22] Z.O. Arzikulov, A. Hasanov, T.G. Ergashev, "Confluent hypergeometric functions and their application to the solution of Dirichlet problem for the Helmholtz equation with three singular coefficients" , Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 29(3) (2025) 407–429.
[23] C. Miranda, Partial Differential Equations of Elliptic Type, Second Revised Edition., (AM-256). Translated from the Italian edition by Z. C. Motteler, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 2, Berlin, Heidelberg and New York: Springer-Verlag, 1970.
[24] I.S. Gradshteyn, I.M. Ryzhik, Table of integrals, series and products, 7 th ed., (AM-1172). Amsterdam, Boston, Heidelberg, London, New York, Oxford, Paris, San Diego, San Fransisco, Singapore, Sydney, Tokyo: Academic Press, 2007.
[25] T.G. Ergashev. "Fundamental solutions of the generalized Helmholtz equation with several singular coefficients and confluent hypergeometric functions of many variables" , Lobachevskii Journal of Mathematics., 41(1) (2020) 15–26.
[26] A.R. Ryskan, Z.O. Arzikulov, T.G. Ergashev, "Particular solutions of multidimensional generalized Euler-Poisson-Darboux equations of various (elliptic or hyperbolic) types" , KazNU Bulletin Mathematics Mechanics Computer Science Series., 1(121) (2024) 76–88.
[27] M.B. Kapilevich, "On an equation of mixed elliptic-hyperbolic type" , Matematicheskiy sbornik., 30(72)(1) (1952) 11–38.
[28] A. Hasanov, M. Ruzhansky, "Hypergeometric expansions of solutions of the degenerating model parabolic equations of the third order" , Lobachevskii Journal of Mathematics., 41(1) (2020) 27–31.
[29] N.J. Komilova, A. Hasanov, T.G. Ergashev, "Expansions of Kampé de Fériet hypergeometric functions", KazNU Bulletin Mathematics Mechanics Computer Science Series., 2(122) (2024) 48–63.
[30] A.S. Berdyshev, A.R.Ryskan. The Neumann and Dirichlet problems for one four-dimensional degenerate elliptic equation. Lobachevskii Journal of Mathematics, 41(6) (2020) 1051–1066. DOI: 10.1134/S1995080220060062
