Email this article Confluent hypergeometric functions and their application to the solution of Dirichlet problem for the Helmholtz equation with three singular coefficients
- Authors: Arzikulov Z.O.1, Hasanov A.2,3, Ergashev T.G.2,4,3
-
Affiliations:
- Fergana State Technical University
- V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Science
- Ghent University
- Tashkent Institute of Irrigation and Agricultural Mechanization Engineers
- Issue: Vol 29, No 3 (2025)
- Pages: 407-429
- Section: Differential Equations and Mathematical Physics
- URL: https://bakhtiniada.ru/1991-8615/article/view/349680
- DOI: https://doi.org/10.14498/vsgtu2156
- EDN: https://elibrary.ru/YWKYZB
- ID: 349680
Cite item
Full Text
Abstract
In the course of a series of studies spanning the fifty-year period from 1889 to 1939, all double hypergeometric series of the second order were systematically investigated. A significant contribution to the study of hypergeometric functions of two variables was made by Horn, who proposed their classification into two types: complete and confluent. Horn’s final list comprised fourteen complete (non-confluent) functions of two variables and twenty distinct confluent functions, which represent limiting cases of the complete ones. In 1985, Srivastava and Karlsson completed the classification of all possible second-order complete hypergeometric functions of three variables, while a similar systematic classification for their confluent counterparts remains incomplete. Thus, the theory of confluent hypergeometric functions of three variables has not yet been fully developed, and the study of functions of four variables represents an area for future research.
This paper investigates certain confluent hypergeometric functions of three and four variables, establishing their new properties and applying them to the solution of the Dirichlet problem for the three-dimensional Helmholtz equation with three singular coefficients.
Fundamental solutions of the aforementioned Helmholtz equation are expressed in terms of a confluent hypergeometric function of four variables, while an explicit solution to the Dirichlet problem in the first octant is constructed using a function of three variables, which is derived as a trace of the four-variable confluent function. A theorem on the computation of limiting values of multivariate functions is proved, and transformation formulas for these functions are established. These results are employed to determine the singularity order of fundamental solutions and to validate the correctness of the solution to the Dirichlet problem.
The uniqueness of the solution to the Dirichlet problem is proved using the maximum principle for elliptic equations.
Full Text
##article.viewOnOriginalSite##About the authors
Zafarjon O. Arzikulov
Fergana State Technical University
Email: zafarbekarzikulov1984@gmail.com
ORCID iD: 0009-0004-2965-4566
https://www.mathnet.ru/rus/person214007
PhD; Senior Lecturer; Dept. of Higher Mathematics
Uzbekistan, 150107, Fergana, Fergana st., 86Anvardjan Hasanov
V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Science; Ghent University
Email: anvarhasanov@yahoo.com
ORCID iD: 0000-0002-9849-4103
https://www.mathnet.ru/eng/person41932
Dr. Phys. & Math. Sci., Professor; Chief Research Fellow, Dept. of Differential Equations and Their Applications; Research Associate, Dept. of Mathematics, Analysis, Logic and Discrete Mathematics
Uzbekistan, 100174, Tashkent, Universitetskaya st., 9; Belgium, 9000, Ghent, Sint-Pietersnieuwstraat, 33Tuhtasin G. Ergashev
V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Science; Tashkent Institute of Irrigation and Agricultural Mechanization Engineers; Ghent University
Author for correspondence.
Email: ergashev.tukhtasin@gmail.com
ORCID iD: 0000-0003-3542-8309
ResearcherId: ABG-9381-2020
https://www.mathnet.ru/rus/person37309
Dr. Phys. & Math. Sci., Professor; Research Associate, Dept. of Differential Equations and Their Applications; Professor, Dept. of Higher Mathematics; Research Associate, Dept. of Mathematics, Analysis, Logic and Discrete Mathematics
Uzbekistan, 100174, Tashkent, Universitetskaya st., 9; 100000, Tashkent, Kari-Niyazi st., 39; Belgium, 9000, Ghent, Sint-Pietersnieuwstraat, 33References
- Vekua I. N. New Methods for Solving Elliptic Equations, North-Holland Series in Applied Mathematics and Mechanics, vol. 1. New York, John Wiley & Sons, 1967, xii+358 pp.
- Vekua I. N. On one expansion of metaharmonic functions, Dokl. Akad. Nauk SSSR, 1945, vol. 48, no. 1, pp. 3–6 (In Russian).
- Kapilevich M. B. On an equation of mixed elliptic-hyperbolic type, Sb. Math., 1952, vol. 72, no. 1, pp. 11–38 (In Russian).
- Frankl F. I. Izbrannye trudy po gazovoi dinamike [Selected Works in Gas Dynamics]. Moscow, Nauka, 1973, 712 pp. (In Russian)
- Pulkin S. P. Some boundary value problems for the equation $u_{xx}+u_{yy}+px^{-1}u_x=0$, Uch. Zap. Kuibysh. Ped. In-ta. Fiz.-mat. Nauki, 1958, vol. 21, pp. 3–54 (In Russian).
- Amanov D. Some boundary value problems for a degenerate elliptic equation in an unbounded domain, Izv. Akad. Nauk UzSSR, Ser. Fiz.-Mat. Nauk, 1984, no. 1, pp. 8–13 (In Russian).
- Lerner M. E., Repin O. A. Nonlocal boundary value problems in a vertical half-strip for a generalized axisymmetric Helmholtz equation, Differ. Equ., 2001, vol. 37, no. 11, pp. 1640–1642. EDN: LGMMOT. DOI: https://doi.org/10.1023/A:1017985319783.
- Moiseev E. I. Solvability of a nonlocal boundary value problem, Differ. Equ., 2001, vol. 37, no. 11, pp. 1643–1646. EDN: LGWFPF. DOI: https://doi.org/10.1023/A:1017937403853.
- Pleshchinsky N. B., Tumakov D. N. Boundary value problems for the Helmholtz equation in a quadrant and in a half-plane formed from two quadrants, Russian Math. (Iz. VUZ), 2004, vol. 48, no. 7, pp. 60–71.
- Radzhabov N. R. Uniqueness theorems and analogues of Poisson’s formula in the first octant for an equation of Helmholtz type with $n$ singular hyperplanes, Sov. Math., Dokl, 1978, vol. 19, no. 4, pp. 111–115.
- Salakhitdinov M. C., Hasanov A. The Tricomi problem for an equation of mixed type with a nonsmooth line of degeneracy, Differ. Uravn., 1983, vol. 19, no. 1, pp. 110–119 (In Russian).
- Gilbert R. P. Function Theoretic Methods in Partial Differential Equations. New York, London, Academic Press, 1969, xviii+311 pp.
- Gakhov F. D. Boundary Value Problems. New York, Dover Publ., 1990, xvii+561 pp.
- Hasanov A. Fundamental solutions bi-axially symmetric Helmholtz equation, Complex Var. Elliptic Equ., 2007, vol. 52, no. 8, pp. 673–683. DOI: https://doi.org/10.1080/17476930701300375.
- Samko S. G., Kilbas A. A., Marichev O. I. Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya [Integrals and Derivatives of Fractional Order and Some of Their Applications]. Minsk, Nauka i Tekhnika, 1987, 688 pp. (In Russian)
- Repin O. A., Lerner M. E. On the Dirichlet problem for the generalized biaxisymmetric Helmholtz equation in the first quadrant, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 1998, no. 6, pp. 5–8 (In Russian). EDN: HKVCIB. DOI: https://doi.org/10.14498/vsgtu1.
- Abashkin A. A. On a weighted boundary-value problem in an infinite half-strip for a bi-axisymmetric Helmholtz equation, Russian Math. (Iz. VUZ), 2013, vol. 57, no. 6, pp. 1–9. EDN: RFHWBR. DOI: https://doi.org/10.3103/S1066369X13060017.
- Urinov A. K., Karimov E. T. On fundamental solutions for 3D singular elliptic equations with a parameter, Appl. Math. Lett., 2011, vol. 24, no. 3, pp. 314–319. DOI: https://doi.org/10.1016/j.aml.2010.10.013.
- Hasanov A. Hypergeometric Functions and Their Applications to Solving Boundary Value Problems for Degenerate Second-Order Differential Equations, D.Sc. (Physics and Mathematics) Thesis. Tashkent, 2009, 240 pp. (In Russian)
- Karimov E. T. On the Dirichlet problem for a three-dimensional elliptic equation with singular coefficients, Dokl. Akad Nauk Uzbekistana, 2010, vol. 2, pp. 9–11 (In Russian).
- Ergashev T. G. Potentials for a three-dimensional elliptic equation with one singular coefficient and their application, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2021, vol. 25, no. 2, pp. 257–285 (In Russian). EDN: HVACIC. DOI: https://doi.org/10.14498/vsgtu1810.
- Niukkanen A. W. Generalised hypergeometric series ${}^NF(x_1,dots,x_N)$ arising in physical and quantum chemical applications, J. Phys. A: Math. Gen., 1983, vol. 16, no. 9, pp. 1813–1825. DOI: https://doi.org/10.1088/0305-4470/16/9/007.
- Bers L. Mathematical Aspects of Subsonic and Transonic Gas Dynamics, Surveys in Applied Mathematics, vol. 3. New York, John Wiley & Sons, 1958, xv+278 pp.
- Appell P. Sur les séries hypergéométriques de deux variables et sur dés équations différentielles linéaires aux dérivés partielles, C. R. Acad. Sci., Paris, 1880, vol. 90, pp. 296–299 (In French).
- Lauricella G. Sulle funzioni ipergeometriche a più variabili, Palermo Rend., 1893, vol. 7, pp. 111–158 (In Italian).
- Ergashev T. G., Tulakova Z. R. The Dirichlet problem for an elliptic equation with several singular coefficients in an infinite domain, Russian Math. (Iz. VUZ), 2021, vol. 65, no. 7, pp. 71–80. EDN: WJAPIG. DOI: https://doi.org/10.3103/S1066369X21070082.
- Ergashev T. G. Generalized Holmgren problem for an elliptic equation with several singular coefficients, Differ. Equ., 2020, vol. 56, no. 7, pp. 842–856. DOI: https://doi.org/10.1134/S0012266120070046.
- Ergashev T. G., Tulakova Z. R. A problem with mixed boundary conditions for a singular elliptic equation in an infinite domain, Russian Math. (Iz. VUZ), 2022, vol. 66, no. 7, pp. 51–63. EDN: GYKGFC. DOI: https://doi.org/10.3103/S1066369X22070039.
- Ergashev T. G. Fundamental solutions of the generalized Helmholtz equation with several singular coefficients and confluent hypergeometric functions of many variables, Lobachevskii J. Math., 2020, vol. 41, no. 1, pp. 15–26. EDN: HRRMPZ. DOI: https://doi.org/10.1134/S1995080220010047.
- Volkodavov V. F., Bisrtova O. K. Construction of Riemann–Hadamard functions for a degenerate equation, Differ. Uravn., 1991, vol. 27, no. 8, pp. 1444–1446 (In Russian).
- Erdélyi A., Magnus W., Oberhettinger F., Tricomi F. G. Higher Transcendental Functions, vol. 1. New York, McGraw-Hill, 1953, xxvi+302 pp.
- Miranda C. Partial Differential Equations of Elliptic Type. Berlin, Springer, 1970, xii+372 pp. DOI: https://doi.org/10.1007/978-3-642-87773-5.
- Gradshtein I. S., Ryzhik I. M. Tablitsy integralov, summ, ryadov i proizvedenii [Tables of Integrals, Sums, Series, and Products]. Moscow, Fizmatlit, 1962, 1100 pp. (In Russian)
- Prudnikov A. P., Brychkov Yu. A., Marichev O. I. Integraly i ryady. Dopolnitel’nye glavy [Integrals and Series: Additional Chapters]. Moscow, Nauka, 1986, 800 pp. (In Russian)
- Marichev O. I. Metod vychisleniya integralov ot spetsial’nykh funktsii (teoriya i tablitsy formul) [Methods for Computing Integrals of Special Functions (Theory and Formula Tables)]. Minsk, Nauka i zhizn’, 1978, 312 pp. (In Russian)
- Ergashev T. G., Tulakova Z. R. The Neumann problem for a multidimensional elliptic equation with several singular coefficients in an infinite domain, Lobachevskii J. Math., 2022, vol. 43, no. 1, pp. 199–206. EDN: SNHKZE. DOI: https://doi.org/10.1134/S1995080222040102.
Supplementary files
