We construct a system of solutions for the coefficients of expansion in the system of trigonometric functions for the solution of the Helmholtz equation in a cylindrical coordinate system in the form of homogeneous polynomials in two biorthogonal systems of functions. Some properties of the biorthogonal systems of functions are proved.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
E. L. Ince, Ordinary Differential Equations, Longmans & Green Co., London (1927).
H. Bateman and A. Erdélyi, Higher Transcendental Functions, Vol. 2: Bessel Functions, Parabolic Cylinder Functions, and Orthogonal Polynomials, McGraw-Hill, New York (1953).
V. S. Vladimirov, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1981).
N. N. Vorob’ev, Theory of Series [in Russian], Nauka, Moscow (1979).
V. F. Zheverzheev, L. A. Kal’nitskii, and N. A. Sapogov, Special Course of Higher Mathematics for Technical Colleges [in Russian], Vysshaya Shkola, Moscow (1970).
E. Kamke, Differentialgleichungen: Lösungsmethoden und Lösungen, Vol. I: Gewöhnliche Differentialgleichungen, Teubner, Leipzig (1977); https://doi.org/10.1007/978-3-663-05925-7.
M. A. Lavrent’ev and B. V. Shabat, Methods of the Theory of Functions of Complex Variable [in Russian], Nauka, Moscow (1987).
A. I. Markushevich, Selected Chapters of the Theory of Analytic Functions [in Russian], Nauka, Moscow (1976).
A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, Vol. 1: Elementary Functions [in Russian], Nauka, Moscow (1981); English translation: P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, Vol. 1: Elementary Functions, Gordon & Breach Sci. Publ., New York (1986).
M. A. Sukhorolsky, “Expansion of functions in a system of polynomials biorthogonal on a closed contour with a system of functions regular at infinitely remote point,” Ukr. Mat. Zh., 62, No. 2, 238–254 (2010); English translation: Ukr. Math. J., 62, No. 2, 268–288 (2010); https://doi.org/10.1007/s11253-010-0350-6.
M. A. Sukhorolsky, “Systems of solutions of the Helmholtz equation,” Visn. Nats. Univ. “L’viv. Politekhnika”. Ser. Fiz.-Mat. Nauky, No. 718, 19–34 (2011).
M. A. Sukhorolsky and V. V. Dostoyna, “One class of biorthogonal systems of functions that arise in the solution of the Helmholtz equation in the cylindrical coordinate system,” Mat. Met. Fiz.-Mekh. Polya, 55, No. 2, 52–62 (2012); English translation: J. Math. Sci., 192, No. 5, 541–554 (2013); https://doi.org/10.1007/s10958-013-1415-5.
M. A. Sukhorolsky, V. V. Dostoina, and O. V. Veselovska, “Biorthogonal systems of solutions of the Helmholtz system in the cylindrical coordinate system,” Visn. Nats. Univ. “L’viv. Politekhnika,” Ser. Fiz.-Mat. Nauky, No. 898, 56–68 (2018).
M. A. Sukhorolsky, I. S. Kostenko, and V. V. Dostoina, “Construction of the solutions of partial differential equations in the form of contour integrals,” Vestn. Kherson. Nats. Univ., No. 2(47), 323–326 (2013).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 64, No. 4, pp. 47–54, October–December, 2021.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Veselovska, O.V., Dostoina, V.V. & Drohomyretska, K.T. Construction of Solutions of the Helmholtz Equation in a Cylindrical Coordinate System in the Form of Homogeneous Polynomials in Two Biorthogonal Systems of Functions. J Math Sci 279, 170–180 (2024). https://doi.org/10.1007/s10958-024-07003-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-024-07003-5