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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Functional difference equations in the problem on the forced oscillations of a fluid in an infinite pool with conical bottom

Author: M. A. Lyalinov
Translated by: the author
Original publication: Algebra i Analiz, tom 29 (2017), nomer 2.
Journal: St. Petersburg Math. J. 29 (2018), 267-287
MSC (2010): Primary 35Q35
Published electronically: March 12, 2018
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Abstract: The model problem under study concerns the stationary forced oscillations of a fluid of small amplitude under the action of the field of gravity in an infinite pool with sources located on its conical bottom with infiltration. A classical solution of that problem is studied in the linear approximation. By the use of the Mellin transform and expansion in spherical functions, the problem is reduced to a set of systems of functional difference equations with meromorphic coefficients that are combinations of associated Legendre functions and their derivatives. Then, the problem on systems of difference equations reduces to singular integral equations. For this, in particular, solutions of some auxiliary first order functional equations with meromorphic coefficients are computed. It is shown that the system of integral equations in question is Fredholm with index zero. Within some assumptions, the classical solution of the problem exists and is unique. Some estimates of the classical solution in the vicinity of the conic point and at infinity are obtained.

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Additional Information

M. A. Lyalinov
Affiliation: St. Petersburg State University, Univesitetskaya nab. 7/9, 199034 St. Petersburg, Russia

Keywords: Forced oscillations of a liquid, functional equations, Fredholm integral equations, conic domain
Received by editor(s): October 10, 2016
Published electronically: March 12, 2018
Additional Notes: Supported in part by RFBR (grant no. 17-01-00668a)
Dedicated: Dedicated to the memory of V. S. Buslaev
Article copyright: © Copyright 2018 American Mathematical Society

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