Oscillatory processes under impulsive perturbations
Abstract
The problem of oscillatory processes under impulsive perturbations at fixed moments in time is considered. The mathematical model consists of a classical partial differential equation describing oscillatory processes, supplemented by boundary and initial conditions, along with additional conditions characterizing the impulsive effects at the specified time instances.
Using the method of separation of variables, the solution is constructed in the form of a Fourier series involving the eigenfunctions of Laplace operator. Conditions under which the resulting solution is classical are discussed. A special case is examined in which the domain is a rectangle, and in this case, an explicit analytic solution is obtained.
The results obtained in this paper may be especially useful in the analysis of mathematical models for various applied problems involving short-term effects, as well as in the further development of the theory of impulsive differential equations, including those involving partial derivatives.
References
R.P. Agarwal, S. Hristova, D. O’Regan, Non-instantaneous impulses in differential equations, Springer International Publishing, Switzerland, 2017. https://doi.org/10.1007/978-3-319-66384-5
M.U. Akhmet, Li-Yorke chaos in the system with impacts, J. Math. Anal. Appl., 351 (2009), no.2, 804–810. https://doi.org/10.1016/j.jmaa.2008.11.015
M.U. Akhmet, Principles of discontinuous dynamical systems, Springer-Verlag, New York, 2010. https://doi.org/10.1007/978-1-4419-6581-3
D. Bainov, P. Simeonov, impulsive differential equations: periodic solutions and applications, Longman Scientific & Technical Group UK Limited, New York, 1993.
M. Benchohra, J. Henderson, S. Ntouyas, Impulsive differential equations and inclusions, contemporary mathematics and its applications, V.2, Hindawi Publ. Corp., New York, 2006.
A. Boichuk, M. Langerove, J. Skorikova, Solutions of linear impulsive differential systems bounded on the entire real axis, Adv. Differ. Equ., (2010), 494379. https://doi.org/10.1155/2010/494379
J.W. Brown, R.V. Churchille, Fourier series and boundary value problems, 8-th ed., McGraw Hill, 2011.
S.M. Chuiko, O.V. Nesmelova, K.S. Shevtsova, Differential-algebraic boundary-value problems with impulsive action, J. Math. Sciences, 273 (2023), 316–331. https://doi.org/10.1007/s10958-023-06500-3
K.K. Elgondiyev, O.O. Kurbanbaev, S.R. Matmuratova, String oscillations with impulsive effects, Karakalpak Scientific J., 3 (2020), no.1, 46–52.
A. Halanay, D. Wexler, Qualitative theory of systems with impulses, Editura Acad. Rep. Soc. Romania, Bucuresti, 1968. (in Romanian)
V.M. Kirilich, A.D. Myshkis, M.V. Prokhorenko, Oscillations of a diaphragm under the action of pulse forces, Ukr. Math. J., 61 (2009), 1357–1363. https://doi.org/10.1007/s11253-010-0281-2
V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of impulsive differential equations, World Scientific, Singapore, 1989.
A.D. Myshkis, A.M. Samoilenko, Systems with shocks at prescribed instants of time, Mathematics of the USSR-Sbornik, 3 (1967), no.2, 187–193.
N.A. Perestyuk, V.A. Plotnikov, A.M. Samoilenko, N.V. Skripnik, Differential equations with impulse effects, Multivalued Right-Hand Sides with Discontinuities, de Gruyter Stud. Math., 40, Walter De Gruyter GmbH & Co, Berlin–Boston, 2011.
A.V. Plotnikov, T.A. Komleva, N.V. Skripnik, Existence of basic solutions of first order linearhomogeneous set-valued differential equations, Mat. Stud., 61 (2024), no.1, 61–78. https://doi.org/10.3097010.30970/ms.61.1.61-78
A.M. Samoilenko, N.A. Perestyuk, Impulsive differential equations, World Scientific, Singapore, 1995.
A.M. Samoilenko, Y.I. Kaplun, V.H. Samoilenko, Singularly perturbed equations with impulse action, Ukr. Math. J., 54 (2002), 1309–1323. https://doi.org/10.1023/A:1023483507636
A.M. Samoilenko, V.G. Samoilenko, V.V. Sobchuk, On periodic solutions of the equation of a nonlinear oscillator with pulse influence, Ukr. Math. J., 51 (1999), 926–933. https://doi.org/10.1007/BF02591979
V.G. Samoilenko, K.K. Elgondyev, On periodic solutions of linear differential equations with pulsed influence, Ukr. Math. J., 49 (1997), 156–164. https://doi.org/10.1007/BF02486623
V.H. Samoilenko, K.K. Yelgondyev, On existence of periodical solutions for differential equations with impulsive effects, Facta Universitatis, Series: Mechanics, automatic control and robotics, 2 (1998), no.2, 635–639.
A.N. Sharkovsky, S.F. Kolyada, A.G. Sivak, V.V. Fedorenko, Dynamics of one-dimensional maps, Springer Dordrecht, 2010. https://doi.org/10.1007/978-94-015-8897-3
N.V. Skripnik, Averaging method for impulsive differential inclusions with fuzzy right-handside, Mat. Stud., 55 (2021), no.1, 76–84. https://doi.org/10.30970/ms.55.1.76-84
A. Tolstonogov, Differential inclusions in a Banach space, Kluwer Academic Publishers, Dordrecht, 2000. https://doi.org/10.1007/978-94-015-9490-5
P. Zamora-Leon et al, Impulsive mathematical model and its practical application in a cytotoxicity assay, J Phys.: Conf. Ser., 3117 (2025), 012003. https://doi.org/10.1088/1742-6596/3117/1/012003
Copyright (c) 2026 K.K. Elgondiev
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Matematychni Studii is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) license.
