ISSN 2079-6900 (Print) 
ISSN 2587-7496 (Online)

Middle Volga Mathematical Society Journal

DOI 10.15507/2079-6900.25.202301.542-553

Original article

ISSN 2079-6900 (Print)

ISSN 2587-7496 (Online)

MSC2020 45K05

Exact Solutions of One Nonlinear Countable-Dimensional System of Integro-Differential Equations

A. E. Rassadin

Higher School of Economics (Nizhny Novgorod, Russian Federation)

Abstract. In the present paper, a nonlinear countable-dimensional system of integrodifferential equations is investigated, whose vector of unknowns is a countable set of functions of two variables. These variables are interpreted as spatial coordinate and time. The nonlinearity of this system is constructed from two simultaneous convolutions: first convolution is in the sense of functional analysis and the second one is in the sense of linear space of double-sided sequences. The initial condition for this system is a doublesided sequence of functions of one variable defined on the entire real axis. The system itself can be written as a single abstract equation in the linear space of double-sided sequences. As the system may be resolved with respect to the time derivative, it may be presented as a dynamical system. The solution of this abstract equation can be interpreted as an approximation of the solution of a nonlinear integro-differential equation, whose unknown function depends not only on time, but also on two spatial variables. General representation for exact solution of system under study is obtained in the paper. Also two kinds of particular examples of exact solutions are presented. The first demonstrates oscillatory spatio-temporal behavior, and the second one shows monotone in time behavior. In the paper typical graphs of the first components of these solutions are plotted. Moreover, it is demonstrated that using some procedure one can generate countable set of new exact system’s solutions from previously found solutions. From radio engineering point of view this procedure just coincides with procedure of upsampling in digital signal processing.

Key Words: Cauchy problem, generating function, Laurent expansion, Fourier transform, Bessel functions of the first kind, modified Bessel functions

For citation: A. E. Rassadin. Exact Solutions of One Nonlinear Countable-Dimensional System of Integro-Differential Equations. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 25:1(2023), 542–553. DOI:

Submitted: 10.11.2022; Revised: 19.12.2022; Accepted: 25.02.2023

Information about the author:

Alexander E. Rassadin, Post-graduate student, National Research University — Higher School of Economics in Nizhny Novgorod (25/12, Bolshaya Pecherskaya Str., Nizhny Novgorod, 603155, Russia), ORCID:,

The author have read and approved the final manuscript.

Conflict of interest: The author declare no conflict of interest.

Creative Commons Attribution 4.0 International License This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International License.