Analytical method to study a mathematical model of wave processes under two­point time conditions

Authors

DOI:

https://doi.org/10.15587/1729-4061.2019.155148

Keywords:

oscillatory systems, mathematical models of wave processes, differential-symbol method, two-point problem, wave equation

Abstract

Research and analysis of dynamic processes in oscillatory systems are closely connected to the establishment of exact or approximate analytical solutions to the problems of mathematical physics, which model such systems. The mathematical models of wave propagation in oscillatory systems under certain initial conditions at a fixed time are well known in the literature. However, wave processes in lengthy structures subject to an external force only and at the assigned states of the process at two points in time have been insufficiently studied. Such processes are modeled by a two-point time problem for the inhomogeneous wave equation in an unbounded domain t>0, x∈ℝs. The model takes into consideration the assignment of a linear combination with unknown amplitude of oscillations and the rate of its change at two points in time. A two-point problem, generally speaking, is the ill-posed boundary value problem, since the respective homogeneous problem has non-trivial solutions. A class of quasi-polynomials has been established as the class of the existence of a single solution to the problem. This class does not contain the non-trivial elements from the problem's kernel, which ensures the uniqueness of solution to the problem. We have proposed a precise method to build the solution in the specified class. The essence of the method is that the problem's solution is represented as the action of a differential expression, whose symbol is the right-hand side of the equation, on some function of parameters. The function is constructed in a special way using the equation and two-point conditions, and has special features associated with zeroes of the denominator – the characteristic determinant of the problem.

The method is illustrated by the description of oscillatory processes within an infinite string and a membrane.

The main practical application of the constructed method is the possibility to adequately mathematically model the oscillatory systems, which takes into consideration a possibility to control the system's parameters. Such a control over parameters makes it possible to perform optimal synthesis and design of parameters for the relevant technical systems in order to analyze and account for special features in the dynamic modes of oscillations

Author Biographies

Zinovii Nytrebych, Lviv Polytechnic National University Bandery str., 12, Lviv, Ukraine, 79013

Doctor of Physical and Mathematical Sciences, Professor, Head of Department

Department of Mathematics

Volodymyr Ilkiv, Lviv Polytechnic National University Bandery str., 12, Lviv, Ukraine, 79013

Lviv Polytechnic National University

Bandery str., 12, Lviv, Ukraine, 79013

Petro Pukach, Lviv Polytechnic National University Bandery str., 12, Lviv, Ukraine, 79013

Doctor of Technical Sciences, Professor, Head of Department

Department of Computational Mathematics and Programming

Oksana Malanchuk, Danylo Galytsky Lviv National Medical University Pekarska str., 69, Lvіv, Ukraine, 79010

PhD

Department of Biophysics

Ihor Kohut, Lviv Polytechnic National University Bandery str., 12, Lviv, Ukraine, 79013

PhD, Associate Professor

Department of Computational Mathematics and Programming

Andriy Senyk, Hetman Petro Sahaidachnyi National Army Academy Heroiv Maidanu str., 32, Lviv, Ukraine, 79012

PhD, Associate Professor

Department of Engineering Mechanics (Weapons and Equipment of Military Engineering Forces)

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Published

2019-02-27

How to Cite

Nytrebych, Z., Ilkiv, V., Pukach, P., Malanchuk, O., Kohut, I., & Senyk, A. (2019). Analytical method to study a mathematical model of wave processes under two­point time conditions. Eastern-European Journal of Enterprise Technologies, 1(7 (97), 74–83. https://doi.org/10.15587/1729-4061.2019.155148

Issue

Vol. 1 No. 7 (97) (2019): Applied mechanics

Section

Applied mechanics

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Copyright (c) 2019 Zinovii Nytrebych, Volodymyr Ilkiv, Petro Pukach, Oksana Malanchuk, Ihor Kohut, Andriy Senyk

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