A class of time-varying differential equations for vibration research and application

Duoduo Zhao; Kai Zhou; Fengming Ye; Xin Xu; Duoduo Zhao; Kai Zhou; Fengming Ye; Xin Xu

doi:10.3934/math.20241396

AIMS Mathematics

2024, Volume 9, Issue 10: 28778-28791. doi: 10.3934/math.20241396

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Research article

A class of time-varying differential equations for vibration research and application

1.
School of Big Data and Artificial Intelligence, Center of Applied Mathematics Research, Chizhou University, Chizhou, Anhui 247100, China
2.
School of Mathematics and Information Science, Wenzhou University, Wenzhou, Zhejiang 325035, China

Received: 27 July 2024 Revised: 02 October 2024 Accepted: 06 October 2024 Published: 11 October 2024
MSC : 34C10

As a pivotal branch within the realm of differential equations, the theory of oscillation holds a crucial position in the exploration of natural sciences and the construction of modern control theory frameworks. Despite the extensive research conducted globally, focusing on individual or combined analyses of key elements such as explicit damping terms, positive and negative coefficients, time-varying delays, and nonlinear neutral terms, systematic investigations into the oscillatory behavior of even-order differential equations that concurrently embody these four complex characteristics remain scarce. This paper, by establishing reasonable assumptions, innovatively presents two crucial criteria, aiming to preliminary delve into the oscillation patterns of even-order differential equations under specific complex settings. In the course of the study, a variety of mathematical techniques, such as Riccati transformation, calculus scaling methods, and partial integration, have been utilized by the researchers to perform the necessary derivations and confirmations.
- even-order differential equations,
- oscillation behavior,
- Riccati transformation,
- positive and negative coefficients,
- variable time delay
Citation: Duoduo Zhao, Kai Zhou, Fengming Ye, Xin Xu. A class of time-varying differential equations for vibration research and application[J]. AIMS Mathematics, 2024, 9(10): 28778-28791. doi: 10.3934/math.20241396

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Abstract

As a pivotal branch within the realm of differential equations, the theory of oscillation holds a crucial position in the exploration of natural sciences and the construction of modern control theory frameworks. Despite the extensive research conducted globally, focusing on individual or combined analyses of key elements such as explicit damping terms, positive and negative coefficients, time-varying delays, and nonlinear neutral terms, systematic investigations into the oscillatory behavior of even-order differential equations that concurrently embody these four complex characteristics remain scarce. This paper, by establishing reasonable assumptions, innovatively presents two crucial criteria, aiming to preliminary delve into the oscillation patterns of even-order differential equations under specific complex settings. In the course of the study, a variety of mathematical techniques, such as Riccati transformation, calculus scaling methods, and partial integration, have been utilized by the researchers to perform the necessary derivations and confirmations.

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