Approximate Symmetries and Conservation Laws for Mechanical Systems Described by Mixed Derivative Perturbed PDEs | Journal of Engineering Sciences

Approximate Symmetries and Conservation Laws for Mechanical Systems Described by Mixed Derivative Perturbed PDEs

Author(s): Shamaoon A.1, Agarwal P.2,3, Cesarano C.4, Jain S.5*

Affiliation(s):
1 Northumbria University, Ellison Pl., Newcastle upon Tyne, NE1 8ST Newcastle, United Kingdom;
2 Anand International College of Engineering, Agra Rd., 303012 Jaipur, India;
3 Nonlinear Dynamic Research Centre, Ajman University, UAE;
4 Università Telematica Internazionale UniNettuno, 39, Corso Vittorio Emanuele II St., 00186 Rome, Italy;
5 Poornima College of Engineering, Rajasthan, 302022 Jaipur, India

*Corresponding Author’s Address: [email protected]

Issue: Volume 10, Issue 2 (2023)

Dates:
Submitted: July 14, 2023
Received in revised form: October 6, 2023
Accepted for publication: October 17, 2023
Available online: November 1, 2023

Citation:
Shamaoon A., Agarwal P., Cesarano C., Jain S. (2023). Approximate symmetries and conservation laws for mechanical systems described by mixed derivative perturbed PDEs. Journal of Engineering Sciences (Ukraine), Vol. 10(2), pp. E8–E15. DOI: 10.21272/jes.2023.10(2).e2

DOI: 10.21272/jes.2023.10(2).e2

Research Area:  MECHANICAL ENGINEERING: Computational Mechanics

Abstract. This article focuses on developing and applying approximation techniques to derive conservation laws for the Timoshenko–Prescott mixed derivatives perturbed partial differential equations (PDEs). Central to our approach is employing approximate Noether-type symmetry operators linked to a conventional Lagrangian one. Within this framework, this paper highlights the creation of approximately conserved vectors for PDEs with mixed derivatives. A crucial observation is that the integration of these vectors resulted in the emergence of additional terms. These terms hinder the establishment of the conservation law, indicating a potential flaw in the initial approach. In response to this challenge, we embarked on the rectification process. By integrating these additional terms into our model, we could modify the conserved vectors, deriving new modified conserved vectors. Remarkably, these modified vectors successfully satisfy the conservation law. Our findings not only shed light on the intricate dynamics of fourth-order mechanical systems but also pave the way for refined analytical approaches to address similar challenges in PDE-driven systems.

Keywords: beams oscillations, traveling-wave reduction, conserved vectors, Noether approach.

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