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

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.