Exploring novel solutions for the generalized q-deformed wave equation

Abstract

Our primary goal is to address the q-deformed wave equation, which serves as a mathematical framework for characterizing physical systems with symmetries that have been violated. By incorporating a q-deformation parameter, this equation expands upon the traditional wave equation, introducing non-commutativity and non linearity to the dynamics of the system. In our investigation, we explore three distinct approaches for solving the generalized q-deformed wave equation: the reduced q-differential transform method (RqDTM) [17], the separation method (SM), and the variational iteration method (VIM). The RqDTM is a modified version of the differential transform method specially designed to handle q-deformed equations. The SM aims to identify solutions that can be expressed as separable variables, while the VIM employs an iterative scheme to refine the solution. We conduct a comparative analysis of the accuracy and efficiency of the solutions obtained through these methods and present numerical results. This comparative analysis enables us to evaluate the strengths and weaknesses of each approach in effectively solving the q-deformed wave equation, providing valuable insights into their applicability and performance. Additionally, this paper introduces a generalization of the q-deformed wave equation, as previously proposed in [13], and investigates its solution using two different analytical methods: RqDTM, SM, and an approximation method is known as VIM.