Novel solutions and stability of quaternion differential equations: symmetry and asymmetry analyses using matrix representation and Lyapunov methods

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The article demonstrates the novel solutions available for several QDEs and offers incredible promise for the investigation of quaternion differential equations (QDEs). This result is likely due to the matrix representation strategy and the use of the Picard–Lindelöf theorem, which is an important consequence of traditional ordinary differential equations (ODE) formulations. Using this numerical calculation aid research, the authors rightly point out that there is a feasible QDE solution. Furthermore, this article goes beyond the direct proof of its existence and digs into the analysis of the stability of the solution, which was obtained. In particular, the authors use the appropriate Lyapunov equations to determine the asymptotic stability of the QDE. This study adds depth to the understanding of QDE solutions and provides fundamental insights into their absorbed dynamics. Furthermore, this particular manuscript examines the symmetry and asymmetry aspects of QDEs, possibly investigating how these properties manifest themselves in solving implicit situations Through examples, architects describe methods that through the behaviours exhibited by QDEs, revealing insights into the elegant mathematical architecture inherent in these situations. By Homotopy Pertubation method and Li–He modified Homotopy Perturbation method, the numerical solutions are provided. In general, this article fully contributes to the conceptual framework associated with QDEs, providing new insights into their unique existence, stability, and symmetry properties