Approximate Analytical Solutions of Nonlinear Jerk Equations Using the Parameter Expansion Method

The Parameter Expansion Method (PEM) is employed to study nonlinear Jerk equations, which are often difficult to solve because of their strong nonlinearity. This method provides higher accuracy and broader applicability, enabling analytical insights and closed-form approximations. This study explores the use of Prof. He’s PEM to derive approximate analytical solutions of the nonlinear third-order Jerk equation, this model is commonly encountered in the analysis of complex dynamical systems across physics and engineering. Owing to the strong nonlinearity inherent in Jerk equations, exact solutions are often unattainable. The PEM provides a simple, effective framework by expanding the solution with respect to an embedding parameter, allowing accurate approximations without the need of small parameters or linearization. The method’s reliability and precision are validated through comparisons with numerical simulations, demonstrating its practicality and robustness in tackling nonlinear problems. The results indicate that PEM provides highly accurate approximations of nonlinear Jerk equation, showcasing greater simplicity and efficiency relative to other analytical methods, along with excellent concordance with numerical simulations. Additionally, the nonlinear Jerk equation demonstrates exact approximate solutions via PEM, closely mirroring numerical results and surpassing several contemporary analytical techniques in efficiency and usability. Furthermore, the study indicates that PEM is a straightforward and effective approach in solving nonlinear Jerk equation. It generates accurate estimates that nearly align with numerical simulations and surpass numerous other analytical methods.