A Novel Analytical Approximate Solution for Strongly Nonlinear Third-Order Jerk Equations Using a Modified Iteration Method

Nonlinear jerk equations, characterized by a third-order time derivative, play a crucial role in modeling various physical phenomena across disciplines like mechanics, circuits, and biology. Accurately solving these equations is essential for understanding and predicting the behavior of such systems. However, obtaining analytical solutions for nonlinear jerk equations can be challenging, necessitating the development of robust and accurate approximation methods. This work explores, for the first time, the application of the modified iteration approach to solve third-order jerk equations. By comparing the obtained approximate solutions with both exact and existing analytical solutions for established engineering problems, we demonstrate the superior accuracy and rapid convergence of the proposed method. The significantly reduced error percentages highlight the effectiveness of the modified iteration approach in providing precise solutions for nonlinear jerk equations, paving the way for its application in a wide range of oscillation problems within nonlinear sciences and engineering.