Analyzing nonlinear oscillations with He's frequency-amplitude method and numerical comparison in jet engine vibration system.

In this study, we describe and successfully solve a jet engine vibration equation using a straightforward tool known as the He's frequency-amplitude Method (HFAM). The jet engine vibration system demonstrates diverse applications across aerospace, power generation, industrial machinery, transportation, marine propulsion, energy optimization, defense, and aviation training. Utilizing HFAM, we derive periodic solutions in a general form for this system, considering various cases dependent on damping and driving forces. The obtained results highlight the effectiveness of HFAM as a distinct and straightforward technique for nonlinear equations. By comparing the solutions with numerical results obtained using the fourth-order Runge-Kutta method, we demonstrate the excellent accuracy of our solutions.

Exact soliton solutions and the significance of time-dependent coefficients in the Boussinesq equation: theory and application in mathematical physics.

This article effectively establishes the exact soliton solutions for the Boussinesq model, characterized by time-dependent coefficients, employing the advanced modified simple equation, generalized Kudryashov and modified sine-Gordon expansion methods. The adaptive applicability of the Boussinesq system  to coastal dynamics, fluid behavior, and wave propagation enriches interdisciplinary research across hydrodynamics and oceanography. The solutions of the system obtained through these significant techniques make a path to understanding nonlinear phenomena in various fields, surpassing traditional barriers and further motivating research and application. Significant impacts of the coefficients of the equation, wave velocity, and related parameters are evident in the profiles of soliton-shaped waves in both 3D and 2D configurations when all these factors are treated as variables, which are not seen in the case for constant coefficients. This study enhances the understanding of the significant role played by nonlinear evolution equations with time-dependent coefficients through careful dynamic explanations and detailed analyses. This revelation opens up an interesting and challenging field of study, with promising insights that resonate across diverse scientific disciplines.