Critical analysis for nonlinear oscillations by least square HPM

Abstract

In this study, a novel adapted homotopy perturbation method (HPM) is used to treat the nonlinear phenomena of free vibration in a system with one degree of freedom. This adaptation involves the integration of HPM with a least-squares optimizer, resulting in a hybrid method called the least square homotopy perturbation method (LSHPM). The LSHPM is tested on various nonlinear problems documented in the existing literature. To evaluate the effectiveness of the proposed approach, the identified problems are also tackled using HPM and the MATLAB built-in function bvp5c, and then the results are compared with those obtained using LSHPM. In addition, a comparative analysis is carried out with the results of the AG method as found in the literature. The results show that LSHPM is a reliable and efficient method suitable for solving more complicated initial value problems in the fields of science and engineering.

Numerical simulation of fractional-order Duffing system with extended Mittag-Leffler derivatives

“…On the other hand, oscillatory systems, which refer to dynamic systems that exhibit periodic motion around an equilibrium point, are found everywhere in the natural world and have appeared in various physical, electrical, mechanical, and biological models [10]. The Duffing equation, a well-known nonlinear differential equation, has been applied in many physical, engineering and even biological problems [11][12][13].…”