A heuristic review on the homotopy perturbation method for non-conservative oscillators

Abstract

The homotopy perturbation method (HPM) was proposed by Ji-Huan. He was a rising star in analytical methods, and all traditional analytical methods had abdicated their crowns. It is straightforward and effective for many nonlinear problems; it deforms a complex problem into a linear system; however, it is still developing quickly. The method is difficult to deal with non-conservative oscillators, though many modifications have appeared. This review article features its last achievement in the nonlinear vibration theory with an emphasis on coupled damping nonlinear oscillators and includes the following categories: (1) Some fallacies in the study of non-conservative issues; (2) non-conservative Duffing oscillator with three expansions; (3)the non-conservative oscillators through the modified homotopy expansion; (4) the HPM for fractional non-conservative oscillators; (5) the homotopy perturbation method for delay non-conservative oscillators; and (6) quasi-exact solution based on He’s frequency formula. Each category is heuristically explained by examples, which can be used as paradigms for other applications. The emphasis of this article is put mainly on Ji-Huan He’s ideas and the present authors’ previous work on the HPM, so the citation might not be exhaustive.

The non‐conservative forced Toda oscillator

“…The calculation for the solution (24) being plotted with the numerical solution of the original Equation (1) as shown in Figures 7-9. These figures illustrate the time history of the solution (24) taking into account the relation (29). Investigating these graphs shows the high agreement of the analytical solution with the numerical solution.…”

“…It was also revealed that the linear damping coefficient plays a dual role in the stability behavior, while the nonlinear counterpart exerted a destabilizing influence on the resonance response. The approximate solution is compared with the numerical solution using the Mathematica software which shows excellent agreements under a specific relation (28) or (29). The effectiveness of the proposed method is also evaluated by the forced Toda model.…”

“…In this case, the classical HPM failed to solve this problem [37]. Accordingly, we expand the solution in the form [26][27][28][29][30] 𝑢(𝑡) = 𝑒 −𝜌𝜑 𝑡 (𝑢 0 (𝑡) + 𝜌𝑢 1 (𝑡) + 𝜌 2 𝑢 2 (𝑡) + ⋯), (13) where 𝑢 𝑖 , 𝑖 = 0, 1, 2, … is an unknown to be determined later and 𝜑 is an unknown representing the full damping rate to be determined. Inserting the expansion (13)…”

“…To solve the high nonlinear Equation ( 37), at the resonance case, the enhanced perturbed solution may be utilized in the form [24,[26][27][28][29]39]…”

“…However, the traditional perturbation techniques become invalid for non-conservative oscillators [24,25]. The suggestion, of a modified HPM that is valid for non-conservative oscillators, is focused on in this work [26][27][28][29].…”