The solution structure of the Düffing oscillator’s transient response and general solution

Zou, Keguan; Nagarajaiah, Satish

doi:10.1007/s11071-015-2015-3

Cited by 10 publications

(2 citation statements)

References 5 publications

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“…To be specific, when solving nonlinear oscillator problems with the HAM, it is straightforward to choose time-related trigonometric functions as base functions to express the steady-state solution [23,24] or exponential and trigonometric functions for transient state [25]. But various single-frequency or multifrequency auxiliary linear operators with or without damping terms can all meet the basic requirements for constructing a homotopy [23][24][25][26][27][28], and there are no universally applicable rules on how to choose the effective ones. Currently, a proper choice on L requires not only sufficient mathematical knowledge, but also extensive experience in applying the HAM, especially for problems with strong and hybrid nonlinearities.…”

Section:   Aq and  mentioning

confidence: 99%

“…However, when implementing abovementioned homotopy methods, there can be infinitely possible auxiliary linear operators that satisfy the three basic rules for a given nonlinear equation and a certain set of base functions, but yet there are no mathematical theorems which can guide us to choose a proper linear operator L that can guarantee a convergent and efficient homotopy. To be specific, when solving nonlinear oscillator problems with the HAM, it is straightforward to choose time-related trigonometric functions as base functions to express the steady-state solution [23,24] or exponential and trigonometric functions for transient state [25]. But various single-frequency or multifrequency auxiliary linear operators with or without damping terms can all meet the basic requirements for constructing a homotopy [23][24][25][26][27][28], and there are no universally applicable rules on how to choose the effective ones.…”

Section:   Aq and  mentioning

confidence: 99%

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Residue Regulating Homotopy Method for Strongly Nonlinear Oscillator

Song

Zhang

Shao

2022

Preprint

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It is highly desired yet challenging to obtain analytical approximate solutions to strongly nonlinear oscillators accurately and efficiently. Here we propose a new approach, which combines the homtopy concept with a “residue-regulating” technique to construct a continuous homotopy from an initial guess solution to a high-accuracy analytical approximation of the nonlinear problems, namely the residue regulating homotopy method (RRHM). In our method, the analytical expression of each order homotopy-series solution is associated with a set of base functions which are pre-selected or generated during the previous order of approximations, while the corresponding coefficients are solved from deformation equations specified by the nonlinear equation itself and auxiliary residue functions. The convergence region, rate and final accuracy of the homotopy are controlled by a residue-regulating vector and an expansion threshold. General procedures of implementing RRHM are demonstrated using the Duffing and Van der Pol-Duffing oscillators, where approximate solutions containing abundant frequency components are successfully obtained, yielding significantly better convergence rate and performance stability compared to the other conventional homotopy-based methods.

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