Newton–harmonic balancing approach for accurate solutions to nonlinear cubic–quintic Duffing oscillators

This paper presents a new approach for solving accurate approximate analytical higher-order solutions for strong nonlinear Duffing oscillators with cubic–quintic nonlinear restoring force. The system is conservative and with odd nonlinearity. The new approach couples Newton’s method with harmonic balancing. Unlike the classical harmonic balance method, accurate analytical approximate solutions are possible because linearization of the governing differential equation by Newton’s method is conducted prior to harmonic balancing. The approach yields simple linear algebraic equations instead of nonlinear algebraic equations without analytical solution. Using the approach, accurate higher-order approximate analytical expressions for period and periodic solution are established. These approximate solutions are valid for small as well as large amplitudes of oscillation. In addition, it is not restricted to the presence of a small parameter such as in the classical perturbation method. Illustrative examples are presented to verify accuracy and explicitness of the approximate solutions. The effect of strong quintic nonlinearity on accuracy as compared to cubic nonlinearity is also discussed.     

联合Duffing方程和Van der Pol方程的非线性分数阶微分方程

本文研究了Adomian分解方法在非线性分数阶微分方程求解中的应用. 利用Riemann-Liouville分数阶导数和Adomian分解方法, 将Duffing方程和Van der Pol方程联合在一个分数阶方程中,并获得了此方程的解析近似解.     

New method based on the HPM and RKHSM for solving forced Duffing equations with integral boundary conditions

This paper investigates the forced Duffing equation with integral boundary conditions. Its approximate solution is developed by combining the homotopy perturbation method (HPM) and the reproducing kernel Hilbert space method (RKHSM). HPM is based on the use of the traditional perturbation method and the homotopy technique. The HPM can reduce nonlinear problems to some linear problems and generate a rapid convergent series solution in most cases. RKHSM is also an analytical technique, which can solve powerfully linear boundary value problems. Therefore, the forced Duffing equation with integral boundary conditions can be solved using advantages of these two methods. Two numerical examples are presented to illustrate the strength of the method.     

An averaging method applied to a duffing equation

We approximate a Duffing equation by an averaged system. We solve the system explicitely and draw bifurcation diagrams in dependence of the forcing term. We discuss the goodness of the averaging method, relative to the behavior of the solutions in dependence of involved parameters, by comparing with results obtained in [6,10] for the original Duffing equation.     

Approximate partial Noether operators and first integrals for cubically coupled nonlinear Duffing oscillators subject to a periodically driven force

The approximate first integrals for a system of two cubically coupled nonlinear Duffing oscillators subject to a periodically driven force are constructed with the help of approximate partial Noether approach. Firstly, approximate partial Noether operators associated with a partial Lagrangian are derived. Then approximate first integrals are obtained for both the resonant and non-resonant cases.     

Iterative homotopy harmonic balancing approach for conservative oscillator with strong odd-nonlinearity

A new approach that the iterative homotopy harmonic balancing is presented for charactering and predicting analytical approximations of conservative oscillator with strong odd-nonlinearity in the paper. The new technique does not depend upon small parameter assumption and incorporates the salient features of both methods of the parameter-expansion and the harmonic balance. The results of applying this procedure to the studied examples that the cubic–quintic Duffing equation and Duffing-harmonic equation show the high accurate, simplicity and efficiency of the approach. The approach can be easily extended to other nonlinear oscillators.     

一类非线性波动方程的显式精确解

本文用直接方法和假设的一种结合求出了一类较广泛的非线性波动方程ｕｔｔ－ａ１ｕｘｘ＋ａ２ｕｔ＋ａ３ｕ＋ａ４ｕＳ＾２＋ａ５ｕ＾３＝０的一些显式精确行波解,贱个有重要的非线性数学物理方程,如φ＾４方程,Ｋｌｅｉｎ－Ｇｏｒｄｏｎ方程,Ｓｉｎｅ－Ｇｏｒｄｏｎ方程,及Ｓｉｎｈ－Ｇｏｒｄｏｎ方程的近似,Ｌａｎｄａｕ－Ｇｉｎｚｂｕｒｇ－Ｈｉｇｇｓ方程,Ｄｕｆｆｉｎｇ方程,非线性电报方程等都可作为该方程的特殊情形得     

Approximation of solutions for an initial and terminal value problem for the forced Duffing equation with non-viscous damping

An algorithm to find explicit approximate solutions of an initial and terminal value problem for the forced Duffing equation with non-viscous damping is accomplished via a generalized quasilinearization method. In fact, we obtain a monotone sequence of approximate solutions converging uniformly and quadratically to a unique solution of the problem.     

Study on asymptotic analytical solutions using HAM for strongly nonlinear vibrations of a restrained cantilever beam with an intermediate lumped mass

Presented herein is to establish the asymptotic analytical solutions for the fifth-order Duffing type temporal problem having strongly inertial and static nonlinearities. Such a problem corresponds to the strongly nonlinear vibrations of an elastically restrained beam with a lumped mass. Taking into consideration of the inextensibility condition and using an assumed single mode Lagrangian method, the single-degree-of-freedom ordinary differential equation can be derived from the governing equations of the beam model. Various parameters of the nonlinear unimodal temporal equation stand for different vibration modes of inextensible cantilever beam. By imposing the homotopy analysis method (HAM), we establish the asymptotic analytical approximations for solving the fifth-order nonlinear unimodal temporal problem. Within this research framework, both the frequencies and periodic solutions of the nonlinear unimodal temporal equation can be explicitly and analytically formulated. For verification, numerical comparisons are conducted between the results obtained by the homotopy analysis and numerical integration methods. Illustrative examples are selected to demonstrate the accuracy and correctness of this approach. Besides, the optimal HAM approach is introduced to accelerate the convergence of solutions.     

Existence and approximation of solutions of forced duffing type integro-differential equations with three-point nonlinear boundary conditions

A generalized quasilinearization technique is developed to obtain an analytic approximation of the solutions of the forced Duffing type integro-differential equation with nonlinear three-point boundary conditions. Monotone sequences of approximate solutions converging uniformly and quadratically to a unique solution of the problem are presented.     

Existence of approximate solutions of the forced Duffing equation with discontinuous type integral boundary conditions

We develop a generalized quasilinearization technique to obtain a sequence of approximate solutions converging monotonically and quadratically to the unique solution of the forced Duffing equation with discontinuous type integral boundary conditions.     

Exact Solutions and Integrability of the Duffing – Van der Pol Equation

The force-free Duffing–Van der Pol oscillator is considered. The truncated expansions for finding the solutions are used to look for exact solutions of this nonlinear ordinary differential equation. Conditions on parameter values of the equation are found to have the linearization of the Duffing–Van der Pol equation. The Painlevé test for this equation is used to study the integrability of the model. Exact solutions of this differential equation are found. In the special case the approach is simplified to demonstrate that some well-known methods can be used for finding exact solutions of nonlinear differential equations. The first integral of the Duffing–Van der Pol equation is found and the general solution of the equation is given in the special case for parameters of the equation. We also demonstrate the efficiency of the method for finding the first integral and the general solution for one of nonlinear second-order ordinary differential equations.     

Static and dynamic stability for geometrically nonlinear governing equations of elastic thin shallow shells

In this study, the governing equations for large deflection of elastic thin shallow shells are deduced into an algebraic cubic equation to determine the unknown coefficient of the assumed deflection by applying Galerkin's method in combination with the algebraic polynomial and Fourier series. For the dynamic problem, the coefficient is replaced by an unknown function of time; after the same process is applied, the governing equations are deduced to be a nonlinear ODE of order two called the Duffing equation, and its analytical solution is known. The combination of the algebraic polynomial and Fourier series gives very rapid convergence in the asymptotic solutions.     

Synchronization of a periodically forced Duffing oscillator with a periodically excited pendulum

In this paper, the analytical conditions for a periodically forced Duffing oscillator synchronized with a chaotic pendulum are developed through the theory of discontinuous dynamical systems. From the analytical conditions, the synchronization invariant domains are developed. For a better understanding of synchronization of two different dynamical systems, the partial and full synchronizations of the Duffing oscillator with the chaotic pendulum are presented for illustrations. The control parameter map is developed from the analytical conditions. Under special parameters, the two systems can be fully and partially synchronized. Since the forced pendulum has librational and rotational chaotic motions, the periodically forced Duffing oscillator can be synchronized only with the librational chaotic motions of the pendulum. It is impossible for the forced Duffing oscillator to be synchronized with the rotational chaotic motions.     

The comparison of two reliable methods for accurate solution of time‐fractional Kaup–Kupershmidt equation arising in capillary gravity waves

In this paper, we compared two different methods, one numerical technique, viz Legendre multiwavelet method, and the other analytical technique, viz optimal homotopy asymptotic method (OHAM), for solving fractional‐order Kaup–Kupershmidt (KK) equation. Two‐dimensional Legendre multiwavelet expansion together with operational matrices of fractional integration and derivative of wavelet functions is used to compute the numerical solution of nonlinear time‐fractional KK equation. The approximate solutions of time fractional Kaup–Kupershmidt equation thus obtained by Legendre multiwavelet method are compared with the exact solutions as well as with OHAM. The present numerical scheme is quite simple, effective, and expedient for obtaining numerical solution of fractional KK equation in comparison to analytical approach of OHAM. Copyright © 2016 John Wiley & Sons, Ltd.     

Lagrange stability for impulsive Duffing equations

This work discusses the boundedness of solutions for impulsive Duffing equation with time-dependent polynomial potentials. By KAM theorem, we prove that all solutions of the Duffing equation with low regularity in time undergoing suitable impulses are bounded for all time and that there are many (positive Lebesgue measure) quasi-periodic solutions clustering at infinity. This result extends some well-known results on Duffing equations to impulsive Duffing equations.     

Nonlinear dynamics and solitons in the presence of rapidly varying periodic perturbations

Problems of nonlinear dynamics and soliton propagation in the presence of rapidly varying periodic perturbations are considered applying a rigorous analytical approach based on asymptotic expansions. The method we develop allows derivation of an effective nonlinear equation for the slowly varying field component in any order of the asymptotic procedure as expansions in the parameter ω⁻¹, ω being the frequency of the rapidly varying (direct or parametric) driving force. The general approach is demonstrated on several examples of different physical nature, including chaos suppression in the parametrically driven Duffing oscillator, dynamics of the sine-Gordon kinks in the presence of rapidly varying direct or parametric driving force, propagation of envelope (nonlinear Schrödinger) solitons in optical fibres with periodic amplification, stability of solitons on rapidly varying spatial periodic potential, and so on.     

Primary resonance of Duffing oscillator with fractional-order derivative

In this paper the primary resonance of Duffing oscillator with fractional-order derivative is researched by the averaging method. At first the approximately analytical solution and the amplitude-frequency equation are obtained. Additionally, the effect of the fractional-order derivative on the system dynamics is analyzed, and it is found that the fractional-order derivative could affect not only the viscous damping, but also the linear stiffness, which is characterized by the equivalent damping coefficient and the equivalent stiffness coefficient. This conclusion is remarkably different from the existing research results about nonlinear system with fractional-order derivative. Moreover, the comparisons of the amplitude-frequency curves by the approximately analytical solution and the numerical integration are fulfilled, and the results certify the correctness and satisfactory precision of the approximately analytical solution. At last, the effects of the two parameters of the fractional-order derivative, i.e. the fractional coefficient and the fractional order, on the amplitude-frequency curves are investigated, which are different from the traditional integer-order Duffing oscillator.     

Accurate analytical solutions to nonlinear oscillators by means of the Hamiltonian approach

The purpose of this paper is to apply the Hamiltonian approach to nonlinear oscillators. The Hamiltonian approach is applied to derive highly accurate analytical expressions for periodic solutions or for approximate formulas of frequency. A conservative oscillator always admits a Hamiltonian invariant, H , which stays unchanged during oscillation. This property is used to obtain approximate frequency–amplitude relationship of a nonlinear oscillator with high accuracy. A trial solution is selected with unknown parameters. Next, the Ritz–He method is used to obtain the unknown parameters. This will yield the approximate analytical solution of the nonlinear ordinary differential equations. In contrast with the traditional methods, the proposed method does not require any small parameter in the equation. Copyright © 2011 John Wiley & Sons, Ltd.