A Volterra integral formulation for determining the periodic solutions of some autonomous, nonlinear, third-order ordinary differential equations

A Volterra integral formulation based on the introduction of a term proportional to the velocity times the square of the (unknown) frequency of oscillation, a new independent variable equal to the original one times the (unknown) frequency of oscillation, the method of variation of parameters and series expansions of both the solution and the frequency of oscillation, is used to determine the periodic solutions to three nonlinear, autonomous, third-order, ordinary differential equations. It is shown that the first term of the series expansion of the frequency of oscillation coincides with that determined from a first-order harmonic balance procedure, whereas the two-term approximation to the frequency of oscillation is shown to be more accurate than that of a parameter perturbation procedure for the second equation considered in this paper. For the third equation, it is shown that the two-term approximation presented in this paper is more accurate than the corresponding one for one of the parameter perturbation methods, and for initial velocities less than one, for the other parameter perturbation approach.     

Approximate methods based on order reduction for the periodic solutions of nonlinear third-order ordinary differential equations

Four approximate methods based on order reduction, the introduction of a book-keeping parameter and power series expansions for the solution and the frequency of oscillation are used to analyze three autonomous, nonlinear, third-order ordinary differential equations which have analytical periodic solutions. The first method introduces the velocity in both sides of the equation if this (linear) term is not present. The second one is based on the first one but employs a new independent variable, whereas, in the third and fourth techniques, a term equal to the velocity times the square of the unknown frequency of oscillation is introduced in both sides of the equation. The third method uses the original independent variable, whereas the fourth one employs a new independent variable which depends linearly on the unknown frequency of oscillation. It is shown that the second method provides accurate solutions only for initial velocities close to unity, whereas the third one is found to yield very accurate results for the first and second equations considered here and only for large initial velocities for the third one. The fourth technique provides as accurate results as or more accurate results than parameter-perturbation techniques which deal with the third-order equations directly and are based on the expansion of certain constants that appear in the differential equations in terms of a book-keeping parameter.     

Solution for an anti-symmetric quadratic nonlinear oscillator by a modified He’s homotopy perturbation method

In this paper He’s homotopy perturbation method has been adapted to calculate higher-order approximate periodic solutions for a nonlinear oscillator with discontinuity for which the elastic force term is an anti-symmetric and quadratic term. We find that He’s homotopy perturbation method works very well for the whole range of initial amplitudes, and the excellent agreement of the approximate frequencies and periodic solutions with the exact ones has been demonstrated and discussed. Just one iteration leads to high accuracy of the solutions with a maximal relative error for the approximate period of less than 0.73% for all values of oscillation amplitude, while this relative error is as low as 0.040% when the second iteration is considered. Comparison of the result obtained using this method with those obtained by the harmonic balance method reveals that the former is very effective and convenient.     

Predicting nonlinear dynamic response of internal cantilever beam system on a steadily rotating ring

This paper is focused on nonlinear dynamic response of internal cantilever beam system on a steadily rotating ring via a nonlinear dynamic model. The analytical approximate solutions to the oscillation motion are obtained by combining Newton linearization with Galerkin's method. Numerical solutions could be obtained by using the shooting method on the exact governing equation. Compared with numerical solutions, the approximate analytical solutions here show excellent accuracy and rapid convergence. Two different kinds of oscillating internal cantilever beam system on a steadily rotating ring are investigated by using the analytical approximate solutions. These include symmetric vibration through three equilibrium points, and asymmetric vibration through the only trivial equilibrium point. The effects of geometric and physical parameters on dynamic response are useful and can be easily applied to design practical engineering structures. In particular, the ring angular velocity plays a significant role on the period and periodic solution of the beam oscillation. In conclusion, the analytical approximate solutions presented here are sufficiently precise for a wide range of oscillation amplitudes.     

Analytical approximate solutions to oscillation of a current-carrying wire in a magnetic field

This paper is concerned with analytical approximate solutions to dynamic oscillation of a current-carrying wire in a magnetic field generated by a fixed current-carrying conductor parallel to the wire. The wire is restrained to a fixed wall by linear elastic springs. The periodic oscillation solutions are obtained by generalizing the Newton-harmonic balance method. The procedure yields rapid convergence with respect to the “exact” solution obtained by numerical integration. In general, the results are valid for small as well as large oscillation amplitude. The method presented in this paper can be applied to other strongly nonlinear oscillators with more general restoring forces of rational form.     

Analytical approximate solutions for the generalized nonlinear oscillator

He's energy balance method (HEBM) is employed in this article to obtain the analytical approximate solution of the generalized nonlinear oscillator. Existence of periodic solutions is analytically verified and consequently the relationship between the natural frequency and the initial amplitude is obtained in an analytical form. A number of numerical simulations are carried out and accuracy of the HEBM is then examined within an error analysis. The exact values of the natural frequency numerically obtained via the elliptic integrals are taken into account as the references bases and the relative error is then evaluated for a range of oscillation amplitudes. Excellent correlation of the approximate frequencies with the exact ones demonstrates that the approximate solutions are quite consistent even for large amplitudes of oscillation.     

Analytical approximations for a conservative nonlinear singular oscillator in plasma physics

A modified variational approach and the coupled homotopy perturbation method with variational formulation are exerted to obtain periodic solutions of a conservative nonlinear singular oscillator in plasma physics. The frequency–amplitude relations for the oscillator which the restoring force is inversely proportional to the dependent variable are achieved analytically. The approximate frequency obtained using the coupled method is more accurate than the modified variational approach and ones obtained using other approximate methods and the discrepancy between the approximate frequency using this coupled method and the exact one is lower than 0.31% for the whole range of values of oscillation amplitude. The coupled method provides a very good accuracy and is a promising technique to a lot of practical engineering and physical problems.     

Global residue harmonic balance method to periodic solutions of a class of strongly nonlinear oscillators

A new approach, namely the global residue harmonic balance method, was advanced to determine the accurate analytical approximate periodic solution of a class of strongly nonlinear oscillators. A class of nonlinear jerk equation containing velocity-cubed and velocity times displacements-squared was taken as a typical example. Unlike other harmonic balance methods, all the former residual errors are introduced in the present approximation to improve the accuracy. Comparison of the result obtained using this approach with the exact one and simplicity and efficiency of the proposed procedure. The method can be easily extended to other strongly nonlinear oscillators.     

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.     

Bifurcating periodic solution for a class of first-order nonlinear delay differential equation after Hopf bifurcation

In this paper, we predict the accurate bifurcating periodic solution for a general class of first-order nonlinear delay differential equation with reflectional symmetry by constructing an approximate technique, named residue harmonic balance. This technique combines the features of the homotopy concept with harmonic balance which leads to easy computation and gives accurate prediction on the periodic solution to the desired accuracy. The zeroth-order solution using just one Fourier term is applied by solving a set of nonlinear algebraic equations containing the delay term. The unbalanced residues due to Fourier truncation are considered iteratively by solving linear equations to improve the accuracy and increase the number of Fourier terms of the solutions successively. It is shown that the solutions are valid for a wide range of variation of the parameters by two examples. The second-order approximations of the periodic solutions are found to be in excellent agreement with those obtained by direct numerical integration. Moreover, the residue harmonic balance method works not only in determining the amplitude but also the frequency of the bifurcating periodic solution. The method can be easily extended to other delay differential equations.     

Periodic wave solutions of coupled integrable dispersionless equations by residue harmonic balance

We introduce the residue harmonic balance method to generate periodic solutions for nonlinear evolution equations. A PDE is firstly transformed into an associated ODE by a wave transformation. The higher-order approximations to the angular frequency and periodic solution of the ODE are obtained analytically. To improve the accuracy of approximate solutions, the unbalanced residues appearing in harmonic balance procedure are iteratively considered by introducing an order parameter to keep track of the various orders of approximations and by solving linear equations. Finally, the periodic solutions of PDEs result. The proposed method has the advantage that the periodic solutions are represented by Fourier functions rather than the sophisticated implicit functions as appearing in most methods.     

Analytical approximations to primary resonance response of harmonically forced oscillators with strongly general nonlinearity

This paper presents an innovative analytical approximate method for constructing the primary resonance response of harmonically forced oscillators with strongly general nonlinearity. A linearization process is introduced prior to harmonic balancing (HB) of the nonlinear system and a linear system is derived by which the accurate analytical approximation procedure is easily and innovatively implemented. The main cutting edge of the proposed method is that complicated and coupled nonlinear algebraic equations obtained by the classical HB method is avoided. With only one iteration, very accurate analytical approximate primary resonance response can be determined, even for significantly nonlinear systems. Another advantage is the direct determination of the maximum oscillation amplitude. This is due to the appropriate form chosen for the approximation with no extra processing required. It is concluded that the result of an initial approximate solution from the two-term (constant plus the first harmonic term) harmonic balance is not reliable especially for strongly nonlinear systems and a correction to the initial approximation is necessary. The proposed method can be applied to general oscillators with mixed nonlinearities, such as the Helmholtz-Duffing oscillator. Two examples are presented to illustrate the applicability and effectiveness of the proposed technique.     

An interior layer in the thermal power-law blown film model

Film blowing is a highly complex industrial process used to manufacture thin sheets of polymer. Models that describe this process are highly nonlinear and numerical instabilities often occur when solving the highly nonlinear differential equations. This paper investigates the structure of typical solutions that arise when the polymer is assumed to be described by a power-law fluid operating under nonisothermal conditions. We consider both a shear-thinning and shear-thickening polymer and use a balance of orders argument to identify the structure of a region of rapid expansion in the radial profile of the film. A mixture of heuristic and singular perturbation techniques is applied to obtain a closed form approximate expression for the radial profile of the film which displays the interior layer phenomenon. We demonstrate how approximate solutions to the highly nonlinear two-point boundary value problem describing this process may be constructed using this expression as an initial estimate in an iterative scheme. Numerical solutions for the radial temperature, velocity and thickness profiles of the film are subsequently obtained by iteration.     

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.     

Nonlinear behavior of a galloping cable

This paper presents the numerical experiment of a discretized loaded cable with periodic forcing. There appeared to be various type of nonlinear oscillations over a wide range of frequencies and amplitudes for the periodic forcing term. The same forcing term can give rise to large or small oscillation by solving initial value problem and observing the solutions after a long time.     

Chaos control for numerical instability of first order reliability method

The HL-RF algorithm of the first order reliability method (FORM) is a kind of popular iterative algorithm for solving the reliability index in structural reliability analysis and reliability-based design optimization. However, there are the phenomena of convergence failure such as periodic oscillation, bifurcation and chaos in the FORM for some nonlinear problems. This paper suggests a novel method to overcome the numerical instabilities of HL-RF algorithm of FORM based on the principle of chaos control. The essential causes of chaotic dynamics for numerical instabilities including periodic oscillation and chaos of iterative solutions of FORM are revealed. Moreover, the geometrical properties of periodic oscillation of the iterative formulas derived from the FORM and performance measure approach are analyzed and compared. Finally, the stability transformation method (STM) of chaos feedback control is proposed to implement the convergence control of FORM. Several numerical examples with explicit or implicit HL-RF iterative formulas illustrate that the STM is effective, simple and versatile, and can control the periodic oscillation, bifurcation and chaos of the FORM iterative algorithm.     

Asymptotic Behaviour and Global Existence of Solutions for Some Classes of Nonlinear Parabolic Equations

The paper deals with the asymptotic behaviour and global existence of solutions for some classes of nonlinear parabolic equations in regard to the monotone properties of the nonlinear term. The asymptotic behaviour of the solutions of initial-boundary value problem for nonlinear parabolic equations is studied via the method of differential inequalities in order to obtain oscillation criterion for the solutions. Existence of extremal solutions of semilinear elliptic and parabolic equations is investigated via monotone iterative methods. The extremal solutions are obtained via monotone iterates.     

一维3×3非线性系统脉冲波的干扰

利用非线性几何光学的方法,讨论一维情形下3×3半线性偏微分方程具有两个初始脉冲波的干扰问题.证明方程组近似解的存在性,得到脉冲波的干扰出现在一阶项中.通过对近似解进行误差分析,得到精确解的一个好的近似.     

Analytical analysis for large-amplitude oscillation of a rotational pendulum system

This paper deals with large amplitude oscillation of a nonlinear pendulum attached to a rotating structure. By coupling of the well-known Maclaurin series expansion and orthogonal Chebyshev polynomials, the governing differential equation with sinusoidal nonlinearity can be reduced to form a cubic-quintic Duffing equation. The resulting Duffing type temporal problem is solved by an analytic iteration approach. Two approximate formulas for the frequency (period) and the periodic solution are established for small as well as large amplitudes of motion. Illustrative examples are selected and compared to those analytical and exact solutions to substantiate the accuracy and correctness of the approximate analytical approach.