Letters to the Editor

OSCILLATIONS IN AN x^{(2m+2)/(2n+1)} POTENTIAL

In this study, Perturbation–Iteration Algorithm, namely PIA, is applied to solve some types of fractional differential equations (FDEs) for the first time. To illustrate the efficiency of the method, numerical solutions are compared with the results published in the literature by considering some FDEs. The results confirm that the PIA is a powerful, efficient and accurate method for solving nonlinear fractional differential equations.

A family of conservative, truly nonlinear, oscillators with integer or non-integer order nonlinearity is considered. These oscillators have only one odd power-form elastic-term and exact expressions for their period and solution were found in terms of Gamma functions and a cosine-Ateb function, respectively. Only for a few values of the order of nonlinearity, is it possible to obtain the periodic solution in terms of more common functions. However, for this family of conservative truly nonlinear oscillators we show in this paper that it is possible to obtain the Fourier series expansion of the exact solution, even though this exact solution is unknown. The coefficients of the Fourier series expansion of the exact solution are obtained as an integral expression in which a regularized incomplete Beta function appears. These coefficients are a function of the order of nonlinearity only and are computed numerically. One application of this technique is to compare the amplitudes for the different harmonics of the solution obtained using approximate methods with the exact ones computed numerically as shown in this paper. As an example, the approximate amplitudes obtained via a modified Ritz method are compared with the exact ones computed numerically.

Generalization to the previous oscillators is done by introducing the nonlinear elastic and damping forces. The mathematical model of the system is a second order differential equation with nonlinear elastic and damping terms whose order is integer and/or noninteger. Cveticanin’s solving procedure is extended for solving such a strong nonlinear differential equation. The approximate solution obtained is a function of initial amplitude and initial phase. A damping coefficient and order of damping interaction with an elastic coefficient and order of elasticity for the generalized oscillators are also determined. Special attention is paid to obtain the relation between initial amplitude and phase, on the one hand, and initial displacement and velocity on the other hand. Correction to the frequency of vibration for the linear oscillators with nonlinear damping and for the pure nonlinear oscillators with linear damping is obtained and analyzed. Analytical results given in this paper are compared with numerically obtained ones and show a good agreement.

In this paper a new analytical method for solving the differential equations which describe the motion of the oscillator with fraction order elastic force is introduced. Using the first integral of motion the exact period of vibration in the form of the Euler beta function is obtained. Based on that value the approximate solution of the strong nonlinear fraction order differential equation is formed. The suggested procedure is applied for various fraction values, but also for the pure quadratic, cubic and quintic oscillators. The advantage of the suggested method is that is valid for all fraction values $\alpha ⩾1$, the accuracy of the approximate solution is very high as the period of vibration is exactly analytically determined and is independent on the time.

In the paper the fraction order differential equation with small linear and nonlinear terms is also considered. The Krylov–Bogolubov method is extended due to the fact that frequency of vibration depends on the amplitude of vibration. The approximate solution is with time variable amplitude, phase and frequency. The linear damped fraction order oscillator is widely discussed. The approximate solution of the differential equation which describes the vibrations of such oscillator is compared with exact numerical one and shows a good agreement.

In this paper the properties of the oscillatory motion of the system with non-polynomial damping is investigated. The two limits for damping are the dry friction and linear viscous damping. The mathematical model of the system is a strong nonlinear differential equation with fraction order velocity terms. Using the modified version of He's homotopy perturbation method the approximate analytic solution is obtained. The generating solution is assumed in the form which corresponds to the system with linear viscous damping. Two special cases are considered: first, when the coefficient of the damping force is small and second, when the damping force is close to dry friction. The obtained analytical solutions are compared with numerical ones. They show good agreement.

An artificial parameter-Linstedt–Poincaré method based on the introduction of a linear stiffness term, a change of independent variable and the expansion of both the solution and the frequency of oscillation in power series of the artificial parameter, and the theory of generalized functions are used to determine the frequency of limit cycles of non-smooth oscillators. The method is applied to three non-smooth oscillators, and it is shown that the first-order approximation coincides with those obtained by means of harmonic balance and parameter-expansion techniques, whereas higher-order ones are given in terms of convergent series.