On Truly Nonlinear Oscillator Equations of Ermakov-Pinney Type

In this paper we present a general class of differential equations of Ermakov-Pinney type which may serve as truly nonlinear oscillators. We show the existence of periodic solutions by exact integration after the phase plane analysis. The related quadratic Lienard type equations are examined to show for the first time that the Jacobi elliptic functions may be solution of second-order autonomous non-polynomial differential equations.

Solution of fractional oscillator equations using ultraspherical wavelets

Fractional oscillator type equations are well-known model equations to describe several phenomenon in mathematical physics, engineering and biology. In this paper, a new method incorporated by the ultraspherical wavelet operational matrix of general order integration and block-pulse functions are adopted to investigate the solution of fractional oscillator type equations. To facilitate this, the ultraspherical wavelets are first presented and the corresponding operational matrix of fractional-order integration is derived by virtue of block pulse functions. The properties of ultraspherical wavelets and block pulse functions are used to transform the underlying problem to a system of algebraic equations which can be easily solved by any of the usual numerical methods. The efficiency and accuracy of the proposed method is demonstrated by presenting several benchmark test problems. Moreover, special attention is given to the comparison of the numerical results obtained by the new algorithm with those found by other known methods.

Limit Cycles of Nonlinear Oscillator Equations with Absolute Value by Means of the Homotopy Analysis Method

An analytic approach based on the homotopy analysis method is proposed to obtain the limit cycles of highly nonlinear oscillating equations with absolute value terms. The non-smoothness of the absolute value terms is handled by means of an iteration approach with Fourier expansion. Two typical examples are employed to illustrate the validity and flexibility of this approach. It has general meanings and thus can be used to solve many other highly nonlinear oscillating systems with this kind of non-smoothness.