Effect of Fractional Derivative Properties on the Periodic Solution of the Nonlinear Oscillations

Abstract

A periodic solution of the time-fractional nonlinear oscillator is derived based on the Riemann–Liouville definition of the fractional derivative. In this approach, the particular integral to the fractional perturbed equation is found out. An enhanced perturbation method is developed to study the forced nonlinear Duffing oscillator. The modified homotopy equation with two expanded parameters and an additional auxiliary parameter is applied in this proposal. The basic idea of the enhanced method is to apply the annihilator operator to construct a simplified equation freeness of the periodic force. This method makes the solution process for the forced problem much simpler. The resulting equation is valid for studying all types of possible resonance states. The outcome shows that this alteration method overcomes all shortcomings of the perturbation method and leads to obtain a periodic solution.

The reducing rank method to solve third‐order Duffing equation with the homotopy perturbation

“…A modification to the HPM with the so-called auxiliary equivalence technique has been demonstrated by Shen and El-Dib [35]. El-Dib and Elgazery [13] applied the method of the so-called reducing rank technique, through the integrating of the fraction auxiliary operator.…”

Gamma function method for the nonlinear cubic-quintic Duffing oscillators

“…Many scholars have made outstanding contributions and many different methods are obtained such as homotopy perturbation method, [1][2][3] variational approach, [4][5][6][7][8][9] variational iteration method, [10][11][12][13][14][15][16] He's frequency formulation, [17][18][19] Hamiltonian approach, 20 Taylor series method, 21 and so on. [22][23][24] The well-known Duffing oscillator equation was named after a German electrical engineer Georg Duffing who first proposed the equation in 1918, 25 and then, it is developed into different forms to describe many physical, mechanical engineering, circuits and biological processes in various areas of science. [26][27][28] Thus, the study of the Duffing oscillator equation is important.…”

Study on the nonlinear vibration of embedded carbon nanotube via the Hamiltonian-based method

“…where l is the length of the CNT, ρ represents the density, E indicates Young's modulus, and A and I mean the crosssectional area and cross-sectional inertia moment, respectively. There are many methods available for obtaining the frequency property of equation (1.1) such as the incremental harmonic balanced method, 6 variational iteration method, 8 Fourier series and Stokes' transformation, 9 homotopy perturbation method, 10,11 and He's frequency formulation. 12 In this study, we will use a new method called the Hamiltonian-based method to determine the frequency property.…”