An Accurate Method for Solving the Undamped Duffing Equation with Cubic Nonlinearity

Syam, Muhammed I.; Raja, Muhammad Asif Zahoor; Syam, Mahmmoud M.; Jaradat, H. M.

doi:10.1007/s40819-018-0502-1

Cited by 8 publications

(3 citation statements)

References 17 publications

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“…There are many methods for solving the fractional Duffing equation. The authors of [10] considered the method of homotopy analysis and the method of finitedifference schemes [5]; article [11] used a modified method of fractional power series. A qualitative analysis of the fractional Duffing oscillator can be found in articles [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29].…”

Section: Introductionmentioning

confidence: 99%

Implicit Finite-Difference Scheme for a Duffing Oscillator with a Derivative of Variable Fractional Order of the Riemann-Liouville Type

2023

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The article considers an implicit finite-difference scheme for the Duffing equation with a derivative of a fractional variable order of the Riemann–Liouville type. The issues of stability and convergence of an implicit finite-difference scheme are considered. Test examples are given to substantiate the theoretical results. Using the Runge rule, the results of the implicit scheme are compared with the results of the explicit scheme. Phase trajectories and oscillograms for a Duffing oscillator with a fractional derivative of variable order of the Riemann–Liouville type are constructed, chaotic modes are detected using the spectrum of maximum Lyapunov exponents and Poincare sections. Q-factor surfaces, amplitude-frequency and phase-frequency characteristics are constructed for the study of forced oscillations. The results of the study showed that the implicit finite-difference scheme shows more accurate results than the explicit one.

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Section: Introductionmentioning

confidence: 99%

Implicit Finite-Difference Scheme for a Duffing Oscillator with a Derivative of Variable Fractional Order of the Riemann-Liouville Type

2023

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“…[13][14][15][16] The numerical methods [17][18][19][20] and analytical perturbation solutions [21][22][23][24] of fractional oscillators are found to be hard work of research. [25][26][27][28] Therefore, the researchers have given full attention to developing strong and novel techniques to handle this complicated class of fractional mathematical tools. 29 In this regard, a new technique is urgent to simplify and reduce the hard work required for obtaining an asymptotic solution that is closer to the exact numerical solution.…”

Section: Introductionmentioning

confidence: 99%

Immediate solution for fractional nonlinear oscillators using the equivalent linearized method

El‐Dib

2022

Journal of Low Frequency Noise, Vibration and Active Control

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The existence of the derivative with a fractional-order in a class of differential equations could lead to complicating the analysis. In this paper, a novel approach has been introduced to facilitate the analysis of the oscillation having fractional-order derivatives and to obtain the analytical solution easily. The present technique is formulated to provide an easy way of understanding. The suggested technique has been utilized to study two examples for illustration. The similitude between the analytical and numerical solution verifies and gives satisfactory precision to the equivalent solution. The new technique is represented to be the best tool for solving the nonlinear oscillation problems in physics and engineering which have a fractional-order.

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“…Succinctly, the oscillating and chaotic nature of this nonlinear model has augmented a miraculous notion in the literature, as it imitates the dynamics of our natural world. [10][11][12][13][14][15][16] In essence, the following equation for Duffing-harmonic oscillation is formulated and studied, i.e. Y 00 t ð Þ þ Y 3 t ð Þ 1 þ Y 2 t ð Þ ¼ 0; Y 0 ð Þ ¼ A; Y 0 0 ð Þ ¼ 0 (1) where A represents the amplitude of the oscillation.…”

Section: Introductionmentioning

confidence: 99%

Intelligent computing for Duffing-Harmonic oscillator equation via the bio-evolutionary optimization algorithm

Khan

Hameed

Razzaq

et al. 2018

Journal of Low Frequency Noise, Vibration and Active Control

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This paper presents a bio-evolutionary metaheuristic approach to study the harmonically oscillating behavior of the Duffing equation. The proposed methodology is an amalgamation of the artificial neural network with the firefly algorithm. A novelty in the activation of neurons of artificial neural network is described using the cosine function with the angular frequency. Chronologically, artificial neural network approximates discretizes the nonlinear functions of the governing problem, which then undergoes an optimization process by the firefly algorithm that then later generates the effective values of the unknown parameters. Generally, the algorithm and implementation of the scheme are assimilated by considering an application of Duffing-harmonic oscillator. Some error measurements, in order to discuss the convergence and accuracy of the scheme, are also visualized through tables and graphs. An effective optimized relationship between the angular frequency and amplitude is derived and its results are depicted in a tabular form. The comparison of the proposed methodology is also deliberated by homotopy perturbation method. Moreover, the geometrical illustration of the trajectories of the dynamic system is also added in the phase plane for different values of amplitude and angular frequency.

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An Accurate Method for Solving the Undamped Duffing Equation with Cubic Nonlinearity

Cited by 8 publications

References 17 publications

Implicit Finite-Difference Scheme for a Duffing Oscillator with a Derivative of Variable Fractional Order of the Riemann-Liouville Type

Implicit Finite-Difference Scheme for a Duffing Oscillator with a Derivative of Variable Fractional Order of the Riemann-Liouville Type

Immediate solution for fractional nonlinear oscillators using the equivalent linearized method

Intelligent computing for Duffing-Harmonic oscillator equation via the bio-evolutionary optimization algorithm

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