Novel Analytical and Numerical Approximations to the Forced Damped Parametric Driven Pendulum Oscillator: Chebyshev Collocation Method

Alharthi, M.R.; Salas, Alvaro H.; Albalawi, Wedad; El-Tantawy, S. A.

doi:10.1155/2022/5454685

Cited by 3 publications

(1 citation statement)

References 33 publications

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“…and some other equations related to this oscillator have been analyzed and investigated using some different effective analytical and numerical techniques, such as the ansatz method [4], He's frequency-amplitude principle [4], He's homotopy perturbation method (HPM) [4], the Krylov-Bogoliúbov Mitropolsky (KBM) method [4], the 4th-order Runge Kutta (RK4), the hybrid Padé-finite difference method [4], the Chebyshev collocation method (CCM) [5], the Galerkin method [6], the ansatz method (AM) and He's frequency formulation [6]. Moreover, the AM and the HPT with the extended KBM were used in the study of the damped cubic nonlinearity Duffing-Mathieu-type oscillator [7].…”

Section: Introductionmentioning

confidence: 99%

Analytical approximations to a generalized forced damped complex Duffing oscillator: multiple scales method and KBM approach

Alhejaili¹,

Salas²,

El-Tantawy³

2023

Commun. Theor. Phys.

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In this investigation, some different approaches including the hybrid homotopy perturbation method (H-HPM) which sometimes is called Krylov-Bogoliubov-Mitropolsky (KBM) method and the multiple scales method (MSM), are implemented for analyzing a generalized forced damped complex Duffing oscillator. All mentioned methods are applied to obtain some accurate and stable approximations to the proposed problem without decoupling for the original problem. All obtained approximations are discussed graphically using different numerical values to the relevant parameters. Moreover, all obtained approximate solution are compared with the 4^{th}-order Runge-Kutta (RK4) numerical approximation. Also, the maximum residual distance error (MRDE) is estimated in order to verify the high-accuracy to the obtained analytic approximations.

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Section: Introductionmentioning

confidence: 99%

Analytical approximations to a generalized forced damped complex Duffing oscillator: multiple scales method and KBM approach

Alhejaili¹,

Salas²,

El-Tantawy³

2023

Commun. Theor. Phys.

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The Krylov–Bogoliubov–Mitropolsky method for modeling a forced damped quadratic pendulum oscillator

Alhejaili,

Salas,

El-Tantawy

2023

AIP Advances

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In the present investigation, a quadratically forced damped pendulum-type equation is solved analytically using several effective and highly accurate approaches. Some different types of pendulum oscillators linked to the forced and damped terms, in addition to the power of the damping term, are analyzed. In the first part, the Krylov–Bogoliubov–Mitropolsky (KBM) technique and the ansatz method are carried out for analyzing the quadratically damped pendulum oscillator. In the second part, the two variants of the KBM technique are implemented for investigating the quadratically forced damped pendulum oscillator. Some symmetric approximations are derived and compared with the fourth-order Runge–Kutta numerical approximation. In addition, the maximum distance error is estimated in the whole study domain for ensuring that all obtained approximations are accurate. The obtained results are illustrated through concrete examples.

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On the feed-forward neural network for analyzing pantograph equations

Az-Zo’bi,

Shah,

Alyousef

et al. 2024

AIP Advances

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Ordinary differential equations (ODEs) are fundamental tools for modeling and understanding a wide range of chemistry, physics, and biological phenomena. However, solving complex ODEs often presents significant challenges, necessitating advanced numerical approaches beyond traditional analytical techniques. Thus, a novel machine learning (ML)-based method for solving and analyzing ODEs is proposed in the current investigation. In this study, we utilize a feed-forward neural network (FNN) with five fully connected layers trained on data samples generated from the exact solutions of specific ODEs. To show the efficacy of our suggested method, we will conduct a thorough evaluation by comparing the anticipated solutions of the FNN with the exact solutions for some ODEs. Furthermore, we analyze the absolute error and present the loss functions for some ODE examples, providing valuable insights into the model’s performance and potential areas for further development.

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Novel Analytical and Numerical Approximations to the Forced Damped Parametric Driven Pendulum Oscillator: Chebyshev Collocation Method

Cited by 3 publications

References 33 publications

Analytical approximations to a generalized forced damped complex Duffing oscillator: multiple scales method and KBM approach

Analytical approximations to a generalized forced damped complex Duffing oscillator: multiple scales method and KBM approach

The Krylov–Bogoliubov–Mitropolsky method for modeling a forced damped quadratic pendulum oscillator

On the feed-forward neural network for analyzing pantograph equations

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