Some Novel Analytical Approximations to the (Un)damped Duffing–Mathieu Oscillators

Alyousef, Haifa A.; Salas, Alvaro H.; Alkhateeb, Sadah A.; El-Tantawy, S. A.

doi:10.1155/2022/2715767

Cited by 9 publications

(2 citation statements)

References 24 publications

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“…35 Many researchers proved that the KBM technique is better than many techniques mentioned in the literature because it is characterized by high accuracy and ease of application. [35][36][37][38][39][40][41] In addition, it does not require to solve a system of differential equations but only solves a system of algebraic equations.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 99%

“…In the present investigation, we follow the recent studies [35][36][37][38][39][40][41] for analyzing the QFDP Eq. (1) using both ansatz and KBM methods.…”

Section: Introductionmentioning

confidence: 99%

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

Alhejaili,

Salas,

El-Tantawy

2023

AIP Advances

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Approximate solution to a generalized Van der Pol equation arising in plasma oscillations

2022

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Motivated by some published theoretical investigations and based on the two-fluid model, nonlinear plasma oscillations are analyzed and discussed in the framework of the generalized Van der Pol equation. This equation is analyzed and solved using two different analytical approaches. In this first approach, the ansatz method is carried out for deriving an approximation in the form of a trigonometric function. In the second approach, the Krylov–Bogoliubov–Mitropolsky (KBM) technique is applied for obtaining a high-accurate approximation. The obtained approximations are compared with the numerical approximation using the Runge–Kutta (RK) method. Moreover, the distance error between the obtained approximations (using the ansatz method and the KBM technique) and the RK numerical approximation is estimated. In our investigation, both the proposed methods and obtained approximations can help many authors investigate several nonlinear oscillations in different plasma models and fluid mechanics.

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On the propagation and interaction of ion-acoustic solitary, periodic, shock, and breather waves in a non-Maxwellian electron–positron–ion magnetoplasma

Raut¹,

Sarkar

Mali

et al. 2023

Physics of Fluids

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This investigation analyzes the propagation of nonlinear ion-acoustic waves (IAWs) in an unmagnetized, collisionless plasma composed of inertial positive ions and inertialess Maxwellian positrons as well as the inertialess non-Maxwellian electrons that obey (r, q)-distribution. To observe the impact of particle trapping on the nonlinear IAWs in an electron–positron–ion plasma, the Korteweg–De Vries (KdV) and modified KdV (mKdV) equations are derived using a reductive perturbation method. In the distribution function, the spectral parameters (r, q) put up their contribution to the flatness and high-energy tails, respectively. An important aspect of this investigation is the determination of well-known quasi-periodic solutions, multi-soliton solutions, breathers, and shocks under the variation of different physical parameters, especially spectral indices (r, q). Finally, the interaction of solitons is also presented for discussion of the complete profile. In addition, a detailed comparison, especially in a periodic wave, is made between the generalized (r, q)-distribution and the limiting cases of Kappa and Maxwellian distributions. The results presented in this study contribute to a better understanding of the characteristics of both high- and low-energy parts of the electron distribution function as well as the formation of periodic, soliton, multi-soliton, breathers, and shocks in space and astrophysical plasmas.

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Some Novel Analytical Approximations to the (Un)damped Duffing–Mathieu Oscillators

Cited by 9 publications

References 24 publications

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

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

Approximate solution to a generalized Van der Pol equation arising in plasma oscillations

On the propagation and interaction of ion-acoustic solitary, periodic, shock, and breather waves in a non-Maxwellian electron–positron–ion magnetoplasma

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