Galerkin’s method to solve a fractional time-delayed jerk oscillator

El‐Dib, Yusry O.; Elgazery, Nasser S.; Alyousef, Haifa A.

doi:10.1007/s00419-023-02455-8

Cited by 6 publications

(1 citation statement)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This solution stands out as it does not rely on free parameters, setting it apart from the two-step process rooted in gradient-based techniques. The Jerk-Duffing oscillator's nonlinearity was curtailed by introducing time-delayed position and velocity [21,22]. The inclusion of this time delay acts as a protective barrier, ensuring the system in question does not undergo nonlinear vibrations [23].…”

Section: Introductionmentioning

confidence: 99%

Stability analysis of a time-delayed Van der Pol–Helmholtz–Duffing oscillator in fractal space with a non-perturbative approach

El-Dib

2024

Commun. Theor. Phys.

Self Cite

View full text Add to dashboard Cite

The time-delayed fractal Van der Pol-Helmholtz-Duffing (VPHD) oscillator is the subject of this paper, which explores its mechanisms and highlights its stability analysis. While time-delayed technologies are currently garnering significant attention, the focus of this research remains crucially relevant. A non-perturbative approach is employed to refine and set the stage for the system under scrutiny. The innovative methodologies introduced yield an equivalent linear differential equation, mirroring the inherent nonlinearities of the system. Notably, the incorporation of quadratic nonlinearity into the frequency formula represents a cutting-edge advancement. The analytical solution's validity is corroborated using a numerical approach. Stability conditions are ascertained through the residual Galerkin method. Intriguingly, it is observed that the delay parameter, in the context of the fractal system, reverses its stabilizing influence, impacting both the amplitude of delayed velocity and position. The analytical solution's precision is underscored by its close alignment with numerical results. Furthermore, the study reveals that fractal characteristics emulate damping behaviors. Given its applicability across diverse nonlinear dynamical systems, this non-perturbative approach emerges as a promising avenue for future research.

show abstract

Section: Introductionmentioning

confidence: 99%

Stability analysis of a time-delayed Van der Pol–Helmholtz–Duffing oscillator in fractal space with a non-perturbative approach

El-Dib

2024

Commun. Theor. Phys.

Self Cite

View full text Add to dashboard Cite

show abstract

A dynamic study of a bead sliding on a wire in fractal space with the non-perturbative technique

El-Dib

2024

Arch Appl Mech

View full text Add to dashboard Cite

Drawing on the principles of fractal properties and nonlinear vibration analysis, this paper delves into the investigation of a moving bead on a vertically rotated parabola. The dynamical nonlinear equation of motion, incorporating fractal derivatives, transforms traditional derivatives within continuous space. Consequently, the equation of motion takes the form of the Duffing-Van der Pol oscillator. Utilizing a non-perturbative approach, the nonlinear oscillator is systematically transformed into a linear one, boasting an exact solution. The analytical solution yields two valid formulas governing the frequency-amplitude relationships. Numerical solutions affirm that these proposed formulas offer highly satisfactory approximations to the analytical solution. Leveraging fractal properties through Galerkin’s method, the paper successfully determines the fractalness parameter of the medium, shedding light on the intricate dynamics of the system.

show abstract