On a new generalized local fractal derivative operator

El-Nabulsi, Rami Ahmad; Golmankhaneh, Alireza Khalili; Agarwal, Praveen

doi:10.1016/j.chaos.2022.112329

Cited by 16 publications

(3 citation statements)

References 37 publications

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“…[40][41][42] Further, Wang 43 used the Fourier series to provide a novel method for a class of discontinuous fractal nonlinear oscillators in the context of microgravity space. El-Nabulsi et al 44 also developed a novel local derivative operator with fractals across Lagrangian/Hamiltonian functions for further research of the dynamical systems. The two-scale fractal theory was developed by Anjum and his school 45,46 to study the mathematical models of both population and tsunami waves.…”

Section: Introductionmentioning

confidence: 99%

An instrumental insight for a periodic solution of a fractal Mathieu–Duffing equation

El‐Dib

Elgazery

Alyousef

2023

Journal of Low Frequency Noise, Vibration and Active Control

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The primary goal of the present study is to investigate how to obtain a periodic solution for a fractal Mathieu–Duffing oscillator. To achieve this, the fractal oscillator in the fractal space has been transformed into a damping Mathieu–Duffing equation in the continuous space by employing a new modification of He’s definition of the fractal derivative. The required analytical periodic solution has been based on the rank upgrade technique (RUT) presented. The RUT successfully generates a periodic solution without sacrificing the damping coefficient by creating an alternate equation, aside from any challenges in managing the impact of the linear damping component. The homotopy perturbation method (HPM) has been used to find the required periodic solution for the alternate equation. A comparison of the numerical solutions of the original equation and the alternative equation showed good agreement. The stability behavior in the non-resonance case as well as in the sub-harmonic resonance case has also been discussed. Further, another method, “the non-perturbative approach”, that deals with the obtained equation has been introduced.

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Section: Introductionmentioning

confidence: 99%

An instrumental insight for a periodic solution of a fractal Mathieu–Duffing equation

El‐Dib

Elgazery

Alyousef

2023

Journal of Low Frequency Noise, Vibration and Active Control

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“…In the field of microgravity space, Wang [60] also used the Fourier series to propose a new technique for a class of fractal nonlinear oscillators that have discontinuities. For more analysis of the dynamical systems, El-Nabulsi et al [61] achieved a novel local derivative operator with fractal via both Hamiltonian and Lagrangian functions. On the other hand, there are a lot of effective contributions to 'analytical approximation' and 'fractional/fractal' modeling in the literature.…”

Section: Introductionmentioning

confidence: 99%

An innovative technique to solve a fractal damping Duffing-jerk oscillator

El‐Dib

Elgazery

Khattab

et al. 2023

Commun. Theor. Phys.

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The idea of the present article is to look into the nonlinear dynamics and vibration of a damping Duffing-jerk oscillator in fractal space exhibiting the non-perturbative approach. Using a new analytical technique, namely, the modification of He's a fractal derivative that converts the fractal derivative to the traditional derivative in continuous space, this study provides an effective and easy-to-apply procedure that is dependent on He's fractal derivative approach. The analytic approximate solution has significant match with the results of the numerical simulation as the fractal parameter is very closer to unity, which proves the reliability of the method. Stability behavior is discussed and illustrated graphically. These new powerful analytical tools are developed in an attempt to obtain effective analytical tools, valid for any fractal nonlinear problems.

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“…The introduction of a novel generalized local fractal derivative operator and its exploration in classical systems via Lagrangian and Hamiltonian formalisms were undertaken. The practical applicability of the variational method in describing dissipative dynamical systems was showcased, and the Hamiltonian approach produced auxiliary constraints without reliance on Dirac auxiliary functions [44]. Furthermore, fractal stochastic differential equations have been defined, with categorizations for processes like fractional Brownian motion and diffusion occurring within mediums with fractal structures [45,46,47,48,49].…”

Section: Introductionmentioning

confidence: 99%

Non-standard analysis for fractal calculus

Golmankhaneh

Welch

Serpa

et al. 2023

J Anal

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In this paper, we delve into the fascinating realm of fractal calculus applied to fractal sets and fractal curves. Our study includes an exploration of the method analogues of the separable method and the integrating factor technique for solving α-order differential equations. Notably, we extend our analysis to solve Fractal Bernoulli differential equations. The applications of our findings are then showcased through the solutions of problems such as fractal compound interest, the escape velocity of the earth in fractal space and time, and the estimation of time of death incorporating fractal time. Visual representations of our results are also provided to enhance understanding.

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On a new generalized local fractal derivative operator

Cited by 16 publications

References 37 publications

An instrumental insight for a periodic solution of a fractal Mathieu–Duffing equation

An instrumental insight for a periodic solution of a fractal Mathieu–Duffing equation

An innovative technique to solve a fractal damping Duffing-jerk oscillator

Non-standard analysis for fractal calculus

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