Modeling attractors of chaotic dynamical systems with fractal–fractional operators

Atangana, Abdon; Qureshi, Sania

doi:10.1016/j.chaos.2019.04.020

Cited by 319 publications

(100 citation statements)

References 25 publications

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“…Therefore, much attention has been given to tackle the problems arising from these fields. For instance, Reserchers in [1,2] used fractional-order derivatives to understand the copmlex behaviour of some chaotic models. In the field of mathematical epidemiology, many authors have recently fractionalyzed classical systems related with various types of diseases and proved their efficacy with the help of real statistical data [3][4][5][6][7].…”

Section: Introductionmentioning

confidence: 99%

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Using Shehu integral transform to solve fractional order Caputo type initial value problems

Qureshi¹,

Kumar²

2019

J. Appl. Math. Comput. Mech.

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In the present research analysis, linear fractional order ordinary differential equations with some defined condition (s) have been solved under the Caputo differential operator having order α > 0 via the Shehu integral transform technique. In this regard, we have presented the proof of finding the Shehu transform for a classical nth order integral of a piecewise continuous with an exponential order function which leads towards devising a theorem to yield exact analytical solutions of the problems under investigation. Varying fractional types of problems are solved whose exact solutions can be compared with solutions obtained through existing transformation techniques including Laplace and Natural transforms.

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Section: Introductionmentioning

confidence: 99%

“…Example 4 Consider the following linear fractional order in-homogeneous initial value problem with α ∈ [1,2]:…”

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confidence: 99%

Using Shehu integral transform to solve fractional order Caputo type initial value problems

Qureshi¹,

Kumar²

2019

J. Appl. Math. Comput. Mech.

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“…Very recently, an African professor Atangana presented his concept of fractal‐fractional differentiation based on Mittage‐Leffler, exponential decay, and power‐law memories in which he described that fractal‐fractional differentiation attracts more nonlocal natural problems that display at the same time fractal behaviors 16 . Atangana and Qureshi 17 captured self‐similarities in the chaotic attractors based on the basis of three numerical schemes for systems of nonlinear differential equations. Their investigated dynamical systems containing the general conditions for the existence and the uniqueness have been explored.…”

Section: Introductionmentioning

confidence: 99%

Mathematical analysis of memristor through fractal‐fractional differential operators: A numerical study

Abro

Atangana

2020

Math Methods in App Sciences

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The newly generalized energy storage component, namely, memristor, which is a fundamental circuit element so called universal charge‐controlled mem‐element, is proposed for controlling the analysis and coexisting attractors. The governing differential equations of memristor are highly nonlinear for mathematical relationships. The mathematical model of memristor is established in terms of newly defined fractal‐fractional differential operators so called Atangana‐Baleanu, Caputo‐Fabrizio, and Caputo fractal‐fractional differential operator. A novel numerical approach is developed for the governing differential equations of memristor on the basis of Atangana‐Baleanu, Caputo‐Fabrizio, and Caputo fractal‐fractional differential operator. We discussed chaotic behavior of memristor under three criteria such as (i) varying fractal order, we fixed fractional order; (ii) varying fractional order, we fixed fractal order; and (ii) varying fractal and fractional orders simultaneously. Our investigated graphical illustrations and simulated results via MATLAB for the chaotic behaviors of memristor suggest that newly presented Atangana‐Baleanu, Caputo‐Fabrizio, and Caputo fractal‐fractional differential operators generate significant results as compared with classical approach.

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“…There exist several classical definitions for fractional derivatives, for example, Caputo, Riemann-Liouville, Marchaud, Hadamard and the Weyl derivatives, among others. In recent years with the intention of solving some problems that the classical fractional derivatives do not achieve and as a consequence of those, new fractional derivatives have been defined combining power law, exponential decay and Mittag-Leffler kernel; among them those of Liouville-Caputo, Atanga-Caputo, Atanga-Gómez and Atanga-Baleanu derivatives, see [2], [3], [5], [6] and [16].…”

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confidence: 99%

“…Here we established fundamental results for fractional calculus in the sense of the distributional Henstock-Kurzweil integral. On the other hand, the Riemann-Liouville fractional derivative seems to be the most suitable according to theoretical and applied studies, see [6]. Nevertheless, many numerical approximations of the fractional derivative were made considering the Caputo derivative and the Lebesgue integral.…”

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confidence: 99%

Riemann-Liouville derivative over the space of integrable distributions

Morales

Došlá

Mendoza

2020

era

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In this paper, we generalize the Riemann-Liouville differential and integral operators on the space of Henstock-Kurzweil integrable distributions, D HK. We obtain new fundamental properties of the fractional derivatives and integrals, a general version of the fundamental theorem of fractional calculus, semigroup property for the Riemann-Liouville integral operators and relations between the Riemann-Liouville integral and differential operators. Also, we achieve a generalized characterization of the solution for the Abel integral equation. Finally, we show relations for the Fourier transform of fractional derivative and integral. These results are based on the properties of the distributional Henstock-Kurzweil integral and convolution.

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Modeling attractors of chaotic dynamical systems with fractal–fractional operators

Cited by 319 publications

References 25 publications

Using Shehu integral transform to solve fractional order Caputo type initial value problems

Using Shehu integral transform to solve fractional order Caputo type initial value problems

Mathematical analysis of memristor through fractal‐fractional differential operators: A numerical study

Riemann-Liouville derivative over the space of integrable distributions

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