Analysis and new applications of fractal fractional differential equations with power law kernel

Akgül, Ali

doi:10.3934/dcdss.2020423

Cited by 13 publications

(7 citation statements)

References 13 publications

(13 reference statements)

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“…Author details 1 Laboratoire d' Analyse Non Linéaire et Mathématiques Appliquées, Université de Tlemcen, Tlemcen, Algeria. 2 Department of Mathematics, Faculty of Exact Sciences and Informatics, Hassiba Benbouali University, Chlef, Algeria.…”

Section: Fundingmentioning

confidence: 99%

“…To get more models of prey and hunter-modeled under the influence of a disease, we encourage interested readers to refer to the relevant literature in [7, 9, 10, 13-21, 23-25, 29, 31, 34-36, 39, 41]. Some other general problems can be found in [1][2][3][4][5][6]28].…”

Section: Introductionmentioning

confidence: 99%

“…Section 4 is devoted to studying the stability of these equilibrium points. A numerical scheme is used in the fifth section for plotting the solution of model (1), where some graphical representations are used for verifying the expected mathematical results. Finally, the concluding section ends the paper.…”

Section: Introductionmentioning

confidence: 99%

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The influence of an infectious disease on a prey-predator model equipped with a fractional-order derivative

Djilali

Ghanbari

2021

Adv Differ Equ

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In this research, we discuss the influence of an infectious disease in the evolution of ecological species. A computational predator-prey model of fractional order is considered. Also, we assume that there is a non-fatal infectious disease developed in the prey population. Indeed, it is considered that the predators have a cooperative hunting. This situation occurs when a pair or group of animals coordinate their activities as part of their hunting behavior in order to improve their chances of making a kill and feeding. In this model, we then shift the role of standard derivatives to fractional-order derivatives to take advantage of the valuable benefits of this class of derivatives. Moreover, the stability of equilibrium points is studied. The influence of this infection measured by the transmission rate on the evolution of predator-prey interaction is determined. Many scenarios are obtained, which implies the richness of the suggested model and the importance of this study. The graphical representation of the mathematical results is provided through a precise numerical scheme. This technique enables us to approximate other related models including fractional-derivative operators with high accuracy and efficiency.

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Section: Fundingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The influence of an infectious disease on a prey-predator model equipped with a fractional-order derivative

Djilali

Ghanbari

2021

Adv Differ Equ

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“…The authors in [5] have proposed an efficient operation matrix method for solving fractal-fractional differential equations. Other valuable work on fractal-fractional differential operators can be found in [29][30][31] and references therein. But most of these methods are based on the finite difference method for temporal discretization, and they encounter an increase of computing cost with advancing time and thus these methods have low efficiency in the simulation of the long time history of fractional differential equations.…”

Section: Introductionmentioning

confidence: 99%

On the Approximation of Fractal-Fractional Differential Equations Using Numerical Inverse Laplace Transform Methods

Kamran¹,

Ahmad²,

Shah³

et al. 2023

Computer Modeling in Engineering &Amp; Sciences

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Laplace transform is one of the powerful tools for solving differential equations in engineering and other science subjects. Using the Laplace transform for solving differential equations, however, sometimes leads to solutions in the Laplace domain that are not readily invertible to the real domain by analytical means. Thus, we need numerical inversion methods to convert the obtained solution from Laplace domain to a real domain. In this paper, we propose a numerical scheme based on Laplace transform and numerical inverse Laplace transform for the approximate solution of fractal-fractional differential equations with order α, β. Our proposed numerical scheme is based on three main steps. First, we convert the given fractal-fractional differential equation to fractional-differential equation in Riemann-Liouville sense, and then into Caputo sense. Secondly, we transform the fractional differential equation in Caputo sense to an equivalent equation in Laplace space. Then the solution of the transformed equation is obtained in Laplace domain. Finally, the solution is converted into the real domain using numerical inversion of Laplace transform. Three inversion methods are evaluated in this paper, and their convergence is also discussed. Three test problems are used to validate the inversion methods. We demonstrate our results with the help of tables and figures. The obtained results show that Euler's and Talbot's methods performed better than Stehfest's method.

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“… [2] studied a SIR model to assess the dynamics of COVID 19 using fractals Fraction operator. By using power-law kernels and new applications, Akgul [3] described some advances in fractal fractional differential equations.…”

Section: Introductionmentioning

confidence: 99%

Fractal–fractional operator for COVID-19 (Omicron) variant outbreak with analysis and modeling

et al. 2022

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Analysis and new applications of fractal fractional differential equations with power law kernel

Cited by 13 publications

References 13 publications

The influence of an infectious disease on a prey-predator model equipped with a fractional-order derivative

The influence of an infectious disease on a prey-predator model equipped with a fractional-order derivative

On the Approximation of Fractal-Fractional Differential Equations Using Numerical Inverse Laplace Transform Methods

Fractal–fractional operator for COVID-19 (Omicron) variant outbreak with analysis and modeling

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