On weighted Atangana–Baleanu fractional operators

Al-Refai, Mohammed

doi:10.1186/s13662-019-2471-z

Cited by 48 publications

(50 citation statements)

References 32 publications

(21 reference statements)

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“…e theory of the stability of FDEs involving fractional derivatives with nonsingular kernels is new, and it requires an important development in order to study the dynamical behaviors of several systems available in the literature and using such derivatives. For these reasons, the main purpose of this paper is to extend the Lyapunov direct method for systems of FDEs involving the new generalized Hattaf fractional (GHF) derivative [12], which covers the most famous fractional derivatives with nonsingular kernels existing in the literature such as the Caputo-Fabrizio fractional derivative [13], the Atangana-Baleanu fractional derivative [14], and the weighted Atangana-Baleanu fractional derivative [15].…”

Section: Introductionmentioning

confidence: 99%

Stability of Fractional Differential Equations with New Generalized Hattaf Fractional Derivative

Hattaf¹

2021

Mathematical Problems in Engineering

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This paper aims to study the stability of fractional differential equations involving the new generalized Hattaf fractional derivative which includes the most types of fractional derivatives with nonsingular kernels. The stability analysis is obtained by means of the Lyapunov direct method. First, some fundamental results and lemmas are established in order to achieve the goal of this study. Furthermore, the results related to exponential and Mittag–Leffler stability existing in recent studies are extended and generalized. Finally, illustrative examples are presented to show the applicability of our main results in some areas of science and engineering.

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Section: Introductionmentioning

confidence: 99%

Stability of Fractional Differential Equations with New Generalized Hattaf Fractional Derivative

Hattaf¹

2021

Mathematical Problems in Engineering

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“…(ii) If w(t) � 1 and β � c � α, then (1) is reduced to the Atangana-Baleanu fractional derivative [2] given by 1) is reduced to the weighted Atangana-Baleanu fractional derivative [3] given by…”

Section: Remarkmentioning

confidence: 99%

“…An extension of [1] was proposed by Atangana and Baleanu [2] by using Mittag-Lefler function with one parameter. Due to the importance of weighted fractional derivatives to solve several types of integral equations with elegant ways, Al-Refai [3] introduced the weighted Atangana-Baleanu fractional operators and he studied their properties. A generalized version of all previous fractional derivative operators with non-singular kernel was recently proposed in [4].…”

Section: Introductionmentioning

confidence: 99%

On Some Properties of the New Generalized Fractional Derivative with Non-Singular Kernel

Hattaf¹

2021

Mathematical Problems in Engineering

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This paper presents some new formulas and properties of the generalized fractional derivative with non-singular kernel that covers various types of fractional derivatives such as the Caputo–Fabrizio fractional derivative, the Atangana–Baleanu fractional derivative, and the weighted Atangana–Baleanu fractional derivative. These new properties extend many recent results existing in the literature. Furthermore, the paper proposes some interesting inequalities that estimate the generalized fractional derivatives of some specific functions. These inequalities can be used to construct Lyapunov functions with the aim to study the global asymptotic stability of several fractional-order systems arising from diverse fields of science and engineering.

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“…Proof If R 0 < 1, we obviously have that E 0 is the unique no-disease point of model (1). Furthermore, suppose that E 0 is not locally asymptotically stable.…”

Section: Local Behaviour Of No-disease Equilibrium Pointmentioning

confidence: 99%

Analysis of Schistosomiasis Global Dynamics with General Incidence Functions and Two Delays

Koutou

Traoré

Sangaré

2021

Int. J. Appl. Comput. Math

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As most communicable diseases, schistosomiasis transmission mechanism involves some delay due to the incubation period. In this study, we seek to investigate the impact of incubation period on schistosomiasis global transmission dynamics. For that, starting from our previous work and using delay differential equations, we have proposed a more sophisticated model in which the new feature we accounted is the latency period in both human and snail hosts. Then, under some suitable assumptions, the mathematical analysis of the model has been done. Specifically, we calculated the basic reproduction number and showed its dependence on the delays in both humans and snails as follows: when the delays in humans and snails increase, R 0 decreases. We also proved that when R 0 < 1, the no-disease point is globally asymptotically stable and when R 0 > 1, the system is uniformly persistent and admits a unique endemic equilibrium. Besides, by formulating a suitable Lyapunov function, we established that the unique endemic steady state of the model is globally asymptotically stable when the threshold parameter R 0 exceeds 1. Furthermore, sensitivity analysis has been carried out to show how parameters variations affect the model global behaviour. We finished by illustrating some numerical results that are well in keeping with our theoretical findings.

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On weighted Atangana–Baleanu fractional operators

Cited by 48 publications

References 32 publications

Stability of Fractional Differential Equations with New Generalized Hattaf Fractional Derivative

Stability of Fractional Differential Equations with New Generalized Hattaf Fractional Derivative

On Some Properties of the New Generalized Fractional Derivative with Non-Singular Kernel

Analysis of Schistosomiasis Global Dynamics with General Incidence Functions and Two Delays

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