Fractional differential equations with Atangana–Baleanu fractional derivative: Analysis and applications

Syam, Muhammed I.; Al-Refai, Mohammed

doi:10.1016/j.csfx.2019.100013

Cited by 62 publications

(35 citation statements)

References 10 publications

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“…From Theorem 2, (15), and (16) it follows that the origin of system (14) is GCES, and we have e ≤ C e 0 E ψ α (-γ , t, t 0 )…”

Section: Theoremmentioning

confidence: 84%

“…with C = λ max (P) λ min (P) , γ = ε 2λ max (P) and e 0 = e(t 0 ). Furthermore, from (16) and the properties of norms we have…”

Section: Theoremmentioning

confidence: 99%

“…Given that the Riemann-Liouville and Caputo derivatives may deal with singularity issues in the kernel, some operators with nonsingular kernel have been proposed, such as Caputo-Fabrizio [11] and Atangana-Baleanu [12] derivatives; these operators are currently being studied extensively, and they have been used for both theoretical results and applications [13][14][15][16][17].…”

Section: Introductionmentioning

confidence: 99%

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General conformable estimators with finite-time stability

2020

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In this paper, some estimators are proposed for nonlinear dynamical systems with the general conformable derivative. In order to analyze the stability of these estimators, some Lyapunov-like theorems are presented, taking into account finite-time stability. Thus, to prove these theorems, a stability function is defined based on the general conformable operator, which implies exponential stability. The performance of the estimators is assessed by means of numerical simulations. Furthermore, a comparison is made between the results obtained with the integer, fractional, and general conformable derivatives.

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“…From Theorem 2, (15), and (16) it follows that the origin of system (14) is GCES, and we have e ≤ C e 0 E ψ α (-γ , t, t 0 )…”

Section: Theoremmentioning

confidence: 84%

“…with C = λ max (P) λ min (P) , γ = ε 2λ max (P) and e 0 = e(t 0 ). Furthermore, from (16) and the properties of norms we have…”

Section: Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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General conformable estimators with finite-time stability

2020

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“…Therefore, the problem of introducing and studying the weighted Atangana-Baleanu fractional operators with respect to another function z(t) and weight function w(t) with their properties is still open. Also, the question whether the new models in the paper can be solved by the available numerical techniques in the literature [25,26,37,38] is a valid question, and this issue has to considered in a future research.…”

Section: Discussionmentioning

confidence: 99%

On weighted Atangana–Baleanu fractional operators

Al-Refai

2020

Adv Differ Equ

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In this paper, we define the weighted Atangana-Baleanu fractional operators of Caputo sense. We obtain the solution of a related linear fractional differential equation in a closed form, and use the result to define the weighted Atangana-Baleanu fractional integral. We then express the weighted Atangana-Baleanu fractional derivative in a convergent series of Riemann-Liouville fractional integrals, and establish commutative results of the weighted Atangana-Baleanu fractional operators.

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“…AB form of FCs features novel aspects in comparison to integer‐order derivative and notable Caputo FCs of arbitrary order because it contains Mittag–Leffler form of the kernel. It is recognized that the MLF is a lot of convenience in expressing the physical phenomenon 17,18 ; consequently AB derivative is extremely appropriate to represent material heterogeneity and a few design or media consisting of individual scales 19 . The nonlocality of the AB form of FCs sanctions improved the enlightenment of the memory within organizations and media consisting of individual scales 20 .…”

Section: Introductionmentioning

confidence: 99%

An analysis for Klein–Gordon equation using fractional derivative having Mittag–Leffler‐type kernel

Kumar¹,

Băleanu

2020

Math Methods in App Sciences

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Within this paper, we present an analysis of the fractional model of the Klein–Gordon (K‐G) equation. K‐G equation is considered as one of the significant equations in mathematical physics that describe the interaction of soliton in a collision less plasma. In a novel aspect of this work, we have used the latest form of fractional derivative (FCs), which contains the Mittag–Leffler type of kernel. The homotopy analysis transform method (HATM) is being taken to solve the fractional model of the K‐G equation. A convergence study of HATM has been studied. The existence and uniqueness of the solution for the fractional K‐G equation are presented. For verifying the obtained numerical outcomes regarding accuracy and competency, we have given different graphical presentations. Figures are reflecting that a novel form of the technique is a good organization in respect of proficiency and accurateness to solve the mentioned fractional problem.

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Fractional differential equations with Atangana–Baleanu fractional derivative: Analysis and applications

Cited by 62 publications

References 10 publications

General conformable estimators with finite-time stability

General conformable estimators with finite-time stability

On weighted Atangana–Baleanu fractional operators

An analysis for Klein–Gordon equation using fractional derivative having Mittag–Leffler‐type kernel

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