On some new properties of fractional derivatives with Mittag-Leffler kernel

Băleanu, Dumitru; Fernandez, Arran

doi:10.1016/j.cnsns.2017.12.003

Cited by 270 publications

(175 citation statements)

References 39 publications

(88 reference statements)

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“…The following result has been proved for example in [34], using Laplace transforms, and also in [26] using only the definition of AB derivatives and integrals. …”

Section: The Mean Value Theoremmentioning

confidence: 99%

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The mean value theorem and Taylor’s theorem for fractional derivatives with Mittag–Leffler kernel

Fernandez

Băleanu

2018

Adv Differ Equ

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“…The following result has been proved for example in [34], using Laplace transforms, and also in [26] using only the definition of AB derivatives and integrals. …”

Section: The Mean Value Theoremmentioning

confidence: 99%

“…In each case the functions and variables used satisfy the following requirements [26]: Certain fundamental results of calculus have already been established in the AB model: Laplace transforms [7], integration by parts [27], the product rule and chain rule [26], etc. But as the idea is still so new, much remains to be done in this area.…”

Section: Introductionmentioning

confidence: 99%

The mean value theorem and Taylor’s theorem for fractional derivatives with Mittag–Leffler kernel

Fernandez

Băleanu

2018

Adv Differ Equ

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“…The optimal control problems and calculus of variation for variational equality with fractional time derivative with nonlocal and nonsingular Mittag-Leffler kernel are studied in many papers (see, for example, [1,2,10,11,21] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, inclusions differential equations seems to be an appropriate model to describe inclusions problems. Several articles have studied the existence of mild solutions and controllability problems for various types of integers (see [19][20][21][22][23][24][25][26][27][28][29][30][31][32]34] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Optimality conditions for fractional differential inclusions with nonsingular Mittag–Leffler kernel

Bahaa

Hamiaz

2018

Adv Differ Equ

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In this paper, by using the Dubovitskii-Milyutin theorem, we consider a differential inclusions problem with fractional-time derivative with nonsingular Mittag-Leffler kernel in Hilbert spaces. The Atangana-Baleanu fractional derivative of order α in the sense of Caputo with respect to time t, is considered. Existence and uniqueness of solution are proved by means of the Lions-Stampacchia theorem. The existence of solution is obtained for all values of the fractional parameter α ∈ (0, 1). Moreover, by applying control theory to the fractional differential inclusions problem, we obtain an optimality system which has also a unique solution. The controllability of the fractional Dirichlet problem is studied. Some examples are analyzed in detail. MSC: 46C05; 49J20; 93C20

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“…Non-local operators are grouped into three classes, including fractional differential operators with singular and non-local kernel, fractional differential operators with non-singular and local kernel, and fractional differential operators with non-singular and non-local kernel. Those with singular kernel were recently introduced and have been applied in several fields of science, technology and engineering, with great success [12][13][14][15][16][17][18]. One of their key properties is the crossover behaviour, described for given-range random walk, deterministic and stochastic behaviour.…”

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confidence: 99%

Focus Point on Modelling Complex Real-World Problems with Fractal and New Trends of Fractional Differentiation

et al. 2018

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The complexities of nature have led researchers to develop more sophisticated and complex mathematical operators to capture, or accurately replicate observed realities [1][2][3][4][5][6][7]. In the recent decades, it has been widely accepted that the differential operators, based on the concept of rate of change, are not always suitable candidates for modelling complex real-world problems, as they cannot capture heterogeneous behaviours [8][9][10][11]. Specific examples are provided by long-range, random walk, anomalous diffusion, non-Markovian processes, fractal behaviours, and groundwater flowing in non-conventional media. To accurately replicate these natural processes, the concept of non-local differential operators was suggested as well as that of local differential operators along with power law setting. Non-local operators are grouped into three classes, including fractional differential operators with singular and non-local kernel, fractional differential operators with non-singular and local kernel, and fractional differential operators with non-singular and non-local kernel. Those with singular kernel were recently introduced and have been applied in several fields of science, technology and engineering, with great success [12][13][14][15][16][17][18]. One of their key properties is the crossover behaviour, described for given-range random walk, deterministic and stochastic behaviour. For instance, those with Mittag-Leffler kernel have crossover behaviour from waiting time distribution, mean square displacement to density probability distribution. These operators do not impose singularities to any model, especially those with no singularities, as in the case of singular kernel operators. The Focus Point was devoted to creating a discussion and to collecting the latest innovative results, where non-singular kernel differential operators and local operators, based on power law setting, are applied to more complex problems arising in physics and other related topics. We invited top researchers in the field and received many submissions from all the continents. Of the total submitted papers, only 22 paper were accepted, providing a well-balanced representation of all continents. Sonal suggested a new numerical scheme for solving ordinary differential equations with power law setting differential operator [1]. A new numerical method for partial differential equations based on the two-step Laplace transform Adams-Bashforth approximation was applied by Jain and Alkahtani [2,3]. Owolabi presented the model of a dynamical system using the Atangana-Baleanu differential operator [4], and Ahmed et al.suggested a modified exponential rational function method for non-linear fractional differential equations [5]. Allwright applied a power-law setting differential operator to the advection dispersion equation and suggested a new numerical scheme [6]. Karaagac presented an analysis of the cable equation using the Atangana-Baleanu differential operator [7]. Atangana and Alqahtani studied a tumour model with intrusive mo...

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On some new properties of fractional derivatives with Mittag-Leffler kernel

Cited by 270 publications

References 39 publications

The mean value theorem and Taylor’s theorem for fractional derivatives with Mittag–Leffler kernel

The mean value theorem and Taylor’s theorem for fractional derivatives with Mittag–Leffler kernel

Optimality conditions for fractional differential inclusions with nonsingular Mittag–Leffler kernel

Focus Point on Modelling Complex Real-World Problems with Fractal and New Trends of Fractional Differentiation

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