On a Differential Equation with Caputo-Fabrizio Fractional Derivative of Order 1 &lt;β ≤ 2 and Application to Mass-Spring-Damper System

Al-Salti, Nasser; Karimov, Erkinjon; Sadarangani, Kishin

doi:10.18576/pfda/020403

Cited by 56 publications

(21 citation statements)

References 15 publications

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“…So the integral in (7) converges if and only if t a f (x)(t − x) α−1 dx converges, i.e. if and only if the RL integral D −α a+ f (t) is well-defined.…”

Section: A New Formula For the Fractional Derivative With Mittag-leffmentioning

confidence: 99%

“…The function E α −α 1−α (t− x) α and its t-derivative, considered as functions of x, are holomorphic at every point in the interval [a, t). And the interval of integration is finite, so the only way the integral could possibly diverge would be due to behaviour near x = t. Thus the conditions for the ABR derivative to be well-defined are exactly that the integral in (7) should behave well as x → t from below. As x → t, we have (t − x) α → 0 and therefore…”

Section: A New Formula For the Fractional Derivative With Mittag-leffmentioning

confidence: 99%

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

Băleanu

Fernandez

2018

Communications in Nonlinear Science and Numerical Simulation

274

161

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We establish a new formula for the fractional derivative with Mittag-Leffler kernel, in the form of a series of Riemann-Liouville fractional integrals, which brings out more clearly the non-locality of fractional derivatives and is easier to handle for certain computational purposes. We also prove existence and uniqueness results for certain families of linear and nonlinear fractional ODEs defined using this fractional derivative. We consider the possibility of a semigroup property for these derivatives, and establish extensions of the product rule and chain rule, with an application to fractional mechanics.

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“…So the integral in (7) converges if and only if t a f (x)(t − x) α−1 dx converges, i.e. if and only if the RL integral D −α a+ f (t) is well-defined.…”

Section: A New Formula For the Fractional Derivative With Mittag-leffmentioning

confidence: 99%

Section: A New Formula For the Fractional Derivative With Mittag-leffmentioning

confidence: 99%

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

Băleanu

Fernandez

2018

Communications in Nonlinear Science and Numerical Simulation

274

161

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“…The basic challenge they were addressing was whether it is possible to construct another type of fractional operator which has nonsingular kernel and which can better describe in some cases the dynamics of nonlocal phenomena. The Caputo-Fabrizio definition has already found applications in areas such as diffusion modeling [39] and mass-spring-damper systems [8]. However, some issues were pointed out against both derivatives, including one in the Caputo sense and one in the Riemann-Liouville sense.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, as far as the optimal control of problem (1.1)-(1.3) is concerned, one can refer to the methods of the Lagrange multiplier technique for the classical Caputo and Riemann-Liouville fractional time derivative presented by different authors (see, e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]26] 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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“…The basic challenge they were addressing was whether it is possible to construct another type of fractional operator which has nonsingular kernel and which can better describe in some cases the dynamics of non-local phenomena. The Caputo-Fabrizio definition has already found applications in areas such as diffusion modelling [34] and mass-spring-damper systems [8]. However, some issues were pointed out against both derivatives, including the one in Caputo sense and the one in Riemann-Liouville sense.…”

mentioning

confidence: 99%

Generalized variational calculus in terms of multi-parameters involving Atangana-Baleanu's derivatives and application

Bahaa¹

2020

Discrete &Amp; Continuous Dynamical Systems - S

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In this paper, the generalized variational calculus in terms of multi-parameters involving Atangana-Baleanu's Derivatives are discussed. We consider the Hilfers generalized fractional derivative that in sense Atangana-Baleanu derivatives. We develop integration by parts formulas for the generalized fractional derivatives which are key to developing fractional variational calculus. It is shown that many derivatives used recently and their variational formulations can be obtained by setting different parameters to different values. The fractional Euler-Lagrange equations of fractional Lagrangians for constrained systems contains a fractional Hilfer-Atangana-Baleanu's derivatives with multi parameters are investigated. We also define fractional generalized momenta and provide fractional Hamiltonian formulations in terms of the new generalized derivatives. An example is presented to show applications of the formulations presented here. Some possible extensions of this research are also discussed. We present a general formulation and a solution scheme for a class of Fractional Optimal Control Problems (FOCPs) for those systems. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of FDEs. The calculus of variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler-Lagrange equations for the FOCP.

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On a Differential Equation with Caputo-Fabrizio Fractional Derivative of Order 1 <β ≤ 2 and Application to Mass-Spring-Damper System

Cited by 56 publications

References 15 publications

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

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

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

Generalized variational calculus in terms of multi-parameters involving Atangana-Baleanu's derivatives and application

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