On some analytic properties of tempered fractional calculus

Fernandez, Arran; Ustaoğlu, Ceren

doi:10.1016/j.cam.2019.112400

Cited by 53 publications

(31 citation statements)

References 25 publications

(45 reference statements)

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“…Remark 2. Note that, in some works (for example, see [8][9][10]), the so-called tempered fractional integral and tempered fractional derivative are applied and defined by the following:…”

Section: Preliminary Resultsmentioning

confidence: 99%

Quadratic Lyapunov Functions for Stability of the Generalized Proportional Fractional Differential Equations with Applications to Neural Networks

Almeida

Agarwal

Hristova

et al. 2021

Axioms

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A fractional model of the Hopfield neural network is considered in the case of the application of the generalized proportional Caputo fractional derivative. The stability analysis of this model is used to show the reliability of the processed information. An equilibrium is defined, which is generally not a constant (different than the case of ordinary derivatives and Caputo-type fractional derivatives). We define the exponential stability and the Mittag–Leffler stability of the equilibrium. For this, we extend the second method of Lyapunov in the fractional-order case and establish a useful inequality for the generalized proportional Caputo fractional derivative of the quadratic Lyapunov function. Several sufficient conditions are presented to guarantee these types of stability. Finally, two numerical examples are presented to illustrate the effectiveness of our theoretical results.

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“…Remark 2. Note that, in some works (for example, see [8][9][10]), the so-called tempered fractional integral and tempered fractional derivative are applied and defined by the following:…”

Section: Preliminary Resultsmentioning

confidence: 99%

Quadratic Lyapunov Functions for Stability of the Generalized Proportional Fractional Differential Equations with Applications to Neural Networks

Almeida

Agarwal

Hristova

et al. 2021

Axioms

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“…In addition to the most popular fractional derivatives, the Liouville-Caputo, Riemann-Liouville, and Hadamard fractional operators, the emergence of new ones, such as the Hilfer, Katugampola, Caputo-Fabrizio, and generalized proportional fractional operators, has enriched the research in the topic; see [12][13][14][15][16][17] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

On the Initial Value Problems for Caputo-Type Generalized Proportional Vector-Order Fractional Differential Equations

Abbas

Hristova

2021

Mathematics

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A generalized proportional vector-order fractional derivative in the Caputo sense is defined and studied. Two types of existence results for the mild solutions of the initial value problem for nonlinear Caputo-type generalized proportional vector-order fractional differential equations are obtained. With the aid of the Leray–Schauder nonlinear alternative and the Banach contraction principle, the main results are established. In the case of a local Lipschitz right hand side part function, the existence of a bounded mild solution is proved. Some examples illustrating the main results are provided.

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“…Moreover, some of the definitions provide useful expansion that reduces the computational complexity of fractional order differential equations. For instance, Laplace expansion of Caputo fractional derivative [9] , tempered fractional derivative [10] and proportional fractional derivative [11] etc., have added a great contribution in fractional calculus. The operators of fractional calculus have been broadly functional in many fields such as, engineering and sciences [11] , [12] , [13] , [14] .…”

Section: Introductionmentioning

confidence: 99%

Behavioral response of population on transmissibility and saturation incidence of deadly pandemic through fractional order dynamical system

et al. 2021

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The world entered in another wave of the SARS-CoV-2 due to non-compliance of standard operating procedures appropriately, initiated by respective governments. Apparently, measures like using face masks and social distancing were not observed by populace that ultimately worsens the situation. The behavioral response of the population induces a change in the dynamical outcomes of the pandemic, which is documented in this paper for all intents and purposes. The innovative perception is executed through a compartmental model with the incorporation of fractional calculus and saturation incident rate. In the first instance, the epidemiological model is designed with proportional fractional definition considering the compartmental individuals of susceptible, social distancing, exposed, quarantined, infected, isolated and recovered populations. By virtue of proportional fractional derivative, effective dynamical outcomes of equilibrium states and basic reproduction number are successfully elaborated with memory effect. The expansion of this derivative greatly simplifies the model to integer order while remaining in the fractional context. Subsequently, the memory effects on the asymptotic profiles are demonstrated through various graphical plots and tabulated values. In addition, the inclusion of saturation incident rate further explains the transmissibility of infection for different behavior of susceptible individuals. Mathematically, the results are also validated through comparative analysis of values with the solutions attained from fractional fourth order Runge-Kutta method (FRK4).

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On some analytic properties of tempered fractional calculus

Cited by 53 publications

References 25 publications

Quadratic Lyapunov Functions for Stability of the Generalized Proportional Fractional Differential Equations with Applications to Neural Networks

Quadratic Lyapunov Functions for Stability of the Generalized Proportional Fractional Differential Equations with Applications to Neural Networks

On the Initial Value Problems for Caputo-Type Generalized Proportional Vector-Order Fractional Differential Equations

Behavioral response of population on transmissibility and saturation incidence of deadly pandemic through fractional order dynamical system

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