Generalization of Caputo-Fabrizio Fractional Derivative and Applications to Electrical Circuits

Alshabanat, Amal; Jleli, Mohamed; Kumar, Sunil; Samet, Bessem

doi:10.3389/fphy.2020.00064

Cited by 116 publications

(66 citation statements)

References 28 publications

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“…On the other hand, fractional calculus and fractional differential equations have recently been applied in various areas of engineering, mathematics, physics and bio-engineering, and other applied sciences. We refer the reader to the monographs [4][5][6][7] and the articles [8,9]. In particular, there are a growing number of research areas in physics which employ fractional calculus [10] and it has many applications among its different branches, ranging from imaging processing to fractional quantum harmonic oscillator [11].…”

Section: Introductionmentioning

confidence: 99%

Basic Control Theory for Linear Fractional Differential Equations With Constant Coefficients

Buedo‐Fernández

Nieto

2020

Front. Phys.

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In this paper we present an analogous result of the famous Kalman controllability criterion for first order linear ordinary differential equations with constant coefficients that applies to the case of linear differential equations of fractional order with constant coefficients. We use the fractional Gramian matrix, the range space and the Kalman matrix as main tools to derive a sufficient and necessary condition for the controllability of the fractional system. Moreover, we provide some simple examples, including a linear fractional harmonic oscillator, to illustrate our results. Finally, several open problems arising from this topic are suggested, including another simple linear system of incommensurate fractional orders.

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Section: Introductionmentioning

confidence: 99%

Basic Control Theory for Linear Fractional Differential Equations With Constant Coefficients

Buedo‐Fernández

Nieto

2020

Front. Phys.

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“…Some works addressed unsteady governing equations for stretching permeable sheets using HAM (Homotopy Analysis Method) [44][45][46][47]. Some other interesting analytical solution for different applicable mathematical problems could be found in [48][49][50][51][52][53][54][55][56].…”

Section: Introductionmentioning

confidence: 99%

Pulsating flow in a channel filled with a porous medium under local thermal non-equilibrium condition: an exact solution

Nazari

Mahmoudi

2020

J Therm Anal Calorim

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The present work investigates analytically the problem of forced convection heat transfer of a pulsating flow, in a channel filled with a porous medium under local thermal non-equilibrium condition. Internal heat generation is considered in the porous medium, and the channel walls are subjected to constant heat flux boundary condition. Exact solutions are obtained for velocity, Nusselt number and temperature distributions of the fluid and solid phases in the porous medium. The influence of pertinent parameters, including Biot number, Darcy number, fluid-to-solid effective thermal conductivity ratio and Prandtl number are discussed. The applied pressure gradient is considered in a sinusoidal waveform. The effect of dimensionless frequency and coefficient of the pressure amplitude on the system's velocity and temperature fields are discussed. The general shape of the unsteady velocity for different times is found to be very similar to the steady data. Results show that the amplitudes of the unsteady temperatures for the fluid and solid phases decrease with the increase in Biot number or thermal conductivity ratio. For large Biot numbers, dimensionless temperatures of the solid and fluid phases are similar and are close to their steady counterparts. Results for the Nusselt number indicate that increasing Biot number or thermal conductivity ratio decreases the amplitude of Nusselt number. Increase in the internal heat generation in the solid phase does not have a significant influence on the ratio of amplitude-to-mean value of the Nusselt number, while internal heat generation in the fluid phase enhances this ratio. Keywords Porous media • Pulsating flow • Convective heat transfer • Local thermal non-equilibrium • Analytical solution List of symbols a n , a j Time-dependent coefficient a sf Interfacial area per unit volume of porous media (m −1) A A defined function of time A 1 ,A 2 Constant coefficients b m,n Time-dependent coefficient B A defined function of time B 1 ,B 2 Constant coefficients Bi Biot number, Bi = a sf h sf H 2 k s,eff c pf Fluid specific heat (J kg −1 K −1) c ps Solid specific heat (J kg −1 K −1) C Constant parameter, C =

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“…In their recent work, Caputo and Fabrizio [7] introduced a new fractional-order derivative with a nonsingular kernel, hereinafter called the fractional Caputo-Fabrizio (CF) derivative. This new fractional derivative is less affected by the past compared to the Caputo fractional derivative, which may exhibit slow stabilization [1,33]. The properties and numerical aspects of the CF derivative and their corresponding fractional integrals been studied in [2,6,8,11,18,30,38].…”

Section: Introductionmentioning

confidence: 99%

Stability analysis of fractional-order linear neutral delay differential–algebraic system described by the Caputo–Fabrizio derivative

Al-Sawoor

2020

Adv Differ Equ

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This paper is concerned with the asymptotic stability of linear fractional-order neutral delay differential–algebraic systems described by the Caputo–Fabrizio (CF) fractional derivative. A novel characteristic equation is derived using the Laplace transform. Based on an algebraic approach, stability criteria are established. The effect of the index on such criteria is analyzed to ensure the asymptotic stability of the system. It is shown that asymptotic stability is ensured for the index-1 problems provided that a stability criterion holds for any delay parameter. Also, asymptotic stability is still valid for higher-index problems under the conditions that the system matrices have common eigenvectors and each pair of such matrices is simultaneously triangularizable so that a stability criterion holds for any delay parameter. An example is provided to demonstrate the effectiveness and applicability of the theoretical results.

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Generalization of Caputo-Fabrizio Fractional Derivative and Applications to Electrical Circuits

Cited by 116 publications

References 28 publications

Basic Control Theory for Linear Fractional Differential Equations With Constant Coefficients

Basic Control Theory for Linear Fractional Differential Equations With Constant Coefficients

Pulsating flow in a channel filled with a porous medium under local thermal non-equilibrium condition: an exact solution

Stability analysis of fractional-order linear neutral delay differential–algebraic system described by the Caputo–Fabrizio derivative

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