scholarly journals A New Generalized Definition of Fractional Derivative with Non-Singular Kernel

Computation ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 49 ◽  
Author(s):  
Khalid Hattaf
Keyword(s):  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Singular Kernel ◽  
Fundamental Properties ◽  
Generalized Fractional Derivatives ◽  
Definition Of

This paper proposes a new definition of fractional derivative with non-singular kernel in the sense of Caputo which generalizes various forms existing in the literature. Furthermore, the version in the sense of Riemann–Liouville is defined. Moreover, fundamental properties of the new generalized fractional derivatives in the sense of Caputo and Riemann–Liouville are rigorously studied. Finally, an application in epidemiology as well as in virology is presented.

scholarly journals On Some Properties of the New Generalized Fractional Derivative with Non-Singular Kernel

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Khalid Hattaf
Keyword(s):  
Asymptotic Stability ◽  
Fractional Derivative ◽  
Lyapunov Functions ◽  
Fractional Derivatives ◽  
Singular Kernel ◽  
Fractional Order Systems ◽  
Science And Engineering ◽  
Generalized Fractional Derivatives ◽  
Derivatives Of ◽  
Generalized Fractional Derivative

This paper presents some new formulas and properties of the generalized fractional derivative with non-singular kernel that covers various types of fractional derivatives such as the Caputo–Fabrizio fractional derivative, the Atangana–Baleanu fractional derivative, and the weighted Atangana–Baleanu fractional derivative. These new properties extend many recent results existing in the literature. Furthermore, the paper proposes some interesting inequalities that estimate the generalized fractional derivatives of some specific functions. These inequalities can be used to construct Lyapunov functions with the aim to study the global asymptotic stability of several fractional-order systems arising from diverse fields of science and engineering.

scholarly journals A remark on local fractional calculus and ordinary derivatives

2016 ◽  
Vol 14 (1) ◽  
pp. 1122-1124 ◽  
Author(s):  
Ricardo Almeida ◽  
Małgorzata Guzowska ◽  
Tatiana Odzijewicz
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Short Note ◽  
General Definition ◽  
Local Fractional Derivative ◽  
Fundamental Properties ◽  
Local Fractional Calculus ◽  
Definition Of

AbstractIn this short note we present a new general definition of local fractional derivative, that depends on an unknown kernel. For some appropriate choices of the kernel we obtain some known cases. We establish a relation between this new concept and ordinary differentiation. Using such formula, most of the fundamental properties of the fractional derivative can be derived directly.

On Finite Part Integrals and Hadamard-Type Fractional Derivatives

2018 ◽  
Vol 13 (9) ◽  
Author(s):  
Li Ma ◽  
Changpin Li
Keyword(s):  
Fractional Derivative ◽  
Singular Integral ◽  
Fractional Derivatives ◽  
Continuous Functions ◽  
Finite Part ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
Fundamental Properties ◽  
Logarithmic Series ◽  
The Right

This paper is devoted to investigating the relation between Hadamard-type fractional derivatives and finite part integrals in Hadamard sense; that is to say, the Hadamard-type fractional derivative of a given function can be expressed by the finite part integral of a strongly singular integral, which actually does not exist. Besides, our results also cover some fundamental properties on absolutely continuous functions, and the logarithmic series expansion formulas at the right end point of interval for functions in certain absolutely continuous spaces.

scholarly journals FRACTIONAL DERIVATIVE AS FRACTIONAL POWER OF DERIVATIVE

2007 ◽  
Vol 18 (03) ◽  
pp. 281-299 ◽  
Author(s):  
VASILY E. TARASOV
Keyword(s):  
Fourier Series ◽  
Fractional Derivative ◽  
Taylor Series ◽  
Fractional Derivatives ◽  
Fractional Power ◽  
Weyl Quantization ◽  
Fractional Powers ◽  
Derivative Operator ◽  
Quantization Procedure ◽  
Definition Of

Definitions of fractional derivatives as fractional powers of derivative operators are suggested. The Taylor series and Fourier series are used to define fractional power of selfadjoint derivative operator. The Fourier integrals and Weyl quantization procedure are applied to derive the definition of fractional derivative operator. Fractional generalization of concept of stability is considered.

scholarly journals A New Conformable Fractional Derivative and Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Ahmed Kajouni ◽  
Ahmed Chafiki ◽  
Khalid Hilal ◽  
Mohamed Oukessou
Keyword(s):  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Conformable Fractional Derivative ◽  
First Order ◽  
Rolle’S Theorem ◽  
Power Rule ◽  
The Mean ◽  
Definition Of

This paper is motivated by some papers treating the fractional derivatives. We introduce a new definition of fractional derivative which obeys classical properties including linearity, product rule, quotient rule, power rule, chain rule, Rolle’s theorem, and the mean value theorem. The definition D α f t = lim h ⟶ 0 f t + h e α − 1 t − f t / h , for all t > 0 , and α ∈ 0,1 . If α = 0 , this definition coincides to the classical definition of the first order of the function f .

A New Look at the Fractional Brownian Motion Definition

2007 ◽  
Author(s):  
Manuel Duarte Ortigueira ◽  
Arnaldo Guimara˜es Batista
Keyword(s):  
Brownian Motion ◽  
Fractional Derivative ◽  
White Noise ◽  
Fractional Brownian Motion ◽  
Fractional Derivatives ◽  
Autocorrelation Functions ◽  
Fractional Noise ◽  
Simulation Procedure ◽  
Definition Of ◽  
Fractional Noises

A reinterpretation of the classic definition of fractional Brownian motion leads to a new definition involving a fractional noise obtained as a fractional derivative of white noise. To obtain this fractional noise, two sets of fractional derivatives are considered: a) the forward and backward and b) the central derivatives. For these derivatives the autocorrelation functions of the corresponding fractional noises have the same representations. The obtained results are used to define and propose a new simulation procedure.

scholarly journals Fractional Mass-Spring-Damper System Described by Generalized Fractional Order Derivatives

2019 ◽  
Vol 3 (3) ◽  
pp. 39 ◽  
Author(s):  
Ndolane Sene ◽  
José Francisco Gómez Aguilar
Keyword(s):  
Asymptotic Stability ◽  
Fractional Derivative ◽  
Analytical Solutions ◽  
Global Asymptotic Stability ◽  
Fractional Derivatives ◽  
Mass Damper ◽  
Generalized Fractional Derivatives ◽  
Fractional Mass ◽  
Mass Spring ◽  
Generalized Fractional Derivative

This paper proposes novel analytical solutions of the mass-spring-damper systems described by certain generalized fractional derivatives. The Liouville–Caputo left generalized fractional derivative and the left generalized fractional derivative were used. The behaviors of the analytical solutions of the mass-spring-damper systems described by the left generalized fractional derivative and the Liouville–Caputo left generalized fractional derivative were represented graphically and the effect of the orders of the fractional derivatives analyzed. We finish by analyzing the global asymptotic stability and the converging-input-converging-state of the unforced mass-damper system, the unforced spring-damper, the spring-damper system, and the mass-damper system.

scholarly journals New Approach for the Analysis of Damped Vibrations of Fractional Oscillators

2009 ◽  
Vol 16 (4) ◽  
pp. 365-387 ◽  
Author(s):  
Yuriy A. Rossikhin ◽  
Marina V. Shitikova
Keyword(s):  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Time Derivative ◽  
Fractional Power ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Derivative Term ◽  
Mechanical Oscillators ◽  
Definition Of ◽  
Nonlinear Elastic

The dynamic behavior of linear and nonlinear mechanical oscillators with constitutive equations involving fractional derivatives defined as a fractional power of the operator of conventional time-derivative is considered. Such a definition of the fractional derivative enables one to analyse approximately vibratory regimes of the oscillator without considering the drift of its position of equilibrium. The assumption of small fractional derivative terms allows one to use the method of multiple time scales whereby a comparative analysis of the solutions obtained for different orders of low-level fractional derivatives and nonlinear elastic terms is possible to be carried out. The interrelationship of the fractional parameter (order of the fractional operator) and nonlinearity manifests itself in full measure when orders of the small fractional derivative term and of the cubic nonlinearity entering in the oscillator's constitutive equation coincide.

Comparative analysis of suitability of fractional derivatives in modelling the practical capacitor

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Rawid Banchuin
Keyword(s):  
Comparative Analysis ◽  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Fractional Power ◽  
Electrolytic Capacitor ◽  
Content Type ◽  
Local Fractional Derivative ◽  
Definition Of ◽  
Double Layer Capacitors ◽  
Derivatives Of

Purpose The purpose of this paper is to compare the suitability of fractional derivatives in the modelling of practical capacitors. Such suitability refers to ability to provide the analytical capacitance function that matches the experimental ones of each fractional derivative. Design/methodology/approach The analytical capacitance functions based on various fractional derivatives of both local and nonlocal types including the author’s have been derived. The derived capacitance functions have been simulated and compared with the experimental ones of aluminium electrolytic and electrical double layer capacitors (EDLCs). Findings This paper has found that any local fractional derivative with fractional power law-based relationship with the conventional one is suitable for modelling the aluminium electrolytic capacitor (AEC) by incorporating with the conventional capacitance definition. On the other hand, the author’s nonlocal fractional derivatives have been found to be more suitable than the others for modelling the EDLC by incorporating with the revisited definition of capacitance. Originality/value The proposed comparative analysis has been originally presented in this work. The criterion for local fractional derivative, to be suitable for modelling the AEC, has been found. The nonlocal fractional operators which are most suitable for modelling the EDLC have been derived where the unsuitable one has been pointed out.

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