scholarly journals Why fractional derivatives with nonsingular kernels should not be used

2020 ◽  
Vol 23 (3) ◽  
pp. 610-634 ◽  
Author(s):  
Kai Diethelm ◽  
Roberto Garrappa ◽  
Andrea Giusti ◽  
Martin Stynes
Keyword(s):  
Fractional Calculus ◽  
Fundamental Theorem ◽  
Fractional Derivatives ◽  
Mathematical Reasoning ◽  
Singular Kernel ◽  
Initial Time ◽  
Left Inverse ◽  
Convolution Integral ◽  
Caputo Fractional Derivatives ◽  
Liouville Derivative

AbstractIn recent years, many papers discuss the theory and applications of new fractional-order derivatives that are constructed by replacing the singular kernel of the Caputo or Riemann-Liouville derivative by a non-singular (i.e., bounded) kernel. It will be shown here, through rigorous mathematical reasoning, that these non-singular kernel derivatives suffer from several drawbacks which should forbid their use. They fail to satisfy the fundamental theorem of fractional calculus since they do not admit the existence of a corresponding convolution integral of which the derivative is the left-inverse; and the value of the derivative at the initial time t = 0 is always zero, which imposes an unnatural restriction on the differential equations and models where these derivatives can be used. For the particular cases of the so-called Caputo-Fabrizio and Atangana-Baleanu derivatives, it is shown that when this restriction holds the derivative can be simply expressed in terms of integer derivatives and standard Caputo fractional derivatives, thus demonstrating that these derivatives contain nothing new.

scholarly journals Fractional derivatives and the fundamental theorem of fractional calculus

2020 ◽  
Vol 23 (4) ◽  
pp. 939-966 ◽  
Author(s):  
Yuri Luchko
Keyword(s):  
Fractional Calculus ◽  
Fundamental Theorem ◽  
Fractional Derivatives ◽  
Finite Interval ◽  
Fractional Integrals ◽  
Important Class ◽  
Left Inverse ◽  
Caputo Fractional Derivatives ◽  
Fractional Integrals And Derivatives ◽  
The One

AbstractIn this paper, we address the one-parameter families of the fractional integrals and derivatives defined on a finite interval. First we remind the reader of the known fact that under some reasonable conditions, there exists precisely one unique family of the fractional integrals, namely, the well-known Riemann-Liouville fractional integrals. As to the fractional derivatives, their natural definition follows from the fundamental theorem of the Fractional Calculus, i.e., they are introduced as the left-inverse operators to the Riemann-Liouville fractional integrals. Until now, three families of such derivatives were suggested in the literature: the Riemann-Liouville fractional derivatives, the Caputo fractional derivatives, and the Hilfer fractional derivatives. We clarify the interconnections between these derivatives on different spaces of functions and provide some of their properties including the formulas for their projectors and the Laplace transforms. However, it turns out that there exist infinitely many other families of the fractional derivatives that are the left-inverse operators to the Riemann-Liouville fractional integrals. In this paper, we focus on an important class of these fractional derivatives and discuss some of their properties.

On Fractional Hamilton Formulation Within Caputo Derivatives

2007 ◽  
Author(s):  
Dumitru Baleanu ◽  
Sami I. Muslih ◽  
Eqab M. Rabei
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivatives ◽  
Variational Principles ◽  
Hamiltonian Dynamics ◽  
Fractional Integration ◽  
Fractional Dynamics ◽  
Classical Dynamics ◽  
Lagrange Equations ◽  
Caputo Fractional Derivatives ◽  
Non Locality

The fractional Lagrangian and Hamiltonian dynamics is an important issue in fractional calculus area. The classical dynamics can be reformulated in terms of fractional derivatives. The fractional variational principles produce fractional Euler-Lagrange equations and fractional Hamiltonian equations. The fractional dynamics strongly depends of the fractional integration by parts as well as the non-locality of the fractional derivatives. In this paper we present the fractional Hamilton formulation based on Caputo fractional derivatives. One example is treated in details to show the characteristics of the fractional dynamics.

scholarly journals On representation and interpretation of Fractional calculus and fractional order systems

2019 ◽  
Vol 22 (2) ◽  
pp. 522-537
Author(s):  
Juan Paulo García-Sandoval
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Extra Dimension ◽  
Fractional Derivatives ◽  
Area Under The Curve ◽  
Geometric Interpretation ◽  
Order Partial Differential Equation ◽  
Fractional Order Systems ◽  
Caputo Fractional Derivatives ◽  
Fractional Derivatives And Integrals

Abstract In this work a relationship between Fractional calculus (FC) and the solution of a first order partial differential equation (FOPDE) is suggested. With this relationship and considering an extra dimension, an alternative representation for fractional derivatives and integrals is proposed. This representation can be applied to fractional derivatives and integrals defined by convolution integrals of the Volterra type, i.e. the Riemann-Liouville and Caputo fractional derivatives and integrals, and the Riesz and Feller potentials, and allows to transform fractional order systems in FOPDE that only contains integer-order derivatives. As a consequence of considering the extra dimension, the geometric interpretation of fractional derivatives and integrals naturally emerges as the area under the curve of a characteristic trajectory and as the direction of a tangent characteristic vector, respectively. Besides this, a new physical interpretation is suggested for the fractional derivatives, integrals and dynamical systems.

scholarly journals APPLICATIONS OF TWO IMPORTANT THEOREMS OF FRACTIONAL CALCULUS

2021 ◽  
Author(s):  
CHII-HUEI YU
Keyword(s):  
Fractional Calculus ◽  
Fundamental Theorem ◽  
Fractional Derivatives ◽  
The Other ◽  
Integration By Parts ◽  
Other Hand

Abstract. In this article, we study the fundamental theorem of fractional calculus and integration by parts for fractional calculus, regarding the Jumarie type of modified Riemann-Liouville fractional derivatives. On the other hand, some examples are proposed to illustrate the applications of these two important theorems.

scholarly journals Towards a combined fractional mechanics and quantization

2012 ◽  
Vol 15 (3) ◽  
Author(s):  
Agnieszka Malinowska ◽  
Delfim Torres
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Fractional Calculus ◽  
Calculus Of Variations ◽  
Fractional Derivatives ◽  
Hamiltonian Formalism ◽  
Partial Differential ◽  
Caputo Fractional Derivatives ◽  
Fractional Mechanics ◽  
Fractional Calculus Of Variations

AbstractA fractional Hamiltonian formalism is introduced for the recent combined fractional calculus of variations. The Hamilton-Jacobi partial differential equation is generalized to be applicable for systems containing combined Caputo fractional derivatives. The obtained results provide tools to carry out the quantization of nonconservative problems through combined fractional canonical equations of Hamilton type.

Lagrangians With Linear Velocities Within Hilfer Fractional Derivative

2011 ◽  
Author(s):  
Dumitru Baleanu ◽  
Om P. Agrawal ◽  
Sami I. Muslih
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Variational Principles ◽  
Major Area ◽  
Lagrange Equations ◽  
Hilfer Fractional Derivative ◽  
Caputo Fractional Derivatives ◽  
Generalized Fractional Derivative

Fractional variational principles started to be one of the major area in the field of fractional calculus. During the last few years the fractional variational principles were developed within several fractional derivatives. One of them is the Hilfer’s generalized fractional derivative which interpolates between Riemann-Liouville and Caputo fractional derivatives. In this paper the fractional Euler-Lagrange equations of the Lagrangians with linear velocities are obtained within the Hilfer fractional derivative.

Solution of inverse coefficient problem for fractional anomalous diffusion equation

2012 ◽  
Vol 9 (2) ◽  
pp. 65-70
Author(s):  
E.V. Karachurina ◽  
S.Yu. Lukashchuk
Keyword(s):  
Anomalous Diffusion ◽  
Fractional Derivatives ◽  
Descent Method ◽  
Extremum Problem ◽  
Steepest Descent Method ◽  
Coefficient Problem ◽  
Caputo Fractional Derivatives ◽  
Inverse Coefficient Problem ◽  
Residual Functional ◽  
Inverse Coefficient

An inverse coefficient problem is considered for time-fractional anomalous diffusion equations with the Riemann-Liouville and Caputo fractional derivatives. A numerical algorithm is proposed for identification of anomalous diffusivity which is considered as a function of concentration. The algorithm is based on transformation of inverse coefficient problem to extremum problem for the residual functional. The steepest descent method is used for numerical solving of this extremum problem. Necessary expressions for calculating gradient of residual functional are presented. The efficiency of the proposed algorithm is illustrated by several test examples.

scholarly journals Stability Analysis and Existence of Solutions for a Modified SIRD Model of COVID-19 with Fractional Derivatives

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1431
Author(s):  
Bilal Basti ◽  
Nacereddine Hammami ◽  
Imadeddine Berrabah ◽  
Farid Nouioua ◽  
Rabah Djemiat ◽  
...  
Keyword(s):  
Mathematical Model ◽  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Fixed Point Theorems ◽  
Existence Of Solutions ◽  
Biological Models ◽  
Uniqueness Of Solutions ◽  
Caputo Fractional Derivatives ◽  
Single Form ◽  
Stability Results

This paper discusses and provides some analytical studies for a modified fractional-order SIRD mathematical model of the COVID-19 epidemic in the sense of the Caputo–Katugampola fractional derivative that allows treating of the biological models of infectious diseases and unifies the Hadamard and Caputo fractional derivatives into a single form. By considering the vaccine parameter of the suspected population, we compute and derive several stability results based on some symmetrical parameters that satisfy some conditions that prevent the pandemic. The paper also investigates the problem of the existence and uniqueness of solutions for the modified SIRD model. It does so by applying the properties of Schauder’s and Banach’s fixed point theorems.

scholarly journals Ulam–Hyers–Mittag-Leffler stability for tripled system of weighted fractional operator with TIME delay

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mohammed A. Almalahi ◽  
Satish K. Panchal ◽  
Fahd Jarad ◽  
Thabet Abdeljawad
Keyword(s):  
Fixed Point ◽  
Time Delay ◽  
Fractional Derivatives ◽  
Fixed Point Theorems ◽  
Sufficient Conditions ◽  
Fractional Operator ◽  
Caputo Fractional Derivatives ◽  
Picard Operator

AbstractThis study is aimed to investigate the sufficient conditions of the existence of unique solutions and the Ulam–Hyers–Mittag-Leffler (UHML) stability for a tripled system of weighted generalized Caputo fractional derivatives investigated by Jarad et al. (Fractals 28:2040011 2020) in the frame of Chebyshev and Bielecki norms with time delay. The acquired results are obtained by using Banach fixed point theorems and the Picard operator (PO) method. Finally, a pertinent example of the results obtained is demonstrated.

Mkhitar Djrbashian and his contribution to Fractional Calculus

2020 ◽  
Vol 23 (6) ◽  
pp. 1797-1809
Author(s):  
Sergei Rogosin ◽  
Maryna Dubatovskaya
Keyword(s):  
Cauchy Problem ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Approximation Theory ◽  
Fractional Derivatives ◽  
Complex Analysis ◽  
Integral Transforms ◽  
Modern Theory ◽  
Survey Paper ◽  
The Cauchy Problem

Abstract This survey paper is devoted to the description of the results by M.M. Djrbashian related to the modern theory of Fractional Calculus. M.M. Djrbashian (1918-1994) is a well-known expert in complex analysis, harmonic analysis and approximation theory. Anyway, his contributions to fractional calculus, to boundary value problems for fractional order operators, to the investigation of properties of the Queen function of Fractional Calculus (the Mittag-Leffler function), to integral transforms’ theory has to be understood on a better level. Unfortunately, most of his works are not enough popular as in that time were published in Russian. The aim of this survey is to fill in the gap in the clear recognition of M.M. Djrbashian’s results in these areas. For same purpose, we decided also to translate in English one of his basic papers [21] of 1968 (joint with A.B. Nersesian, “Fractional derivatives and the Cauchy problem for differential equations of fractional order”), and were invited by the “FCAA” editors to publish its re-edited version in this same issue of the journal.

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