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.

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.

scholarly journals An introduction to fractional calculus

2020 ◽  
Vol 9 (1) ◽  
pp. 19-30
Author(s):  
A. Persechino
Keyword(s):  
Fractional Calculus ◽  
High Frequency ◽  
Dielectric Response ◽  
Numerical Models ◽  
Fractional Integration ◽  
Classical Electrodynamics ◽  
Fractional Dynamics ◽  
Theoretical Discussion ◽  
Comprehensive Review ◽  
Non Locality

The aim of this work is to introduce the main concepts of Fractional Calculus, followed by one of its application to classical electrodynamics, illustrating how non-locality can be interpreted naturally in a fractional scenario. In particular, a result relating fractional dynamics to high frequency dielectric response is used as motivation. In addition to the theoretical discussion, a comprehensive review of two numerical procedures for fractional integration is carried out, allowing one immediately to build numerical models applied to high frequency electromagnetics and correlated fields.

scholarly journals Continuous and discrete Noether's fractional conserved quantities for restricted calculus of variations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jacky Cresson ◽  
Fernando Jiménez ◽  
Sina Ober-Blöbaum
Keyword(s):  
Fractional Calculus ◽  
Calculus Of Variations ◽  
Fractional Derivatives ◽  
Dissipative Systems ◽  
Alternative Formulation ◽  
Conserved Quantities ◽  
Lagrange Equations ◽  
Discrete Counterpart ◽  
Fractional Calculus Of Variations

<p style='text-indent:20px;'>We prove a Noether's theorem of the first kind for the so-called <i>restricted fractional Euler-Lagrange equations</i> and their discrete counterpart, introduced in [<xref ref-type="bibr" rid="b26">26</xref>,<xref ref-type="bibr" rid="b27">27</xref>], based in previous results [<xref ref-type="bibr" rid="b11">11</xref>,<xref ref-type="bibr" rid="b35">35</xref>]. Prior, we compare the restricted fractional calculus of variations to the <i>asymmetric fractional calculus of variations</i>, introduced in [<xref ref-type="bibr" rid="b14">14</xref>], and formulate the restricted calculus of variations using the <i>discrete embedding</i> approach [<xref ref-type="bibr" rid="b12">12</xref>,<xref ref-type="bibr" rid="b18">18</xref>]. The two theories are designed to provide a variational formulation of dissipative systems, and are based on modeling irreversbility by means of fractional derivatives. We explicit the role of time-reversed solutions and causality in the restricted fractional calculus of variations and we propose an alternative formulation. Finally, we implement our results for a particular example and provide simulations, actually showing the constant behaviour in time of the discrete conserved quantities outcoming the Noether's theorems.</p>

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.

Fractional Mechanics on the Extended Phase Space

2009 ◽  
Author(s):  
Dumitru Baleanu ◽  
Sami I. Muslih ◽  
Alireza K. Golmankhaneh ◽  
Ali K. Golmankhaneh ◽  
Eqab M. Rabei
Keyword(s):  
Phase Space ◽  
Variational Principles ◽  
Fractional Dynamics ◽  
Classical Dynamics ◽  
Extended Phase Space ◽  
Science And Engineering ◽  
Extended Phase ◽  
Potential Applications ◽  
Fractional Mechanics ◽  
Complex Phenomena

Fractional calculus has gained a lot of importance and potential applications in several areas of science and engineering. The fractional dynamics and the fractional variational principles started to be used intensively as an alternative tool in order to describe the physical complex phenomena. In this paper we have discussed the fractional extension of the classical dynamics. The fractional Hamiltonian is constructed and the fractional generalized Poisson’s brackets on the extended phase space is established.

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.

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.

Formulation of Euler–Lagrange Equations for Multidelay Fractional Optimal Control Problems

2018 ◽  
Vol 13 (6) ◽  
Author(s):  
Sohrab Effati ◽  
Seyed Ali Rakhshan ◽  
Samane Saqi
Keyword(s):  
Optimal Control ◽  
Numerical Scheme ◽  
Fractional Derivatives ◽  
Optimal Control Problems ◽  
Fractional Integration ◽  
Control Problems ◽  
Algebraic Equations ◽  
Lagrange Equations ◽  
Fractional Optimal Control ◽  
Fractional Optimal Control Problems

In this paper, a new numerical scheme is proposed for multidelay fractional order optimal control problems where its derivative is considered in the Grunwald–Letnikov sense. We develop generalized Euler–Lagrange equations that results from multidelay fractional optimal control problems (FOCP) with final terminal. These equations are created by using the calculus of variations and the formula for fractional integration by parts. The derived equations are then reduced into system of algebraic equations by using a Grunwald–Letnikov approximation for the fractional derivatives. Finally, for confirming the accuracy of the proposed approach, some illustrative numerical examples are solved.

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.

Initialization of Riemann-Liouville and Caputo Fractional Derivatives

2011 ◽  
Author(s):  
Jean-Claude Trigeassou ◽  
Nezha Maamri ◽  
Alain Oustaloup
Keyword(s):  
Numerical Simulations ◽  
State Vector ◽  
Fractional Derivatives ◽  
Initial Conditions ◽  
Fractional Integration ◽  
Initial State ◽  
Caputo Fractional Derivatives ◽  
Infinite Dimensional ◽  
Fractional Integrator ◽  
Initial Conditions Problem

Riemann-Liouville and Caputo fractional derivatives are fundamentally related to fractional integration operators. Consequently, the initial conditions of fractional derivatives are the frequency distributed and infinite dimensional state vector of fractional integrators. The paper is dedicated to the estimation of these initial conditions and to the validation of the initialization problem based on this distributed state vector. Numerical simulations applied to Riemann-Liouville and Caputo derivatives demonstrate that the initial conditions problem can be solved thanks to the estimation of the initial state vector of the fractional integrator.

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