scholarly journals Approximate construction of new conservative physical magnitudes through the fractional derivative of polynomialtype functions: a particular case in semiconductors of type AlxGa1-xAs

2020 ◽  
Vol 66 (6 Nov-Dec) ◽  
pp. 874
Author(s):  
J. C. Campos-García ◽  
M. E. Molinar-Tabares ◽  
C. Figueroa-Navarro ◽  
L. Castro-Arce
Keyword(s):  
Fractional Calculus ◽  
Potential Energy ◽  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Natural Sciences ◽  
Semiconductor Structures ◽  
Approximate Construction ◽  
Self Similar ◽  
The Many ◽  
Derivatives Of

The fractional calculus has a very large diversification as it relates to applications from physical interpretations to experimental facts to modeling of new problems in the natural sciences. Within the framework of a recently published article, we obtained the fractional derivative of the variable concentration x (z), the effective mass of the electron dependent on the position m (z) and the potential energy V (z), produced by the confinement of the electron in a semiconductor of type AlxGa1-xAs, with which we can intuit a possible geometric and physical interpretation. As a consequence, it is proposed the existence of three physical and geometric conservative quantities approximate character, associated with each of these parameters of the semiconductor, which add to the many physical magnitudes that already exist in the literature within the context of fractional variation rates. Likewise, we find that the fractional derivatives of these magnitudes, apart from having a common critical point, manifest self-similar behavior, which could characterize them as a type of fractal associated with the type of semiconductor structures under study.

scholarly journals Fractional Derivatives and Projectile Motion

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 297
Author(s):  
Anastasios K. Lazopoulos ◽  
Dimitrios Karaoulanis
Keyword(s):  
Experimental Data ◽  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Projectile Motion ◽  
The Many

Projectile motion is studied using fractional calculus. Specifically, a newly defined fractional derivative (the Leibniz L-derivative) and its successor (Λ-fractional derivative) are used to describe the motion of the projectile. Experimental data were analyzed in this study, and conclusions were made. The results of well-established fractional derivatives were also compared with those of L-derivative and Λ-fractional derivative, showing the many advantages of these new derivatives.

scholarly journals Time-Domain Analysis of Fractional Electrical Circuit Containing Two Ladder Elements

Electronics ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 475
Author(s):  
Ewa Piotrowska ◽  
Krzysztof Rogowski
Keyword(s):  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Electric Circuit ◽  
Theoretical Models ◽  
Time Domain Analysis ◽  
Electrical Circuit ◽  
State Variable ◽  
State Space System ◽  
Derivatives Of ◽  
Different Order

The paper is devoted to the theoretical and experimental analysis of an electric circuit consisting of two elements that are described by fractional derivatives of different orders. These elements are designed and performed as RC ladders with properly selected values of resistances and capacitances. Different orders of differentiation lead to the state-space system model, in which each state variable has a different order of fractional derivative. Solutions for such models are presented for three cases of derivative operators: Classical (first-order differentiation), Caputo definition, and Conformable Fractional Derivative (CFD). Using theoretical models, the step responses of the fractional electrical circuit were computed and compared with the measurements of a real electrical system.

Properties of Spatial-Rotation Transformations of Fractional Derivatives

2020 ◽  
pp. 198-223
Author(s):  
Ehab Malkawi
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivatives ◽  
Dimensional Space ◽  
Scalar Fields ◽  
Two Dimensional ◽  
Spatial Rotation ◽  
Essential Step ◽  
Derivatives Of

The transformation properties of the fractional derivatives under spatial rotation in two-dimensional space and for both the Riemann-Liouville and Caputo definitions are investigated and derived in their general form. In particular, the transformation properties of the fractional derivatives acting on scalar fields are studied and discussed. The study of the transformation properties of fractional derivatives is an essential step for the formulation of fractional calculus in multi-dimensional space. The inclusion of fractional calculus in the Lagrangian and Hamiltonian dynamical formulation relies on such transformation. Specific examples on the transformation of the fractional derivatives of scalar fields are discussed.

Fractional derivatives of multidimensional Colombeau generalized stochastic processes

2013 ◽  
Vol 16 (4) ◽  
Author(s):  
Danijela Rajter-Ćirić ◽  
Mirjana Stojanović
Keyword(s):  
Stochastic Process ◽  
Fractional Derivative ◽  
Stochastic Processes ◽  
Compact Support ◽  
Caputo Fractional Derivative ◽  
Fractional Derivatives ◽  
One Dimensional ◽  
Compactly Supported ◽  
Derivatives Of ◽  
Do So

AbstractWe consider fractional derivatives of a Colombeau generalized stochastic process G defined on ℝn. We first introduce the Caputo fractional derivative of a one-dimensional Colombeau generalized stochastic process and then generalize the procedure to the Caputo partial fractional derivatives of a multidimensional Colombeau generalized stochastic process. To do so, the Colombeau generalized stochastic process G has to have a compact support. We prove that an arbitrary Caputo partial fractional derivative of a compactly supported Colombeau generalized stochastic process is a Colombeau generalized stochastic process itself, but not necessarily with a compact support.

scholarly journals A Note on Fractional Order Derivatives and Table of Fractional Derivatives of Some Special Functions

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Abdon Atangana ◽  
Aydin Secer
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Special Functions ◽  
Advantages And Disadvantages ◽  
Derivatives Of

The purpose of this note is to present the different fractional order derivatives definition that are commonly used in the literature on one hand and to present a table of fractional order derivatives of some functions in Riemann-Liouville sense On other the hand. We present some advantages and disadvantages of these fractional derivatives. And finally we propose alternative fractional derivative definition.

Initialization Issues of the Caputo Fractional Derivative

2005 ◽  
Author(s):  
B. N. Narahari Achar ◽  
Carl F. Lorenzo ◽  
Tom T. Hartley
Keyword(s):  
Differential Equation ◽  
Fractional Calculus ◽  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Fractional Derivatives ◽  
Past History ◽  
The Past ◽  
History Of ◽  
Fractional Differential ◽  
Generally True

The importance of proper initialization in taking into account the history of a system whose time evolution is governed by a differential equation of fractional order, has been established by Lorenzo and Hartley, who also gave the method of properly incorporating the effect of the past (history) by means of an initialization function for the Riemann-Liouville and the Grunwald formulations of fractional calculus. The present work addresses this issue for the Caputo fractional derivative and cautions that the commonly held belief that the Caputo formulation of fractional derivatives properly accounts for the initialization effects is not generally true when applied to the solution of fractional differential equations.

scholarly journals New approach to the fractional derivatives

2003 ◽  
Vol 2003 (5) ◽  
pp. 315-325 ◽  
Author(s):  
Kostadin Trenčevski
Keyword(s):  
Positive Integer ◽  
Fractional Derivative ◽  
Taylor Expansion ◽  
Fractional Derivatives ◽  
Analytical Continuation ◽  
New Approach ◽  
Analytical Functions ◽  
Summation Of Divergent Series ◽  
Summation Of Series ◽  
Derivatives Of

We introduce a new approach to the fractional derivatives of the analytical functions using the Taylor series of the functions. In order to calculate the fractional derivatives off, it is not sufficient to know the Taylor expansion off, but we should also know the constants of all consecutive integrations off. For example, any fractional derivative ofexisexonly if we assume that thenth consecutive integral ofexisexfor each positive integern. The method of calculating the fractional derivatives very often requires a summation of divergent series, and thus, in this note, we first introduce a method of such summation of series via analytical continuation of functions.

scholarly journals Special Functions of Fractional Calculus in the Form of Convolution Series and Their Applications

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2132
Author(s):  
Yuri Luchko
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Special Functions ◽  
Analytic Solutions ◽  
Fractional Integrals ◽  
Function Type ◽  
Fractional Differential ◽  
Derivatives Of

In this paper, we first discuss the convolution series that are generated by Sonine kernels from a class of functions continuous on a real positive semi-axis that have an integrable singularity of power function type at point zero. These convolution series are closely related to the general fractional integrals and derivatives with Sonine kernels and represent a new class of special functions of fractional calculus. The Mittag-Leffler functions as solutions to the fractional differential equations with the fractional derivatives of both Riemann-Liouville and Caputo types are particular cases of the convolution series generated by the Sonine kernel κ(t)=tα−1/Γ(α),0<α<1. The main result of the paper is the derivation of analytic solutions to the single- and multi-term fractional differential equations with the general fractional derivatives of the Riemann-Liouville type that have not yet been studied in the fractional calculus literature.

scholarly journals On inequalities of Kolmogorov type for fractional derivatives of functions defined on the real domain

2021 ◽  
Vol 16 ◽  
pp. 28
Author(s):  
V.F. Babenko ◽  
M.S. Churilova
Keyword(s):  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Kolmogorov Type ◽  
The Real ◽  
Higher Derivative ◽  
Real Domain ◽  
Higher Derivatives ◽  
Derivatives Of ◽  
New Inequalities

We obtain new inequalities that generalize known result of Geisberg, which was obtained for fractional Marchaud derivatives, to the case of higher derivatives, at that the fractional derivative is a Riesz one. The inequality with second higher derivative is sharp.

Sign in / Sign up

Export Citation Format

Share Document