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 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 Some New Fractional-Calculus Connections between Mittag–Leffler Functions

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 485 ◽  
Author(s):  
Hari M. Srivastava ◽  
Arran Fernandez ◽  
Dumitru Baleanu
Keyword(s):  
Experimental Data ◽  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Integral ◽  
Integral Expression ◽  
Potential Applications ◽  
The One ◽  
Two Parameter

We consider the well-known Mittag–Leffler functions of one, two and three parameters, and establish some new connections between them using fractional calculus. In particular, we express the three-parameter Mittag–Leffler function as a fractional derivative of the two-parameter Mittag–Leffler function, which is in turn a fractional integral of the one-parameter Mittag–Leffler function. Hence, we derive an integral expression for the three-parameter one in terms of the one-parameter one. We discuss the importance and applications of all three Mittag–Leffler functions, with a view to potential applications of our results in making certain types of experimental data much easier to analyse.

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.

A Modified Fractional Calculus Approach to Model Hysteresis

2010 ◽  
Vol 77 (3) ◽  
Author(s):  
Mohammed Rabius Sunny ◽  
Rakesh K. Kapania ◽  
Ronald D. Moffitt ◽  
Amitabh Mishra ◽  
Nakhiah Goulbourne
Keyword(s):  
Experimental Data ◽  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Integral ◽  
Conductive Polymer ◽  
Polymer Nanocomposite ◽  
Hysteretic Behavior ◽  
Integral Model ◽  
Strain Variation ◽  
Integer Order

This paper describes the development of a fractional calculus approach to model the hysteretic behavior shown by the variation in electrical resistances with strain in conductive polymers. Experiments have been carried out on a conductive polymer nanocomposite sample to study its resistance-strain variation under strain varying with time in a triangular manner. A combined fractional derivative and integer order integral model and a fractional integral model (with two submodels) have been developed to simulate this behavior. The efficiency of these models has been discussed by comparing the results, obtained using these models, with the experimental data. It has been shown that by using only a few parameters, the hysteretic behavior of such materials can be modeled using the fractional calculus with some modifications.

scholarly journals Fractional Calculus and Shannon Wavelet

2012 ◽  
Vol 2012 ◽  
pp. 1-26 ◽  
Author(s):  
Carlo Cattani
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Rate Of Convergence ◽  
Analytical Formula ◽  
Approximation Error ◽  
Wavelet Method ◽  
Shannon Wavelet ◽  
The Many ◽  
Shannon Wavelets ◽  
Wavelet Series

An explicit analytical formula for the any order fractional derivative of Shannon wavelet is given as wavelet series based on connection coefficients. So that for anyL2(ℝ)function, reconstructed by Shannon wavelets, we can easily define its fractional derivative. The approximation error is explicitly computed, and the wavelet series is compared with Grünwald fractional derivative by focusing on the many advantages of the wavelet method, in terms of rate of convergence.

scholarly journals What is the best fractional derivative to fit data?

2017 ◽  
Vol 11 (2) ◽  
pp. 358-368 ◽  
Author(s):  
Ricardo Almeida
Keyword(s):  
Differential Equation ◽  
Experimental Data ◽  
Fractional Derivative ◽  
Least Squares ◽  
Fractional Differential Equation ◽  
Fractional Derivatives ◽  
Least Squares Fitting ◽  
Different Types ◽  
Fractional Differential

The aim of this work is to show, based on concrete data observation, that the choice of the fractional derivative when modelling a problem is relevant for the accuracy of a method. Using the least squares fitting technique, we determine the order of the fractional differential equation that better describes the experimental data, for different types of fractional derivatives.

scholarly journals Local Fractional Sumudu Transform with Application to IVPs on Cantor Sets

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
H. M. Srivastava ◽  
Alireza Khalili Golmankhaneh ◽  
Dumitru Baleanu ◽  
Xiao-Jun Yang
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Analytical Solutions ◽  
Initial Value Problems ◽  
Fractional Derivatives ◽  
Cantor Sets ◽  
Coupling Method ◽  
Initial Value ◽  
Local Fractional Calculus ◽  
Sumudu Transform

Local fractional derivatives were investigated intensively during the last few years. The coupling method of Sumudu transform and local fractional calculus (called as the local fractional Sumudu transform) was suggested in this paper. The presented method is applied to find the nondifferentiable analytical solutions for initial value problems with local fractional derivative. The obtained results are given to show the advantages.

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 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.

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