Leibniz Rule and Fractional Derivatives of Power Functions

2015 ◽  
Vol 11 (3) ◽  
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Power Function ◽  
Fractional Order ◽  
Special Form ◽  
Fractional Derivatives ◽  
Algebraic Approach ◽  
Leibniz Rule ◽  
Order Derivative ◽  
Power Functions ◽  
Fractional Order Derivative ◽  
Derivatives Of

In this paper, we prove that unviolated simple Leibniz rule and equation for fractional-order derivative of power function cannot hold together for derivatives of orders α≠1. To prove this statement, we use an algebraic approach, where special form of fractional-order derivatives is not applied.

Geometric interpretation of fractional-order derivative

2016 ◽  
Vol 19 (5) ◽  
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Differential Geometry ◽  
Fractional Order ◽  
Infinite Series ◽  
Fractional Derivatives ◽  
Geometric Interpretation ◽  
Order Derivative ◽  
Jet Bundles ◽  
Fractional Order Derivative ◽  
Integer Order ◽  
Derivatives Of

AbstractA new geometric interpretation of the Riemann-Liouville and Caputo derivatives of non-integer orders is proposed. The suggested geometric interpretation of the fractional derivatives is based on modern differential geometry and the geometry of jet bundles. We formulate a geometric interpretation of the fractional-order derivatives by using the concept of the infinite jets of functions. For this interpretation, we use a representation of the fractional-order derivatives by infinite series with integer-order derivatives. We demonstrate that the derivatives of non-integer orders connected with infinite jets of special type. The suggested infinite jets are considered as a reconstruction from standard jets with respect to order.

COMMENTS ON "RIEMANN–CHRISTOFFEL TENSOR IN DIFFERENTIAL GEOMETRY OF FRACTIONAL ORDER APPLICATION TO FRACTAL SPACE-TIME", [FRACTALS 21 (2013) 1350004]

Fractals ◽  
2015 ◽  
Vol 23 (02) ◽  
pp. 1575001 ◽  
Author(s):  
VASILY E. TARASOV
Keyword(s):  
Differential Geometry ◽  
Fractional Derivative ◽  
Power Function ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Space Time ◽  
Leibniz Rule ◽  
Integer Order ◽  
Fractal Space ◽  
Derivatives Of

We prove that main properties represented by Eq. (4.2) for fractional derivative of power function and the non-fractional Leibniz rule in the form (4.3) of the considered paper, cannot hold together for derivatives of non-integer order. As a result, we prove that the usual Leibniz rule (4.3) cannot hold for fractional derivatives.

Digital Fractional Order Savitzky-Golay Differentiator and Its Application

2011 ◽  
Author(s):  
Dali Chen ◽  
Dingyu Xue ◽  
YangQuan Chen
Keyword(s):  
Fractional Order ◽  
Least Square ◽  
One Dimension ◽  
Order Derivative ◽  
Noisy Signal ◽  
Fractional Order Derivative ◽  
Least Square Fitting ◽  
Image Enhancing ◽  
The One ◽  
Derivatives Of

Firstly the one-dimension digital fractional order Savitzky-Golay differentiator (1-D DFOSGD), which generalizes the Savitzky-Golay filter from the integer order to the fractional order, is proposed to estimate the fractional order derivative of the noisy signal. The polynomial least square fitting technology and the Riemann-Liouville fractional order derivative definition are used to ensure robust and accuracy. Experiments demonstrate that 1-D DFOSGD can estimate the fractional order derivatives of both ideal signal and noisy signal accurately. Secondly, the two-dimension DFOSGD is obtained from 1-D DFOSGD by defining a group of direction operators, and a new image enhancing method based on 2-D DFOSGD is presented. Experiments demonstrate that 2-D DFOSGD has very good performance on image enhancement.

scholarly journals A new general fractional derivative Goldstein-Kac-type telegraph equation

2020 ◽  
Vol 24 (6 Part B) ◽  
pp. 3893-3898
Author(s):  
Ping Cui ◽  
Yi-Ying Feng ◽  
Jian-Gen Liu ◽  
Lu-Lu Geng
Keyword(s):  
Fractional Order ◽  
Telegraph Equation ◽  
Order Derivative ◽  
Mathematical Tool ◽  
Fractional Order Derivative ◽  
Derivative Formula ◽  
The Laplace Transform ◽  
The One ◽  
First Time ◽  
Derivatives Of

In this paper, we consider the Riemann-Liouville-type general fractional derivatives of the non-singular kernel of the one-parametric Lorenzo-Hartley function. A new general fractional-order-derivative Goldstein-Kac-type telegraph equation is proposed for the first time. The analytical solution of the considered model with the graphs is obtained with the aid of the Laplace transform. The general fractional-order-derivative formula is as a new mathematical tool proposed to model the anomalous behaviors in complex and power-law phenomena.

scholarly journals A comparison study of steady-state vibrations with single fractional-order and distributed-order derivatives

Open Physics ◽  
2017 ◽  
Vol 15 (1) ◽  
pp. 809-818 ◽  
Author(s):  
Jun-Sheng Duan ◽  
Cui-Ping Cheng ◽  
Lian Chen
Keyword(s):  
Steady State ◽  
Weight Function ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Order Derivative ◽  
Similar Response ◽  
Driving Frequency ◽  
Fractional Order Derivative ◽  
The One ◽  
Distributed Order

AbstractWe conduct a detailed study and comparison for the one-degree-of-freedom steady-state vibrations under harmonic driving with a single fractional-order derivative and a distributed-order derivative. For each of the two vibration systems, we consider the stiffness contribution factor and damping contribution factor of the term of fractional derivatives, the amplitude and the phase difference for the response. The effects of driving frequency on these response quantities are discussed. Also the influences of the orderαof the fractional derivative and the parameterγparameterizing the weight function in the distributed-order derivative are analyzed. Two cases display similar response behaviors, but the stiffness contribution factor and damping contribution factor of the distributed-order derivative are almost monotonic change with the parameterγ, not exactly like the case of single fractional-order derivative for the orderα. The case of the distributed-order derivative provides us more options for the weight function and parameters.

scholarly journals Novel Approaches for Getting the Solution of the Fractional Black–Scholes Equation Described by Mittag-Leffler Fractional Derivative

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Ndolane Sene ◽  
Babacar Sène ◽  
Seydou Nourou Ndiaye ◽  
Awa Traoré
Keyword(s):  
Analytical Solution ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Numerical Scheme ◽  
Fractional Derivatives ◽  
Fractional Integration ◽  
Order Derivative ◽  
Graphical Representations ◽  
Fractional Order Derivative ◽  
Black Scholes

The value of an option plays an important role in finance. In this paper, we use the Black–Scholes equation, which is described by the nonsingular fractional-order derivative, to determine the value of an option. We propose both a numerical scheme and an analytical solution. Recent studies in fractional calculus have included new fractional derivatives with exponential kernels and Mittag-Leffler kernels. These derivatives have been found to be applicable in many real-world problems. As fractional derivatives without nonsingular kernels, we use a Caputo–Fabrizio fractional derivative and a Mittag-Leffler fractional derivative. Furthermore, we use the Adams–Bashforth numerical scheme and fractional integration to obtain the numerical scheme and the analytical solution, and we provide graphical representations to illustrate these methods. The graphical representations prove that the Adams–Bashforth approach is helpful in getting the approximate solution for the fractional Black–Scholes equation. Finally, we investigate the volatility of the proposed model and discuss the use of the model in finance. We mainly notice in our results that the fractional-order derivative plays a regulator role in the diffusion process of the Black–Scholes equation.

The properties of Bessel-type fractional derivatives and integrals

2016 ◽  
pp. 3973-3982
Author(s):  
V. R. Lakshmi Gorty
Keyword(s):  
Fractional Order ◽  
Real Line ◽  
Computer Algebra System ◽  
Fractional Derivatives ◽  
Graphical Representation ◽  
Fractional Integrals ◽  
The Real ◽  
Fractional Derivatives And Integrals ◽  
Half Axis ◽  
Derivatives Of

The fractional integrals of Bessel-type Fractional Integrals from left-sided and right-sided integrals of fractional order is established on finite and infinite interval of the real-line, half axis and real axis. The Bessel-type fractional derivatives are also established. The properties of Fractional derivatives and integrals are studied. The fractional derivatives of Bessel-type of fractional order on finite of the real-line are studied by graphical representation. Results are direct output of the computer algebra system coded from MATLAB R2011b.

GENERALIZED THEORY OF MAGNETO-THERMO-VISCOELASTIC SPHERICAL CAVITY PROBLEM UNDER FRACTIONAL ORDER DERIVATIVE: STATE SPACE APPROACH

2020 ◽  
Vol 9 (11) ◽  
pp. 9769-9780
Author(s):  
S.G. Khavale ◽  
K.R. Gaikwad
Keyword(s):  
State Space ◽  
Fractional Order ◽  
Order Derivative ◽  
Laplace Inversion ◽  
State Space Approach ◽  
Spherical Region ◽  
Fractional Order Derivative ◽  
Numerical Laplace Inversion ◽  
Mathcad Software ◽  
Space Approach

This paper is dealing the modified Ohm's law with the temperature gradient of generalized theory of magneto-thermo-viscoelastic for a thermally, isotropic and electrically infinite material with a spherical region using fractional order derivative. The general solution obtained from Laplace transform, numerical Laplace inversion and state space approach. The temperature, displacement and stresses are obtained and represented graphically with the help of Mathcad software.

scholarly journals Novel hyperbolic and exponential ansatz methods to the fractional fifth-order Korteweg–de Vries equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Choonkil Park ◽  
R. I. Nuruddeen ◽  
Khalid K. Ali ◽  
Lawal Muhammad ◽  
M. S. Osman ◽  
...  
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Mathematical Physics ◽  
Order Derivative ◽  
Conformable Fractional Derivative ◽  
Fractional Order Derivative ◽  
De Vries ◽  
Fifth Order ◽  
2D And 3D

Abstract This paper aims to investigate the class of fifth-order Korteweg–de Vries equations by devising suitable novel hyperbolic and exponential ansatze. The class under consideration is endowed with a time-fractional order derivative defined in the conformable fractional derivative sense. We realize various solitons and solutions of these equations. The fractional behavior of the solutions is studied comprehensively by using 2D and 3D graphs. The results demonstrate that the methods mentioned here are more effective in solving problems in mathematical physics and other branches of science.

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