FUNCTIONS THAT HAVE NO FIRST ORDER DERIVATIVE MIGHT HAVE FRACTIONAL DERIVATIVES OF ALL ORDERS LESS THAN ONE

1994 ◽  
Vol 20 (1) ◽  
pp. 140 ◽  
Author(s):  
Ross ◽  
Samko ◽  
Love
Keyword(s):  
Fractional Derivatives ◽  
Order Derivative ◽  
First Order ◽  
Derivatives Of

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.

NORMALIZED VERTICAL DERIVATIVES IN THE EDGE ENHANCEMENT OF MAXIMUM-EDGE-RECOGNITION METHODS IN POTENTIAL FIELDS

Geophysics ◽  
2021 ◽  
pp. 1-88
Author(s):  
Yingjie Zhu ◽  
wanyin wang ◽  
Colin Farquharson ◽  
Jinming Huang ◽  
Minghua Zhang ◽  
...  
Keyword(s):  
Extreme Values ◽  
Order Derivative ◽  
Magnetic Data ◽  
Recognition Method ◽  
Potential Field Data ◽  
First Order ◽  
Horizontal Derivative ◽  
Total Horizontal Derivative ◽  
Edge Recognition ◽  
Derivatives Of

Gravity and magnetic data have unique advantages for studying the lateral extents of geological bodies. There is a class of methods for edge recognition called the maximum-edge-recognition methods that use their extreme values to locate the edges of geological bodies. These methods include the total horizontal derivative, the analytic signal amplitude, the theta map, and the normalized standard deviation. These are all first-order derivative-based techniques. There are also higher-order derivative-based methods that are derived from the first-order filters, for example, the total horizontal derivative of the tilt angle. We present an edge recognition filter that is based on the idea of the normalized vertical derivatives of existing methods. For each maximum-edge-recognition method, we first calculate its nth-order vertical derivative and then use thresholding to locate its peaks. The peak values are subsequently normalized by the values of the original maximum-edge-recognition method. Testing on synthetic and real data shows that the normalized vertical derivatives of the maximum-edge-recognition methods have higher accuracy, better lateral resolution and are more interpretable than existing techniques, and thus are a worthwhile addition to the set of edge-detection tools for potential-field data.

A Numerical Scheme for Dynamic Systems Containing Fractional Derivatives

1998 ◽  
Author(s):  
Lixia Yuan ◽  
Om P. Agrawal
Keyword(s):  
Differential Equations ◽  
Dynamic Systems ◽  
Numerical Scheme ◽  
Fractional Derivatives ◽  
Integral Formula ◽  
Analytical Technique ◽  
First Order ◽  
Numerical Studies ◽  
Fractional Differential ◽  
Derivatives Of

Abstract This paper presents a numerical scheme for dynamic analysis of mechanical systems subjected to damping forces which are proportional to fractional derivatives of displacements. In this scheme, a fractional differential equation governing the dynamic of a system is transformed into a set of differential equations with no fractional derivative terms. Using Laguerre integral formula, this set is converted to a set of first order ordinary differential equations, which are integrated using a numerical scheme to obtain the response of the system. Numerical studies show that the solution converges as the number of Laguerre node points are increased. Further, results obtained using this scheme agree well with those obtained using an analytical technique.

scholarly journals ON A SECOND-ORDER DIFFERENTIAL PROBLEM WITH FRACTIONAL DERIVATIVES OF ORDER GREATER THAN ONE

2013 ◽  
Vol 18 (1) ◽  
pp. 53-65
Author(s):  
Nasser-eddine Tatar
Keyword(s):  
Existence And Uniqueness ◽  
Present Author ◽  
Fractional Derivatives ◽  
Second Order ◽  
Classical Solutions ◽  
Differential Problem ◽  
Abstract Problem ◽  
Integer Order ◽  
First Order ◽  
Derivatives Of

A second-order abstract problem with derivatives of non-integer order is investigated. The nonlinearity involves fractional derivatives between 1 and 2. Existence and uniqueness of mild and classical solutions are established in appropriate spaces. This work extends similar works with or without a derivative of first order and also a work of the present author, where the order of the derivatives were between 0 and 1.

scholarly journals Simultaneous Determination of Prasugrel and Aspirin by Second Order and Ratio First Order Derivative Ultraviolet Spectrophotometry

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Shahabuddin N. Alvi ◽  
Mehul N. Patel ◽  
Prakash B. Kathiriya ◽  
Bhavna A. Patel ◽  
Shraddha J. Parmar
Keyword(s):  
Simultaneous Determination ◽  
Second Order ◽  
Order Derivative ◽  
Zero Crossing ◽  
First Order ◽  
Spectrophotometric Methods ◽  
Relative Standard ◽  
Derivative Method ◽  
Derivatives Of

Two simple, accurate, and precise UV derivative spectrophotometric methods for the simultaneous determination of Prasugrel and Aspirin in synthetic mixture form have been developed. The first method involves measurement of second order derivative spectra of Prasugrel and Aspirin. The zero crossing wavelengths 267.62 nm and 252.40 nm were selected for estimation of Prasugrel and Aspirin, respectively. In the second method, the first order derivatives of ratio spectra were calculated and used for the determination of Prasugrel and Aspirin by measuring the peak intensity at 268 nm and 290 nm, respectively. The methods were validated as per the ICH guideline Q2 (R1). Beer’s law is followed in the range of 5–45 μg/mL for Prasugrel and 25–150 μg/mL for Aspirin by second order derivative method and 6–22 μg/mL for Prasugrel and 45–165 μg/mL for Aspirin by ratio first order derivative method. The recovery studies confirmed the accuracy of the methods. Relative standard deviations for repeatability and inter- and intraday assays were less than 2%. Hence, the described derivative spectrophotometric methods are simple, accurate, precise, and excellent alternatives to sophisticated chromatographic techniques and can be potentially used for the simultaneous determination of Prasugrel and Aspirin in combined dosage form.

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.

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.

Complete monotonicity of entropy production in chemical reaction

1981 ◽  
Vol 46 (2) ◽  
pp. 452-456
Author(s):  
Milan Šolc
Keyword(s):  
Entropy Production ◽  
Equilibrium Constant ◽  
Chemical Reaction ◽  
Relative Entropy ◽  
Order Reaction ◽  
Complete Monotonicity ◽  
First Order Reaction ◽  
First Order ◽  
Completely Monotonic ◽  
Derivatives Of

The successive time derivatives of relative entropy and entropy production for a system with a reversible first-order reaction alternate in sign. It is proved that the relative entropy for reactions with an equilibrium constant smaller than or equal to one is completely monotonic in the whole definition interval, and for reactions with an equilibrium constant larger than one this function is completely monotonic at the beginning of the reaction and near to equilibrium.

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