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

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.

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On inequalities for $L_p$-norms of fractional derivatives on the real domain

Researches in Mathematics ◽

10.15421/240704 ◽

2021 ◽

Vol 15 ◽

pp. 26

Author(s):

V.F. Babenko ◽

M.S. Churilova

Keyword(s):

Fractional Derivatives ◽

The Real ◽

Real Domain ◽

Derivatives Of ◽

New Inequalities

We obtain new inequalities for fractional Marchaud derivatives of functions defined on the whole real domain, in integral metric ($1 \leqslant p < \infty$); for $p = 1$ we establish the sharpness of obtained inequalities.

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The properties of Bessel-type fractional derivatives and integrals

JOURNAL OF ADVANCES IN PHYSICS ◽

10.24297/jap.v11i8.203 ◽

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.

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

Electronics ◽

10.3390/electronics10040475 ◽

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.

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Fractional derivatives of multidimensional Colombeau generalized stochastic processes

Fractional Calculus and Applied Analysis ◽

10.2478/s13540-013-0058-z ◽

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.

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Two compartmental fractional derivative model with fractional derivatives of different order

Communications in Nonlinear Science and Numerical Simulation ◽

10.1016/j.cnsns.2013.01.004 ◽

2013 ◽

Vol 18 (9) ◽

pp. 2507-2514 ◽

Cited By ~ 12

Author(s):

Jovan K. Popović ◽

Stevan Pilipović ◽

Teodor M. Atanacković

Keyword(s):

Fractional Derivative ◽

Fractional Derivatives ◽

Fractional Derivative Model ◽

Derivatives Of ◽

Different Order

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

Abstract and Applied Analysis ◽

10.1155/2013/279681 ◽

2013 ◽

Vol 2013 ◽

pp. 1-8 ◽

Cited By ~ 60

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.

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New approach to the fractional derivatives

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171203206050 ◽

2003 ◽

Vol 2003 (5) ◽

pp. 315-325 ◽

Cited By ~ 6

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.

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