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 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.

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.

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.

Comparative analysis of suitability of fractional derivatives in modelling the practical capacitor

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Rawid Banchuin
Keyword(s):  
Comparative Analysis ◽  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Fractional Power ◽  
Electrolytic Capacitor ◽  
Content Type ◽  
Local Fractional Derivative ◽  
Definition Of ◽  
Double Layer Capacitors ◽  
Derivatives Of

Purpose The purpose of this paper is to compare the suitability of fractional derivatives in the modelling of practical capacitors. Such suitability refers to ability to provide the analytical capacitance function that matches the experimental ones of each fractional derivative. Design/methodology/approach The analytical capacitance functions based on various fractional derivatives of both local and nonlocal types including the author’s have been derived. The derived capacitance functions have been simulated and compared with the experimental ones of aluminium electrolytic and electrical double layer capacitors (EDLCs). Findings This paper has found that any local fractional derivative with fractional power law-based relationship with the conventional one is suitable for modelling the aluminium electrolytic capacitor (AEC) by incorporating with the conventional capacitance definition. On the other hand, the author’s nonlocal fractional derivatives have been found to be more suitable than the others for modelling the EDLC by incorporating with the revisited definition of capacitance. Originality/value The proposed comparative analysis has been originally presented in this work. The criterion for local fractional derivative, to be suitable for modelling the AEC, has been found. The nonlocal fractional operators which are most suitable for modelling the EDLC have been derived where the unsuitable one has been pointed out.

scholarly journals Fractional Derivative Regularization in QFT

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Euclidean Space ◽  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Differential Operators ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Regularization Procedure ◽  
Derivatives Of ◽  
Massless Fields

We propose in this paper a new regularization, where integer-order differential operators are replaced by fractional-order operators. Regularization for quantum field theories based on application of the Riesz fractional derivatives of noninteger orders is suggested. The regularized loop integrals depend on parameter that is the order α>0 of the fractional derivative. The regularization procedure is demonstrated for scalar massless fields in φ4-theory on n-dimensional pseudo-Euclidean space-time.

Numerical solution of fractionally damped beam by homotopy perturbation method

Open Physics ◽  
2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Diptiranjan Behera ◽  
Snehashish Chakraverty
Keyword(s):  
Fractional Derivative ◽  
Perturbation Method ◽  
Numerical Solution ◽  
Fractional Derivatives ◽  
Homotopy Perturbation Method ◽  
Analytic Solutions ◽  
Step Function ◽  
Homotopy Perturbation ◽  
Derivatives Of ◽  
Appl Math

AbstractThis paper investigates the numerical solution of a viscoelastic continuous beam whose damping behaviours are defined in term of fractional derivatives of arbitrary order. The Homotopy Perturbation Method (HPM) is used to obtain the dynamic response. Unit step function response is considered for the analysis. The obtained results are depicted in various plots. From the results obtained it is interesting to note that by increasing the order of the fractional derivative the beam suffers less oscillation. Similar observations have also been made by keeping the order of the fractional derivative constant and varying the damping ratios. Comparisons are made with the analytic solutions obtained by Zu-feng and Xiao-yan [Appl. Math. Mech. 28, 219 (2007)] to show the effectiveness and validation of this method.

scholarly journals Theory Analysis of Left-Handed Grünwald-Letnikov Formula with0<α<1to Detect and Locate Singularities

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Shaoxiang Hu ◽  
Ping Liang
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Theory Analysis ◽  
Left Handed ◽  
Local Extrema ◽  
Derivative Formula ◽  
Derivatives Of

We study fractional-order derivatives of left-handed Grünwald-Letnikov formula with0<α<1to detect and locate singularities in theory. The widely used four types of ideal singularities are analyzed by deducing their fractional derivative formula. The local extrema of fractional derivatives are used to locate the singularities. Theory analysis indicates that fractional-order derivatives of left-handed Grünwald-Letnikov formula with0<α<1can detect and locate four types of ideal singularities correctly, which shows better performance than classical 1-order derivatives in theory.

scholarly journals Fractional derivatives in spaces of generalized functions

2011 ◽  
Vol 14 (1) ◽  
Author(s):  
Mirjana Stojanović
Keyword(s):  
Continuous Function ◽  
Fractional Derivatives ◽  
Generalized Functions ◽  
Analytical Continuation ◽  
Heaviside Function ◽  
Absolutely Continuous ◽  
Discontinuous Functions ◽  
Absolutely Continuous Function ◽  
Derivatives Of

AbstractWe generalize the two forms of the fractional derivatives (in Riemann-Liouville and Caputo sense) to spaces of generalized functions using appropriate techniques such as the multiplication of absolutely continuous function by the Heaviside function, and the analytical continuation. As an application, we give the two forms of the fractional derivatives of discontinuous functions in spaces of distributions.

Sign in / Sign up

Export Citation Format

Share Document