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 Fractional derivatives of Colombeau generalized stochastic processes defined on R+

2011 ◽  
Vol 5 (2) ◽  
pp. 283-297 ◽  
Author(s):  
Danijela Rajter-Ciric
Keyword(s):  
Stochastic Process ◽  
Cauchy Problem ◽  
Stochastic Processes ◽  
Fractional Derivatives ◽  
Subject Classification ◽  
2000 Mathematics Subject Classification ◽  
Derivatives Of

We consider Caputo and Riemann-Liouville fractional derivatives of a Colombeau generalized stochastic process G defined on R+. We give proper definitions and prove that both are Colombeau generalized stochastic processes themselves. We also give a solution to a certain Cauchy problem illustrating the application of the theory. 2000 Mathematics Subject Classification: 46F30, 60G20, 60H10, 26A33

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.

scholarly journals A Convergent Collocation Approach for Generalized Fractional Integro-Differential Equations Using Jacobi Poly-Fractonomials

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 979
Author(s):  
Sandeep Kumar ◽  
Rajesh K. Pandey ◽  
H. M. Srivastava ◽  
G. N. Singh
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Collocation Method ◽  
Caputo Fractional Derivative ◽  
Fractional Derivatives ◽  
Special Cases ◽  
Collocation Approach ◽  
Linear And Nonlinear ◽  
Simulation Results ◽  
Theoretical Results

In this paper, we present a convergent collocation method with which to find the numerical solution of a generalized fractional integro-differential equation (GFIDE). The presented approach is based on the collocation method using Jacobi poly-fractonomials. The GFIDE is defined in terms of the B-operator introduced recently, and it reduces to Caputo fractional derivative and other fractional derivatives in special cases. The convergence and error analysis of the proposed method are also established. Linear and nonlinear cases of the considered GFIDEs are numerically solved and simulation results are presented to validate the theoretical results.

scholarly journals The Dirihlet problem for the fractional Poisson’s equation with Caputo derivatives: A finite difference approximation and a numerical solution

2012 ◽  
Vol 16 (2) ◽  
pp. 385-394 ◽  
Author(s):  
V.D. Beibalaev ◽  
R.P. Meilanov
Keyword(s):  
Monte Carlo ◽  
Finite Difference ◽  
Fractional Derivative ◽  
Monte Carlo Methods ◽  
Caputo Fractional Derivative ◽  
Fractional Derivatives ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Difference Problem ◽  
The Stability

A finite difference approximation for the Caputo fractional derivative of the 4-?, 1 < ? ? 2 order has been developed. A difference schemes for solving the Dirihlet?s problem of the Poisson?s equation with fractional derivatives has been applied and solved. Both the stability of difference problem in its right-side part and the convergence have been proved. A numerical example was developed by applying both the Liebman and the Monte-Carlo methods.

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 Some new Caputo fractional derivative inequalities for exponentially $ (\theta, h-m) $–convex functions

2021 ◽  
Vol 7 (2) ◽  
pp. 3006-3026
Author(s):  
Imran Abbas Baloch ◽  
◽  
Thabet Abdeljawad ◽  
Sidra Bibi ◽  
Aiman Mukheimer ◽  
...  
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Convex Functions ◽  
Integral Identity ◽  
Fractional Derivatives ◽  
Order Derivative ◽  
Special Cases

<abstract><p>Firstly, we obtain some inequalities of Hadamard type for exponentially $ (\theta, h-m) $–convex functions via Caputo $ k $–fractional derivatives. Secondly, using integral identity including the $ (n+1) $–order derivative of a given function via Caputo $ k $-fractional derivatives we prove some of its related results. Some new results are given and known results are recaptured as special cases from our results.</p></abstract>

scholarly journals Distributions that are convolvable with generalized Poisson kernel of solvable extensions of homogeneous Lie groups

2009 ◽  
Vol 105 (1) ◽  
pp. 31 ◽  
Author(s):  
Ewa Damek ◽  
Jacek Dziubanski ◽  
Philippe Jaming ◽  
Salvador Pérez-Esteva
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Poisson Kernel ◽  
Boundary Behavior ◽  
One Dimensional ◽  
Generalized Poisson ◽  
Poisson Kernels ◽  
Derivatives Of ◽  
Do So

In this paper, we characterize the class of distributions on a homogeneous Lie group $\mathfrak N$ that can be extended via Poisson integration to a solvable one-dimensional extension $\mathfrak S$ of $\mathfrak N$. To do so, we introduce the $\mathcal S'$-convolution on $\mathfrak N$ and show that the set of distributions that are $\mathcal S'$-convolvable with Poisson kernels is precisely the set of suitably weighted derivatives of $L^1$-functions. Moreover, we show that the $\mathcal S'$-convolution of such a distribution with the Poisson kernel is harmonic and has the expected boundary behavior. Finally, we show that such distributions satisfy some global weak-$L^1$ estimates.

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.

scholarly journals On Some Inequalities Involving Liouville–Caputo Fractional Derivatives and Applications to Special Means of Real Numbers

Mathematics ◽  
2018 ◽  
Vol 6 (10) ◽  
pp. 193 ◽  
Author(s):  
Bessem Samet ◽  
Hassen Aydi
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Fractional Derivatives ◽  
Real Numbers ◽  
Caputo Fractional Derivatives ◽  
Parallel Development ◽  
Special Means ◽  
The Right ◽  
Class Of Functions

We are concerned with the class of functions f ∈ C 1 ( [ a , b ] ; R ) , a , b ∈ R , a < b , such that c D a α f is convex or c D b α f is convex, where 0 < α < 1 , c D a α f is the left-side Liouville–Caputo fractional derivative of order α of f and c D b α f is the right-side Liouville–Caputo fractional derivative of order α of f. Some extensions of Dragomir–Agarwal inequality to this class of functions are obtained. A parallel development is made for the class of functions f ∈ C 1 ( [ a , b ] ; R ) such that c D a α f is concave or c D b α f is concave. Next, an application to special means of real numbers is provided.

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