scholarly journals Study on a fractional-order controllers based on best rational approximation of fractional calculus operators

2016 ◽  
Vol 18 (5) ◽  
pp. 3412-3424
Author(s):  
Wu Deng ◽  
Huimin Zhao ◽  
Xinhua Yang ◽  
Yu Xue
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Rational Approximation ◽  
Fractional Calculus Operators ◽  
Best Rational Approximation

scholarly journals Rational Implementation of Fractional Calculus Operator Based on Quadratic Programming

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Zhisong Xu ◽  
Mingqiu Li
Keyword(s):  
Fractional Calculus ◽  
Quadratic Programming ◽  
Frequency Domain ◽  
Frequency Band ◽  
Rational Approximation ◽  
Least Squares Method ◽  
Approximation Accuracy ◽  
Approximation Function ◽  
Fractional Calculus Operators ◽  
Quadratic Programming Method

When fractional calculus operators and models are implemented rationally, there exist some problems such as low approximation accuracy of rational approximation function, inability to specify arbitrary approximation frequency band, or poor robustness. Based on the error criterion of the least squares method, a quadratic programming method based on the frequency-domain error is proposed. In this method, the frequency-domain error minimization between the fractional operator s ± r and its rational approximation function is transformed into a quadratic programming problem. The results show that the construction method of the optimal rational approximation function of fractional calculus operator is effective, and the optimal rational approximation function constructed can effectively approximate the fractional calculus operator and model for the specified approximation frequency band.

scholarly journals General fractional calculus operators containing the generalized Mittag-Leffler functions applied to anomalous relaxation

2017 ◽  
Vol 21 (suppl. 1) ◽  
pp. 317-326 ◽  
Author(s):  
Xiaojun Yang
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Integral Transforms ◽  
Kernel Functions ◽  
Order Derivative ◽  
Fractional Order Derivative ◽  
Laplace Integral ◽  
Fractional Calculus Operators ◽  
Anomalous Relaxation ◽  
First Time

In this paper, we address a family of the general fractional calculus operators of Wiman and Prabhakar types for the first time. The general Mittag-Leffler function to structure the kernel functions of the fractional order derivative operators and their Laplace integral transforms are considered in detail. The formulations are as the mathematical tools proposed to investigate the anomalous relaxation.

scholarly journals Composition Formulae for the k-Fractional Calculus Operators Associated with k-Wright Function

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
D. L. Suthar
Keyword(s):  
Fractional Calculus ◽  
Hypergeometric Function ◽  
Fractional Order ◽  
Wright Function ◽  
Special Cases ◽  
Fractional Calculus Operators ◽  
Fractional Order Integral

In this article, the k-fractional-order integral and derivative operators including the k-hypergeometric function in the kernel are used for the k-Wright function; the results are presented for the k-Wright function. Also, some of special cases related to fractional calculus operators and k-Wright function are considered.

Mkhitar Djrbashian and his contribution to Fractional Calculus

2020 ◽  
Vol 23 (6) ◽  
pp. 1797-1809
Author(s):  
Sergei Rogosin ◽  
Maryna Dubatovskaya
Keyword(s):  
Cauchy Problem ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Approximation Theory ◽  
Fractional Derivatives ◽  
Complex Analysis ◽  
Integral Transforms ◽  
Modern Theory ◽  
Survey Paper ◽  
The Cauchy Problem

Abstract This survey paper is devoted to the description of the results by M.M. Djrbashian related to the modern theory of Fractional Calculus. M.M. Djrbashian (1918-1994) is a well-known expert in complex analysis, harmonic analysis and approximation theory. Anyway, his contributions to fractional calculus, to boundary value problems for fractional order operators, to the investigation of properties of the Queen function of Fractional Calculus (the Mittag-Leffler function), to integral transforms’ theory has to be understood on a better level. Unfortunately, most of his works are not enough popular as in that time were published in Russian. The aim of this survey is to fill in the gap in the clear recognition of M.M. Djrbashian’s results in these areas. For same purpose, we decided also to translate in English one of his basic papers [21] of 1968 (joint with A.B. Nersesian, “Fractional derivatives and the Cauchy problem for differential equations of fractional order”), and were invited by the “FCAA” editors to publish its re-edited version in this same issue of the journal.

The bouncing ball and the Grünwald-Letnikov definition of fractional derivative

2021 ◽  
Vol 24 (4) ◽  
pp. 1003-1014
Author(s):  
J. A. Tenreiro Machado
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Restitution Coefficient ◽  
Classical Physics ◽  
Bouncing Ball ◽  
Mechanical Experiment ◽  
Fractional Models ◽  
Definition Of

Abstract This paper proposes a conceptual experiment embedding the model of a bouncing ball and the Grünwald-Letnikov (GL) formulation for derivative of fractional order. The impacts of the ball with the surface are modeled by means of a restitution coefficient related to the coefficients of the GL fractional derivative. The results are straightforward to interpret under the light of the classical physics. The mechanical experiment leads to a physical perspective and allows a straightforward visualization. This strategy provides not only a motivational introduction to students of the fractional calculus, but also triggers possible discussion with regard to the use of fractional models in mechanics.

On some Hardy-type inequalities for fractional calculus operators

2017 ◽  
Vol 11 (2) ◽  
pp. 438-457 ◽  
Author(s):  
Sajid Iqbal ◽  
Josip Pečarić ◽  
Muhammad Samraiz ◽  
Zivorad Tomovski
Keyword(s):  
Fractional Calculus ◽  
Hardy Type Inequalities ◽  
Hardy Type ◽  
Fractional Calculus Operators
Sign in / Sign up

Export Citation Format

Share Document