scholarly journals Riesz potential and Riemann-Liouville fractional integrals and derivatives of Jacobi polynomials

1997 ◽  
Vol 10 (1) ◽  
pp. 103-108 ◽  
Author(s):  
I. Podlubny
Keyword(s):  
Jacobi Polynomials ◽  
Riesz Potential ◽  
Fractional Integrals ◽  
Fractional Integrals And Derivatives ◽  
Derivatives Of

scholarly journals Katugampola Fractional Calculus With Generalized k−Wright Function

2021 ◽  
Vol 1 (1) ◽  
pp. 34-44
Author(s):  
Ahmad Y. A. Salamooni ◽  
D. D. Pawar
Keyword(s):  
Fractional Calculus ◽  
Fractional Integrals ◽  
Wright Function ◽  
Fractional Integrals And Derivatives ◽  
Derivatives Of

In this article, we present some properties of the Katugampola fractional integrals and derivatives. Also, we study the fractional calculus properties involving Katugampola Fractional integrals and derivatives of generalized k−Wright function nΦkm(z).

scholarly journals Finding the fractional value of 2pi

2020 ◽  
Author(s):  
Anurag Vaidya
Keyword(s):  
Fractional Derivatives ◽  
Fractional Integrals ◽  
Explorative Study ◽  
Detection Technology ◽  
Fractional Integrals And Derivatives ◽  
Definition Of ◽  
Rational Order ◽  
Derivatives Of ◽  
Sine And Cosine Functions ◽  
Cosine Functions

Rational order integral and derivative of a myriad of functions—ln(x); e^(ax) and t^(ax)—are known. Nevertheless, the investigation focuses on rational order integrals and derivatives of sine and cosine as these functions follow a cyclic order and determiningwhether properties of sine and cosine extend to their fractional integrals and derivativescould expand current applications of sine and cosine, which are extensively in engineeringand economics. For example, Mehdi and Majid outline in, "Applications of FractionalCalculus" how fractional integrals and derivatives come handy while modeling ultrasonicwave propagation in human cancellous bones and bettering edge detection technology. Thus, this explorative study investigates the properties of fractional derivatives and fractional integrals of standard sine and cosine functions. THe study also looks at how to generalize the definition of the 2pi using fractional calculus.

scholarly journals An Expansion Formula with Higher-Order Derivatives for Fractional Operators of Variable Order

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Ricardo Almeida ◽  
Delfim F. M. Torres
Keyword(s):  
Differential Equations ◽  
Approximation Method ◽  
Fractional Derivatives ◽  
Fractional Integrals ◽  
Variable Order ◽  
Fractional Operators ◽  
Expansion Formula ◽  
Approximation Formulas ◽  
Fractional Integrals And Derivatives ◽  
Derivatives Of

We obtain approximation formulas for fractional integrals and derivatives of Riemann-Liouville and Marchaud types with a variable fractional order. The approximations involve integer-order derivatives only. An estimation for the error is given. The efficiency of the approximation method is illustrated with examples. As applications, we show how the obtained results are useful to solve differential equations, and problems of the calculus of variations that depend on fractional derivatives of Marchaud type.

scholarly journals General Fractional Integrals and Derivatives of Arbitrary Order

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 755
Author(s):  
Yuri Luchko
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivatives ◽  
Arbitrary Order ◽  
Fractional Integrals ◽  
Basic Properties ◽  
Suitable Generalization ◽  
Fractional Integrals And Derivatives ◽  
Fundamental Theorems ◽  
Derivatives Of

In this paper, we introduce the general fractional integrals and derivatives of arbitrary order and study some of their basic properties and particular cases. First, a suitable generalization of the Sonine condition is presented, and some important classes of the kernels that satisfy this condition are introduced. Whereas the kernels of the general fractional derivatives of arbitrary order possess integrable singularities at the point zero, the kernels of the general fractional integrals can—depending on their order—be both singular and continuous at the origin. For the general fractional integrals and derivatives of arbitrary order with the kernels introduced in this paper, two fundamental theorems of fractional calculus are formulated and proved.

Modulating Functions Based Fast and Robust Estimation for a Class of Fractional Order Vibration Systems

2021 ◽  
Author(s):  
Zhi-Bo Wang ◽  
Da-Yan Liu ◽  
Driss Boutat ◽  
Yang Tian ◽  
Hao-Ran Liu
Keyword(s):  
Fractional Order ◽  
Original Problem ◽  
Initial Conditions ◽  
Main Idea ◽  
Fractional Integrals ◽  
Initial Values ◽  
Integral Formulas ◽  
Fractional Integrals And Derivatives ◽  
Modulating Functions ◽  
Derivatives Of

Abstract This paper aims to fast and robustly estimate the fractional integrals and derivatives of positions from noisy accelerations for a class of fractional order vibration systems defined by the Caputo fractional derivative. The main idea is to convert the original issue into the estimation of the fractional integrals of accelerations and the ones of the unknown initial conditions, on the basis of the additive index law. Being proper integrals, the fractional integrals of accelerations can be estimated via a numerical method. Consequently, solving the original problem boils down to estimating the unknown initial values. To this end, the modulating functions method is adopted. By constructing appropriate modulating functions, the unknown initial values are exactly given in terms of algebraic integral formulas in different situations. Finally, two illustrations are presented to verify the correctness and robustness of the proposed estimators.

scholarly journals Evaluation of Fractional Integrals and Derivatives of Elementary Functions: Overview and Tutorial

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 407 ◽  
Author(s):  
Roberto Garrappa ◽  
Eva Kaslik ◽  
Marina Popolizio
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Fractional Integrals ◽  
Full Understanding ◽  
Elementary Functions ◽  
Practical Applications ◽  
Integer Order ◽  
Fractional Integrals And Derivatives ◽  
Derivatives Of

Several fractional-order operators are available and an in-depth knowledge of the selected operator is necessary for the evaluation of fractional integrals and derivatives of even simple functions. In this paper, we reviewed some of the most commonly used operators and illustrated two approaches to generalize integer-order derivatives to fractional order; the aim was to provide a tool for a full understanding of the specific features of each fractional derivative and to better highlight their differences. We hence provided a guide to the evaluation of fractional integrals and derivatives of some elementary functions and studied the action of different derivatives on the same function. In particular, we observed how Riemann–Liouville and Caputo’s derivatives converge, on long times, to the Grünwald–Letnikov derivative which appears as an ideal generalization of standard integer-order derivatives although not always useful for practical applications.

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