scholarly journals Maximal Domains for Fractional Derivatives and Integrals

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1107
Author(s):  
R. Hilfer ◽  
T. Kleiner
Keyword(s):  
Fractional Derivatives ◽  
Short Communication ◽  
Fractional Derivatives And Integrals

The purpose of this short communication is to announce the existence of fractional calculi on precisely specified domains of distributions. The calculi satisfy desiderata proposed above in Mathematics 7, 149 (2019). For the desiderata (a)–(c) the examples are optimal in the sense of having maximal domains with respect to convolvability of distributions. The examples suggest to modify desideratum (f) in the original list.

The properties of Bessel-type fractional derivatives and integrals

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.

scholarly journals A study on multiterm hybrid multi-order fractional boundary value problem coupled with its stability analysis of Ulam–Hyers type

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ahmed Nouara ◽  
Abdelkader Amara ◽  
Eva Kaslik ◽  
Sina Etemad ◽  
Shahram Rezapour ◽  
...  
Keyword(s):  
Stability Analysis ◽  
Boundary Value Problem ◽  
Fractional Derivatives ◽  
Boundary Value ◽  
Research Work ◽  
Existence Results ◽  
Fractional Boundary Value Problem ◽  
Numerical Examples ◽  
Fractional Derivatives And Integrals ◽  
Fractional Differential

AbstractIn this research work, a newly-proposed multiterm hybrid multi-order fractional boundary value problem is studied. The existence results for the supposed hybrid fractional differential equation that involves Riemann–Liouville fractional derivatives and integrals of multi-orders type are derived using Dhage’s technique, which deals with a composition of three operators. After that, its stability analysis of Ulam–Hyers type and the relevant generalizations are checked. Some illustrative numerical examples are provided at the end to illustrate and validate our obtained results.

Fractional Order Derivative and Integral Computation with a Small Number of Discrete Input Values Using Grünwald–Letnikov Formula

2019 ◽  
Vol 17 (05) ◽  
pp. 1940006
Author(s):  
Dariusz W. Brzeziński
Keyword(s):  
Input Data ◽  
Fractional Derivatives ◽  
Difficult Problem ◽  
Computational Method ◽  
Problem Solution ◽  
Fractional Derivatives And Integrals ◽  
Discrete Values ◽  
Arbitrary Precision ◽  
Modern Programming Language ◽  
Accuracy Of Approximation

High-accuracy computer approximation of fractional derivatives and integrals by applying Grünwald–Letnikov formula generally requires a large number of input values. If required amount cannot be supplied, accuracy of approximation drops drastically. In this paper, we solve a difficult problem in this scope, i.e., when input data consists only of a small number of discrete values. Furthermore, some of the values may be unusable for computational purposes. Our problem solution includes an appropriate method of input data preprocessing, an interpolation algorithm with extrapolation abilities, a central point function discretization schema, recurrent computational method of coefficients and application of Horner’s schema for the core of the Grünwald–Letnikov method: coefficients and function’s values multiplication. Numerical method presented in the paper enables to compute fractional derivatives and integrals of complicated functions with much higher accuracy than it is possible when application of the default approach to Grünwald–Letnikov method computer implementation is applied. This new method usually takes only 10% of function’s values required by the default approach for the same computations and is much less restrictive for their quality. The general novelty of the method is an efficient configuration of existing numerical methods and enhancement of their abilities by applying modern programming language — Python and arbitrary precision arithmetic for computations.

scholarly journals Desiderata for Fractional Derivatives and Integrals

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 149 ◽  
Author(s):  
Rudolf Hilfer ◽  
Yuri Luchko
Keyword(s):  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Fractional Integral ◽  
Special Issue ◽  
Fractional Derivatives And Integrals

The purpose of this brief article is to initiate discussions in this special issue by proposing desiderata for calling an operator a fractional derivative or a fractional integral. Our desiderata are neither axioms nor do they define fractional derivatives or integrals uniquely. Instead they intend to stimulate the field by providing guidelines based on a small number of time honoured and well established criteria.

scholarly journals Recent applications of fractional calculus to science and engineering

2003 ◽  
Vol 2003 (54) ◽  
pp. 3413-3442 ◽  
Author(s):  
Lokenath Debnath
Keyword(s):  
Dynamical Systems ◽  
Fractional Calculus ◽  
Control Theory ◽  
Numerical Computation ◽  
Fractional Derivatives ◽  
Fluid Flows ◽  
Voltage Divider ◽  
Science And Engineering ◽  
Electrical Circuits ◽  
Fractional Derivatives And Integrals

This paper deals with recent applications of fractional calculus to dynamical systems in control theory, electrical circuits with fractance, generalized voltage divider, viscoelasticity, fractional-order multipoles in electromagnetism, electrochemistry, tracer in fluid flows, and model of neurons in biology. Special attention is given to numerical computation of fractional derivatives and integrals.

scholarly journals Nonlocal Fractional Hybrid Boundary Value Problems Involving Mixed Fractional Derivatives and Integrals via a Generalization of Darbo’s Theorem

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Ayub Samadi ◽  
Sotiris K. Ntouyas ◽  
Jessada Tariboon
Keyword(s):  
Boundary Value Problem ◽  
Theoretical Result ◽  
Boundary Value Problems ◽  
Fractional Derivatives ◽  
Existence Result ◽  
Boundary Value ◽  
Measures Of Noncompactness ◽  
Fractional Derivatives And Integrals ◽  
Mixed Fractional Derivatives

In this work, a new existence result is established for a nonlocal hybrid boundary value problem which contains one left Caputo and one right Riemann–Liouville fractional derivatives and integrals. The main result is proved by applying a new generalization of Darbo’s theorem associated with measures of noncompactness. Finally, an example to justify the theoretical result is also presented.

scholarly journals On representation and interpretation of Fractional calculus and fractional order systems

2019 ◽  
Vol 22 (2) ◽  
pp. 522-537
Author(s):  
Juan Paulo García-Sandoval
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Extra Dimension ◽  
Fractional Derivatives ◽  
Area Under The Curve ◽  
Geometric Interpretation ◽  
Order Partial Differential Equation ◽  
Fractional Order Systems ◽  
Caputo Fractional Derivatives ◽  
Fractional Derivatives And Integrals

Abstract In this work a relationship between Fractional calculus (FC) and the solution of a first order partial differential equation (FOPDE) is suggested. With this relationship and considering an extra dimension, an alternative representation for fractional derivatives and integrals is proposed. This representation can be applied to fractional derivatives and integrals defined by convolution integrals of the Volterra type, i.e. the Riemann-Liouville and Caputo fractional derivatives and integrals, and the Riesz and Feller potentials, and allows to transform fractional order systems in FOPDE that only contains integer-order derivatives. As a consequence of considering the extra dimension, the geometric interpretation of fractional derivatives and integrals naturally emerges as the area under the curve of a characteristic trajectory and as the direction of a tangent characteristic vector, respectively. Besides this, a new physical interpretation is suggested for the fractional derivatives, integrals and dynamical systems.

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