scholarly journals The best polynomial approximation, derivatives of fractional order, and widths of classes of functions in $L_2$

2016 ◽  
Vol 24 ◽  
pp. 10
Author(s):  
S.B. Vakarchuk ◽  
M.B. Vakarchuk
Keyword(s):  
Fractional Order ◽  
Polynomial Approximation ◽  
Fractional Derivatives ◽  
Periodic Functions ◽  
Best Polynomial Approximation ◽  
Alpha Beta ◽  
Derivatives Of

On the classes of $2\pi$-periodic functions ${\mathcal{W}}^{\alpha} (K_{\beta}, \Phi)$, where $\alpha, \beta \in (0;\infty)$, defined by $K$-functionals $K_{\beta}$, fractional derivatives of order $\alpha$, and majorants $\Phi$, the exact values of different $n$-widths have been computed in the space $L_2$.

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.

Wave equation involving fractional derivatives of real and complex fractional order

2019 ◽  
pp. 327-352
Author(s):  
Teodor M. Atanacković ◽  
Sanja Konjik ◽  
Stevan Pilipović
Keyword(s):  
Wave Equation ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Derivatives Of

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.

Geometric interpretation of fractional-order derivative

2016 ◽  
Vol 19 (5) ◽  
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Differential Geometry ◽  
Fractional Order ◽  
Infinite Series ◽  
Fractional Derivatives ◽  
Geometric Interpretation ◽  
Order Derivative ◽  
Jet Bundles ◽  
Fractional Order Derivative ◽  
Integer Order ◽  
Derivatives Of

AbstractA new geometric interpretation of the Riemann-Liouville and Caputo derivatives of non-integer orders is proposed. The suggested geometric interpretation of the fractional derivatives is based on modern differential geometry and the geometry of jet bundles. We formulate a geometric interpretation of the fractional-order derivatives by using the concept of the infinite jets of functions. For this interpretation, we use a representation of the fractional-order derivatives by infinite series with integer-order derivatives. We demonstrate that the derivatives of non-integer orders connected with infinite jets of special type. The suggested infinite jets are considered as a reconstruction from standard jets with respect to order.

Numerical solution of time-fractional three-species food chain model arising in the realm of mathematical ecology

2020 ◽  
Vol 13 (02) ◽  
pp. 2050011 ◽  
Author(s):  
Ved Prakash Dubey ◽  
Rajnesh Kumar ◽  
Devendra Kumar
Keyword(s):  
Fractional Order ◽  
Food Chain ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Time Derivative ◽  
Chain Model ◽  
Homotopy Analysis ◽  
Food Chain Model ◽  
Derivatives Of

This research paper implements the fractional homotopy analysis transform technique to compute the approximate analytical solution of the nonlinear three-species food chain model with time-fractional derivatives. The offered technique is a fantastic blend of homotopy analysis method (HAM) and Laplace transform (LT) operator and has been used fruitfully in the numerical computation of various fractional differential equations (FDEs). This paper involves the fractional derivatives of Caputo style. The numerical solutions of this selected fractional-order food chain model are evaluated by making use of the associated initial conditions. It is revealed by the adopting procedure that the more desirable estimation of the solution can be easily acquired through the calculation of some number of iteration terms only — a fact which authenticates the easiness and soundness of the suggested hybrid scheme. The variations of fractional order of time derivative on the solutions for different specific cases have been depicted through graphical presentations. The outcomes demonstrated through the graphs expound that the adopted scheme is very fantastic and accurate.

scholarly journals Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat transfer problems

2017 ◽  
Vol 21 (3) ◽  
pp. 1161-1171 ◽  
Author(s):  
Xiao-Jun Yang
Keyword(s):  
Heat Transfer ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Complex Phenomenon ◽  
Anomalous Relaxation ◽  
Comparative Results ◽  
Transfer Problems ◽  
First Time ◽  
Derivatives Of ◽  
Relaxation Models

In this paper, we address a class of the fractional derivatives of constant and variable orders for the first time. Fractional-order relaxation equations of constants and variable orders in the sense of Caputo type are modeled from mathematical view of point. The comparative results of the anomalous relaxation among the various fractional derivatives are also given. They are very efficient in description of the complex phenomenon arising in heat transfer.

scholarly journals On the General Solutions of Some Non-Homogeneous Div-Curl Systems with Riemann–Liouville and Caputo Fractional Derivatives

2021 ◽  
Vol 5 (3) ◽  
pp. 117
Author(s):  
Briceyda B. Delgado ◽  
Jorge E. Macías-Díaz
Keyword(s):  
Fractional Order ◽  
Fractional Derivatives ◽  
Decomposition Theorem ◽  
Helmholtz Decomposition ◽  
Fractional Operators ◽  
Gradient Vector ◽  
Caputo Fractional Derivatives ◽  
General Solutions ◽  
Derivatives Of ◽  
Fractional Space

In this work, we investigate analytically the solutions of a nonlinear div-curl system with fractional derivatives of the Riemann–Liouville or Caputo types. To this end, the fractional-order vector operators of divergence, curl and gradient are identified as components of the fractional Dirac operator in quaternionic form. As one of the most important results of this manuscript, we derive general solutions of some non-homogeneous div-curl systems that consider the presence of fractional-order derivatives of the Riemann–Liouville or Caputo types. A fractional analogous to the Teodorescu transform is presented in this work, and we employ some properties of its component operators, developed in this work to establish a generalization of the Helmholtz decomposition theorem in fractional space. Additionally, right inverses of the fractional-order curl, divergence and gradient vector operators are obtained using Riemann–Liouville and Caputo fractional operators. Finally, some consequences of these results are provided as applications at the end of this work.

scholarly journals On the best polynomial approximation of $2\pi$-periodic functions in the $L_2$ space

2015 ◽  
Vol 23 ◽  
pp. 19
Author(s):  
S.B. Vakarchuk
Keyword(s):  
Polynomial Approximation ◽  
Finite Differences ◽  
Jackson Type ◽  
Periodic Functions ◽  
Best Polynomial Approximation ◽  
Exact Constants

On the classes $L^r_2$, where $r\in {\mathbb{Z}}_+$, exact constants of Jackson type inequalities have been obtained for the characteristics of smoothness ${\Delta}_k (f)$, $k\in \mathbb{N}$, which are defined by the averaged $k$-th order finite differences of functions $f\in L_2$.

Viscoelasticity of Fractional Order: New Restrictions on Constitutive Equations with Applications

2020 ◽  
Vol 20 (13) ◽  
pp. 2041011
Author(s):  
Teodor M. Atanacković ◽  
Marko B. Janev ◽  
Stevan Pilipović ◽  
Dora Seleši
Keyword(s):  
Fractional Order ◽  
Constitutive Equations ◽  
Linear Viscoelasticity ◽  
Fractional Derivatives ◽  
Second Law Of Thermodynamics ◽  
Weak Form ◽  
Second Law ◽  
Isothermal Conditions ◽  
Complex Order ◽  
Derivatives Of

In this paper, we analyze the restrictions on the coefficients in the constitutive equations of linear Viscoelasticity that follow from the Second Law of Thermodynamics under isothermal conditions. Especially, we analyze the constitutive equations in which fractional derivatives of real and complex order appear. We present the conditions that follow after application of the Bochner–Schwartz theorem. Conditions derived here, representing in certain cases a weak form of the Second law of Thermodynamics, are more general (weaker) than the classical Bagley–Torvik conditions widely used in Viscoelasticity Theory. Several examples that illustrate the theory are presented.

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