scholarly journals Advanced ordinary and fractional approximation by positive sublinear operators

Filomat ◽  
2021 ◽  
Vol 35 (6) ◽  
pp. 1899-1913
Author(s):  
George Anastassiou
Keyword(s):  
Fractional Derivatives ◽  
Initial Conditions ◽  
Integral Operators ◽  
Jackson Type ◽  
Generalized Convolution ◽  
Approximation Of Functions ◽  
Sublinear Operators ◽  
Fractional Differentiability ◽  
Derivatives Of ◽  
Fractional Approximation

Here we consider the ordinary and fractional approximation of functions by sublinear positive operators with applications to generalized convolution type operators expressed by sublinear integrals such as of Choquet and Shilkret ones. The fractional approximation is under fractional differentiability of Caputo, Canavati and Iterated-Caputo types. We produce Jackson type inequalities under basic initial conditions. So our way is quantitative by producing inequalities with their right hand sides involving the modulus of continuity of ordinary and fractional derivatives of the function under approximation. We give also an application related to Picard singular integral operators.

FRACTIONAL APPROXIMATION BY NORMALIZED BELL AND SQUASHING TYPE NEURAL NETWORK OPERATORS

2013 ◽  
Vol 09 (01) ◽  
pp. 43-63 ◽  
Author(s):  
GEORGE A. ANASTASSIOU
Keyword(s):  
Neural Network ◽  
Fractional Derivatives ◽  
Jackson Type ◽  
Moduli Of Continuity ◽  
Caputo Fractional Derivatives ◽  
Fractional Rate ◽  
The Right ◽  
Derivatives Of ◽  
Fractional Approximation

This article deals with the determination of the fractional rate of convergence to the unit of some neural network operators, namely, the normalized bell and "squashing" type operators. This is given through the moduli of continuity of the involved right and left Caputo fractional derivatives of the approximated function and they appear in the right-hand side of the associated Jackson type inequalities.

scholarly journals Approximation by Max-Product Operators

2018 ◽  
Vol 60 (1) ◽  
pp. 5-28 ◽  
Author(s):  
George A. Anastassiou
Keyword(s):  
Modulus Of Continuity ◽  
Initial Conditions ◽  
High Order ◽  
Order Derivative ◽  
Jackson Type ◽  
Approximation Of Functions ◽  
High Order Derivative ◽  
Right Hand ◽  
Sublinear Operators

Abstract Here we study the approximation of functions by a great variety of Max-Product operators under differentiability. These are positive sublinear operators. Our study is based on our general results about positive sublinear operators. We produce Jackson type inequalities under initial conditions. So our approach is quantitative by producing inequalities with their right hand sides involving the modulus of continuity of a high order derivative of the function under approximation. We improve known related results which do not use smoothness of functions..

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.

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 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 Fractional Diffusion Equations for Lattice and Continuum: Grünwald-Letnikov Differences and Derivatives Approach

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Fractional Derivatives ◽  
Three Dimensional ◽  
Lattice Models ◽  
Fractional Diffusion ◽  
Lattice Diffusion ◽  
Diffusion Equations ◽  
Difference Operators ◽  
Fractional Diffusion Equations ◽  
Nonlocal Media ◽  
Derivatives Of

Fractional diffusion equations for three-dimensional lattice models based on fractional-order differences of the Grünwald-Letnikov type are suggested. These lattice fractional diffusion equations contain difference operators that describe long-range jumps from one lattice site to another. In continuum limit, the suggested lattice diffusion equations with noninteger order differences give the diffusion equations with the Grünwald-Letnikov fractional derivatives for continuum. We propose a consistent derivation of the fractional diffusion equation with the fractional derivatives of Grünwald-Letnikov type. The suggested lattice diffusion equations can be considered as a new microstructural basis of space-fractional diffusion in nonlocal media.

scholarly journals Necessary conditions for local and global existence to a reaction-diffusion system with fractional derivatives

2006 ◽  
Vol 2006 ◽  
pp. 1-8 ◽  
Author(s):  
Kamel Haouam ◽  
Mourad Sfaxi
Keyword(s):  
Global Existence ◽  
Fractional Derivatives ◽  
Necessary Conditions ◽  
Initial Conditions ◽  
Reaction Diffusion ◽  
Single Equation ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Existence Of A Solution

We give some necessary conditions for local and global existence of a solution to reaction-diffusion system of type (FDS) with temporal and spacial fractional derivatives. As in the case of single equation of type (STFE) studied by M. Kirane et al. (2005), we prove that these conditions depend on the behavior of initial conditions for large|x|.

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