and Applied Analysis 3 Let Γ(⋅) denote the gamma function. For any positive integer n and real number θ (n − 1 < θ < n), there are different definitions of fractional derivatives with order θ in [8]. During this paper, we consider the left, (right) Caputo derivative and left (right) Riemann-Liouville derivative defined as follows: (i) the left Caputo derivative: C 0 D θ t V (t) = 1 Γ (n − θ) ∫ ...

2. Preliminaries Fractional differintegration’s set foot in control compass, also consistency of Caputo definition among the other fractional derivative’s definitions with engineering applications, is caused to devote this section to the basic definition of fractional calculus and its properties (Table 1). The Riemann–Liouville (RL) and the Caputo are the most popular fractional derivatives def...

Here we state the main properties of the Caputo, Riemann-Liouville and the Caputo via Riemann-Liouville fractional derivatives and give some general notes on these properties. Some properties given in some recent literatures and used to solve fractional nonlinear partial differential equations will be proved that they are incorrect by giving some counter examples.

This study examines the nature of media bias within the news media (for this study, newspapers and television news), the trajectories and methods through which these biases are spread and how a hegemony is formed, and looks at the qualities of this hegemony. The thesis begins with a review of some of the literature covering bias in the news media, approaching the issue from a number of angles a...

This paper develops appropriate boundary conditions for the two-sided fractional diffusion equation, where the usual second derivative in space is replaced by a weighted average of positive and negative fractional derivatives. Mass preserving, reflecting boundary conditions for two-sided fractional diffusion involve a balance of left and right fractional derivatives at the boundary. Stable, con...

In this paper, we present a numerical computational approach for solving Caputo type fractional differential equations. This method is based on approximation of Caputo derivative in terms of integer order derivatives and waveform relaxation method. The utility of the method is shown by applying it to several examples. A comparative study indicates that our approach is more efficient and accurat...

There has been a recent increase in interest in the use of distributed order differential equations (particularly in the case where the derivatives are given in the Caputo sense) to model various phenomena. Recent papers have provided insights into the numerical approximation of the solution, and some results on existence and uniqueness have been proved. In each case, the representation of the ...

In this paper we will prove certain Hadamard and Fejer-Hadamard inequalities for the functions whose nth derivatives are convex by using Caputo k-fractional derivatives. These results have some relationship with inequalities for Caputo fractional derivatives.

*Correspondence: [email protected] 1School of Mathematics and Physics, China University of Geosciences, Lumo 388, Wuhan, 430074, China Full list of author information is available at the end of the article Abstract In this paper we start by giving a new definition of weak Caputo derivative in the sense of distributions, and we give a variational formulation to a fractional diffusion equa...

In this paper, the solution to Fuzzy sequential Fractional Initial Value Problem [FFIVP] under Caputo type fuzzy fractional derivatives by a modified fractional Euler method is presented. The Caputo-type fuzzy fractional derivatives are defined based on Hukuhara difference and strongly generalized fuzzy differentiability. The modified fractional Euler method based on a generalized Taylors formu...