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In this work I consider the abstract Cauchy problems with Caputo fractional time derivative of order $\alpha\in(0,1]$, and discuss the continuity of the respective solutions regarding the parameter $\alpha$. I also present a study about the…
The Gr\"unwald and shifted Gr\"unwald formulas for the function $y(x)-y(b)$ are first order approximations for the Caputo fractional derivative of the function $y(x)$ with lower limit at the point $b$. We obtain second and third order…
We present a method to solve fractional optimal control problems, where the dynamic depends on integer and Caputo fractional derivatives. Our approach consists to approximate the initial fractional order problem with a new one that involves…
Taylor series is a useful mathematical tool when describing and constructing a function. With the series representation, some properties of fractional calculus can be revealed clearly. This paper investigates two typical applications:…
We set the main concepts for multiplicative fractional calculus. We define Caputo, Riemann and Letnikov multiplicative fractional derivatives and multiplicative fractional integrals and study some of their properties. Finally, the…
The aim of the present paper is to make some notes to the newly introduced conformable derivative as a type local fractional derivative and to present a surprising result about the relation between the conformable derivatives and the usual…
The application of the approximation-operational approach to solving linear differential equations of fractional order with variable coefficients is considered. It is shown that the method can also be applied to solving differential…
We generalize the fractional Caputo derivative to the fractional derivative ${{^CD}^{\alpha,\beta}_{\gamma}}$, which is a convex combination of the left Caputo fractional derivative of order $\alpha$ and the right Caputo fractional…
Fractional variation is defined as the limit of the difference quotient of the increments of a function and its argument raised to a fractional power. Fractional velocity can be suitable for characterizing singular behavior of derivatives…
A new calculus based on fractal subsets of the real line is formulated. In this calculus, an integral of order $\alpha, 0 < \alpha \leq 1$, called $F^\alpha$-integral, is defined, which is suitable to integrate functions with fractal…
We consider a conflict-controlled dynamical system described by a nonlinear ordinary fractional differential equation with the Caputo derivative of an order $\alpha \in (0, 1).$ Basing on the finite-difference Gr\"{u}nwald-Letnikov…
We study two generalizations of fractional variational problems by considering higher-order derivatives and a state time delay. We prove a higher-order integration by parts formula involving a Caputo fractional derivative of variable order…
The Caputo time-derivative is usually defined pointwise for well-behaved functions, say, for continuously differentiable functions. Accordingly, in the theory of the partial fractional differential equations with the Caputo derivatives, the…
This article analysis differential equations which represents damped and fractional oscillators. First, it is shown that prior to using physical quantities in fractional calculus, it is imperative that they are turned dimensionless.…
Two approximations, derived from continuous expansions of Riemann-Liouville fractional derivatives into series involving integer order derivatives, are studied. Using those series, one can formally transform any problem that contains…
When $0<\alpha<1$, the approximation for the Caputo derivative $$y^{(\alpha)}(x) = \frac{1}{\Gamma(2-\alpha)h^\alpha}\sum_{k=0}^n \sigma_k^{(\alpha)} y(x-kh)+O\bigl(h^{2-\alpha}\bigr),$$ where $\sigma_0^{(\alpha)} = 1, \sigma_n^{(\alpha)} =…
In this paper, we first provide a short summary of the main properties of the so-called general fractional derivatives with the Sonin kernels introduced so far. These are integro-differential operators defined as compositions of the first…
We examine the fractional derivative of composite functions and present a generalization of the product and chain rules for the Caputo fractional derivative. These results are especially important for physical and biological systems that…
A generalized fractional derivative (GFD) definition is proposed in this work. For a differentiable function that can be expanded by Taylor series, we show that D^Elafa*D^Beta f(t)=D^(Elafa+Beta)f(t). GFD is applied for some functions in…
In the present paper, a generalized local Taylor formula with the local fractional derivatives (LFDs) is proposed based on the local fractional calculus (LFC). From the fractal geometry point of view, the theory of local fractional…