Related papers: On Multiplicative Fractional Calculus
Many different types of fractional calculus have been proposed, which can be organised into some general classes of operators. For a unified mathematical theory, results should be proved in the most general possible setting. Two important…
This article provides an accessible introduction to fractional derivatives, a concept that extends classical calculus by allowing derivatives of non-integer order. It explores both the fundamental definitions and some of the most relevant…
This short study consists of two parts, firstly we obtain some inequalities on Caputo Fractional derivatives using the elementary inequalities. Secondly we establish several new inequalities including Caputo fractional derivatives for…
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 derivative…
The thesis deals with applications of fractional calculus to fractals. It introduces the notion of local fractional derivative (LFD). Fractal and multifractal functions have been studied in the thesis using LFD. New kind of equations are…
The fractional calculus of variations is now a subject under strong research. Different definitions for fractional derivatives and integrals are used, depending on the purpose under study. In this paper the fractional operators are defined…
We introduce the definition of conformable derivative on time scales and develop its calculus. Fundamental properties of the conformable derivative and integral on time scales are proved. Linear conformable differential equations with…
We review some recent results of the fractional variational calculus. Necessary optimality conditions of Euler-Lagrange type for functionals with a Lagrangian containing left and right Caputo derivatives are given. Several problems are…
It is well known that the Leibniz rule for the integer derivative of order one does not hold for the fractional derivative case when the fractional order lies between 0 and 1. Thus it poses a great difficulty in the calculation of…
The paper presents two representative classes of Impulsive Fractional Differential Equations defined with generalized Caputo\'s derivative, with fixed lower limit and changing lower limit, respectively. Memory principle is studied and…
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…
In this paper we prove that local fractional derivatives of differentiable functions are integer-order derivative or zero operator. We demonstrate that the local fractional derivatives are limits of the left-sided Caputo fractional…
We obtain a new decomposition of the Riemann-Liouville operators of fractional integration as a series involving derivatives (of integer order). The new formulas are valid for functions of class $C^n$, $n \in \mathbb{N}$, and allow us to…
In the present article, a new method for the evaluation of fractional derivatives of arbitrary real order is proposed. Numerous but inequivalent formulations have been given in the past. Some of them exhibit unsatisfactory properties such…
In recent years, the theory for Leibniz integral rule in the fractional sense has not been able to get substantial development. As an urgent problem to be solved, we study a Leibniz integral rule for Riemann-Liouville and Caputo type…
A generalization of exterior calculus is considered by allowing the partial derivatives in the exterior derivative to assume fractional orders. That is, a fractional exterior derivative is defined. This is found to generate new vector…
We introduce the linear operators of fractional integration and fractional differentiation in the framework of the Riemann-Liouville fractional calculus. Particular attention is devoted to the technique of Laplace transforms for treating…
In this note we analyse the propagation of a small density perturbation in a one-dimensional compressible fluid by means of fractional calculus modelling, replacing thus the ordinary time derivative with the Caputo fractional derivative in…
In this study the general formula for differential and integral operations of fractional calculus via fractal operators by the method of cumulative diminution and cumulative growth is obtained. The under lying mechanism in the success of…
It has been recognized recently that fractional calculus is useful for handling scaling structures and processes. We begin this survey by pointing out the relevance of the subject to physical situations. Then the essential definitions and…