Related papers: Generalized Fractional Order Derivatives, Its Prop…
The fractional calculus of variations and fractional optimal control are generalizations of the corresponding classical theories, that allow problem modeling and formulations with arbitrary order derivatives and integrals. Because of the…
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…
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…
The Riemann-Liouville formula for fractional derivatives and integrals (differintegration) is used to derive formulae for matrix order derivatives and integrals. That is, the parameter for integration and differentiation is allowed to…
In a recent paper, Cohl and Costas-Santos derived a number of interesting multi-derivative and multi-integral relations for associated Legendre and Ferrers functions in which the orders of those functions are changed in integral steps.…
Designing effective neural networks requires tuning architectural elements. This study integrates fractional calculus into neural networks by introducing fractional order derivatives (FDO) as tunable parameters in activation functions,…
In order to describe more complex problem using the concept of fractional derivatives, we introduce in this paper the concept of fractional derivatives with orders. The new definitions are based upon the concept of power law together with…
We obtain approximation formulas for fractional integrals and derivatives of Riemann-Liouville and Marchaud types with a variable fractional order. The approximations involve integer-order derivatives only. An estimation for the error is…
Over the last decade, it has been demonstrated that many systems in science and engineering can be modeled more accurately by fractional-order than integer-order derivatives, and many methods are developed to solve the problem of fractional…
The fractional order grey models (FGM) have appealed considerable interest of research in recent years due to its higher effectiveness and flexibility than the conventional grey models and other prediction models. However, the definitions…
This paper is devoted to the study of quasi-periodic properties of fractional order integrals and derivatives of periodic functions. Considering Riemann-Liouville and Caputo definitions, we discuss when the fractional derivative and when…
The paper presents derivation and interpretation of one type of variable order derivative definitions. For mathematical modelling of considering definition the switching and numerical scheme is given. The paper also introduces a numerical…
Contrary to integer order derivative, the fractional-order derivative of a non-constant periodic function is not a periodic function with the same period, as a consequence of this property the time-invariant fractional order system does not…
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…
We introduce a new fractional derivative that generalizes the so-called alternative fractional derivative recently proposed by Katugampola. We denote this new differential operator by $\mathscr{D}_{M}^{\alpha,\beta }$, where the parameter…
There are many possible definitions of derivatives, here we present some and present one that we have called generalized that allows us to put some of the others as a particular case of this but, what interests us is to determine that there…
This chapter presents some numerical methods to solve problems in the fractional calculus of variations and fractional optimal control. Although there are plenty of methods available in the literature, we concentrate mainly on approximating…
The goal of this communication is to propose a generalized notion of the "traditional derivative". This generalization includes the fractional derivatives such as the Riemann-Liouville, Gruenwald-Letnikov, Weyl, Riesz, Caputo, Marchaud…
This paper presents a modified numerical scheme for a class of Fractional Optimal Control Problems (FOCPs) formulated in Agrawal (2004) where a Fractional Derivative (FD) is defined in the Riemann-Liouville sense. In this scheme, the entire…
Distributed optimization is fundamental to modern machine learning applications like federated learning, but existing methods often struggle with ill-conditioned problems and face stability-versus-speed tradeoffs. We introduce fractional…