7 research outputs found
Duality for the left and right fractional derivatives
We prove duality between the left and right fractional derivatives, independently on the type of fractional operator. Main result asserts that the right derivative of a function is the dual of the left derivative of the dual function or, equivalently, the left derivative of a function is the dual of the right derivative of the dual function. Such duality between left and right fractional operators is useful to obtain results for the left operators from analogous results on the right operators and vice versa. We illustrate the usefulness of our duality theory by proving a fractional integration by parts formula for the right Caputo derivative and by proving a Tonelli-type theorem that ensures the existence of minimizer for fractional variational problems with right fractional operators. (C) 2014 Elsevier B.V. All rights reserved
Time-Fractional Optimal Control of Initial Value Problems on Time Scales
We investigate Optimal Control Problems (OCP) for fractional systems
involving fractional-time derivatives on time scales. The fractional-time
derivatives and integrals are considered, on time scales, in the
Riemann--Liouville sense. By using the Banach fixed point theorem, sufficient
conditions for existence and uniqueness of solution to initial value problems
described by fractional order differential equations on time scales are known.
Here we consider a fractional OCP with a performance index given as a
delta-integral function of both state and control variables, with time evolving
on an arbitrarily given time scale. Interpreting the Euler--Lagrange first
order optimality condition with an adjoint problem, defined by means of right
Riemann--Liouville fractional delta derivatives, we obtain an optimality system
for the considered fractional OCP. For that, we first prove new fractional
integration by parts formulas on time scales.Comment: This is a preprint of a paper accepted for publication as a book
chapter with Springer International Publishing AG. Submitted 23/Jan/2019;
revised 27-March-2019; accepted 12-April-2019. arXiv admin note: substantial
text overlap with arXiv:1508.0075
Solutions of systems with the Caputo-Fabrizio fractional delta derivative on time scales
Caputo-Fabrizio fractional delta derivatives on an arbitrary time scale are
presented. When the time scale is chosen to be the set of real numbers, then
the Caputo-Fabrizio fractional derivative is recovered. For isolated or partly
continuous and partly discrete, i.e., hybrid time scales, one gets new
fractional operators. We concentrate on the behavior of solutions to initial
value problems with the Caputo-Fabrizio fractional delta derivative on an
arbitrary time scale. In particular, the exponential stability of linear
systems is studied. A necessary and sufficient condition for the exponential
stability of linear systems with the Caputo-Fabrizio fractional delta
derivative on time scales is presented. By considering a suitable fractional
dynamic equation and the Laplace transform on time scales, we also propose a
proper definition of Caputo-Fabrizio fractional integral on time scales.
Finally, by using the Banach fixed point theorem, we prove existence and
uniqueness of solution to a nonlinear initial value problem with the
Caputo-Fabrizio fractional delta derivative on time scales.Comment: This is a preprint of a paper whose final and definite form is with
'Nonlinear Analysis: Hybrid Systems', ISSN: 1751-570X, available at
[http://www.journals.elsevier.com/nonlinear-analysis-hybrid-systems].
Submitted 21/May/2018; Revised 07/Oct/2018; Accepted for publication
01-Dec-201
On a Fractional Oscillator Equation with Natural Boundary Conditions
We prove existence of solutions for a nonlinear fractional oscillator equation with both left Riemann–Liouville and right
Caputo fractional derivatives subject to natural boundary conditions. The proof is based on a transformation of the problem into an
equivalent lower order fractional boundary value problem followed by the use of an upper and lower solutions method. To succeed with
such approach, we first prove a result on the monotonicity of the right Caputo derivative
The Variable-Order Fractional Calculus of Variations
This book intends to deepen the study of the fractional calculus, giving
special emphasis to variable-order operators. It is organized in two parts, as
follows. In the first part, we review the basic concepts of fractional calculus
(Chapter 1) and of the fractional calculus of variations (Chapter 2). In
Chapter 1, we start with a brief overview about fractional calculus and an
introduction to the theory of some special functions in fractional calculus.
Then, we recall several fractional operators (integrals and derivatives)
definitions and some properties of the considered fractional derivatives and
integrals are introduced. In the end of this chapter, we review integration by
parts formulas for different operators. Chapter 2 presents a short introduction
to the classical calculus of variations and review different variational
problems, like the isoperimetric problems or problems with variable endpoints.
In the end of this chapter, we introduce the theory of the fractional calculus
of variations and some fractional variational problems with variable-order. In
the second part, we systematize some new recent results on variable-order
fractional calculus of (Tavares, Almeida and Torres, 2015, 2016, 2017, 2018).
In Chapter 3, considering three types of fractional Caputo derivatives of
variable-order, we present new approximation formulas for those fractional
derivatives and prove upper bound formulas for the errors. In Chapter 4, we
introduce the combined Caputo fractional derivative of variable-order and
corresponding higher-order operators. Some properties are also given. Then, we
prove fractional Euler-Lagrange equations for several types of fractional
problems of the calculus of variations, with or without constraints.Comment: The final authenticated version of this preprint is available online
as a SpringerBrief in Applied Sciences and Technology at
[https://doi.org/10.1007/978-3-319-94006-9]. In this version some typos,
detected by the authors while reading the galley proofs, were corrected,
SpringerBriefs in Applied Sciences and Technology, Springer, Cham, 201
Cálculo das variações fracionário
Doutoramento em Matemática e AplicaçõesO cálculo de ordem não inteira, mais conhecido por cálculo fracionário,
consiste numa generalização do cálculo integral e diferencial de ordem
inteira. Esta tese é dedicada ao estudo de operadores fracionários
com ordem variável e problemas variacionais específicos, envolvendo
também operadores de ordem variável. Apresentamos uma
nova ferramenta numérica para resolver equações diferenciais envolvendo
derivadas de Caputo de ordem fracionária variável. Consideram-
-se três operadores fracionários do tipo Caputo, e para cada um deles
é apresentada uma aproximação dependendo apenas de derivadas de
ordem inteira. São ainda apresentadas estimativas para os erros de
cada aproximação. Além disso, consideramos alguns problemas variacionais,
sujeitos ou não a uma ou mais restrições, onde o funcional
depende da derivada combinada de Caputo de ordem fracionária variável.
Em particular, obtemos condições de otimalidade necessárias
de Euler–Lagrange e sendo o ponto terminal do integral, bem como o
seu correspondente valor, livres, foram ainda obtidas as condições de
transversalidade para o problema fracionário.The calculus of non–integer order, usual known as fractional calculus,
consists in a generalization of integral and differential integer-order calculus.
This thesis is devoted to the study of fractional operators with
variable order and specific variational problems involving also variable
order operators. We present a new numerical tool to solve differential
equations involving Caputo derivatives of fractional variable order.
Three Caputo-type fractional operators are considered, and for each
one of them, an approximation formula is obtained in terms of standard
(integer-order) derivatives only. Estimations for the error of the
approximations are also provided. Furthermore, we consider variational
problems subject or not to one or more constraints, where the functional
depends on a combined Caputo derivative of variable fractional
order. In particular, we establish necessary optimality conditions of
Euler–Lagrange. As the terminal point in the cost integral, as well the
terminal state, are free, thus transversality conditions are obtained
A Variant of Chebyshev's Method with 3 alpha th-Order of Convergence by Using Fractional Derivatives
[EN] In this manuscript, we propose several iterative methods for solving nonlinear equations whose common origin is the classical Chebyshev's method, using fractional derivatives in their iterative expressions. Due to the symmetric duality of left and right derivatives, we work with right-hand side Caputo and Riemann-Liouville fractional derivatives. To increase as much as possible the order of convergence of the iterative scheme, some improvements are made, resulting in one of them being of 3 alpha-th order. Some numerical examples are provided, along with an study of the dependence on initial estimations on several test problems. This results in a robust performance for values of alpha close to one and almost any initial estimation.This research was partially supported by Ministerio de Ciencia, Innovacion y Universidades under grants PGC2018-095896-B-C22 and by Generalitat Valenciana PROMETEO/2016/089.Cordero Barbero, A.; Girona, I.; Torregrosa Sánchez, JR. (2019). A Variant of Chebyshev's Method with 3 alpha th-Order of Convergence by Using Fractional Derivatives. Symmetry (Basel). 11(8):1-11. https://doi.org/10.3390/sym11081017S111118Altaf Khan, M., Ullah, S., & Farhan, M. (2019). The dynamics of Zika virus with Caputo fractional derivative. AIMS Mathematics, 4(1), 134-146. doi:10.3934/math.2019.1.134Akgül, A., Cordero, A., & Torregrosa, J. R. (2019). A fractional Newton method with 2αth-order of convergence and its stability. Applied Mathematics Letters, 98, 344-351. doi:10.1016/j.aml.2019.06.028Caputo, M. C., & Torres, D. F. M. (2015). Duality for the left and right fractional derivatives. Signal Processing, 107, 265-271. doi:10.1016/j.sigpro.2014.09.026Odibat, Z. M., & Shawagfeh, N. T. (2007). Generalized Taylor’s formula. Applied Mathematics and Computation, 186(1), 286-293. doi:10.1016/j.amc.2006.07.102Magreñán, Á. A. (2014). A new tool to study real dynamics: The convergence plane. Applied Mathematics and Computation, 248, 215-224. doi:10.1016/j.amc.2014.09.06