9,402 research outputs found
The Generalized Fractional Calculus of Variations
We review the recent generalized fractional calculus of variations. We
consider variational problems containing generalized fractional integrals and
derivatives and study them using indirect methods. In particular, we provide
necessary optimality conditions of Euler-Lagrange type for the fundamental and
isoperimetric problems, natural boundary conditions, and Noether type theorems.Comment: This is a preprint of a paper whose final and definite form will
appear in Southeast Asian Bulletin of Mathematics (2014
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
Fractional variational calculus of variable order
We study the fundamental problem of the calculus of variations with variable
order fractional operators. Fractional integrals are considered in the sense of
Riemann-Liouville while derivatives are of Caputo type.Comment: Submitted 26-Sept-2011; accepted 18-Oct-2011; withdrawn by the
authors 21-Dec-2011; resubmitted 27-Dec-2011; revised 20-March-2012; accepted
13-April-2012; to 'Advances in Harmonic Analysis and Operator Theory', The
Stefan Samko Anniversary Volume (Eds: A. Almeida, L. Castro, F.-O. Speck),
Operator Theory: Advances and Applications, Birkh\"auser Verlag
(http://www.springer.com/series/4850
A Generalized Fractional Calculus of Variations
We study incommensurate fractional variational problems in terms of a
generalized fractional integral with Lagrangians depending on classical
derivatives and generalized fractional integrals and derivatives. We obtain
necessary optimality conditions for the basic and isoperimetric problems,
transversality conditions for free boundary value problems, and a generalized
Noether type theorem.Comment: This is a preprint of a paper whose final and definitive form will
appear in Control and Cybernetics. Paper submitted 01-Oct-2012; revised
25-March-2013; accepted for publication 17-April-201
Fractional Noether's theorem in the Riesz-Caputo sense
We prove a Noether's theorem for fractional variational problems with
Riesz-Caputo derivatives. Both Lagrangian and Hamiltonian formulations are
obtained. Illustrative examples in the fractional context of the calculus of
variations and optimal control are given.Comment: Accepted (25/Jan/2010) for publication in Applied Mathematics and
Computatio
The generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative
This paper presents necessary and sufficient optimality conditions for
problems of the fractional calculus of variations with a Lagrangian depending
on the free end-points. The fractional derivatives are defined in the sense of
Caputo.Comment: Accepted (19 February 2010) for publication in Computers and
Mathematics with Application
Fractional isoperimetric Noether's theorem in the Riemann-Liouville sense
We prove Noether-type theorems for fractional isoperimetric variational
problems with Riemann-Liouville derivatives. Both Lagrangian and Hamiltonian
formulations are obtained. Illustrative examples, in the fractional context of
the calculus of variations, are discussed.Comment: Submitted 12-Oct-2012; revised 05-Jan-2013; accepted 23-Jan-2013; for
publication in Reports on Mathematical Physics. arXiv admin note: substantial
text overlap with arXiv:1001.450
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