20 research outputs found
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 variational problems with the Riesz-Caputo derivative
In this paper we investigate optimality conditions for fractional variational problems, with a Lagrangian depending on the Riesz-Caputo derivative. First we prove a generalized Euler-Lagrange equation for the case when the interval of integration of the functional is different from the interval of the fractional derivative. Next we consider integral dynamic constraints on the problem, for several different cases. Finally, we determine optimality conditions for functionals depending not only on the admissible functions, but on time also, and we present a necessary condition for a pair function-time to be an optimal solution to the problem. 漏 2011 Elsevier Ltd. All rights reserved.FCTCIDM
Calculus of Variations with Classical and Fractional Derivatives
We give a proper fractional extension of the classical calculus of
variations. Necessary optimality conditions of Euler-Lagrange type for
variational problems containing both classical and fractional derivatives are
proved. The fundamental problem of the calculus of variations with mixed
integer and fractional order derivatives as well as isoperimetric problems are
considered.Comment: This is a preprint of a paper whose final and definite form has been
published in: Math. Balkanica 26 (2012), no 1-2, 191--202. It was first
announced at the IFAC Workshop on Fractional Derivatives and Applications
(IFAC FDA'2010), held in University of Extremadura, Badajoz, Spain, October
18-20, 2010; then subsequently at conference TMSF'201
Fractional Euler-Lagrange differential equations via Caputo derivatives
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 considered: with fixed or free boundary conditions, and in
presence of integral constraints that also depend on Caputo derivatives.Comment: This is a preprint of a paper whose final and definite form will
appear as Chapter 9 of the book Fractional Dynamics and Control, D. Baleanu
et al. (eds.), Springer New York, 2012, DOI:10.1007/978-1-4614-0457-6_9, in
pres
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
Isoperimetric problems of the calculus of variations with fractional derivatives
In this paper we study isoperimetric problems of the calculus of variations
with left and right Riemann-Liouville fractional derivatives. Both situations
when the lower bound of the variational integrals coincide and do not coincide
with the lower bound of the fractional derivatives are considered.Comment: Submitted 02-Oct-2009; revised 30-Jun-2010; accepted 10-May-2011; for
publication in the journal Acta Mathematica Scienti
Fractional Euler-Lagrange differential equations via Caputo derivatives
We review some recent results of the fractional variational calculus. Necessary optimality conditions of Euler鈥揕agrange type for functionals with a Lagrangian containing left and right Caputo derivatives are given. Several problems
are considered: with 铿亁ed or free boundary conditions, and in presence of integral
constraints that also depend on Caputo derivatives.CIDMAFCTFEDER/POCI 2010Bia艂ystok University of TechnologyS/WI/00/201