26 research outputs found
Fractional Calculus of Variations for Double Integrals
We consider fractional isoperimetric problems of calculus of variations with
double integrals via the recent modified Riemann-Liouville approach. A
necessary optimality condition of Euler-Lagrange type, in the form of a
multitime fractional PDE, is proved, as well as a sufficient condition and
fractional natural boundary conditions.Comment: Submitted 07-Sept-2010; revised 25-Nov-2010; accepted 07-Feb-2011;
for publication in Balkan Journal of Geometers and Its Applications (BJGA
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
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
Variable Order Fractional Variational Calculus for Double Integrals
We introduce three types of partial fractional operators of variable order.
An integration by parts formula for partial fractional integrals of variable
order and an extension of Green's theorem are proved. These results allow us to
obtain a fractional Euler-Lagrange necessary optimality condition for variable
order two-dimensional fractional variational problems.Comment: This is a preprint of a paper whose final and definite form will be
published in: 51st IEEE Conference on Decision and Control, December 10-13,
2012, Maui, Hawaii, USA. Article Source/Identifier: PLZ-CDC12.1240.d4462b33.
Submitted 07-March-2012; accepted 17-July-201
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