8,990 research outputs found
Discrete Direct Methods in the Fractional Calculus of Variations
Finite differences, as a subclass of direct methods in the calculus of
variations, consist in discretizing the objective functional using appropriate
approximations for derivatives that appear in the problem. This article
generalizes the same idea for fractional variational problems. We consider a
minimization problem with a Lagrangian that depends on the left
Riemann-Liouville fractional derivative. Using the Grunwald-Letnikov
definition, we approximate the objective functional in an equispaced grid as a
multi-variable function of the values of the unknown function on mesh points.
The problem is then transformed to an ordinary static optimization problem. The
solution to the latter problem gives an approximation to the original
fractional problem on mesh points.Comment: This work was partially presented 16-May-2012 by Shakoor Pooseh, at
FDA'2012, who received a 'Best Oral Presentation Award'. Submitted
26-Aug-2012; revised 25-Jan-2013; accepted 29-Jan-2013; for publication in
Computers and Mathematics with Application
Calculus of Variations on Time Scales and Discrete Fractional Calculus
We study problems of the calculus of variations and optimal control within
the framework of time scales. Specifically, we obtain Euler-Lagrange type
equations for both Lagrangians depending on higher order delta derivatives and
isoperimetric problems. We also develop some direct methods to solve certain
classes of variational problems via dynamic inequalities. In the last chapter
we introduce fractional difference operators and propose a new discrete-time
fractional calculus of variations. Corresponding Euler-Lagrange and Legendre
necessary optimality conditions are derived and some illustrative examples
provided.Comment: PhD thesis, University of Aveiro, 2010. Supervisor: Delfim F. M.
Torres; co-supervisor: Martin Bohner. Defended 26/July/201
Discrete-Time Fractional Variational Problems
We introduce a discrete-time fractional calculus of variations on the time
scale , . First and second order necessary optimality
conditions are established. Examples illustrating the use of the new
Euler-Lagrange and Legendre type conditions are given. They show that solutions
to the considered fractional problems become the classical discrete-time
solutions when the fractional order of the discrete-derivatives are integer
values, and that they converge to the fractional continuous-time solutions when
tends to zero. Our Legendre type condition is useful to eliminate false
candidates identified via the Euler-Lagrange fractional equation.Comment: Submitted 24/Nov/2009; Revised 16/Mar/2010; Accepted 3/May/2010; for
publication in Signal Processing
Leitmann's direct method for fractional optimization problems
Based on a method introduced by Leitmann [Internat. J. Non-Linear Mech. {\bf
2} (1967), 55--59], we exhibit exact solutions for some fractional optimization
problems of the calculus of variations and optimal control.Comment: Submitted June 16, 2009 and accepted March 15, 2010 for publication
in Applied Mathematics and Computation
A survey on fractional variational calculus
Main results and techniques of the fractional calculus of variations are
surveyed. We consider variational problems containing Caputo derivatives and
study them using both indirect and direct methods. In particular, we provide
necessary optimality conditions of Euler-Lagrange type for the fundamental,
higher-order, and isoperimetric problems, and compute approximated solutions
based on truncated Gr\"{u}nwald--Letnikov approximations of Caputo derivatives.Comment: This is a preprint of a paper whose final and definite form is in
'Handbook of Fractional Calculus with Applications. Vol 1: Basic Theory', De
Gruyter. Submitted 29-March-2018; accepted, after a revision, 13-June-201
Numerical solution of fractional Sturm-Liouville equation in integral form
In this paper a fractional differential equation of the Euler-Lagrange /
Sturm-Liouville type is considered. The fractional equation with derivatives of
order in the finite time interval is
transformed to the integral form. Next the numerical scheme is presented. In
the final part of this paper examples of numerical solutions of this equation
are shown. The convergence of the proposed method on the basis of numerical
results is also discussed.Comment: 14 pages, 3 figures, 2 table
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