Advances in some fractional variational problems with Caputo derivatives.

MELANI BARRIOS; GABRIELA REYERO; DEMIAN GOOS

Edimburgo

Workshop; 13th Europt Workshop on Advances in Continuous Optimization 2015 - Edinburgh; 2015

EURO - The Association of European Operational Research Societies

Derivatives and integrals of fractional order have recently gained more attention due to their successful application to non local phenomena. <br />Motivated by numerous applications in physics and other scientific areas, fractional calculus of variations finds itself in fast development.<br />In this work we consider variational problems of the following kind: find y in E(a,b,alfa,beta), so that it maximizes or minimizes the functional <br />J(y)=int_{a}^{b} L(x, y, D^{alpha}y,D^{beta}y) dx<br />where  L is a Langrangian function, D^{alpha}, D^{beta} denote the left and right fractional derivatives in Caputo´s sense of order alpha, beta in (0,1).<br />Different necessary and sufficient conditions of optimality are considered, particularly some of fractional Euler-Lagrange equation.<br />Several fractional variational problems are presented: with fixed or free boundary conditions, in presence of integral constraints that also depend on Caputo derivatives and some examples are given.