Nuno R. O. Bastos

NECESSARY OPTIMALITY CONDITIONS FOR FRACTIONAL DIFFERENCE PROBLEMS OF THE CALCULUS OF VARIATIONS

We introduce a discrete-time fractional calculus of variations. 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 the solutions of the fractional problems coincide with the solutions of the corresponding non-fractional variational problems when the order of the discrete derivatives is an integer value.

Modeling some real phenomena by fractional differential equations

- This work was supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (FCT-Fundacao para a Ciencia e a Tecnologia), within project UID/MAT/04106/2013; third author by the ALGORITMI R&D Center and project PEst-UID/CEC/00319/2013. The authors are very grateful to two referees for many constructive comments and remarks.