Calculus of variations with fractional derivatives and fractional integrals

We prove Euler-Lagrange fractional equations and sufficient optimality conditions for problems of the calculus of variations with functionals containing both fractional derivatives and fractional integrals in the sense of Riemann-Liouville.Comment: Accepted (July 6, 2009) for publication in Applied Mathematics Letter

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

Deciding KAT and Hoare Logic with Derivatives

Kleene algebra with tests (KAT) is an equational system for program verification, which is the combination of Boolean algebra (BA) and Kleene algebra (KA), the algebra of regular expressions. In particular, KAT subsumes the propositional fragment of Hoare logic (PHL) which is a formal system for the specification and verification of programs, and that is currently the base of most tools for checking program correctness. Both the equational theory of KAT and the encoding of PHL in KAT are known to be decidable. In this paper we present a new decision procedure for the equivalence of two KAT expressions based on the notion of partial derivatives. We also introduce the notion of derivative modulo particular sets of equations. With this we extend the previous procedure for deciding PHL. Some experimental results are also presented.Comment: In Proceedings GandALF 2012, arXiv:1210.202