Calculus of Variations with Classical and Fractional Derivatives

Abstract

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.

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'2011

References in corpus (10)

• A Formulation of Noether's Theorem for Fractional Problems of the Calculus of Variations
• Calculus of variations with fractional derivatives and fractional integrals
• Variational Problems with Fractional Derivatives: Euler-Lagrange Equations
• Necessary Optimality Conditions for Fractional Action-Like Integrals of Variational Calculus with Riemann-Liouville Derivatives of Order $(\alpha ,\beta )$
• Fractional Variational Calculus with Classical and Combined Caputo Derivatives
• The generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative
• Fractional variational problems depending on indefinite integrals
• Fractional variational calculus for nondifferentiable functions
• Leitmann's direct method for fractional optimization problems
• The isoperimetric problem for Holderian curves

Cited by in corpus (3)

• Fractional Variational Calculus with Classical and Combined Caputo Derivatives
• Composition Functionals in Fractional Calculus of Variations
• Fractional isoperimetric Noether's theorem in the Riemann-Liouville sense