Generalized fractional calculus with applications to the calculus of variations

Abstract

We study operators that are generalizations of the classical Riemann-Liouville fractional integral, and of the Riemann-Liouville and Caputo fractional derivatives. A useful formula relating the generalized fractional derivatives is proved, as well as three relations of fractional integration by parts that change the parameter set of the given operator into its dual. Such results are explored in the context of dynamic optimization, by considering problems of the calculus of variations with general fractional operators. Necessary optimality conditions of Euler-Lagrange type and natural boundary conditions for unconstrained and constrained problems are investigated. Interesting results are obtained even in the particular case when the generalized operators are reduced to be the standard fractional derivatives in the sense of Riemann-Liouville or Caputo. As an application we provide a class of variational problems with an arbitrary kernel that give answer to the important coherence embedding problem. Illustrative optimization problems are considered.

Submitted 22-Dec-2011; revised 26-Jan-2012; accepted 27-Jan-2012; for publication in Computers and Mathematics with Applications

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• Fractional Calculus of Variations in Terms of a Generalized Fractional Integral with Applications to Physics
• Discrete Direct Methods in the Fractional Calculus of Variations
• Green's Theorem for Generalized Fractional Derivatives
• Fractional isoperimetric Noether's theorem in the Riemann-Liouville sense
• The DuBois-Reymond Fundamental Lemma of the Fractional Calculus of Variations and an Euler-Lagrange Equation Involving only Derivatives of Caputo
• Fractional Calculus of Variations of Several Independent Variables
• Conformal and Non-Minimal Couplings in Fractional Cosmology
• Variable Order Fractional Variational Calculus for Double Integrals
• A Generalized Fractional Calculus of Variations
• Fractional Einstein-Gauss-Bonnet scalar field cosmology
• Generalized fractional operators for nonstandard Lagrangians
• Fractional Order Version of the HJB Equation
• An approximation formula for the Katugampola integral
• A Non-Differentiable Quantum Variational Embedding in Presence of Time Delays
• Fractional Noether's Theorem with Classical and Riemann-Liouville Derivatives
• A Discretization Method to Solve Fractional Variational Problems with Dependence on Hadamard Derivatives