A Fractional Calculus of Variations for Multiple Integrals with Application to Vibrating String

Abstract

We introduce a fractional theory of the calculus of variations for multiple integrals. Our approach uses the recent notions of Riemann-Liouville fractional derivatives and integrals in the sense of Jumarie. Main results provide fractional versions of the theorems of Green and Gauss, fractional Euler-Lagrange equations, and fractional natural boundary conditions. As an application we discuss the fractional equation of motion of a vibrating string.

Accepted for publication in the Journal of Mathematical Physics (14/January/2010)

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Cited by in corpus (28)

• Discrete-Time Fractional Variational Problems
• Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives
• Fractional Variational Iteration Method for Fractional Nonlinear Differential Equations
• Fractional Noether's theorem in the Riesz-Caputo sense
• 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 calculus of variations for a combined Caputo derivative
• Generalized fractional calculus with applications to the calculus of variations
• Non-Standard Extensions of Gradient Elasticity: Fractional Non-Locality, Memory and Fractality
• A formulation of the fractional Noether-type theorem for multidimensional Lagrangians
• Fractional variational problems depending on indefinite integrals
• Fractional Calculus of Variations in Terms of a Generalized Fractional Integral with Applications to Physics
• Fractional variational calculus of variable order
• Approximation of fractional integrals by means of derivatives
• Duality for the left and right fractional derivatives
• Fractional variational calculus for nondifferentiable functions
• Leitmann's direct method for fractional optimization problems
• Multiobjective fractional variational calculus in terms of a combined Caputo derivative
• Higher-order Hahn's quantum variational calculus
• Fractional Euler-Lagrange differential equations via Caputo derivatives
• Green's Theorem for Generalized Fractional Derivatives
• Generalized Transversality Conditions in Fractional Calculus of Variations
• Isoperimetric problems of the calculus of variations with fractional derivatives
• Fractional Calculus of Variations of Several Independent Variables
• Generalized Tu Formula and Hamilton Structures of Fractional Soliton Equation Hierarchy
• Variable Order Fractional Variational Calculus for Double Integrals
• Lie group classifications and exact solutions for time-fractional Burgers equation
• Nondifferentiable variational principles in terms of a quantum operator