Formulation of Euler–Lagrange equations for fractional variational problems
TLDR
In this article, the Euler-Lagrange type necessary conditions which must be satisfied for the given functional to be extremum were developed for systems containing fractional derivatives, where the fractional derivative is described in the Riemann-Liouville sense.About:
This article is published in Journal of Mathematical Analysis and Applications.The article was published on 2002-08-01 and is currently open access. It has received 866 citations till now. The article focuses on the topics: Fractional calculus & Euler–Lagrange equation.read more
Citations
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Optimality conditions and a solution scheme for fractional optimal control problems
TL;DR: In this article, the authors formulated necessary conditions for optimality in optimal control problems with dynamics described by differential equations of fractional order (derivatives of arbitrary real order) and proposed a new solution scheme.
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Stochastic response determination of nonlinear oscillators with fractional derivatives elements via the Wiener path integral
TL;DR: In this paper, the concept of the Wiener path integral in conjunction with a variational formulation is utilized to derive an approximate closed form solution for the system response non-stationary PDF.
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A New Feature of the Fractional Euler-Lagrange Equations for a Coupled Oscillator Using a Nonsingular Operator Approach
TL;DR: In this paper, the free motion of a coupled oscillator is investigated and a fully description of the system under study is formulated by considering its classical Lagrangian, and as a result, the classical Euler-Lagrange equations of motion are constructed.
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Variational problems with fractional derivatives: Invariance conditions and N\"{o}ther's theorem
TL;DR: In this paper, a variational principle for Lagrangian densities containing derivatives of real order is formulated and the invariance of this principle is studied in two characteristic cases: necessary and sufficient conditions for an infinitesimal transformation group (basic Nother's identity) are obtained.
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A new method of finding the fractional Euler-Lagrange and Hamilton equations within Caputo fractional derivatives
Dumitru Baleanu,Juan I. Trujillo +1 more
TL;DR: In this paper, the fractional Caputo derivative of a composition function was investigated for constrained systems and the obtained results were applied to investigate fractional Euler-Lagrange and Hamilton equations.
References
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Book
An Introduction to the Fractional Calculus and Fractional Differential Equations
Kenneth S. Miller,Bertram Ross +1 more
TL;DR: The Riemann-Liouville Fractional Integral Integral Calculus as discussed by the authors is a fractional integral integral calculus with integral integral components, and the Weyl fractional calculus has integral components.
Book
Fractional Integrals and Derivatives: Theory and Applications
TL;DR: Fractional integrals and derivatives on an interval fractional integral integrals on the real axis and half-axis further properties of fractional integral and derivatives, and derivatives of functions of many variables applications to integral equations of the first kind with power and power-logarithmic kernels integral equations with special function kernels applications to differential equations as discussed by the authors.
Book
Applications Of Fractional Calculus In Physics
TL;DR: An introduction to fractional calculus can be found in this paper, where Butzer et al. present a discussion of fractional fractional derivatives, derivatives and fractal time series.
BookDOI
Fractals and fractional calculus in continuum mechanics
TL;DR: Panagiotopoulos, O.K.Carpinteri, B. Chiaia, R. Gorenflo, F. Mainardi, and R. Lenormand as mentioned in this paper.