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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A Fractional Spectral Method with Applications to Some Singular Problems
Dianming Hou,Chuanju Xu +1 more
TL;DR: In this article, a fractional spectral method for non-smooth solutions of integro-differential equations with weakly singular kernels is presented, and a basic numerical analysis is given, together with a series of numerical examples to verify the efficiency.
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Bending analysis of functionally graded nanobeams based on the fractional nonlocal continuum theory by the variational Legendre spectral collocation method
TL;DR: In this article, the size-dependent bending behavior of nanobeams made of functionally graded materials is studied through a numerical variational approach, where the nonlocal effects are captured in the context of fractional calculus.
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Noether symmetries and conserved quantities for fractional Birkhoffian systems with time delay
TL;DR: The Noether symmetries and the conserved quantities for fractional Birkhoffian systems with time delay in terms of Riemann–Liouville fractional derivatives are proposed and studied.
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The DuBois–Reymond Fundamental Lemma of the Fractional Calculus of Variations and an Euler–Lagrange Equation Involving Only Derivatives of Caputo
TL;DR: In this paper, a new approach to the fractional calculus of variations by generalizing the DuBois-Reymond lemma and showing how Euler-Lagrange equations involving only Caputo derivatives can be obtained.
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.