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J Optim Theory Appl (2013) 156:5667
DOI 10.1007/s10957-012-0203-6
Matheus J. Lazo Delm F. M. Torres
Received: 26 July 2012 / Accepted: 3 October 2012 / Published online: 19 October 2012 Springer Science+Business Media New York 2012
Abstract Derivatives and integrals of noninteger order were introduced more than three centuries ago but only recently gained more attention due to their application on nonlocal phenomena. In this context, the Caputo derivatives are the most popular approach to fractional calculus among physicists, since differential equations involving Caputo derivatives require regular boundary conditions. Motivated by several applications in physics and other sciences, the fractional calculus of variations is currently in fast development. However, all current formulations for the fractional variational calculus fail to give an EulerLagrange equation with only Caputo derivatives. In this work, we propose a new approach to the fractional calculus of variations by generalizing the DuBoisReymond lemma and showing how EulerLagrange equations involving only Caputo derivatives can be obtained.
Keywords Fractional calculus Fractional calculus of variations
DuBoisReymond lemma EulerLagrange equations in integral and differential
forms
1 Introduction
The fractional calculus with derivatives and integrals of noninteger order started more than three centuries ago, with lHpital and Leibniz, when the derivative of order 1/2 was suggested [1]. This subject was then considered by several mathematicians
M.J. Lazo
Instituto de Matemtica, Estatstica e Fsica, FURG, Rio Grande, RS, Brazil e-mail: mailto:[email protected]
Web End [email protected]
D.F.M. Torres ( )
Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugale-mail: mailto:[email protected]
Web End [email protected]
The DuBoisReymond Fundamental Lemma of the Fractional Calculus of Variationsand an EulerLagrange Equation Involving Only Derivatives of Caputo
J Optim Theory Appl (2013) 156:5667 57
like Euler, Fourier, Liouville, Grunwald, Letnikov, Riemann, and many others up to nowadays. Although the fractional calculus is almost as old as the usual integer-order calculus, only in the last three decades it has gained more attention due to its many applications in various elds of science, engineering, economics, biomechanics, etc. (see [26] for a review).
Fractional derivatives are nonlocal operators and are historically applied in the study of nonlocal or time-dependent processes. The rst and well-established application of fractional calculus in physics was in the framework of anomalous diffusion, which is related...