© 2015 Khorshidi et al., published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License. Open Math. 2015; 13: 940–947 Open Mathematics Open Access Research Article Maryam Khorshidi, Mehdi Nadjafikhah*, and Hossein Jafari Fractional derivative generalization of Noether’s theorem DOI 10.1515/math-2015-0086 Received May 25, 2015; accepted August 8, 2015. Abstract: The symmetry of the Bagley–Torvik equation is investigated by using the Lie group analysis method. The Bagley–Torvik equation in the sense of the Riemann–Liouville derivatives is considered. Then we prove a Noether- like theorem for fractional Lagrangian densities with the Riemann-Liouville fractional derivative and few examples are presented as an application of the theory. Keywords: Fractional derivatives, Symmetry, Fractional variational calculus, Fractional Euler–Lagrange equations, Conservation laws, Noether’s theorem MSC: 70H33, 34K17, 70G65 1 Introduction Fractional differentiation is a significant tool to describe and obtain mathematical model of real phenomena in various field of sciences [1, 2]. During the last four decades several analytical and numerical methods were presented for solving fractional differential equations (FDE) [1–4]. However there are some limitation for using those methods for solving different classes of FDE. Symmetry is an important property of nature and all of the equations that are able to describe physical, biological or chemical phenomena have symmetry properties which follow from some fundamental rules [5–7]. Gazizov et al. [8], generalized the prolongation formulas for fractional derivatives and adapted the method of Lie group for symmetry analysis of FDEs. The concept conservation laws or first integrals of the Euler-Lagrange equations is well known in Physics. The general principle relating to symmetry groups and conservation laws was first introduced by Noether (1918) [9, 10]. Riewe [11, 12] studied Euler-Lagrange equations for problems of the calculus of variations with fractional derivatives. Agrawal [13] presented extensions to traditional calculus of variations for systems containing fractional derivatives. Accordingly, the Euler-Lagrange equations were used by Frederico and Torres to prove a Noether-type theorem and Fractional Noether’s theorem in the Riesz–Caputo sense [14, 15]. The Bagley-Torvik equation formulae was originally considered in the studies on properties of real material by using fractional calculus, especially in 1=2 or 3=2 order derivatives [16, 17]. The paper is organized as follows. Section 2 includes several properties pertaining to the fractional derivative Symmetries and prolongation. Then we investigate a symmetry for Bagley-Torvik equation. In Section 3 our aim in Maryam Khorshidi: Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran, E-mail: maryam_khorshidii@yahoo.com *Corresponding Author: Mehdi Nadjafikhah: Department of Pure Mathematics,School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, 1684613114, Iran, E-mail: mnadjafikhah@gmail.com Hossein Jafari: Department of Mathematical Sciences, University of South Africa, PO Box 392, UNISA 0003, South Africa and Department of Mathematics,University of Mazandaran, Babolsar, Iran, E-mail: jafarh@unisa.ac.za Unauthenticated Download Date | 1/9/16 1:42 PM