Fractional derivative generalization of Noether’s theorem

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 Noetherlike theorem for fractional Lagrangian densities with the Riemann-Liouville fractional derivative and few examples are presented as an application of the theory.


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][2][3][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][6][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 the present work is to discuss the fractional Euler-Lagrange equations and generalized Noether's Theorem and prove the theorems in a different way.

Preliminaries
In the sequel we briefly recall some basic facts from the fractional calculus. Let f 2 L 1 .OEa; b/,˛> 0 and t 2 OEa; b, the left, resp. right Riemann-Liouville integrals are defined in the following way: where is the Euler gamma function. Left, resp. right Riemann-Liouville fractional derivative of order n 1 Ä˛< n is well defined for f 2 AC.OEa; b/ and t 2 OEa; b as Applied problems require definitions of fractional derivatives allowing the utilization of physically interpretable initial conditions, so the Caputo fractional derivatives was proposed by Caputo first in his paper [18] as / n ˛ 1 f n . /d ; and according to [1] a D? t f .t / D a Dt f .t / D d n dt n f .t /; if˛D n 2 N; a D? t f .t / D a Dt f .t /; if f .k/ .a/ D 0 k D 0; 1; 2; :::; n 1; (1) Similarly to integer-order differentiation, fractional differentiation is a linear operation [1]. Generalized Leibnitz rule gives [8] We refer the reader interested in supplementary information on fractional derivatives to the encyclopedia book of Samko et al. and Podlubny [1,19]. Consider manifold M D X U ' R p R q to be total space and whose members are p independent and q dependent variables. For m Ä˛< m C 1,˛-th jet space of M M .˛/ D M U 1 ::: U m U˛; whose coordinates of U k represent the derivatives of the dependent variables with respect to X of order k D 1; 2; :::; m;˛. Now let p D 2, q D 1 and be a system over M .˛/ . For continuous parameter " the invertible transformations of the variables M x D '.x; t; u; "/; t D .x; t; u; "/; u D .x; t; u; "/; (3) are said to be symmetry transformations of , if has the same form in the new variables x; t ; u. All such transformations forms one parameter symmetry group G, that is When transformation (3) is applied to usual partial derivatives u x , u t , u xx ; , according to [5,20] the prolongation of (4) on derivatives of u respect to x is By extending transformation (3) to the operator of Riemann-Liouville fractional Dt u and t D˛u, we have where 0 is given by prolongation formulae and (2) 0 can be calculated in a similar operation [8]. The invariance condition cause to 0 .t; x; u/ j tD0 D 0 [8].
If P r˛.v/ denotes the prolongation of the operator v of the˛-order derivative, the classical Lie theory can be written in a more compact form that has to extend v up to appearance derivative orders. The obtained equations define all infinitesimal symmetries of .
subject to the initial conditions and boundary condition as After splitting (7) with respect to u t , u t t , u 2 t ; ; D˛ n t u.t /, for n D 0; 1; which are considered independent, we arrive to over determined system of linear fractional differential equations. Solution of this system gives so Á D 2c 1 x and v D c 1 @ t C 2c 1 x@ u is a symmetry of (6).

Noether's theorem for fractional problems of the calculus of variations
In this section t D .t 1 ; t 2 ; :::; t p /, u D .u 1 ; u 2 ; :::; u q / and u .n/ shows the derivatives of u up to n-order, and m Ä˛;ˇ< m C 1. Now we note an equivalent useful way to writing down the prolongation formulae. Suppose that the q-tuple Q D .Q 1 ; :::; Q q / is characteristic of the vector field v, then where and the summation extends to over multi-indices J D .j 1 ; ; j k /, with 1 Ä j k Ä p and k 0 [5,6]. for some m 2 N, and some functions P 1 i and P 2 i , where each pair P 1 i and P 2 i satisfy  [14].
In continuation, @ i L is the notation of the partial derivative of L with respect to its i th argument.
Definition 3.3. For 1 Ä Ä Ä q , the Ä-th Euler operator is given by For˛;ˇ2 N, we have a Dt D d dt and t Db D d dt and (9) reduces to the standard Euler-Lagrange equation [5,13]. Then a necessary condition for L.u/ to have an extreme for a given function u.t / is that u.t / satisfies the following Euler-Lagrange equation [5,13]: Definition 3.5. A local group of transformations G acting on M Â C U is a vari at i onal sy mmet ry group of (10), if whenever Â C , u D f .t / defined over C whose graph lies in M and g 2 G is such that u D f .t/ D g:f .t/, then Z L.t; P r .n/ f .t /; a Dt u; t Db u/dt D Z L.t ; P r .n/ f .t /; a Dt u; t Db u/d t : Note that for each g 2 G, the group transformation can be regarded as a change of variables, so we can rewrite (12) in the form Z L.t ; P r .n/ f .t /; a Dt u; t Db u/detJ g .t; u 1 /dt; where the Jacobi an matrix has entries for every infinitesimal generator of G, Div denotes the total divergence of the p-tuple D . 1 ; 2 ; :::; p /.
Theorem 3.7. Suppose G is a vari at i onal sy mmet ry group of the (10).
be the infinitesimal generator of G, then Q D .Q 1 ; :::; Q q /is the characteristic of a conservation law for Euler-Lagrange equations E.L/ D 0, in other words, there is a p-tuple P .x; u .˛;ˇ/ / D .P 1 ; P 2 ; :::; P p / such that Proof. By substituting the prolongation formula (14) into (8), we find Q˛E˛.L/ C DivP: So we have proved P D Q˛@ L @uJ C L is a conservation law for Euler-Lagrange equations.  t Db Á@ t Db u /.L/ Á.@ 2 L C t Db @ 3 L C a Dť @ 4 L/ D a Dt Á@ 3 L t Db Á@ 4 L Á t Db @ 3 L Á a Dť @ 4 L D Dt .@ 3 L; Á/ C Dť .@ 4 L; Á/: Example 3.10. If v D @t C Á@u is vari at i onal sy mmet ry of the functional L D R b a L.t; u; a Dt u; t Db u/dt, then Q D Á @u @t ; P r .˛;ˇ/ v.L/ C LDiv Q:E.L/ D @ 1 L C @ 2 L @u @t C @ 3 L a D˛C 1 t u C @ 4 L t DˇC 1 b u C L @ @t Á t Db @ 3 L C t Dą Á@ 3 L Á a Dť @ 4 L C t Dǎ Á@ 4 L C @u @t t Db @ 3 L @ 3 L a Dt .

Conclusion
The classical Lie point symmetry method has been investigated for the Bagley-Torvik equation in the sense of the Riemann-Liouville derivatives. The general form of Noether-like theorem for fractional Lagrangian densities is proved. Few examples are presented as an application of the theory.