REFLECTION SYMMETRIC FORMULATION OF GENERALIZED FRACTIONAL VARIATIONAL CALCULUS

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1 RESEARCH PAPER REFLECTION SYMMETRIC FORMULATION OF GENERALIZED FRACTIONAL VARIATIONAL CALCULUS Ma lgorzata Klimek 1, Maria Lupa 2 Abstract We define generalized fractional derivatives GFDs symmetric and anti-symmetric w.r.t. the reflection symmetry in a finite interval. Arbitrary functions are split into parts with well defined reflection symmetry properties in a hierarchy of intervals [,b/2 m ], m N. For these parts - -projections of function, we derive the representation formulas for generalized fractional operators GFOs and examine integration properties. It appears that GFOs can be reduced to operators determined in subintervals [,b/2 m ]. The results are applied in the derivation of Euler-Lagrange equations for action dependent on Riemann-Liouville type GFDs. We show that for Lagrangian being a sum finite or not of monomials, the obtained equations of motion can be localized in arbitrary short subinterval [,b/2 m ]. MSC 21 : Primary 26A33; Secondary 34A8, 49S5, 7H3 Key Words and Phrases: fractional calculus, generalized fractional integrals and derivatives, fractional mechanics, Euler-Lagrange equations, localization 1. Introduction Fractional mechanics - mechanics based on actions dependent on nonlocal derivatives was introduced by Riewe in [27, 28] and extended in papers [1, 16, 17]. These early results demonstrate the main difficulties one meets c 213 Diogenes Co., Sofia pp , DOI: /s x

2 244 M.Klimek,M.Lupa when decribing physical phenomena via minimum action principle and including memory effects into a model compare results and comments in [2]- [13],[18],[23]-[26]. The obtained Euler-Lagrange equations are non-local and mix derivatives determined by the left and right neighbourhood of the given time point. Such equations are difficult to solve explicitly and this area of fractional differential equations theory still requires further investigations compare [19, 2] and references therein. In the paper we propose a new formalism for fractional variational calculus for actions including the generalized fractional derivatives GFDs. We shall show that GFVC in the developed formulation leads again to systems of non-local equations but that a wide class of models can be localized. The obtained results apply to fractional variational calculus with standard fractional derivatives. The aim of our paper is twofold. First, we shall examine properties of generalized fractional calculus w.r.t. reflections in finite intervals. Then, we focus on Euler-Lagrange equations for actions dependent on generalized fractional derivatives. These equations were derived in the paper by Agrawal [3] in interval [,b]. We shall demonstrate that trajectories can be split into parts for which we will be able to derive Euler-Lagrange equations in subintervals. The obtained systems of equations of motion are the main result of the paper together with corollaries which describe a novel phenomenon for nonlocal fractional differential equations. Namely, the considered Euler-Lagrange systems can be localized in subintervals of the main time interval.we shall begin with some preliminaries. 2. Preliminaries The following left and right operators, generated by Laplace convolution with power function, are constructed in fractional calculus [15, 29]. Definition 2.1. Let α>. The formulas below define the left and right Riemann-Liouville R-L integrals Ia+ft α = 1 t t s α 1 fsds, t > a, 2.1 Γα a Ib α ft = 1 s t α 1 fsds, t < b, 2.2 Γα t where Γ is the Euler gamma function. Property 2.2. the relations The left and right Riemann-Liouville integrals fulfill

3 REFLECTION SYMMETRIC FORMULATION I α a+ = T ai α + T a, 2.3 I α b = Q [a,b]i α a+q [a,b] = Q [a,b] T a I α +T a Q [a,b], 2.4 where T a and Q are respectively the translation and reflection operators acting as follows on function f: T a ft :=ft + a, Q [a,b] ft :=fa + b t. 2.5 Definition 2.3. Let α n 1,n. The left and right Riemann- Liouville derivatives of order α are defined as Da,+ft α =D n Ia,+ 1 α ft, t > a, 2.6 Db, α ft = Dn I 1 α b, ft, t < b, 2.7 where we denoted the first order derivative as D d dt. Analogous formulas yield the left- and right-sided Caputo derivatives of order α: c Da,+ α ft =I1 α a,+ Dn ft, t > a, 2.8 c Db, α ft =I1 α b, Dn ft, t < b. 2.9 In the paper [3], Agrawal introduced generalized fractional derivatives of Riemann-Liouville and Caputo type starting from the following integral determined by kernel k α. In the definitions below we recall generalized fractional integrals GFIs and generalized fractional derivatives GFDs of Riemann-Liouville and Caputo type. Definition 2.4. Let α >. Operator Ka,b;p,q α is a generalized fractional integral GFI given by the formula [ Ka,b;p,q α 1 t ] ft := p k α t, sfsds + q k α s, tfsds. 2.1 Γα a Definition 2.5. Let α n 1,n. The following operators are generalized R-L and Caputo type derivatives of order α A α a,b;p,q := Dn K n α a,b;p,q, 2.11 where again D d dt. B α a,b;p,q t := Kn α a,b;p,q Dn, 2.12 The properties of the GFIs and GFDs are analogous to those of standard fractional operators. We shall describe them briefly in the case when kernel k α t, s =k α t s.

4 246 M.Klimek,M.Lupa Property Let us assume that kernel depends only on the difference of variables: k α t, s k α t s. Then, the following relations hold for GFIs: K,b;1, α ft =ft k αt Γα Ka,b;1, α = T ak,b;1, α T a K α a,b;,1 = Q [a,b]k α a,b;1, Q [a,b] K α a,b;p,q ft =pkα a,b;1, ft+qq [a,b]k α a,b;1, Q [a,b]ft. The T a and Q [a,b] operators are respectively the translation and reflection operators given in formula Let us assume that α n 1,n and k n α t, s k n α t s. Then, the following relations hold for GFDs of Riemann-Liouville and Caputo type [ A α,b;1, ft =Dn ft k ] αt Γα B,b;1, α ft =[Dn ft] k αt Γα A α a,b;1, = T aa α,b;1, T a Ba,b;1, α = T ab,b;1, α T a A α a,b;,1 = 1n Q [a,b] A α a,b;1, Q [a,b] B α a,b;,1 = 1n Q [a,b] B α a,b;1, Q [a,b] A α a,b;p,q ft =paα a,b;1, ft+qaα a,b;,1 ft B α a,b;p,q ft =pbα a,b;1, ft+qbα a,b;,1 ft. Let us point out that in the case k α t =t α 1 we recover fractional calculus of integrals and derivatives from Definitions 2.1 and 2.3. Example 2.7. As an example of the generalized fractional calculus let us discuss the fractional calculus based on operators defined using Mittag- Leffler functions. They were introduced in [14], extended in [3, 31] and applied in papers [32, 33]. For each admissible set of parameters γ,ω,βwe have in our notation A, B, K operators determined by the following kernel k α t, s =k α t s =t s α 1 E γ β,α ωt sβ, 2.13 K α,γ,b;1, ft :=ft k αt Γα, 2.14 where the function E γ β,α - the generalized Mittag-Leffler function called also as Prabhakar function, 1971; [15] is defined by the series given below: E γ β,α := γ l Γβl + α zl l! l=

5 REFLECTION SYMMETRIC FORMULATION Then, the K integrals have the semigroup property [14] K α 1,γ 1,b;1, Kα 2,γ 2,b;1, = Kα 1+α 2,γ 1 +γ 2,b;1, 2.16 and reduce to the classical and fractional integral in the special case K 1,,b;1, ft =ft 1 Γ1 = I1 +ft, 2.17 K α, tα 1,b;1, ft =ft Γα = Iα +ft The left and right derivatives of Riemann-Liouville and Caputo type from Definition 2.5 look as follows A α,γ,b;1, = Dn K n α, γ,b;1,, B α,γ,b;1, = Kn α, γ,b;1, D n, 2.19 A α,γ,b;,1 = QAα,γ,b;1, Q, Bα,γ,b;,1 = QBα,γ,b;1, Q. 2.2 The derivative given in 2.19 is the left inverse operator for the K-integral. Using semi-group property 2.16 we get for α n 1,nrelation A α,γ,b;1, Kα,γ ft =ft 2.21,b;1, valid for any t [,b] for function f C[,b] and a.e. in [,b]inthecase when f L p,b. Next, assuming that f C[,b]andK α n+1, γ,b;1, f = we obtain the composition rule for the B-derivative B α,γ,b;1, Kα,γ,b;1, ft =ft, 2.22 valid for any t [,b]. Summarizing, for each fixed γ>andα>we have generalized fractional calculus as defined in [3]. Now, we go back to the general kernel and assume k α t, s =k α t s. In the paper [3] the integration by parts formula was derived. Property 2.8. The generalized fractional differential operators satisfy the following relation with boundary terms given explicitly when order α, 1 a fxaα a,b;p,q gxdx = a gxbα a,b;q,p fxdx + b.t b.t. = fxk 1 α a,b;p,q gx b x=a 2.24 for k α L 1 a, b and sufficiently smooth functions f,g.

6 248 M.Klimek,M.Lupa In the reflection symmetric formulation of fractional variational calculus we shall apply the GFDs determined by the Riesz-type potentials dependent on kernel k α which we define below. Definition 2.9. Let α n 1,nanda =. The symmetric and respectively anti-symmetric generalized RL type derivatives in interval [,b] are defined as follows ] A α [,b] [A := α,b;p,q + Aα,b;q,p /2p + q =D n K n α [,b], 2.25 ] Ā α [,b] [A := α,b;p,q Aα,b;q,p /2p q =D n Kn α [,b], 2.26 where integrals K n α [,b] and K n α [,b] K n α [,b] ft := 1 Γn α K n α [,b] ft := 1 Γn α are Riesz type potentials in interval [,b]: k α t s fsds, 2.27 k α t s sgnt sfsds The symmetric and respectively anti-symmetric generalized Caputo type derivatives in interval [,b]aregivenas: ] B[,b] [B α :=,b;p,q α + Bα,b;q,p /2p + q =K n α [,b] Dn, 2.29 [ ] B [,b] α := B,b;p,q α Bα n α,b;q,p /2p q = K [,b] Dn Reflection symmetry in generalized fractional calculus In this section we shall describe properties of the introduced symmetric and anti-symmetric GFDs w.r.t. reflection symmetries in finite subintervals of [,b]. We start by the splitting of function f into respective projections which were introduced in [21, 22]. Definition 3.1. Let f be an arbitrary function determined in [,b] and vector =[j 1,...,j m ]havecomponentsj l {, 1}. The following recursive formulas define the [j]- and respectively [J, j m+1 ]-projections of function f: f [j] t := j Q 2 [,b] ft f [J,jm+1 ]t :={ j m+1 Q [,b/2 m ] f t, t b/2 m 1 2 f t, t>b/2 m. 3.2 The function f can be split into the respective projections

7 REFLECTION SYMMETRIC FORMULATION ft = 1 f t, f t = f [J,j] t, 3.3 {,1} m j= where the summation is taken over all m-component vectors with coordinates in the two-element set {, 1}. For each given vector =[j 1,..., j m ], projection f can be represented as follows in subinterval [,b/2 m ] f = ft, 3.4 where the operator is ordered product of the projection operators: := 2 m 1+ 1 jm Q [,b/2 m 1 ] j 1 Q [,b]. 3.5 Let us note that each projection f for vector =[j 1,...,j m ] - is fully determined by its explicit form on subinterval [,b/2 m ]. Our aim is to apply this observation in fractional variational calculus. We shall describe an action in terms of projections of trajectories and demonstrate that the minimum action principle leads to a system of localized Euler-Lagrange equations. This phenomenon was also discussed in [21, 22] for some models dependent on standard fractional operators of orders in range, 1 1, 2. Property 3.2. Let =[j 1,...,j m ] and f [j],f be the projections defined above. 1 The following relations are valid: Q [,b] f [j] t t [,b] = f [j] b t = 1 j f [j] t, 3.6 Q [,b/2 m ]f [j1,...,j m+1 ]t t [,b/2 m ]= 1 j m+1 f [j1,...,j m+1 ]t The projections fulfill orthogonality relations f [j] tg [k] tdt = f [j] tg [j] tdt δ j,k, 3.8 b/2 m b/2 m f [J,j] tg [J,k] tdt = f [J,j] tg [J,j] tdt δ j,k, 3.9 where δ j,k denotes the Kronecker delta. The properties of projections w.r.t. the reflection symmetry and integration rules lead to the following representation of the symmetric Riesz type integral.

8 25 M.Klimek,M.Lupa Proposition 3.3. Let f be the =[j 1,..., j m ]-projection of function f. Its symmetric K-integrals of order α in interval [,b] can be represented as follows K α [,b] f t =2 m K α [,b/2 m ] f t, 3.1 where we denote as the ordered product of the projection operators := 2 m 1+ 1 j 1 Q [,b] j m Q [,b/2 ] m Proof.Letusbeginwithm =1and =[j]: K[,b] α f [j]t = 1 k α t s f Γα [j] sds [ = 1 b/2 ] b k α t s f Γα [j] sds + k α t s f [j] sds. Applying substitution s = b s to the second integral and Property 3.2, we obtain K[,b] α f [j]t = 1 Γα + 1 Γα /2 /2 b/2 k α t s f [j] sds k α b t s 1 j f [j] s ds = 1+ 1 j Q [,b] K α [,b/2] f [j] t. To prove formula 3.1 in the general case we use the mathematical induction principle and assume that it is valid for vector =[j 1,..., j m ]. Then, for vector [J, j m+1 ]wehave K α [,b] f [J,j m+1 ]t =2 m K α [,b/2 m ] f [J,j m+1 ]t 1+ 1 j m+1 Q [,b] K α [,b/2 m+1 ] f [J,j m+1]t =2 m =2 m+1 [J,j m+1 ] K α [,b/2 m+1 ] f [J,j m+1 ]t. Clearly, the above formula coincides with 3.1 for vector [J, j m+1 ]. This ends the proof. An analogous property is valid for the anti-symmetric K-integrals.

9 REFLECTION SYMMETRIC FORMULATION Proposition 3.4. Let f be the =[j 1,..., j m ]-projection of function f. Its anti-symmetric K-integrals of order α in interval [,b] can be represented as follows K [,b] α f t =2 m K [,b/2 α m ] f t, 3.12 [E] where we denoted m-component vector [E] :=[1,..., 1] and vector sum is defined below [E] k := 1+ 1 j k /2 [] k := j k P r o o f. Similarly to the previous proof, we begin with m =1and vector =[j]: K [,b] α f [j]t = 1 Γα = 1 Γα + 1 Γα /2 b/2 k α t s sgnt sf [j] sds k α t s sgnt sf [j] sds k α t s sgnt sf [j] sds. Applying substitution s = b s and Property 3.2 to the second integral, we obtain K [,b] α f [j]t = 1 Γα + 1 Γα = 1 Γα + 1 Γα /2 /2 /2 /2 k α t s sgnt sf [j] sds k α b t s sgnt b + s 1 j f [j] s ds k α t s sgnt sf [j] sds k α b t s sgnb t s 1 j+1 f [j] s ds = 1+ 1 j+1 Q [,b] Kα [,b/2] f [j] t. Formula 3.12 follows from the above relation and the mathematical induction principle similar to the previous proof. The analogous representation formulas for the A, Ā and B, B derivatives result from the properties of integrals demonstrated above and from the fact that for any m N Q [,b/2 m ]Dft = DQ [,b/2 m ]ft. 3.14

10 252 M.Klimek,M.Lupa Proposition 3.5. Let f be the =[j 1,..., j m ]-projection of function f. Its symmetric A-derivatives of order α n 1,n in interval [,b] can be represented as follows A α [,b] f t =2 m A α [,b/2 m ] f t, 3.15 Δ n[e] where we denoted m-component vector [E] :=[1,..., 1], Δ n :=1+ 1 n+1 /2 and vector sum for arbitrary pair of vectors, [K] {, 1} m is given as [K] l := 1+ 1 j l+k l +1 / The symmetric B-derivatives of order α n 1,n in interval [,b] can be represented as follows B[,b] α f t =2 m B[,b/2 α m ] f t Δ n[e] Proposition 3.6. Let f be the =[j 1,..., j m ]-projection of function f. Its anti-symmetric Ā-derivatives of order α n 1,n in interval [,b] can be represented as follows Ā α [,b] f t =2 m Ā α [,b/2 m ] f t, 3.18 Δ n+1 [E] where we denoted m-component vector [E] := [1,..., 1], Δ n+1 := n /2 and vector sum is given in The anti-symmetric B-derivatives of order α n 1,n in interval [,b] can be represented as follows B [,b] α f t =2 m B [,b/2 α m ] f t Δ n+1 [E] Now we shall express integrals of products of functions in terms of their projections. Property 3.7. Let α n 1,n. The following integration formula is valid for any pair of functions such that f L 1,b and A α [,b] g L 1,b, respectively =2 1 fta α [,b] gt dt = 1 j= j= f [j] ta α [,b/2] g [j] Δ n[1]t dt. f [j] ta α [,b] g [j] Δ n[1]t dt 3.2

11 REFLECTION SYMMETRIC FORMULATION P r o o f. Formula 3.2 results from the following property of integration in a finite interval: 1 F tgtdt = Q [,b] F tgt dt = F [j] tg [j] tdt Applying this property and the reflection symmetry of the A α [,b] - derivative: we obtain j= Q [,b] A α [,b] = 1Δn A α [,b] Q [,b], 1 fta α [,b] gt dt = = 1 j= j= From Proposition 3.5 it follows that = f [j] t A α [,b] gt [j] f [j] ta α [,b] g [j] Δ n[1]t dt. f [j] ta α [,b] g [j] Δ n[1]t dt f [j] t1 + 1 j+δn+δn Q [,b] A α [,b/2] g [j] Δ n[1]t dt =2 and this proves formula 3.2. f [j] ta α [,b/2] g [j] Δ n[1]t dt Applying Proposition 3.5 and the mathematical induction principle, we obtain the the following general integration formula. Property 3.8. Let α n 1,n. The following integration formula is valid for any pair of functions such that f L 1,b and A α [,b] g L 1,b respectively: m 1 fta α [,b] gt dt = 2 k k= [j 1,...j k+1 ] /2 k f [j1,...,j k+1 ]ta α b/2 k+1 [,b/2 k+1 ] g [j 1,...,j k+1 ] Δ n[e]t dt + /2 m 2 m+1 f [j1,...,j m+1 ]ta α [,b/2 m+1 ] g [j 1,...,j m+1 ] Δ n[e]t dt. [j 1,...,j m+1 ] Analogous formula is valid for the B α [,b] - derivative. dt

12 254 M.Klimek,M.Lupa From Proposition 3.6 it follows that the respective integrals containing the antisymmetric Āα [,b], B [,b] α -derivatives can also be expressed in terms of projections. Property 3.9. Let α n 1,n. The following integration formula is valid for any pair of functions such that f L 1,b and Āα [,b] g L 1,b, respectively m 1 ftāα [,b] gt dt = k= [j 1,...j k+1 ] 2 k /2 k f [j1,...,j ]tāα k+1 b/2 k+1 [,b/2 k+1 ] g [j 1,...,j k+1 ] Δ n+1 [E]t dt + /2 m 2 m+1 f [j1,...,j ]tāα m+1 [,b/2 m+1 ] g [j 1,...,j m+1 ] Δ n+1 [E]t dt. [j 1,...,j m+1 ] An analogous formula is valid for the B α [,b] -derivative. From Properties 3.8 and 3.9 we infer that in generalized fractional variational calculus we can use the integration formulas: = ηtb[,b] α gt dt /2 m 2 m+1 η tb[,b/2 α m+1 ] g Δ n[e]t dt + T 1 b/2 m,b, ηt B [,b] α gt dt = /2 m 2 m+1 η t B [,b/2 α m+1 ] g Δ n+1 [E]t dt + T 2 b/2 m,b, where =[j 1,..., j m+1 ] are arbitrary vectors with components in {, 1}, η is a variation and terms T 1,T 2 involve integrals in subintervals of [b/2 m,b]. Similar rules are valid for the A α, Āα -derivatives. 4. Reflection-symmetric formulation of generalized fractional variational calculus We shall apply all the introduced and proved representation and integration properties to derive the explicit form of Euler-Lagrange equations in subintervals [,b/2 m ] of initial interval [,b]. In our study we shall discuss the action

13 REFLECTION SYMMETRIC FORMULATION S = L t, x, A α,b;p,q x dt, 4.1 which depends on scalar trajectory x, itsa-derivative of arbitrary order α n 1,n and parameters p, q R. The theorem below describes the set of Euler-Lagrange equations generated for the -projections of an extremal in subinterval [,b/2 m ]. Theorem 4.1. Let α n 1,n and L C 1 [,b] R 2.Thenfor each m N a system of Euler-Lagrange equations for action 4.1 is given as here {, 1} m+1 1 L 2 m+1 +p + qb[,b/2 α L x m+1 ] A α,b;p,q x 4.2 Δ n[e] +q p B [,b/2 α L m+1 ] A α,b;p,q x = Δ n+1 [E] and fulfilled in subinterval [,b/2 m ], provided regularity conditions 4.6, 4.7 are valid. P r o o f. Let us assume that variation η is bounded and has η k derivatives continuous in set [,b]\{b/2 l ; l =1,..., m +1} and vanishing at the ends: η k = η k b =fork =,..., n 1. Then, action variation looks as follows [ δsη = η L x + Aα,b;p,q η ] L A α,b;p,q x dt. 4.3 Using the integration by parts formula from Property 3.8, wecanrewrite the variation as follows: [ ] b L δsη = η x + L Bα,b;q,p A α,b;p,q x dt + b.t. 4.4 [ ] b L = η p x + + qb[,b] α +q p B [,b] α L A α,b;p,q x dt + b.t., where the explicit form of boundary terms denoted as b.t. depends on range of order α n 1,nD d dt : n 1 b.t. = η k D n k 1 K n α L,b;q,p A α,b;p,q x t=b t=. 4.5 k= Assume for k =,..., n 1,

14 256 M.Klimek,M.Lupa D n k 1 D n k 1 K n α,b;q,p K n α,b;q,p L A α,b;p,q x t= <, 4.6 L A α,b;p,q x t=b <. 4.7 When the above regularity conditions are fulfilled, the boundary terms vanish at ends t =,b. Then, the variation can be split into parts dependent on projections of variation - η, =[j 1,..., j m+1 ] according to Properties 3.8 and 3.9, δsη = /2 m L η dt 4.8 x + b/2 m 2 m+1 p + qη B[,b/2 α m+1 ] + b/2 m 2 m+1 q pη Bα [,b/2 m+1 ] + T b/2 m,b, L A α,b;p,q x L A α,b;p,q x Δ n[e] Δ n+1 [E] where the term T contains integrals over subintervals of [b/2 m,b] and yields Euler-Lagrange equations for time t>b/2 m. Choosing respective projections η we arrive at a system of Euler-Lagrange equations fulfilled in interval [,b/2 m ]: 1 2 m+1 L x +p + qb α [,b/2 m+1 ] +q p B [,b/2 α L m+1 ] A α,b;p,q x and this ends the proof. L A α,b;p,q x Δ n+1 [E] Δ n[e] =, dt dt 4.9 Analyzing the above proof, we observe that Theorem 4.1 can be extended to models with Lagrangians dependent on vector trajectories and/or involving several GFDs. However, deriving analogues of system 4.2 one should correctly determine the respective regularity conditions. We present here the result for action with derivatives of orders in range, 1 as in this case the regularity conditions are very simple. We omit its proof similar to the proof presented above in detail.

15 REFLECTION SYMMETRIC FORMULATION Theorem 4.2. Let orders α i, 1, i = 1,..., I, function L C 1 [,b] R I+1 and action S look as follows: S = L t, x, A α 1,b;p 1,q 1 x,..., Aα I,b;p I,q I x dt. 4.1 Then, for each m N a system of Euler-Lagrange equations for the above action is given as {, 1} m+1 1 L I 2 m+1 + p i + q i B α L i x [,b/2 m+1 ] A α i i=1,b;p i,q i x 4.11 [E] I + q i p i B α L i [,b/2 m+1 ] A α = i i=1,b;p i,q i x and fulfilled in subinterval [,b/2 m ], provided the regularity conditions I K 1 α L i,b;q i,p i A α i i=1,b;p i,q i x t= < 4.12 I K 1 α L i,b;q i,p i A α i i=1,b;p i,q i x t=b < 4.13 are fulfilled. Let us note that in the considered model 4.1, terms L x and L A α,b;p,q x from Euler-Lagrange equations system 4.2 depend on x and derivative A α,b;p,q x. First, we calculate explicitly the -projections for derivatives: A α,b;p,q p x = + qa α [,b] x +p qāα [,b] x 4.14 = [K] [ J] p + qa α [,b] x [K] +p qāα [,b] x [K] = p + q Aα [,b] x [K] +q p Āα [,b] x [K] [K] =p + q2 m+1 A α [,b/2 m+1 ] x [K]+ [K] [K] Δ n[e] +p q2 m+1 A α [,b/2 m+1 ] x [K]. [K] [K] Δ n+1 [E] Next, we observe that for monomials of x, the-projections look as follows for arbitrary l N: x l = x [K1 ]... x [Kl ], 4.15 [K 1 ]... [K l ]=

16 258 M.Klimek,M.Lupa where vectors [K 1 ],..., [K l ] {, 1} m+1 and summation was defined for arbitrary pair of vectors in formula Finally, for any product g 1 g 2... g l,whereg j are x or derivative A α,b;p,q x,weobtain g 1 g 2... g l = g[k 1 1 ]... gl [K l ] [K 1 ]... [K l ]= Summarizing, in the case when terms L x and L A α x from system 4.2,b;p,q are finite or infinite sums of monomials given above, Euler-Lagrange equations can be localized in subintervals [,b/2 m ]foreachm N. The same observation and calculations apply to 4.1,4.11. These are our final results formulated below. Corollary 4.3. Let the assumptions of Theorem 4.1 be fulfilled and the terms L x and L A α x from system 4.2 be finite or infinite sums,b;p,q pointwise convergent in [,b] of monomials. Then, the Euler-Lagrange equations 4.2 are localized in [,b/2 m ] for each m N. This means that system of equations 4.2, fulfilled in subinterval [,b/2 m ], involving only the -projections of x, fully determines the trajectory in [,b] according to 3.3. Corollary 4.4. the terms L x and Let the assumptions of Theorem 4.2 be fulfilled and L A α i from system 4.11 be finite or infinite sums x,b;p i,q i pointwise convergent in [,b] of monomials. Then, the Euler-Lagrange equations 4.11 are localized in [,b/2 m ] for each m N. 5. Final remarks In this paper we defined the symmetric and anti-symmetric generalized fractional operators and proved that for functions split into respective projections they can be reduced to operators defined in an arbitrarily short subinterval [,b/2 m ]. We also formulated and demonstrated the integration formulas which we then applied in the generalized fractional variational calculus GFVC. In this framework, actions contain GFDs introduced in [3]. The Euler-Lagrange equations in interval [,b] are the initial step. Applying the representation and integration formulas we obtained the system of Euler-Lagrange equations for parts of function with well-defined symmetry properties w.r.t to the sequence of reflections. The crucial property of the reflection symmetric GFVC, demonstrated in the case of Lagrangians polynomial or pointwise convergent series, is the localization of Euler-Lagrange equations in subintervals. This fact was proved for GFC with operators defined using translation invariant kernels. Other types of GFC and resulting GFVC are under investigation.

17 REFLECTION SYMMETRIC FORMULATION References [1] O.P. Agrawal, Formulation of Euler-Lagrange equations for fractional variational problem. J. Math. Anal. Appl. 272, No 1 22, [2] O.P. Agrawal, Fractional variational calculus in terms of Riesz fractional derivatives. J.Phys.A:Math.Theor.4, No 24 27, [3] O.P. Agrawal, Generalized variational problems and Euler-Lagrange equations. Comput. Math. Appl. 59, No [4] O.P. Agrawal, Some generalized fractional calculus operators and their applications in integral equations. Fract. Calc. Appl. Anal. 15, No 4 212, 7 711; DOI:1.2478/s ; [5] R. Almeida, D.F.M. Torres, Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives. Commun. Nonlinear Sci. Numer. Simulat. 16, No , [6] R. Almeida, S. Pooseh, D.F.M. Torres, Fractional variational problems depending on indefinite integrals. Nonlinear Anal. TMA 75, No 3212, [7] T.M. Atanackovic, B. Stankovic, B. On a differential equation with left and right fractional fractional derivatives. Fract. Calc. Appl. Anal. 1, No 2 27, ; at fcaa. [8] T.M. Atanackovic, P. Konjik, S. and Pilipovic, Variational problems with fractional derivatives: Euler-Lagrange equations. J. Phys. A: Math. Theor. 41, No 9 28, [9] D. Baleanu, T. Avkar, Lagrangians with linear velocities within Riemann-Liouville fractional derivatives. Nuovo Cim. B 119, No 1 24, [1] D. Baleanu, S. Muslish, Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives. Physica Scripta 72, No , [11] D. Baleanu, O.P. Agrawal, Fractional Hamilton formalism within Caputo s derivative. Czech. J. Phys. 56, No , [12] D. Baleanu, J.J. Trujillo, A new method of finding the fractional Euler- Lagrange and Hamilton equations within fractional Caputo derivatives. Commun. Nonlinear Sci. Numer. Simulat. 15, No 5 21, [13] J. Cresson, Fractional embedding of differential operators and Lagrangian systems. J. Math. Phys. 48, No 327, [14] A. A. Kilbas, M. Saigo, Generalized Mittag-Leffler function and generalized fractional calculus operators. Integral Transform. Spec. Func. 15, No 1 24,

18 26 M.Klimek,M.Lupa [15] A.A. Kilbas, H.M. Srivastawa, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam 26. [16] M. Klimek, Fractional sequential mechanics - models with symmetric fractional derivative. Czech. J. Phys. 51, No 12 21, [17] M. Klimek, Lagrangean and Hamiltonian fractional sequential mechanics. Czech. J. Phys. 52, No 11 22, [18] M. Klimek, Lagrangian fractional mechanics - a non-commutative approach. Czech. J. Phys. 55, No 11 25, [19] M. Klimek, On Solutions of Linear Fractional Differential Equations of a Variational Type. The Publishing Office of the Czestochowa University of Technology, Czestochowa Poland 29. [2] M. Klimek, Existence-uniqueness result for a certain equation of motion in fractional mechanics. Bull. Pol. Acad. Sci.: Tech. Sciences 58, No 4 21, [21] M. Klimek, M. Lupa, On reflection symmetry in fractional mechanics. Scientific Research of the Institute of Mathematics and Computer Science 1, No 2 211, ; at [22] M. Klimek, On reflection symmetry and its application to the Euler- Lagrange equations in fractional mechanics. Phil.Trans.R.Soc.A. To appear, 213. [23] A.B. Malinowska, D.F.M. Torres, Fractional calculus of variations for a combined Caputo derivative. Fract. Calc. Appl. Anal. 14, No 4 211, ; DOI:1.2478/s ; [24] A.B. Malinowska, D.F.M. Torres, Towards a combined fractional mechanics and quantization. Fract. Calc. Appl. Anal. 15, No 3 212, ; DOI:1.2478/s ; [25] T. Odzijewicz, A.B. Malinowska, D.F.M. Torres, Fractional variational calculus with classical and combined Caputo derivatives. Nonlinear Anal. TMA 75, No 3 212, [26] T. Odzijewicz, A.B. Malinowska, D.F.M. Torres, Generalized fractional calculus with applications to the calculus of variations. Comput. Math. Appl. 64, No 1 212, [27] F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E 53, No , [28] F. Riewe, Mechanics with fractional derivatives. Phys. Rev. E 55, No , [29] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications. Gordon & Breach Science Publ., New York 1993.

19 REFLECTION SYMMETRIC FORMULATION [3] H. M. Srivastava, Z. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel. Appl. Math. Comput. 211, No 1 29, [31] Z. Tomovski, R. Hilfer, H.M. Srivastava, Fractional and operational calculus with generalized fractional derivative operators and Mittag- Leffler type functions. Integral Transform. Spec. Func. 21, No 11 21, [32] Z. Tomovski, T. Sandev, Fractional wave equation with a frictional memory kernel of Mittag-Leffler type. Appl. Math. Comput. 218, No , [33] Z. Tomovski, T. Sandev, Effects of a fractional friction with powerlaw memory kernel on string vibrations. Comp. Math. Appl. 62, No 3 211, Institute of Mathematics Czestochowa University of Technology Dabrowskiego 73 Czestochowa 42-2, POLAND Received: September 13, 212 s: 1 mklimek@im.pcz.pl, 2 maria.lupa@im.pcz.pl Please cite to this paper as published in: Fract. Calc. Appl. Anal., Vol. 16, No 1 213, pp ; DOI: /s x