## Abstract

We study the properties of fractional differentiation with respect to the reflection symmetry in a finite interval. The representation and integration formulae are derived for symmetric and anti-symmetric fractional derivatives, both of the Riemann–Liouville and Caputo type. The action dependent on the left-sided Caputo derivatives of orders in the range (1,2) is considered and we derive the Euler–Lagrange equations for the symmetric and anti-symmetric part of the trajectory. The procedure is illustrated with an example of the action dependent linearly on fractional velocities. For the obtained Euler–Lagrange system, we discuss its localization resulting from the subsequent symmetrization of the action.

## 1. Introduction

For the past few decades, fractional differential equations have become an important part of mathematical modelling. The fractional derivatives appear in differential equations describing many processes in physics, mechanics, control theory, biochemistry, bioengineering and economics. The theory of fractional differential equations is an area intensively developed during the last decades and established as an interesting field of pure and applied mathematics. The monographs [1–7] enclose a review of methods of solving that are an extension of procedures from differential equation theory. Recently, equation including both left and right fractional derivatives have been discussed [8–13]. Ordinary differential equations mixing both types of derivatives naturally emerge in fractional mechanics whenever a standard variational calculus is applied in the derivation of Euler–Lagrange equations. This approach was started in 1996 by Riewe [14,15], developed by Agrawal [16] and Klimek [17,18] and investigated ever since (compare [19–31] and references therein). The known existence results for the equations with the left- and right-sided derivatives lead to the solution subject to some restrictions, including the parameters of the problem such as the order of derivatives and the length of the time interval [8,9,11,12]. Recently, a detailed analysis of the existence conditions for the solution of the fractional oscillator equation was given in [13]. The aim of the present paper is twofold. First, we study reflection symmetry in fractional calculus. Then, the obtained results are applied to formulate the reflection-symmetric framework for fractional mechanics. Owing to the integration properties, only the symmetric part of the action variation yields the Euler–Lagrange equations. Using the reflection symmetry, we also split the trajectories into symmetric and anti-symmetric parts in interval [0,*b*]. The same operation is repeated in a sequence of intervals: . It is clear that the full trajectory is a sum of its projections. The novelty of our approach is that at least for the presented example, we will be able to derive the Euler–Lagrange equations for the respective projection for which it is localized on an interval shorter than the initial [0,*b*].

The paper is organized as follows. In the next section, we recall the basic definitions of fractional calculus of both integrals and derivatives in finite interval. Then, for a function determined in such an interval, we define the symmetric and anti-symmetric derivatives of the Riemann–Liouville and Caputo type. The main part of the preliminary results is the study of the reflection operators in finite intervals and of the reflection-symmetry properties of fractional differentiation. The obtained results are applied to a basic model of fractional mechanics dependent on the Caputo derivative of order *α*∈(1,2). We discuss a symmetrization of the action and derive Euler–Lagrange equations for the reflection symmetric and anti-symmetric parts of the trajectory. Finally, we apply the proposed procedure to an example of a Lagrangian dependent linearly on fractional velocity. Here, we describe the subsequent symmetrized versions of the Lagrangian and the respective equations of motion. It appears that the first system of equations for the symmetric and anti-symmetric parts of the trajectory in interval [0,*b*] is in fact [0,*b*/2] localized. This means that its solution is fully determined by the part of the solution derived for *t*∈[0,*b*/2]. The subsequent symmetrization leads to the Euler–Lagrange equations localized in interval [0,*b*/2^{2}]. The paper is closed by a short discussion of a possible extension of the obtained results and their further application.

## 2. Preliminaries

First, we recall the basic definitions from fractional calculus [3,32].

### Definition 2.1

Let *Re*(*α*)>0. The left- and right-sided Riemann–Liouville integrals of order *α* are given by the formulae
2.1
and
2.2
respectively, where *Γ* denotes the Euler gamma function.

### Definition 2.2

Let *Re*(*α*)∈(*n*−1,*n*). The left- and right-sided Riemann–Liouville derivatives of order *α* are defined as
2.3
and
2.4
Analogous formulae yield the left- and right-sided Caputo derivatives of order *α*,
2.5
and
2.6

In this paper, we shall study models of fractional mechanics dependent on derivatives of real order. The left and right derivatives produce the symmetric and anti-symmetric derivatives in a finite interval.

### Definition 2.3

Let *α*∈(*n*−1,*n*). The symmetric and anti-symmetric Riemann–Liouville derivatives in interval [0,*b*] are, respectively, given as follows:
2.7
and
2.8
The symmetric and anti-symmetric Caputo derivatives in interval [0,*b*] are, respectively, given as
2.9
and
2.10

The above symmetric and anti-symmetric operators are a particular case of the combined Caputo and Riemann–Liouville derivatives discussed in [24,33]. First introduced in [17,18], they can also be expressed using the notion of Riesz potentials in a finite interval [32], where *β*∈(0,1),
2.11
and
2.12
Using the above Riesz potentials in interval [0,*b*], we can rewrite derivatives (2.7)–(2.10) as follows:
2.13
2.14
Thus, we note that one could also refer to derivatives (2.7)–(2.10) as Riesz derivatives in finite interval of, respectively, Riemann–Liouville or Caputo type. In this paper, we leave the notion of the symmetric derivative (which corresponds to the Riesz derivative) and the anti-symmetric one (corresponding to modified Riesz derivative).

We shall also consider the symmetric and anti-symmetric derivatives (2.7) and (2.8) in doubled intervals. When order *α*∈(1,2) and *t*∈[0,*b*], they are given by formulae (2.7) and (2.8). For *t*∈[*b*,2*b*], they act as
2.15
and
2.16
Similar formulae hold for Caputo derivatives ((2.9) and (2.10)).

In this paper, we shall study the properties and applications of the reflection operators. We define operators *Q*_{[0,b/2m]} acting as follows on arbitrary function *f* determined in interval [−*b*,*b*]:
2.17
Having defined the respective reflection operators, we note that they obey the following general formula:
2.18
where *T*_{c} for *c*∈*R* denotes the translation operator acting as follows: *T*_{c}*f*(*t*):=*f*(*t*+*c*). Now, we are ready to construct the [*J*]-projections of function *f*.

### Definition 2.4

Let *f* be an arbitrary function determined in [0,*b*] and vector [*J*]=[*j*_{1},…,*j*_{m}] have components *j*_{l}∈{0,1} . The following recursive formulae define, respectively, the [*j*]- and [*J*,*j*_{m+1}]-projections of function *f*:
2.19

For any *m*∈*N* function, *f* can be split into the respective projections
2.20
and
2.21
where the summation is taken over all *m*-component vectors with coordinates in the two-element set {0,1}. We note that the *f*_{[0]} and *f*_{[1]} projections are the reflection symmetric and anti-symmetric parts of function *f* in interval [0,*b*]. It is easy to check that the projections defined above obey the set of equations given in the property below.

### Property 2.5

Let [*J*]=[*j*_{1},…,*j*_{m}] and *f*_{[j]},*f*_{[J]} be the projections given in definition 2.4.

The following relations are valid: 2.22 and 2.23

The projections fulfil orthogonality relations 2.24 and 2.25

In the following, we study and apply the reflection-symmetry properties of the symmetric and anti-symmetric fractional derivatives of order *α*∈(1,2).

### (a) Representation formulae for symmetric and anti-symmetric derivatives

In this section, we shall discuss the representation properties of the symmetric and anti-symmetric fractional derivatives. It appears that acting on the [*J*]-projections of function *f*, they can be expressed as operators dependent on values of the function in a relatively short interval obtained as a result of the corresponding partitions of [0,*b*].

### Proposition 2.6

*Let**f*_{[j]}*be the*[*j*]-*projection of function**f**given in equation*(2.19).*Its symmetric derivatives of order**α*∈(1,2)*in interval*[0,*b*]*can be represented as follows*: 2.26*and*2.27*Let**f*_{[J]}*be the*[*J*]-*projection of function**f**given in equation*(2.20)*for vector*[*J*]=[*j*_{1},…,*j*_{m}],*j*_{l}∈{0,1}.*Its symmetric derivatives of order**α*∈(1,2)*in interval*[0,*b*]*can be represented as follows*: 2.28*and*2.29*where we denoted as**the ordered composition of the projection operators*2.30

### Proof.

First, we check equation (2.26) using the integration properties and the reflection properties of the second-order derivative,
Let us observe that equation (2.26) remains valid when we replace *b* by *b*/2^{m} and take *t*∈[0,*b*/2^{m}],
2.31
Now, we can prove equation (2.28) by means of the mathematical induction principle. Indeed, as equations (2.26) and (2.34) are valid, we obtain
The calculations for properties (2.27) and (2.29) of Caputo derivatives are similar to those presented in detail for the Riemann–Liouville derivatives.

Similar results can be obtained for anti-symmetric derivatives acting on the respective [*J*]- projections. We omit the proof, analogous to the proof of proposition 2.6 given above.

### Proposition 2.7

*Let**f*_{[j]}*be the*[*j*]-*projection of function**f**given in equation*(2.19).*Its anti-symmetric derivatives of order**α*∈(1,2)*in interval*[0,*b*]*can be represented as follows*: 2.32*and*2.33*Let**f*_{[J]}*be the*[*J*]-*projection of function**f**given in equation*(2.20)*for vector*[*J*]=[*j*_{1},…,*j*_{m}]*with components in set*{0,1}.*Its anti-symmetric derivatives of order**α*∈(1,2)*in interval*[0,*b*]*can be represented as follows*: 2.34*and*2.35*where**denotes the ordered composition of the projection operators*2.36

Analysing the proof of proposition 2.6, we conclude that representation formulae (2.28), (2.29), (2.34) and (2.35) can be easily extended to derivatives of order *α*∈(*n*−1,*n*). Below, we present the propositions describing representation properties in this case, omitting the proof which is analogous to that of proposition 2.6.

### Proposition 2.8

*Let* *f*_{[J]} *be the* [*J*]-*projection of function* *f* *given in equation* (2.20) *for vector* [*J*]=[*j*_{1},…,*j*_{m}], *j*_{l}∈{0,1}. *Its symmetric derivatives in interval* [0,*b*] *can be represented as follows*:
2.37
*and*
2.38
*where we denoted as* *the ordered composition of the projection operators*
2.39

### Proposition 2.9

*Let* *f*_{[J]} *be the* [*J*]-*projection of function* *f* *given in equation* (2.20) *for vector* [*J*]=[*j*_{1},…,*j*_{m}] *with components in set* {0,1}. *Its anti-symmetric derivatives in interval* [0,*b*] *can be represented as follows*:
2.40
*and*
2.41
*where* *denotes the ordered composition of the projection operators*
2.42

### (b) Integration properties for symmetric and anti-symmetric derivatives

Now, we apply the representation properties of symmetric and anti-symmetric fractional derivatives of order *α*∈(1,2) to prove some integration formulae. Using orthogonality properties (2.24) and (2.25), we arrive at formulae containing products of functions and derivatives.

### Proposition 2.10

*Let* *α*∈(1,2). *The following integration formulae are valid for any pair of functions such that* *f*∈*L*_{1}(0,*b*) *and* *or* *respectively*,
2.43
and
2.44

### Proof.

We apply the reflection property of the definite integral over interval [0,*b*], the representation formulae from proposition 2.6 as well as equation (2.18) and obtain
The proof of equation (2.44) for the Caputo derivative is analogous.

Let us note that according to the proved proposition, the integral for the product of a function and of a symmetric derivative in interval [0,*b*] can be rewritten using the [*j*]-projections of the respective functions and the symmetric derivative in halved interval [0,*b*/2]. We quote the corresponding proposition describing analogous integrals for the anti-symmetric derivatives below.

### Proposition 2.11

*Let* *α*∈(1,2). *The following integration formulae are valid for any pair of functions such that* *f*∈*L*_{1}(0,*b*) *and* *or* *respectively*,
2.45
*and*
2.46
*where* .

The above propositions describing integrals (2.43)–(2.46) can be further reformulated using the fractional operators in interval [0,*b*/2^{2}]. We quote this result for Riemann–Liouville derivatives. Analogous integration formulae hold for the symmetric and anti-symmetric Caputo derivatives.

### Proposition 2.12

*The following integration formula is valid for any pair of functions such that* *f*∈*L*_{1}(0,*b*) *and*
2.47
*The following integration formula is valid for any pair of functions such that* *f*∈*L*_{1}(0,*b*) *and*
2.48
*where* .

## 3. Reflection-symmetric formulation of fractional variational calculus

We shall derive Euler–Lagrange equations for an action dependent on trajectory *x* and its Caputo derivative of order *α*∈(1,2). Euler–Lagrange and Hamiltonian equations for such systems were obtained earlier [27,29], but our aim is to apply in variational calculus all the symmetry and integration properties studied in the first part of our paper. For *x* being a real-valued function determined in interval [0,*b*], the action is
3.1
The substitution property of integral calculus implies that in fact the considered action depends on a reflection-symmetric part of the Lagrangian and is equal to
3.2
where is given as
3.3
The Euler–Lagrange equations for the symmetrized action are described in the following theorem.

### Theorem 3.1

*Let α∈(1,2). Then, the Euler–Lagrange equation for action (*3.2*) is
*
3.4

### Proof.

Let us introduce the space of symmetrized variations *η*_{[j]}, *j*=0,1-continuous functions fulfilling the boundary conditions,
3.5
and
3.6
Any variation of action (3.2) resulting from the variation of coordinate *x* under *η* fulfilling (3.5) and (3.6) can be split into parts dependent on [*j*]-projections of variations *η*,
3.7
The projections of the Caputo derivatives of variation *η* are given explicitly by the following formula:
3.8
with .

Using the above result, the symmetry properties of integrals and the integration by parts formula of fractional calculus, we rewrite the variation of action (3.2) in the form
3.9
with
3.10
The above relation and condition *b*.*t*.=0 require assumptions (3.5) and (3.6) for the space of variations *η*_{[j]}, *j*=0,1. Additionally, we should have at end *t*=0 and *j*=0,1,
3.11
3.12
3.13
3.14
with analogous conditions fulfilled at end *t*=*b*. Then, the minimum action principle yields Euler–Lagrange equations for projections *x*_{[j]} in the form
3.15
This ends the proof.

Let us note that the left-sided Caputo derivative is the sum of the symmetric and anti-symmetric operators 3.16 and this fact leads to the following reformulation of theorem 3.1.

### Corollary 3.2

*Let* *α*∈(1,2). *Then, the Euler–Lagrange equation for action* (3.2) *is*
3.17

The described model is rather simple, but it is clear that this procedure of deriving equations of motion can be extended to models dependent on vector functions, including derivatives of orders *α*_{1},…,*α*_{k}, both left- and right-sided. Let us quote the result for action dependent on the left-sided Caputo derivatives of a scalar function.

### Theorem 3.3

*Let α*_{k}*∈(1,2), k=1,…,m. Then, the Euler–Lagrange equation for action
*
3.18
*is
*
3.19

Using the description of the one-sided derivative given in equation (3.16), we arrive at the following formulation of the above theorem.

### Corollary 3.4

*Let* *α*_{k}∈(1,2), *k*=1,…,*m*. *Then, the Euler–Lagrange equation for action*
3.20
*is*
3.21

Now, we shall discuss an example, in which we show an interesting phenomenon of the localization of Euler–Lagrange equations.

## 4. Example: Euler–Lagrange equations for action linear in fractional derivative

Let us apply the properties and theorems given previously to the case of a simple Lagrangian dependent linearly on fractional velocities. We consider the following action containing the Caputo derivative of order *α*∈(1,2):
4.1
After the symmetrization, the Lagrangian is rewritten using the [*j*]-projections of trajectory *x* and action (4.1),
4.2
where . We note that in fact it does depend only on the symmetric derivative in interval [0,*b*] as the terms including the anti-symmetric derivative depend only on boundary values of *x*_{[j]} and yield b.t.
where
4.3
Analysing the above term and applying the symmetry properties of its components, namely,
4.4
4.5
4.6
4.7
we conclude that this term does not yield any terms in the respective Euler–Lagrange equations.

Thus, we derive the symmetrized equations of motion applying theorem 3.1 to the action with the Lagrangian given below in an explicit form,
4.8
The first set of Euler–Lagrange equations for projections *x*_{[j]} in interval [0,*b*] () is
4.9
On the other hand, we may apply proposition 2.6 and rewrite the Euler–Lagrange equation as
4.10
An interesting feature of system (4.10) appears when we write it explicitly for subintervals
4.11
and
4.12
System (4.11), (4.12) is [0,*b*/2]-localized in the sense that the solution of the subsystem for [*j*]-projections in interval [0,*b*/2] yields via equation (4.12) the part of solutions in [*b*/2,*b*]. From propositions 2.10 and 2.12, it follows that we can rewrite the action given by equation (4.2) in the form of
4.13
Now, applying variational calculus, we obtain Euler–Lagrange equations in the form of a [0,*b*/2^{2}]-localized system of equations for the corresponding projections of trajectory *x*. The system looks a bit more complicated than that given in equations (4.11) and (4.12), but its advantage is the property that solution in interval [*b*/2^{2},*b*] is fully determined by its part in [0,*b*/2^{2}] given by the first equation in the system below,
4.14
4.15
4.16
Clearly, we can continue the symmetrization procedure. In the next step, we shall arrive at the Lagrangian including the [*j*]-projections for *t*∈[*b*/2,*b*], [*j*,*k*]-projections for *t*∈[*b*/2^{2},*b*/2] and finally the [*j*,*k*,*l*]-projections for interval [0,*b*/2^{3}]. We expect by analogy with results (4.11), (4.12) and (4.14)–(4.16) that the generated system of Euler–Lagrange equations will be [0,*b*/2^{3}]-localized.

## 5. Final remarks

In this paper, we investigated the reflection-symmetry properties of the fractional operators in a finite interval. Starting from the known commutation relation of the one-sided Riemann–Liouville and Caputo derivatives, we formulated the respective laws for the symmetric and anti-symmetric derivatives and the integrals containing such derivatives.

The reflection-symmetry properties of the fractional calculus were then applied to fractional mechanics dependent on Caputo derivatives with orders in the range (1,2). From the fundamental properties of the integration, we inferred that the action depends only on the part of the Lagrangian symmetric with respect to the reflection in the finite interval. We derived separate Euler–Lagrange equations for the symmetric and anti-symmetric component of the trajectory.

In the discussion of the example—the action dependent linearly on fractional velocities—we showed that at least in this case, the subsequent symmetrization of the Lagrangian leads to the equations of motion localized in an arbitrarily short interval of time. Namely, applying the symmetrization *n* times and then deriving the Euler–Lagrange system, we obtain equations localized in interval [0,*b*/2^{n}]. From this example, it follows that at least in some cases, the symmetrization of the action and the corresponding localization of the equations of motion should lead to the existence results for the equations of motion in fractional mechanics.

Let us observe that the introduced reflection-symmetric formulation of fractional variational calculus can be extended to models dependent on derivatives of arbitrary order, and this problem will be further investigated. In addition, analysing the proofs of the representation theorems, we conclude that they result from the fact that the basic left fractional integral is constructed as a Laplace convolution with the power-function kernel. Therefore, our considerations can and will be generalized to the fractional variational calculus of models dependent on derivatives constructed in the paper by Agrawal [22], where kernel (*t*−*s*)^{α−1}/*Γ*(*α*) is replaced with the general one: *k*_{α}(*t*−*s*).

## Footnotes

One contribution of 14 to a Theme Issue ‘Fractional calculus and its applications’.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.