An Euler-Lagrange Equation Only Depending on Derivatives of Caputo for Fractional Variational Problems with Classical Derivatives

In this paper we present advances in fractional variational problems with a Lagrangian depending on Caputo fractional and classical derivatives. New formulations of the fractional Euler-Lagrange equation are shown for the basic and isoperimetric problems, one in an integral form, and the other that depends only on the Caputo derivatives. The advantage is that Caputo derivatives are more appropriate for modeling problems than the Riemann-Liouville derivatives and makes the calculations easier to solve because, in some cases, its behavior is similar to the behavior of classical derivatives. Finally, a new exact solution for a particular variational problem is obtained.


Introduction
The fractional variational calculus is a recent field, started in 1997, where classical variational problems are considered, but in the presence of fractional derivatives or integrals. In the last years numerous works have been developed tending to extend the theory of the variational calculus [35] in order to be able to be applied to problems of fractional variational calculus [1,8,10,11,18,23]. This is fundamentally due, on the one hand, to an important development of the fractional calculus both from the mathematical point of view and its applications in other areas (electricity, magnetism, mechanics, dynamics of fluids, medicine, etc) [7,17,19,20,22]; which has led to an overwhelming growth of its study in recent decades. On the other hand, the fractional differential equations establish models far superior to those that use differential equations with integer derivatives because they incorporate into the model issues of memory or later effects that are neglected in the models with classical derivative [9,15,24,33].
Following [31], we consider here that the highest derivative in the Lagrangian is of integer order. The main advantage of our formulation, with respect to the pure fractional approach adopted in the literature, is that the classical results of variational calculus can now be obtained as particular cases. We recall that the only possibility to obtain the classical derivative y ′ from a fractional derivative D α y, α ∈ (0, 1), is to take the limit when α tends to one. However, in general such a limit does not exist [34]. Different from [31], where the fractional derivative are considered in the sense of Riemann-Liouville, here we investigate problems of the calculus of variations with

Preliminaries on Fractional Calculus
In this section, we present some definitions and properties of the Caputo and Riemann-Liouville fractional calculus. For more details on the subject and applications, we refer the reader to [15,24].

Definition 1
The left and right Riemann-Liouville fractional integral operators of order α ∈ R + are defined, respectively, by and where a, b ∈ R with a < b. For α = 0, we set a I 0 x = x I 0 b = I, the identity operator. The Riemann-Liouville fractional integrals coincide with the classical definition of these in the case n ∈ N. Moreover, in the case α ≥ 1 it is obvious that the integrals exist for any integrable function f and for every x ∈ [a, b]. In the case 0 < α < 1 though, the situation is less clear at first sight. However, it is possible to prove the convergence (see [15]).

Remark 1
Let n, m ∈ N, with n > m, we have This property motivates the following definition.  (4) and and

Definition 3
The left and right Caputo fractional derivative operators of order α ∈ R + are defined, respectively, by and and Now we will see some different properties of the Riemann-Liouville and Caputo derivatives.

Remark 2
and and we have that

Remark 3
An important difference between Riemann-Liouville derivatives and Caputo derivatives is that, being K an arbitrary constant, is In this sense, the Caputo fractional derivatives are similar to the classical derivatives. In some cases it is easier to solve differential equations with Caputo derivatives instead of equations involving both Caputo and Riemann-Liouville derivatives. Therefore in this work, we will state and prove equations with only derivatives of Caputo.
almost everywhere. and Moreover and This theorem is known as Integration by parts and its proof is in [24].

Fractional Variational Calculus with Classical Derivatives
We begin with the following problem of the fractional calculus of variations: find a function y ∈ α a E ′ that optimizes (minimizes or maximizes) the functional We now state the Euler-Lagrange equation for this problem, its proof is in [27].

Theorem 3
If y is a local optimizer to the above problem, then y satisfies the next Euler-Lagrange equation:

Remark 4
We call these equations involving both Caputo and Riemann-Liouville derivatives.

Fractional Euler-Lagrange Equation with Only Caputo Derivatives
You can find the following lemma in [25], made by Lazo M. and Torres D. (2013).
This lemma is known as the Dubois-Reymond Fundamental Lemma of the Fractional Calculus of Variations.

Remark 5
Using the definition of the Riemann-Liouville fractional integral (2), we can write the functional J as Inspired on the work of [25], we have made the following generalization considering a Langrangian that depends on the Caputo derivatives and the classical derivative y ′ .

Theorem 4
Let J be the next functional defined in the class of functions y ∈ α a E ′ , and where we assume , differentiable with respect all of its arguments. If y is an optimizer of J, then y satisfies the following fractional Euler-Lagrange integral equation: Proof.: Let y * be an optimizer of J. We define the family of functions where ϵ is a constant, and η ∈ C 1 ([a, b]) is an arbitrary function satisfying the boundary conditions η(a) = η(b) = 0. From (30), η(a) = η(b) = 0, y * (a) = y a and y * (b) = y b , we have that the function y is admissible, that is, y ∈ C 1 ([a, b]), y(a) = y a and y(b) = y b . From now on, we denote L[y] = L(x, y, y ′ , C a D α x y). Because y * is an optimizer of the functional J, the Gateaux derivative δJ[y * ] needs to be identically null.
dx . Using the fact that x D α b x I α b ≡ I, the homogeneity conditions on η and integration by parts we get Similarly, And we get, Writing this otherwise, we obtain for the first variation the following expression: We define we have that f is bounded in a sufficiently small neighborhood of a and taking β = 0 > −α, the hypotheses of the Lemma are verified so it exists K constant such that ( and we obtain the desired equation, In the following theorem we will see a new version of the Euler-Lagrange equation. Its proof follows the same guidelines of the paper [25].

Theorem 5 Let y be an optimizer of
subject to boundary conditions y(a) = y a , y(b) = y b , then y satisfies the fractional Euler-Lagrange differential equation

The Fractional Isoperimetric Problem
We now consider the following isoperimetric problem of the fractional calculus of variations: find a function y ∈ α a E ′ that optimizes the functional Such variational problems have found a broad class of important applications throughout the centuries, with numerous useful implications in astronomy, geometry, algebra, and engineering. For references and recent advancements on the subject, we refer the reader to [16,30].
We now state the Euler-Lagrange equation for this problem, its proof is in [27].

Theorem 6
If y is a local optimizer to the above problem, and suppose further that y does not anulate the Gateaux's derivative of the functional I. Then there exists a constant λ such that y satisfies the next Euler-Lagrange equation: where F = L + λG.

Theorem 7
Let J and I be the above functionals, defined in the class of functions y ∈ α a E ′ , and where L, G ∈ C 1 ([a, b) × R 3 ) are differentiable with respect all of its arguments. If y is an optimizer of J and it does not anulate the Gateaux's derivative of the functional I then y satisfies the following fractional Euler-Lagrange integral equation: for all x ∈ [a, b), where K is a constant and F = L + λG.
subject to boundary conditions y(a) = y a y(b) = y b , then if y does not anulate the Gateaux's derivative of the functional I, it satisfies the fractional Euler-Lagrange differential equation: where F = L + λG.

A Lagrangian with classical higher order derivatives
Let us consider the following problem: find a function y ∈ α a E n that optimizes the functional subject to the boundary conditions:

Theorem 9
Let J be the following functional defined in the class of functions y ∈ α a E n , and where we assume L ∈ C 1 ([a, b) × R n+2 ), differentiable with respect all of its arguments. If y is an optimizer of J, then y satisfies the following fractional Euler-Lagrange integral equation: for all x ∈ [a, b), where K is a constant.

Theorem 10
Let y be an optimizer of J in α a E n with L ∈ C 2 ( [a, b] × R n+2 ) subject to boundary conditions (38), then y satisfies the fractional Euler-Lagrange differential equation

Remark 6
The above theorems can be performed analogous to isoperimetric problems. Its proofs are similar to the case with n = 1.

An Example
Consider the following fractional isoperimetric problem J(y) Using the property We can write And consider x β−α to obtain the solution Using the definition of the Hypergeometric function of parameters a, b and c: x α Γ(1 + α) 2 2 F 1 (1, −α, 1 + α, x) + c 1 x α + c 2 .
When taking the limit α → 1, we have y(x) = −6x 2 + 6x, which is the solution to the classical problem

Conclusion
The generalized version of the fundamental lemma of the variational calculus of DuBois-Reymond enabled us to prove a fractional Euler-Lagrange equation in integral form only containing fractional derivatives of Caputo's type. We have done this for variational and isoperimetric problems in which the Lagrangian depends not only on the fractional derivatives of Caputo but also on derivatives in the classical sense. Furthermore, we also showed that when the Lagrangian is a C 2 function, we can obtain a fractional Euler-Lagrange differential equation only depending on Caputo derivatives. The advantage of this is that Caputo derivatives are superior to Riemann-Liouville derivatives in some cases. Finally, we obtained an exact solution of a particular isoperimetric problem, where the classical solution is recovered when α → 1.