Abstract and Applied AnalysisVolume 2012 (2012), Article ID 871912, 24 pagesdoi:10.1155/2012/871912

Research Article

Fractional Calculus of Variations in Terms of a Generalized Fractional Integral with Applications to Physics

Tatiana Odzijewicz,¹ Agnieszka B. Malinowska,² and Delfim F. M. Torres¹

¹Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal²Faculty of Computer Science, Białystok University of Technology, 15-351 Białystok, Poland

Received 1 January 2012; Revised 25 February 2012; Accepted 27 February 2012

Academic Editor: Muhammad Aslam Noor

Copyright © 2012 Tatiana Odzijewicz et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study fractional variational problems in terms of a generalized fractional integral with Lagrangians depending on classical derivatives, generalized fractional integrals and derivatives. We obtain necessary optimality conditions for the basic and isoperimetric problems, as well as natural boundary conditions for free-boundary value problems. The fractional action-like variational approach (FALVA) is extended and some applications to physics discussed.

1. Introduction

The calculus of variations is a beautiful and useful field of mathematics that deals with problems of determining extrema (maxima or minima) of functionals [1–3]. It starts with the simplest problem of finding a function extremizing (minimizing or maximizing) an integral subject to boundary conditions and . In the literature, many generalizations of this problem were proposed, including problems with multiple integrals, functionals containing higher-order derivatives, and functionals depending on several functions [4–6]. Of our interest is an extension proposed by Riewe in 1996-1997, where fractional derivatives (real or complex order) are introduced in the Lagrangian [7, 8].

During the last decade, fractional problems have increasingly attracted the attention of many researchers. As mentioned in [9], Science Watch of Thomson Reuters identified the subject as an Emerging Research Front area. Fractional derivatives are nonlocal operators and are historically applied in the study of nonlocal or time-dependent processes [10]. The first and well-established application of fractional calculus in physics was in the framework of anomalous diffusion, which is related to features observed in many physical systems. Here we can mention the report [11] demonstrating that fractional equations work as a complementary tool in the description of anomalous transport processes. Within the fractional approach, it is possible to include external fields in a straightforward manner. As a consequence, in a short period of time, the list of applications expanded. Applications include chaotic dynamics [12], material sciences [13], mechanics of fractal and complex media [14, 15], quantum mechanics [16, 17], physical kinetics [18], long-range dissipation [19], and long-range interaction [20, 21], just to mention a few. One of the most remarkable applications of fractional calculus appears, however, in the fractional variational calculus, in the context of classical mechanics. Riewe [7, 8] shows that a Lagrangian involving fractional time derivatives leads to an equation of motion with nonconservative forces such as friction. It is a remarkable result since frictional and nonconservative forces are beyond the usual macroscopic variational treatment and, consequently, beyond the most advanced methods of classical mechanics [22]. Riewe generalizes the usual variational calculus, by considering Lagrangians that are dependent on fractional derivatives, in order to deal with nonconservative forces. Recently, several different approaches have been developed to generalize the least action principle and the Euler-Lagrange equations to include fractional derivatives. Results include problems depending on Caputo fractional derivatives and Riemann-Liouville fractional derivatives [23–35].

A more general unifying perspective to the subject is, however, possible, by considering fractional operators depending on general kernels [36–38]. In this work, we follow such an approach, developing a generalized fractional calculus of variations. We consider very general problems, where the classical integrals are substituted by generalized fractional integrals, and the Lagrangians depend not only on classical derivatives but also on generalized fractional operators. Problems of the type considered here, for particular kernels, are important in physics [39]. Here, we obtain general necessary optimality conditions, for several types of variational problems, which are valid for rather arbitrary operators and kernels. By choosing particular operators and kernels, one obtains the recent results available in the literature of mathematical physics [39–44].

The paper is organized as follows. In Section 2, we introduce the generalized fractional operators and prove some of their basic properties. Section 3 is dedicated to prove integration by parts formulas for the generalized fractional operators. Such formulas are then used in later sections to prove necessary optimality conditions (Theorems 4.2 and 6.3). In Sections 4, 5, and 6 we study three important classes of generalized variational problems: we obtain fractional Euler-Lagrange conditions for the fundamental (Section 4) and generalized isoperimetric problems (Section 6), as well as fractional natural boundary conditions for generalized free-boundary value problems (Section 5). Finally, two illustrative examples are discussed in detail in Section 7, while applications to physics are given in Section 8: in Section 8.1, we obtain the damped harmonic oscillator in quantum mechanics; in Section 8.2, we show how results from FALVA physics can be obtained. We end with Section 9 of conclusion, pointing out an important direction of future research.

2. Preliminaries

In this section, we present definitions and properties of generalized fractional operators. As particular cases, by choosing appropriate kernels, these operators are reduced to standard fractional integrals and fractional derivatives. Other nonstandard kernels can also be considered as particular cases. For more on the subject of generalized fractional calculus and applications, we refer the reader to [37]. Throughout the text, denotes a real number between zero and one. Following [45], we use round brackets for the arguments of functions and square brackets for the arguments of operators. By definition, an operator receives and returns a function.

Definition 2.1 (generalized fractional integral). The operator   is given by where is the parameter set (-set for brevity), , are real numbers, and is a kernel which may depend on . The operator is referred to as the operator   (-op for simplicity) of order and -set , while is called the operation   (or -opn) of of order and -set .Note that if we define then the operator can be written in the form This is a particular case of one of the oldest and most respectable class of operators, the so-called Fredholm operators [46, 47].

Theorem 2.2 (cf. Example  6 of [46]). Let and . If is a square integrable function on the square , then is well-defined, linear, and bounded operator.

Theorem 2.3. Let be a difference kernel, that is, let with . Then, is a well defined bounded and linear operator.

Proof. Obviously, the operator is linear. Let , , and . Define for all . Since is measurable on the square , we have It follows from Fubini’s theorem that is integrable on the square . Moreover, Hence, and .

Remark 2.4. The  -op reduces to the left and the right Riemann-Liouville fractional integrals from a suitably chosen kernel    and  -set  . Let  :(i)if , then is the standard left Riemann-Liouville fractional integral of of order ;(ii)if , then is the standard right Riemann-Liouville fractional integral of of order .

Corollary 2.5. Operators are well defined, linear and bounded.

The generalized fractional derivatives and are defined in terms of the generalized fractional integral -op.

Definition 2.6 (generalized Riemann-Liouville fractional derivative). Let be a given parameter set and . The operator is defined by , where denotes the standard derivative operator, and is referred to as the operator (-op) of order and -set , while , for a function such that , is called the operation (-opn) of of order and -set .

Definition 2.7 (generalized Caputo fractional derivative). Let be a given parameter set and . The operator is defined by , where denotes the standard derivative operator, and is referred to as the operator (-op) of order and -set , while , for a function , is called the operation (-opn) of of order and -set .

Remark 2.8. The standard Riemann-Liouville and Caputo fractional derivatives are easily obtained from the generalized operators   and , respectively. Let  :(i)if , then is the standard left Riemann-Liouville fractional derivative of of order , while is the standard left Caputo fractional derivative of of order ;(ii)if , then is the standard right Riemann-Liouville fractional derivative of of order , while is the standard right Caputo fractional derivative of of order .

3. On Generalized Fractional Integration by Parts

We now prove integration by parts formulas for generalized fractional operators.

Theorem 3.1 (fractional integration by parts for the -op). Let , , be a square-integrable function on , and . The generalized fractional integral satisfies the integration by parts formula where .

Proof. Define for all . Applying Holder’s inequality, we obtain By Fubini’s theorem, functions and belong to for almost all . Therefore, Hence, we can use again Fubini’s theorem to change the order of integration:

Theorem 3.2. Let and . If , , and , then the operator satisfies the integration by parts formula (3.1).

Proof. Define for all . Since is a continuous function on , it is bounded on , that is, there exists a real number such that for all . Therefore, Hence, we can use Fubini’s theorem to change the order of integration in iterated integrals.

Theorem 3.3 (generalized fractional integration by parts). Let and . If functions , and we are in conditions to use formula (3.1) (Theorem 3.1 or Theorem 3.2), then where .

Proof. From Definition 2.7, we know that . It follows that By relation (3.1), and the standard integration by parts formula implies (3.8):

Corollary 3.4 (cf. [48]). Let . If , then

4. The Generalized Fundamental Variational Problem

By , we denote the partial derivative of a function with respect to its th argument. We consider the problem of finding a function , , that gives an extremum (minimum or maximum) to the functional subject to the boundary conditions where , , and , . For simplicity of notation, we introduce the operator defined by With the new notation, one can write (4.1) simply as . The operator has kernel , and operators and have kernels and , respectively. In the sequel, we assume the following:(H1)Lagrangian ,(H2)functions , , , and are continuous on ,(H3)functions , ;(H4)kernels , , and are such that we are in conditions to use Theorems 3.1 or 3.2, and Theorem 3.3.

Definition 4.1. A function is said to be admissible for the fractional variational problem (4.1)-(4.2), if functions and exist and are continuous on the interval , and satisfies the given boundary conditions (4.2).

Theorem 4.2. If is a solution to problem (4.1)-(4.2), then satisfies the generalized Euler-Lagrange equation for all .

Proof. Suppose that is an extremizer of . Consider the value of at a nearby function , where is a small parameter, and is an arbitrary function with continuous -op and -op. We require that . Let A necessary condition for to be an extremizer is given by Using classical and generalized fractional integration by parts formulas (Theorems 3.1 or 3.2, and Theorem 3.3), where . Because , (4.6) simplifies to We obtain (4.4) by application of the fundamental lemma of the calculus of variations (see, e.g., [49, Section 2.2]).

The following corollary gives an extension of the main result of [28].

Corollary 4.3. If is a solution to the problem of minimizing or maximizing in the class subject to the boundary conditions where , , and has continuous Riemann-Liouville fractional derivative , then for all .

Proof. Choose , , and . Then, the -op, the -op, and the -op reduce to the left fractional integral, the left Riemann-Liouville, and the left Caputo fractional derivatives, respectively. Therefore, problem (4.9)-(4.10) is a particular case of problem (4.1)-(4.2) and (4.11) follows from (4.4) with .

The following result is the Caputo analogous to the main result of [50] done for the Riemann-Liouville fractional derivative.

Corollary 4.4. Let . If is a solution to the problem then holds for all .

Proof. The Euler-Lagrange equation (4.13) follows from (4.4) by choosing -sets , , and kernels , , and .

Remark 4.5. In the particular case when the Lagrangian of Corollary 4.4 does not depend on the fractional integral and the classical derivative, one obtains from (4.13) the Euler-Lagrange equation of [51].

5. Generalized Free-Boundary Variational Problems

Assume now that, in problem (4.1)-(4.2), the boundary conditions (4.2) are substituted by

Theorem 5.1. If is a solution to the problem of extremizing functional (4.1) with (5.1) as boundary conditions, then satisfies the Euler-Lagrange equation (4.4). Moreover, the extra natural boundary condition holds.

Proof. Under the boundary conditions (5.1), we do not require in the proof of Theorem 4.2 to vanish at . Therefore, following the proof of Theorem 4.2, we obtain for every admissible with . In particular, condition (5.3) holds for those that fulfill . Hence, by the fundamental lemma of the calculus of variations, (4.4) is satisfied. Now, let us return to (5.3) and let again be arbitrary at point . Inserting (4.4), we obtain the natural boundary condition (5.2).

Corollary 5.2. Let be the functional given by Let be a minimizer of satisfying the boundary condition . Then, satisfies the Euler-Lagrange equation and the natural boundary condition

Proof. Let functional (4.1) be such that it does not depend on the classical (integer) derivative and on the -op. If , , and , then the -op reduces to the left fractional Caputo derivative and we deduce (5.5) and (5.6) from (4.4) and (5.2), respectively.

Corollary 5.3 (cf. Theorem  3.17 of [38]). Let be the functional given by If is a minimizer to satisfying the boundary condition , then satisfies the Euler-Lagrange equation and the natural boundary condition

Proof. Choose, in the problem defined by (4.1) and (5.1), . Then, (5.8) and (5.9) follow from (4.4) and (5.2), respectively.

6. Generalized Isoperimetric Problems

Let . Among all functions satisfying boundary conditions and an isoperimetric constraint of the form we look for the one that extremizes (i.e., minimizes or maximizes) a functional Operators , , and , as well as function , are the same as in problem (4.1)-(4.2). Moreover, we assume that functional (6.2) satisfies hypotheses (H1)–(H4).

Definition 6.1. A function is said to be admissible for problem (6.1)–(6.3) if functions and exist and are continuous on , and satisfies the given boundary conditions (6.1) and the given isoperimetric constraint (6.2).

Definition 6.2. An admissible function   is said to be an extremal for    if it satisfies the Euler-Lagrange equation (4.4) associated with functional in (6.2), that is, where , , and .

Theorem 6.3. If is a solution to the isoperimetric problem (6.1)–(6.3) and is not an extremal for , then there exists a real constant such that for all , where and , .

Proof. Consider a two-parameter family of the form , where, for each , we have . First, we show that we can select such that satisfies (6.2). Consider the quantity . Looking to as a function of , we define . Thus, . On the other hand, applying integration by parts formulas (Theorems 3.1 or 3.2, and Theorem 3.3), we obtain that where , . We assume that is not an extremal for . Hence, the fundamental lemma of the calculus of variations implies that there exists a function such that . According to the implicit function theorem, there exists a function defined in a neighborhood of 0 such that . Let . Function has an extremum at subject to , and we have proved that . The Lagrange multiplier rule asserts that there exists a real number such that . Because one has From the fundamental lemma of the calculus of variations (see, e.g., [49], Section 2.2), it follows that is, with .

Corollary 6.4. Let be a minimizer to the isoperimetric problem If is not an extremal of , then there exists a constant such that satisfies for all , where .

Proof. Let , , , and . Then, the -op and the -op reduce to the left fractional integral and the left fractional Caputo derivative, respectively. Therefore, problem (6.11) is a particular case of problem (6.1)–(6.3), and (6.12) follows from (6.5) with .

Corollary 6.5 (cf. Theorem  3.22 of [38]). Let be a minimizer to If is not an extremal of , then there exists a constant such that satisfies for all , where .

Proof. Let, in problem (6.1)–(6.3), and kernel . Then, the generalized fractional integral becomes the classical integral and (6.14) follows from (6.5).

7. Illustrative Examples

We illustrate our results through two examples with different kernels: one of a fundamental problem (4.1)–(4.2) (Example 7.1), the other an isoperimetric problem (6.1)–(6.3) (Example 7.2).

Example 7.1. Let  , , , and  . Consider the following problem: with kernel such that and . Here, the resolvent is related to the kernel by , , where and are the direct and the inverse Laplace operators, respectively. We apply Theorem 4.2 with Lagrangian given by . Because is the solution to the Volterra integral equation of first kind (see, e.g., Equation 16, page 114 of [47]) it satisfies our generalized Euler-Lagrange equation (4.4), that is, In particular, for the kernel , the boundary conditions are and , and the solution is (cf. [47], page 22).In the next example, we make use of the Mittag-Leffler function of two parameters: if , then the Mittag-Leffler function is defined by This function appears naturally in the solution of fractional differential equations, as a generalization of the exponential function [52].

Example 7.2. Let  , , and  . Consider the following problem: which is an example of (6.1)–(6.3) with -sets and and kernels and . Function of Theorem 6.3 is given by . One can easily check (see [52], page 324) that (i)is not an extremal for ,(ii)satisfies .Moreover, (7.7) satisfies (6.5) for , that is, for all . We conclude that (7.7) is an extremal for problem (7.6).

8. Applications to Physics

If the functional (4.1) does not depend on -op and -op, then Theorem 4.2 gives the following result: if is a solution to the problem of extremizing subject to and , where , then We recognize on the right hand side of (8.2) the generalized weak dissipative parameter

8.1. Quantum Mechanics of the Damped Harmonic Oscillator

As a first application, let us consider kernel and the Lagrangian where is the potential energy and stands for mass. The Euler-Lagrange equation (8.2) gives the following second-order ordinary differential equation: Equation (8.5) coincides with (14) of [44], obtained by modification of Hamilton’s principle.

8.2. Fractional Action-Like Variational Approach (FALVA)

We now extend some of the recent results of [39, 41–43], where the fractional action-like variational approach (FALVA) was proposed to model dynamical systems. FALVA functionals are particular cases of (8.1), where the fractional time integral introduces only one parameter . Let us consider the Caldirola-Kanai Lagrangian [39, 40, 42] which describes a dynamical oscillatory system with exponentially increasing time-dependent mass, where is the frequency and , . Using our generalized FALVA Euler-Lagrange equation (8.2) with kernel to Lagrangian (8.6), we obtain We study two particular kernels.If we choose kernel defined in [53], then the Euler-Lagrange equation is In particular, when , (8.8) becomes the kernel of the Riemann-Liouville fractional integral, and equation (8.9) gives which is the Euler-Lagrange equation proved in [42]. For , we have Therefore, both at the very early time and at very large time, dissipation disappears. Moreover, if , then This shows that, at the origin of time, the time-dependent dissipation becomes stationary and that, at very large time, no dissipation, of any kind, exists.If we choose kernel , then In contrast with previous case, item 1, here dissipation does not disappear at late-time dynamics.

We note that there is a small inconsistence in [42], regarding the coefficient of in (8.7), and a small inconsistence in [39], regarding a sign of (8.13).

9. Conclusion

In this paper, we unify, subsume, and significantly extend the necessary optimality conditions available in the literature of the fractional calculus of variations. It should be mentioned, however, that since fractional operators are nonlocal, it can be extremely challenging to find analytical solutions to fractional problems of the calculus of variations, and, in many cases, solutions may not exist. In our paper, we give two examples with analytic solutions, and many more can be found borrowing different kernels from [47]. On the other hand, one can easily choose examples for which the fractional Euler-Lagrange differential equations are hard to solve, and, in that case, one needs to use numerical methods [54–57]. The question of existence of solutions to fractional variational problems is a complete open area of research. This needs attention. Indeed, in the absence of existence, the necessary conditions for extremality are vacuous: one cannot characterize an entity that does not exist in the first place. For solving a problem of the fractional calculus of variations, one should proceed along the following three steps: (i) first, prove that a solution to the problem exists; (ii) second, verify the applicability of necessary optimality conditions; (iii) finally, apply the necessary conditions which identify the extremals (the candidates). Further elimination, if necessary, identifies the minimizer(s) of the problem. All three steps in the above procedure are crucial. As mentioned by Young in [58], the calculus of variations has born from the study of necessary optimality conditions, but any such theory is “naive” until the existence of minimizers is verified. The process leading to the existence theorems was introduced by Tonelli in 1915 by the so-called direct method [59]. During two centuries, mathematicians were developing “the naive approach to the calculus of variations”. There were, of course, good reasons why the existence problem was only solved in the beginning of XX century, two hundred years after necessary optimality conditions began to be studied: see [60, 61] and references therein. Similar situation happens now with the fractional calculus of variations: the subject is only fifteen years old and is still in the “naive period”. We believe time has come to address the existence question, and this will be considered in a forthcoming paper.

Acknowledgments

This work was supported by FEDER funds through COMPETE—Operational Programme Factors of Competitiveness (Programa Operacional Factores de Competitividade) and by Portuguese funds through the Center for Research and Development in Mathematics and Applications (University of Aveiro) and the Portuguese Foundation for Science and Technology (FCT—Fundação para a Ciência e a Tecnologia), within project PEst-C/MAT/UI4106/2011 with COMPETE no. FCOMP-01-0124-FEDER-022690. T. Odzijewicz was also supported by FCT through the Ph.D. fellowship SFRH/BD/33865/2009; A. B. Malinowska by Białystok University of Technology Grant S/WI/02/2011; D. F. M. Torres by FCT through the Portugal-Austin (USA) cooperation project UTAustin/MAT/0057/2008. This paper is part of the first author's Ph.D., which is carried out at the University of Aveiro under the Doctoral Programme Mathematics and Applications of Universities of Aveiro and Minho.

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