On Fractional Derivatives and Primitives of Periodic Functions

Abstract

We prove that the fractional derivative or the fractional primitive of a -periodic function cannot be a -periodic function, for any period , with the exception of the zero function.

1. Introduction

Periodic functions [1, Ch. 3, pp. 58–92] play a central role in mathematics since the seminal works of Fourier [2, 3]. Nowadays, periodic functions appear in applications ranging from electromagnetic radiation to blood flow and of course in control theory in linear time-varying systems driven by periodic input signals [4]. Linear time-varying systems driven by periodic input signals are ubiquitous in control systems, from natural sciences to engineering, economics, physics, and the life science [4, 5]. Periodic functions also appear in automotive engine applications [6], optimal periodic scheduling of sensor networks [7, 8], or cyclic gene regulatory networks [9], to give some applications.

It is an obvious fact that the classical derivative, if it exists, of a periodic function is also a periodic function of the same period. Also the primitive of a periodic function may be periodic (e.g., as primitive of ).

The idea of integral or derivatives of noninteger order goes back to Riemann and Liouville [3, 10]. Probably the first application of fractional calculus was made by Abel in the solution of the integral equation that arises in the formulation of the tautochrone problem [11]. Fractional calculus appears in many different contexts as speech signals, cardiac tissue electrode interface, theory of viscoelasticity, or fluid mechanics. The asymptotic stability of positive fractional-order nonlinear systems has been proved in [12] by using the Lyapunov function. We do not intend to give a full list of applications but to show the wide range of them.

In this paper we prove that periodicity is not transferred by fractional integral or derivative, with the exception of the zero function. Although this property seems to be known [10, 13, 14], in Section 3 we give a different proof by using the Laplace transform. Our approach relies on the classical concepts of fractional calculus and elementary analysis. Moreover, by using a similar argument as in [15], in Section 4 we prove that the fractional derivative or primitive of a -periodic function cannot be -periodic for any period . A particular but nontrivial example is explicitly given. Finally, as a consequence we show in Section 5 that an autonomous fractional differential equation cannot have periodic solutions with the exception of constant functions.

2. Preliminaries

Let . If is periodic and , then the derivative is also -periodic. However, the primitive of is not, in general, -periodic. Just take so that is not -periodic for any . The necessary and sufficient condition for to be -periodic is that

The purpose of this note is to show that the fractional derivative or the fractional primitive of a -periodic function cannot be -periodic function with the exception, of course, of the zero function. We use the notation and note that but and does not coincide with unless .

We recall some elements of fractional calculus. Let and . We point out that is not necessarily continuous. The fractional integral of of order is defined by [16] provided the right-hand side is defined for a.e. . If, for example, , then the fractional integral (6) is well defined and , for any . Moreover, the fractional operator is linear and bounded.

The fractional Riemann-Liouville derivative of order of is defined as [16, 17] This is well defined if, for example, .

There are many more fractional derivatives. We are not giving a complete list but recall the Caputo derivative [16, 17] which is well defined, for example, for absolutely continuous functions.

As in the integer case we have but or are not, in general, equal to . Indeed and (see [17, (2.113), p. 71]) Also [16, (2.4.4), p. 91]

3. The Fractional Derivative or Primitive of a -Periodic Function Cannot Be -Periodic

We prove the following result in Section 3.1 below.

Theorem 1. Let be a nonzero -periodic function with . Then cannot be -periodic for any .

Corollary 2. Let be a nonzero -periodic function such that . Then the Caputo derivative cannot be -periodic for any . The same result holds for the fractional derivative .

Proof. Suppose that is -periodic. Then by Theorem 1, cannot be -periodic. However, is -periodic. In relation to the fractional Riemann-Liouville derivative, suppose that is -periodic and consider the function which is also -periodic. Then cannot be -periodic.

3.1. Proof of Theorem 1

Let and . By reduction to the absurd, in this section we suppose that is -periodic. Then that is,

Lemma 3. Assume is -periodic. If is also -periodic, then

Proof. For the latter equality reduces to (17). For , The proof follows by induction on . Assume that (18) is valid for some . Then and, by periodicity, Moreover, for , by hypothesis of induction since . Hence,

Lemma 4. Under the hypothesis of Lemma 3,

Proof. Let and be the positive and negative parts of , Equation (18) implies that If or , then from (18) we get . We consider the case For large or equivalently Hence, which is a contradiction.
The case is analogous.
Therefore,

Lemma 5. Under the hypothesis of Lemma 3,

Proof. If and , the equation reduces to (17) and (18), respectively. Let . By using the periodicity of we get (33).

Lemma 6. Under the hypothesis of Lemma 3,

Proof. For or , , relation (35) is true. Let , so that . Then Now, using the additive property of the integral, we have Let us compute separately the integrals in the right-hand side. In all the integrals depending on , we use the (linear) change of variable and rename to obtain For the last integral we use the (linear) change of variable to get By induction on , as in Lemma 3, the proof follows.

Lemma 7. Let be a continuous and -periodic function, . Let be fixed. Assuming that then .

Proof. Since then and therefore we can define . If then .
Let us define From the hypothesis, we have that at any . Therefore, its integral is also zero. Let us integrate with respect to from to for where we have assumed , . Thus, which implies that is a constant function.
Moreover, since where in view of (24) and we have that
Let If we define then the convolution of and is given by Therefore, if we apply the Laplace transform [18, Chapter 17] to the above equality it yields Since where denotes the incomplete gamma function [19, Section 6.5], then which implies that and therefore , that is, , on .

4. The Fractional Derivative or Primitive of a -Periodic Function Cannot Be -Periodic for any Period

Let be a -periodic function and consider such that Then and therefore Let us assume that is a -periodic function. Then by using some basic properties of the Laplace transform it yields Therefore, Let us consider so that is also -periodic and . The above equality becomes or equivalently Thus, Since by using and , the limit as of the left-hand side is zero, which implies Then If we consider in the latter expression we get and therefore By induction, we obtain that Therefore, and there are no nonzero -periodic -solutions of the problem.

Example 8. Let and . The Caputo-fractional derivative of is given by where the hypergeometric series is defined as ([20, 21], Chapter 15) and the Pochhammer symbol , with .
Since we have that is not a -periodic function for any positive and . Plotting both functions and , this last function seems to be periodic but it is not according to our results. Notice that Kaslik and Sivasundaram [10] gave the following alternate representation: in terms of the two-parameter Mittag-Leffler function ([20, 21], Chapter 10)

5. Periodic Solutions of Fractional Differential Equations

In this section we show how Theorem 1 can be used to give a nonexistence result of periodic solutions for fractional differential equations.

Consider the first order ordinary differential equation where is continuous. An important question is the existence of periodic solutions [22–24].

If is a -periodic solution of (73) then obviously One can find -periodic solutions of (73) by solving the equation only on the interval and then checking the values and . If (74) holds, then extending by -periodicity the function , , to we have a -periodic solution of (73).

However, this is not possible for a fractional differential equation. Consider, for , the equation If is a solution of (75), let . Then In the case that is a -periodic solution of (75) we have that is also -periodic. According to Theorem 1, cannot be -periodic unless it is the zero function and we have the following relevant result.

Theorem 9. The fractional equation (75) cannot have periodic solutions with the exception of constant functions , , with .

Remark 10. It is possible to consider the periodic boundary value problem as in, for example, [25], but one cannot extend the solution of that periodic boundary value problem on to a -periodic solution on (unless is a constant function, as indicated in Theorem 9).

Remark 11. The same applies to the Riemann-Liouville fractional differential equation taking into account that

Example 12. Considering the fractional equation with defined by we have that is a -periodic solution of (80). This shows that the result of Theorem 9 is not valid for a nonautonomous fractional differential equation as (80).

6. Conclusion

By using the classical concepts of fractional calculus and elementary analysis, we have proved that periodicity is not transferred by fractional integral or derivative, with the exception of the zero function. We have also proved that the fractional derivative or primitive of a -periodic function cannot be -periodic for any period . As a consequence we have showed that an autonomous fractional differential equation cannot have periodic solutions with the exception of constant functions.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The referees and editor deserve special thanks for careful reading and many useful comments and suggestions which have improved the paper. The work of I. Area has been partially supported by the Ministerio de Economía y Competitividad of Spain under Grant MTM2012-38794-C02-01, cofinanced by the European Community fund FEDER. J. J. Nieto also acknowledges partial financial support by the Ministerio de Economía y Competitividad of Spain under Grant MTM2010-15314, cofinanced by the European Community fund FEDER.