On Fractional Derivatives and Primitives of Periodic Functions

Abstract

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

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I. Area 1 and J. Losada 2 and J. J. Nieto 2,3

Academic Editor:Hari M. Srivastava

1, Departamento de Matemática Aplicada II, E.E. Telecomunicación, Universidade de Vigo, 36310 Vigo, Spain
2, Facultad de Matemáticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain
3, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 23 May 2014; Revised 8 July 2014; Accepted 8 July 2014; 14 August 2014

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.

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., cos ... t as primitive of sin t ).

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 T -periodic function cannot be T ~ -periodic for any period T ~ . 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 T > 0 . If f : R [arrow right] R is T periodic and f ∈ ^{C 1} ( R ) , then the derivative ^{f [variant prime]} is also T -periodic. However, the primitive of f [figure omitted; refer to PDF] is not, in general, T -periodic. Just take f ( t ) = 1 so that F ( t ) = t is not ^{T [variant prime]} -periodic for any ^{T [variant prime]} > 0 . The necessary and sufficient condition for F to be T -periodic is that [figure omitted; refer to PDF]

The purpose of this note is to show that the fractional derivative or the fractional primitive of a T -periodic function cannot be T -periodic function with the exception, of course, of the zero function. We use the notation [figure omitted; refer to PDF] and note that [figure omitted; refer to PDF] but [figure omitted; refer to PDF] and ^{I 1} ( ^{D 1} f ) does not coincide with f unless f ( 0 ) = 0 .

We recall some elements of fractional calculus. Let α ∈ ( 0,1 ) and f : R [arrow right] R . We point out that f is not necessarily continuous. The fractional integral of f of order α is defined by [16] [figure omitted; refer to PDF] provided the right-hand side is defined for a.e. t ∈ R . If, for example, f ∈ ^{L 1} ( R ) , then the fractional integral (6) is well defined and ^{I α} f ∈ ^{L 1} ( 0 , T ) , for any T > 0 . Moreover, the fractional operator [figure omitted; refer to PDF] is linear and bounded.

The fractional Riemann-Liouville derivative of order α of f is defined as [16, 17] [figure omitted; refer to PDF] This is well defined if, for example, f ∈ ^{L loc ... 1} ( R ) .

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

As in the integer case we have [figure omitted; refer to PDF] but ^{I α} ( ^{D α} f ) or ^{I α} ( ^{c D α} f ) are not, in general, equal to f . Indeed [figure omitted; refer to PDF] and (see [17, (2.113), p. 71]) [figure omitted; refer to PDF] Also [16, (2.4.4), p. 91] [figure omitted; refer to PDF]

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

We prove the following result in Section 3.1 below.

Theorem 1.

Let f : R [arrow right] R be a nonzero T -periodic function with f ∈ ^{L loc ... 1} ( R ) . Then ^{I α} f cannot be T -periodic for any α ∈ ( 0,1 ) .

Corollary 2.

Let f : R [arrow right] R be a nonzero T -periodic function such that f ∈ ^{L loc ... 1} ( R ) . Then the Caputo derivative ^{c D α} f cannot be T -periodic for any α ∈ ( 0,1 ) . The same result holds for the fractional derivative ^{D α} f .

Proof.

Suppose that ^{c D α} f is T -periodic. Then by Theorem 1, ^{I α ( c D α} f ) cannot be T -periodic. However, [figure omitted; refer to PDF] is T -periodic. In relation to the fractional Riemann-Liouville derivative, suppose that ^{D α} f is T -periodic and consider the function f ^ = f - f ( 0 ) which is also T -periodic. Then [figure omitted; refer to PDF] cannot be T -periodic.

3.1. Proof of Theorem 1

Let α ∈ ( 0,1 ) and T > 0 . By reduction to the absurd, in this section we suppose that ^{I α} f is T -periodic. Then [figure omitted; refer to PDF] that is, [figure omitted; refer to PDF]

Lemma 3.

Assume f ∈ ^{L loc ... 1} ( R ) is T -periodic. If ^{I α} f is also T -periodic, then [figure omitted; refer to PDF]

Proof.

For n = 1 the latter equality reduces to (17). For n = 2 , [figure omitted; refer to PDF] The proof follows by induction on n . Assume that (18) is valid for some n ∈ N . Then [figure omitted; refer to PDF] and, by periodicity, [figure omitted; refer to PDF] Moreover, for j = 1,2 , ... , n , [figure omitted; refer to PDF] by hypothesis of induction since 1 ...4; n + 1 - j ...4; n . Hence, [figure omitted; refer to PDF]

Lemma 4.

Under the hypothesis of Lemma 3, [figure omitted; refer to PDF]

Proof.

Let ^{f +} and ^{f -} be the positive and negative parts of f , [figure omitted; refer to PDF] Equation (18) implies that [figure omitted; refer to PDF] If ^{∫ 0 T} ... ^{f +} ( s ) d s = 0 or ^{∫ 0 T} ... ^{f -} ( s ) d s = 0 , then from (18) we get f = 0 . We consider the case [figure omitted; refer to PDF] For n large [figure omitted; refer to PDF] or equivalently [figure omitted; refer to PDF] Hence, [figure omitted; refer to PDF] which is a contradiction.

The case [figure omitted; refer to PDF] is analogous.

Therefore, [figure omitted; refer to PDF]

Lemma 5.

Under the hypothesis of Lemma 3, [figure omitted; refer to PDF]

Proof.

If δ = 0 and δ = T , the equation reduces to (17) and (18), respectively. Let 0 < δ < T . [figure omitted; refer to PDF] By using the periodicity of ^{I α} f we get (33).

Lemma 6.

Under the hypothesis of Lemma 3, [figure omitted; refer to PDF]

Proof.

For t ∈ [ 0 , T ] or t = n T , n = 1,2 , ... , relation (35) is true. Let t = n T + δ , so that T + t = ( n + 1 ) T + δ . Then [figure omitted; refer to PDF] Now, using the additive property of the integral, we have [figure omitted; refer to PDF] Let us compute separately the integrals in the right-hand side. In all the integrals depending on j , we use the (linear) change of variable r = s - j T and rename ^{t [variant prime]} = ( n - j ) T + δ to obtain [figure omitted; refer to PDF] For the last integral we use the (linear) change of variable r = s - ( n + 1 ) T to get [figure omitted; refer to PDF] By induction on n , as in Lemma 3, the proof follows.

Lemma 7.

Let f be a continuous and T -periodic function, T > 0 . Let 0 < α < 1 be fixed. Assuming that [figure omitted; refer to PDF] then f ...1; 0 .

Proof.

Since ^{∫ 0 T} ... f ( s ) d s = 0 then 0 = ^{∫ 0 T} ... f ( s ) d s = ^{∫ 0 T} ... ( ^{f +} ( s ) - ^{f -} ( s ) ) d s and therefore we can define c = ^{∫ 0 T} ... ^{f +} ( s ) d s = ^{∫ 0 T} ... ^{f - s} ( s ) d s > 0 . If c = 0 then f = 0 .

Let us define [figure omitted; refer to PDF] From the hypothesis, we have that [varphi] ( t ) = 0 at any t ∈ R . Therefore, its integral is also zero. Let us integrate with respect to t from a to b for 0 ...4; a ...4; b ...4; T [figure omitted; refer to PDF] where we have assumed 0 ...4; a < b , s < T . Thus, [figure omitted; refer to PDF] which implies that [figure omitted; refer to PDF] is a constant function.

Moreover, since [figure omitted; refer to PDF] where [figure omitted; refer to PDF] in view of (24) and [figure omitted; refer to PDF] we have that [figure omitted; refer to PDF]

Let [figure omitted; refer to PDF] If we define [figure omitted; refer to PDF] then the convolution of [straight phi] and f ~ is given by [figure omitted; refer to PDF] Therefore, if we apply the Laplace transform [18, Chapter 17] to the above equality it yields [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] where Γ ( a , z ) denotes the incomplete gamma function [19, Section 6.5], then L [ [straight phi] ] ...0; 0 which implies that L [ f ~ ] = 0 and therefore f ~ = 0 , that is, f = 0 , on [ 0 , T ] .

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

Let f be a T -periodic function and consider u such that [figure omitted; refer to PDF] Then [figure omitted; refer to PDF] and therefore [figure omitted; refer to PDF] Let us assume that u is a T ~ -periodic function. Then by using some basic properties of the Laplace transform it yields [figure omitted; refer to PDF] Therefore, [figure omitted; refer to PDF] Let us consider v = u - _{u 0} so that v is also T ~ -periodic and v ( 0 ) = 0 . The above equality becomes [figure omitted; refer to PDF] or equivalently [figure omitted; refer to PDF] Thus, [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] by using 0 < α < 1 and i ...5; 0 , the limit as λ [arrow right] ^{0 +} of the left-hand side is zero, which implies [figure omitted; refer to PDF] Then [figure omitted; refer to PDF] If we consider λ [arrow right] ^{0 +} in the latter expression we get [figure omitted; refer to PDF] and therefore [figure omitted; refer to PDF] By induction, we obtain that [figure omitted; refer to PDF] Therefore, f = u = 0 and there are no nonzero T ~ -periodic ^{L ∞} -solutions of the problem.

Example 8.

Let f ( t ) = sin ( t ) and 0 < α < 1 . The Caputo-fractional derivative of f ( t ) is given by [figure omitted; refer to PDF] where the hypergeometric series _{1 F 2} ( a ; b , c ; d ) is defined as ([20, 21], Chapter 15) [figure omitted; refer to PDF] and the Pochhammer symbol ( A _{) j} = A ( A + 1 ) ... ( A + j - 1 ) , with ( A _{) 0} = 1 .

Since [figure omitted; refer to PDF] we have that ^{c D α} f ( t ) is not a T ~ -periodic function for any positive T ~ and α ∈ ( 0,1 ) . Plotting both functions sin ( t ) and ^{c D α} sin ( t ) , 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: [figure omitted; refer to PDF] in terms of the two-parameter Mittag-Leffler function ([20, 21], Chapter 10) [figure omitted; refer to PDF]

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 [figure omitted; refer to PDF] where [straight phi] : R [arrow right] R is continuous. An important question is the existence of periodic solutions [22-24].

If u : R [arrow right] R is a T -periodic solution of (73) then obviously [figure omitted; refer to PDF] One can find T -periodic solutions of (73) by solving the equation only on the interval [ 0 , T ] and then checking the values u ( 0 ) and u ( T ) . If (74) holds, then extending by T -periodicity the function u ( t ) , t ∈ [ 0 , T ] , to R we have a T -periodic solution of (73).

However, this is not possible for a fractional differential equation. Consider, for α ∈ ( 0,1 ) , the equation [figure omitted; refer to PDF] If u is a solution of (75), let f ( t ) = [straight phi] ( u ( t ) ) . Then [figure omitted; refer to PDF] In the case that u is a T -periodic solution of (75) we have that f is also T -periodic. According to Theorem 1, ^{I α} f cannot be T -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 u ( t ) = _{u 0} , t ∈ R , with [straight phi] ( _{u 0} ) = 0 .

Remark 10.

It is possible to consider the periodic boundary value problem [figure omitted; refer to PDF] as in, for example, [25], but one cannot extend the solution of that periodic boundary value problem on [ 0 , T ] to a T -periodic solution on R (unless u is a constant function, as indicated in Theorem 9).

Remark 11.

The same applies to the Riemann-Liouville fractional differential equation [figure omitted; refer to PDF] taking into account that [figure omitted; refer to PDF]

Example 12.

Considering the fractional equation [figure omitted; refer to PDF] with ψ : ^{R 2} [arrow right] R defined by [figure omitted; refer to PDF] we have that u ( t ) = sin ( t ) is a 2 π -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 T -periodic function cannot be T ~ -periodic for any period T ~ . As a consequence we have showed that an autonomous fractional differential equation cannot have periodic solutions with the exception of constant functions.

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.

Conflict of Interests

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

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