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Commutative and associative properties of the Caputo fractional derivative and its generalizing convolution operator

https://doi.org/10.1016/j.cnsns.2020.105338Get rights and content

Highlights

  • Caputo fractional derivative of analytic functions (not necessarily of exponential order).

  • Commutative and semigroup properties of Caputo derivative, under less restrictive requirements.

  • Properties of Caputo-type convolution operator defined by Bernstein functions.

Abstract

While for the integer-order derivatives, the commutative and semigroup (or associative) properties hold, the same is not true, in general, for the fractional derivatives. We show that, when the function is analytic, the Caputo derivative enjoys the above mentioned properties, at least when the fractional indices are smaller than one. This result is proved under assumptions on the derivatives (evaluated in zero) that are much less restrictive than the usual requirement of vanishing in the origin. Finally, we study the same properties for a generalization of the above fractional derivative, i.e. the Caputo-type convolution operator defined in [12] and [25]. In this more general setting (which includes the fractional, as particular case), we prove that the commutative property holds, while the associative property must be formulated accordingly.

Introduction

Fractional integrals and derivatives, which generalize the classical integer-order counterparts, have been introduced and widely used, over the last decades, in order to take into account memory effects in many real phenomena. The applications of fractional differential equations range over many areas of science including physics, engineering, hydrology and finance, among others. One of the most widely used fractional derivatives was introduced by Caputo in 1967 and defined as follows (see [3]). Let f:[a,b]R, for a,bR+ and such that a < b, be a function with continuous derivatives up to order m on [a, b] and such that f(m1)(t):=dm1dtm1f(t)AC[a,b], mN{0}, (where AC[ · ] is the space of absolutely continuous functions). We will denote, for a,bR+, ACm[a,b]:={f:[a,b]R,s.t.f(m1)(t)AC[a,b]}, for m=1,2,.Then, we define, for f ∈ ACm[a, b] and αR+,Da+αf(t):={1Γ(mα)atf(m)(u)1(tu)αm+1du,αN{0},m=α+1dmdtmf(t),α=mN{0}where ⌊ · ⌋ denotes the integer part (or floor function). For general reference on the Caputo derivative and its properties, see [11], [22] and [1].

Since its introduction, the Caputo derivative has been studied and applied by many authors: see, among others, [5], [15] for the first rheological applications [4], [17], [18]; for the main physical applications; [2], [16], [19], [20], [21] for some stochastic applications.

It is well-known that its success is mainly due to the fact that initial value problems with Caputo fractional derivatives involves initial conditions with standard derivatives, instead of fractional ones, as happens when using the Riemann-Liouville definition. These conditions have clear physical interpretation, when modelling viscoelastic materials in mechanics, in terms of initial position, velocity, acceleration and so on (see e.g. [10]).

While for the integer-order derivatives, the commutative and associative (or semigroup) properties hold, the same is not true, in general, for the fractional derivatives. For details on the Riemann-Liouville case, see, for example, [11], p.83. In the Caputo case, we show here that when f(t) is a polynomial or, in general, an analytic function, the operator defined in Eq. (1) has both the commutative and the semigroup properties, at least when the fractional indices are less than one. For the sake of simplicity, we will consider the case a=0 and denote Dα:=D0+α. We will prove that the following equalities holdDβDαf=DαDβfandDα+βf=DαDβf,for α, β ∈ (0, 1], even when α+β>1. The previous results are proved under the additional assumption that the derivatives of order nR of the function f, evaluated in zero, increase at most as a power, for increasing n, i.e. |f(n)(0)| < Kn, for some constant K > 0. Clearly, this condition is much less restrictive than the usual hypothesis of being null (see Eq. (3) below).

Examples of analytic functions satisfying this assumption are the polynomials, as well as f(t)=ectp, for any cR and pN. On the other hand, f(t)=log(t) does not fulfill the hypothesis. For the trigonometric functions sine, cosine and tangent the assumption is satisfied, while it is not true for the cotangent (since the first derivative in zero diverges).

We start by recalling the following well-known results on the concatenation properties of two Caputo derivatives of different orders. It has been proved that, for αR+ and where m=α+1, given a function f ∈ ACm[0, T], thenDαDnf=DnDαf=Dα+nf,αR+,n=0,1,.,under the following assumption on the derivatives of ff(k)(t)|t=0=0,k=m,,n,(see, e.g. [22], p.81). Moreover, the following useful result was presented in [6]: let f ∈ Ck[0, T], for some kN, and let α, β > 0 be such that there exists some lN, with l ≤ k; thenDαDβf=Dα+βf,forα,α+β[l1,l].The proof is based on the expression of the Caputo fractional derivative as Dαf:=JmαDmf, where Jα is the Riemann-Liouville fractional integral (see [6], p. 56 for more details). Note that Eq. (4) does not require the condition Eq. (3) and holds for two fractional indices (instead of one integer and one fractional, as in Eq. (2)). On the contrary it holds under the restriction that β ∈ (0, 1) and that both α and α+β are included between two successive integers. Thus, the commutative property does not follow from Eq. (4) when the last condition is not satisfied.

We propose here a different way of proving the commutative and semigroup properties, which does not require the existence of the Laplace transform and thus holds also for functions that are not of exponential order, i.e. such that |f(x)| < Cekx for some C, k > 0 and for any xR. Thus, for example, also the case f(t)=ectp, for any cR and p=2,3,., is included in our results.

Finally, we study the same properties for a generalization of the above fractional derivative, i.e. the Caputo-type convolution operator defined in [12] and [25]. For interesting analysis and applications of the general fractional calculus, see also [8] and [14]. Convolution-type operators, defined as integrals with memory kernels, have been treated recently by many authors: see, among the others, [24] and [7] for physical and stochastic applications. In this case, the existence of the Laplace transform is necessary and thus we restrict to the case of an exponential order function f:R+R, for which we denote by f˜(s):=L{f;s}:=0+estf(t)dt the Laplace transform. In this more general setting (which includes the fractional, as particular case), we prove that the commutative property holds, while the associative property must be formulated accordingly.

Let us recall the definition of Bernstein function (see, for example, [23], p.21): a function g:R+R is a Bernstein function if it is of class C, g(x) ≥ 0, for any x > 0, and such that (1)ng(n)(x)0, for any nN and x > 0. It is well-known also that any Bernstein function g admits the following representationg(x)=a+bx+0+(1esx)ν¯(ds),for a, b ≥ 0 and where ν¯(·) denotes a non-negative measure on (0,+) such that0+(x1)ν¯(dx)<.The triplet (a,b,ν¯) is called the Lévy triplet of the Bernstein function g. Finally, we recall that, for any Bernstein function g, the ratio g(x)/x is completely monotone (i.e. it is positive, of class C and such that (1)ndndxn(g(x)/x)0) and the following representation holds: g(x)/x=b+0+esxν(s)ds, where ν(s):=a+ν¯(s,+) is the tail Lévy measure, under the assumption that it is absolutely continuous (see [23], p.24). We will consider here the case a=b=0, so that ν(s)=s+ν¯(dz).

We now define the Caputo-type convolution operator on the positive half-axes as followsDνf(t):=0tddsf(ts)ν(s)ds,t>0,for f:[0,T]R in AC1[0, T] and such that |f(t)| < Cekt for some C, k > 0 and for any t (see [25], Def.2.4, for b=0). The definition given in Eq. (7) can easily be seen to coincide with that used in [13] (see Eq. (1), p.111), by applying the result of Prop.2.7 in [25]. Thus, we can conclude that the operator Dν is the left inverse of the following generalized fractional integral operatorI(ν)f(t):=0tf(s)ϰ(ts)ds,t>0,where ϰ( · ) is a completely monotone function linked to the Bernstein function g and to the tail Lévy measure ν by the following relationships, respectively:1g(x)=0+exsϰ(s)ds0+ν(ts)ϰ(s)ds=1.We have that DνI(ν)f=f, if f is a locally bounded measurable function on (0, ∞), while I(ν)Dνf(t)=f(t)f(0), if f is an absolutely continuous function on [0, ∞) (see Theorem 1 in [13], for details).

The Laplace transform of Dν is given by0+estDνf(t)dt=g(s)f˜(s)g(s)sf(0),R(s)>s0,(see [25], Lemma 2.5). It is easy to check that, in the trivial case where g(s)=s, the convolution-type derivative coincides with the standard first-order derivative. When g(s)=sα, for α ∈ (0, 1), this Bernstein function coincides with the Laplace exponent of an α -stable subordinator (i.e. of a non-decreasing Lévy process). In this case, the tail Lévy measure reduces to ν(s)=sα/Γ(1α) and thus Eq. (7) coincides with the fractional Caputo derivative of order α.

Section snippets

Properties of the Caputo derivative of analytic functions

Now we restrict ourselves to the case where f:[0,T]R is an analytic function; under this assumption, we prove the following preliminary result.

Lemma 1

Let f:[0,T]R be an analytic function in AC[0, T] with power series expansion f(t)=n=0antn, (i.e. an ≔ f(n)(0)/n!, for nN{0}). If f is such that, for any nN, |f(n)(0)| < Kn, for some constant K > 0, then we have thatDαf(t)=n=1Γ(n+1)antnαΓ(n+1α),t[0,T],α(0,1].

Proof

For α ∈ (0, 1], we can writeDαf(t)=Dαn=0antn=f(0)Dα1+Dαn=1antn=1Γ(1α)0t1(tu)αdd

Properties of the Caputo-type convolution operator

We now analyze the commutative and associative properties of the Caputo-type convolution operator defined in (7). We remark that the proofs of the following results rely on the existence of the Laplace transform for the function f and thus we must add the condition that f is of exponential order. On the other hand, the hypothesis of being analytic is not necessary, in this general setting.

Theorem 4

Let Dν be the operator defined in Eq.(7) and let f:[0,T]R be in AC1[0, T] and such that |f(t)| < Cekt for

CRediT authorship contribution statement

Luisa Beghin: Conceptualization, Methodology, Writing - original draft, Writing - review & editing. Michele Caputo: Conceptualization, Methodology, Writing - original draft, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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