On the sub–diffusion fractional initial value problem with time variable order

Definition. 1

Given f ∈ L¹([0, +∞)), we define

where k:[0, +∞) → ℝ is given in terms of the Laplace transform as

where 𝓓′ is a certain complex domain such that Ca+⊂D′ , Γ is a complex path according to the Bronwich formula for the inversion of the Laplace transform whose precise choice is discussed below, and k(t) vanishes for t < 0. Note that in the case of α(t)=α=constant, the definition (2.1) matches the very well known definition of fractional integral of order α in the sense of Riemann–Liouville, this is why this definition is often known as the fractional integral of variable order in the sense of Riemann–Liouville.

Now we are in a position to define the fractional derivative of variable order α(t) (FDVO). To this end there are typically two different ways, both in terms of the definition of the fractional integral (2.1)–(2.2).

In spite of the regularity does not a matter here, to be precise the functions involved in the following results are required to belong to a convenient Sobolev space. In fact denote

Now, given f ∈ W^1,1([0, +∞)), and α:[0, +∞) → (0, 1):

1st.-We define the FDVO as

Note that α(t) − 1<0, therefore, inside the brackets, we apply definition (2.1)–(2.2) with order 1−α(t) instead of α(t), to have

where f′ stands for the first time derivative of f, and k(t) is given by

and Γ is a suitable path in the complex plane. The choice of Γ is precisely discussed later. The equality (2.4) has been already noticed for α(t)=α=constant (e.g. formula (2.70), pp. 35 in [10], among many others references). In order to keep the notation as simple as possible and while not confusing, we denote again the Laplace transform of the convolution kernel by K(z).

2nd.-Alternatively one may define the FDVO, for 0<α(t) < 1 for f differentiable, as

where k is defined by (2.5).

Several comments related to the previous proposed definition for the FDVO can be made.

First of all note that, unlike what happens with the definition (2.3), with the definition (2.6) if f(t)=constant, then the fractional derivative turns out to be zero. Moreover, if f (0)=0, then both definitions coincide. Although both definitions show some differences, the treatment turns out to be quite similar for both, as we will show below.

Moreover, based on a fact that has already been observed in [6,17, 18], we can assert that (2.2) and (2.5) are meaningful under very weak restriction on α˜ . In particular, let K(z) be a complex–valued or operator–valued function, analytic outside of a complex sector of angle θ, θ < π/2, denoted by

and such that there exist M > 0 and β ∈ ℝ, satisfying

As explained in [6,17, 18], K(z) stands for the Laplace transform of a distribution k(t) in the real line, so that k(t)=0, for t < 0, whose singular support is empty, or merely concentrated at t=0 (e.g. if β<0), and which is analytic for t > 0. In that case the inverse (distributional) Laplace transform, i.e. k(t), admits the integral representation

for a convenient path Γ in the complex plane running outside the sector of analyticity S_θ, e.g. one running parallel to the boundary of S_θ, and with increasing imaginary part. We will precisely state below one of these paths, according to our interest for the proofs.

In this context, let us recall the following lemma which has already been proven in [6], Lemma 2.1; it provides useful bounds for k(t).

Lemma. 1

Let K(z) be a analytic function outside a complex sector S_θ, 0<θ<π/2, satisfying (2.8) for certain M > 0 and β ∈ ℝ.

Then there exists C > 0 depending solely on M, β and θ (but not on t) such that the inverse Laplace transform of K(z), k(t), is bounded as follows

Observe that bound (2.9) suggests the lack of regularity of k(t) at t=0, if β<1. Moreover, note that k(t) is locally integrable is β>0.

A more general case might be considered if instead of the sector S_θ of analyticity, one considers a shifted sector a+S_θ={a+z: z ∈ S_θ}, for a=0. However, since no noticeable additional difficulties arise in that case, and for the sake of the simplicity of the presentation of results, we merely consider the case a=0, i.e. S_θ.

Before going ahead in the paper, we make some assumptions which will be required when studying the solutions of the sub–diffusion initial value problems of variable order α(t) in Section 4. These assumptions are related to sectoriality angle θ, which will also affect the linear operators considered there as it will be discussed in that section, and the fractional order α(t), and more particularly its Laplace transform α˜(z) . In fact, assume the following:

[H0]. The Laplace transform of α(t), α˜(z):=L(α)(z) , exists outside a sector S_θ, for 0<θ<π/2.

[H1]. Denote

There exist R > 0 large enough, and 0 < m₁, m₂, C₁ < 1, satisfying

such that g_I(z) is bounded, for ∣z∣ ⩽ R,

and

[H2]. The analyticity angle θ satisfies

Note that, by hypothesis [H1], we have that 2 − m₂/(1 − C₁)⩽1, so this hypothesis is meaningful.

On the other hand note that, under Hypotheses [H0–H2] we have for (2.5),

Therefore these inequalities suggest a lack of regularity of k(t) as t → 0⁺, and that the function k(t) is locally integrable in ℝ⁺. The same can be said for (2.2) related to the regularity and integrability since

Finally, before ending this section we provide some examples of functions α(t) which satisfy [H0–H1],

Theorem. 1

Let f, g be two functions belonging to W^1,1([0, +∞)). According to Definition (2.3), there holds

and according to Definition (2.6)

where

Proof Consider first the Definition (2.3) On the one hand,

on the other hand

Therefore,

Moreover by (2.4) we have

and the proof of the equality (2.3) ends.

For the Definition (2.6), we merely recall that

from which (3.2) straightforwardly follows, and the proof of the theorem ends. □

Here we have to highlight several facts.

1. The terms ∂^α(t) (h(t, ·))(t) − k(t)f(t)g(t) in (3.2) and ∂^α(t) (h(t, ·))(t) in (3.2) stand for the difference (in the form of an additional term) with respect to the classical product rule, i.e. the case α(t) ≡ 1. Moreover, the formulation (3.1) and (3.2) of the fractional product rules according to the definitions (2.3) and (2.6), respectively, may represent the simplest representations if compared to the ones achieved by considering other definitions of FDVO.

2. It is straightforward to check that equality (3.1) (analogously (3.2)) is satisfied in the particular case of α(t) ≡ α, 0<α<1. It can be easily illustrated by taking for example f(t)=t^p, g(t)=t^q, p, q ∈ \cZ⁺, and any 0<α<1. In particular, in the constant case α(t) ≡ α, 0<α<1, the product rule achieved here perfectly matches with the one derived in Eq. 3.12, [2],

   (3.3) ∂α(fg)(t)=f(t)∂αg(t)+g(t)∂αf(t)−αΓ(1−α)∫0t(f(s)−f(t))(g(s)−g(t))(t−s)α+1 ds−t−αf(t)g(t)Γ(1−α),

   (∂^α stands for the fractional derivative in the sense of Riemann–Liouville of order α>0).

3. Another relevant fact is that both equalities (3.1) and (3.2)) are coherent with the classical product rule, namely the classical product rule turns out to be the limit case as α(t) tends to the constant function α(t) ≡ 1. In that case the corresponding limit of the convolution kernel in the sense of distributions, and always according to definition (2.5), turns out to be k(t)=δ₀(t) where δ₀(t) stands for Dirac’s delta distribution with density concentrated at t=0. In that manner, since k(t)=0, for t < 0, then k(t)f(t)g(t)=0, for t < 0. Moreover, according to the definition (2.3)

   ∂α(t)(h(t,⋅))(t)=ddt∂α(t)−1h(t,⋅)(t),

   and in the limit

   ∂α(t)−1h(t,⋅)(t)=∫0tk(t−s)h(t,s)ds=∫0tδ0(t−s)h(t,s)ds=h(t,t)=0,t>0.

   Therefore it comes ∂^α(t) (h(t, ·))(t)=0. Similarly, according to the definition (2.6)

   ∂α(t)(h(t,⋅))(t)=∂α(t)−1ddsh(t,⋅)(t),

   and since in the limit

   ∂α(t)(h(t,⋅))(t)=∫0tδ0(t−s)ddsh(t,s)ds=ddsh(t,s)s=t=0,t>0,

   both equalities (3.1) and (3.2) boil into the usual Leibniz rule

   (f(t)g(t))′=f(t)g′(t)+f′(t)g(t),t>0,

   as conjectured above.

4. Finally, in spite of the inequality

   ∂α(f2(t))≤2f(t)∂α(f(t)),t>0,(which impliesf(t)∂αf(t)≥0)

   has been proven in [2] from the equality (3.3) for the case of fractional derivatives of constant order α>0, in the case of the derivatives with time variable order considered here, the inequality

   (3.4) ∂α(t)(f2(t))≤2f(t)∂α(t)(f(t)),

   is not satisfied in general. Only if we can guarantee that ∂^α(t) (h(t, ·))(t) − k(t)f(t)g(t) ⩽ 0, (or simply ∂^α(t) (h(t, ·))(t)=0 depending on the definition one adopts) for t > 0, the inequality (3.4) could become valid, although this is not obvious at all for a so general class of functions α(t) we are here considering, i.e., α(t) under the hypotheses made in Section 2.

From Theorem 1 several integration–by–parts formulas for FDVO can be straightforwardly derived. We show two of such formulas in the following corollaries.

Corollary. 2

Let f, g be two functions belonging to W^1,1([0, +∞)). According to the Definition (2.3) we have the following integration–by–parts formula for FDVO

and according to the Definition (2.6)

where k(t) is in both cases the kernel defined in (2.5).

Proof Let φ be a function belonging to L¹([0, +∞)).

First of all, we show an equality that will be useful for the proof. In fact if we consider the Definition (2.3), then applying the Laplace transform we straightforwardly have that

Note that this equality means that ∂^−α(t) is the inverse operator of ∂^α(t). Moreover if α(t) ≡ 1 and φ (0)=0, then this equality coincides with the classical one.

In the same manner if one takes the definition (2.6) there is a subtle difference, in fact applying again the Laplace transform we have

This formula perfectly matches with the classical one.

Now consider Definition (2.3). Thanks to Theorem 1 and equality (3.7), we have

and the equality (3.5) follows (recall that h(t, t)=0, for t=0).

In the same fashion (3.6) is obtained. □

Notice that, by the same arguments as in Theorem 1, as α(t) tends to the constant function α(t) ≡ 1, the kernel k(t) tends in the distributional sense to Dirac’s delta distribution δ₀(t), and therefore the term ∂^−α(t)\Big(k(t)f(t)g(t)\Big) tends to δ₀(t)f (0)g (0)=0 therefore the equality stated in Corollary 2 reads

Therefore, if f (0)g (0)=0 (in particular if f (0)=g (0)=0), then (3.9) is coherent with the classical integration–by–parts formula.

However according to the Definition (2.6) although f (0)=g (0)=0, the formulation is not coherent with classical case since h(t, 0)≠ 0.

An alternative form of the integration–by–parts formula for FDVO in given in the following corollary. This time the corollary is stated only for the Definition (2.3) since we did not find the related formula for the Definition (2.6) coherent with the classical formula i.e. for α(t) ≡ 1.

Proof By Theorem 1, one has

If one considers the Definition (2.3), then

and since

the equality (3.10) straightforwardly follows. □

The equality (3.10) turns out to be coherent with the classical integration–by–parts formula as α(t) tends to the function 1 since as above k(t) tends in the distributional sense to Dirac’s delta distribution δ₀(t) and therefore,

and

In that case, that is if α(t) tends to the constant function α(t) ≡ 1, then we have again the classical integration–by–parts formula