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On a robust stability criterion in the Cattaneo–Hristov diffusion equation

p
expressed by the Caputo–Fabrizio fractional de- and by L (0, T) we denote the space of func-
rivative; see. 17,18 Other heat diffusion equations tions whose p-th power of its absolute value is
with non-singular kernels are presented in. 19 Lebesgue-integrable on (0, T). We consider the
The Cattaneo–Hristov diffusion equation has Sobolev space W 1,p (0, T) and use the particular
1
been analyzed in different contexts. In 20 a frac- case: H (0, T) = W 1,2 (0, T). The validity of
1
tional diffusion equation is solved and extended by the following inclusion is well known C [0, T] ⊂
1
employing the concept of derivative with a non- H (0, T); see e.g. 27 We are particularly interested
local and non-singular kernel using the Atangana– in functions belonging to the following set
Baleanu fractional derivative. In 21 analytical so- Y = {y ∈ H (0, T) : y(0) = 0}.
1
lutions of the Cattaneo–Hristov diffusion equation
with particular boundary conditions in one- and The Caputo fractional derivative of a function
28
1
two-dimensional spaces are proposed. In 22 the y ∈ H (0, T) of order α ∈ (0, 1) is defined by :
optimal control problem of the Cattaneo–Hristov 1 Z t
α
c D y(t) = (t − η) −α ˙ y(η)dη;
diffusion equation is formulated for a certain rigid 0 t Γ(1 − α) 0
thermal conductor whose length is finite. In 23 the 8
Caputo and Fabrizio proposed in a fractional de-
behavior of the Cattaneo–Hristov equation mov- 1
rivative (in Caputo sense) for y ∈ H (0, T) of or-
ing in an interval under the influence of initial
der α ∈ (0, 1) in the form
conditions and in the presence of a specific source
is investigated. In 24 the Cattaneo–Hristov dif- cf D y(t) = B(α) Z t e −µ α(t−η) ˙ y(η)dη,
α
fusion equation is investigated in one- and two- 0 t 1 − α 0
dimensional spaces. Finally, in 25 the Cattaneo– where µ α = α and B(α) is some normaliza-
1−α
Hristov diffusion model defined in a half-plane tion function satisfying the following conditions
with two different boundary conditions is ana- B(0) = B(1) = 1. Losada and Nieto propose
lyzed and solved. an improvement to the definition of the Caputo–
1
In this paper, we assume that the Cattaneo– Fabrizio fractional derivative for y ∈ H (0, T) and
Hristov diffusion model has heat sources that ad- α ∈ (0, 1) described by
mit representations via Fourier series and whose 1 Z t
α
coefficients belong to a particular class of bounded cf D y(t) = e −µ α(t−η) ˙ y(η)dη,
0 t 1 − α
and continuous functions. Our main aim in this 0
paper is to obtain a new robust stability crite- see. 29 We consider this last expression as the
rion for the Cattaneo–Hristov diffusion model by Caputo–Fabrizio fractional derivative. Further-
employing a generalization of a definition of sta- more, we observe that if y ∈ Y, then
bility under constant-action perturbations due to lim cf D y(t) = y(t), lim cf D y(t) = ˙y(t).
α
α
Duboshin and Malkin, and which was often used α→0 0 t α→1 0 t
for systems of ordinary differential equations; see On the other hand, the Caputo–Fabrizio frac-
e.g. 26 tional integral operator that is associated with the
1
The structure of this document is organized into operator cf D α is defined for y ∈ H (0, T) as
t
0
the following sections. Section 2 presents a review Z t
cf α
I y(t) = (1 − α)y(t) + α
of the main definitions and results that are used. 0 t y(η)dη.
0
In Section 3 the Cattaneo–Hristov diffusion equa-
I
tion is presented and the robust stability problem The operators cf 0 D α and cf α satisfy some well-
0 t
t
is posed. Section 4 justifies the use of the Fourier known properties that can be consulted in. 29–33
method and presents the main properties of the For example, from the definition we can see that
Fourier coefficients. In Section 5 we establish our if y ≡ c with c a constant, then the equality
cf α
main result: the obtaining of the robust stabil- 0 D y ≡ 0 is satisfied. Additionally, it is known
t
ity criterion. Finally, in Section 6 we present the that for α ∈ (0, 1) and y ∈ Y, the following
8
conclusions of the paper. equalities are satisfied:
α
cf α cf D y(t) = y(t), cf D α cf α
I
I y(t) = y(t).
0 t 0 t 0 t 0 t
2. Preliminaries
3. Statement of the main result
We present the definition and the properties of the
Caputo–Fabrizio fractional derivative that will be Before stating our main result, we introduce the
used in what follows. Let p ∈ R with 1 ≤ p ≤ ∞, notation that will be used. For each δ > 0, we
k ∈ N ∪ {∞} and T > 0. We use the notation denote by D δ the set of sequences {δ j } j∈N of pos-
k
j=1 j = δ, and for
C [0, T] to denote the space of real-valued func- itive real numbers satisfying P ∞ δ
tions whose k-derivative is continuous in [0, T] each {δ j } j∈N ∈ D δ , we consider the sequence of
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