Page 100 - IJOCTA-15-1

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R. Temoltzi-Avila, J. Temoltzi-Avila / IJOCTA, Vol.15, No.1, pp.92-102 (2025)
δ
}
sets {V δ j j∈N that are defined by where v ∈ V . In the equation (1) we add the
Ω
following initial and boundary conditions:
= {v ∈ C[0, T] : |v(t)| ≤ δ j , t ∈ [0, T]} ,
V δ j
y(x, 0) = 0, x ∈ [0, R], (2)
We define the set
∞ y(0, t) = y(R, t) = 0, t ∈ [0, T], (3)
X
δ
V = sin(β j x)v j (t) : v j ∈ V δ j , As an initial case, we assume that the thermal
Ω
j=1
conductivity constant κ 1 > 0 and the elastic con-
{δ j } j∈N ∈ D δ , (x, t) ∈ Ω , ductivity constant κ 2 ≥ 0 satisfy the inequality
R
where {β j } j∈N is defined by β j = π j for j ∈ N ≤ κ 1 + κ 2 . (4)
R 2π
and Ω = [0, R] × [0, T] with R ∈ (0, π) and T > 0
finite. There are other cases of interest that are deter-
mined by the thermal conductivity and elastic
conductivity constants. Such cases are discussed
δ
We introduce the following norm on V : later in Remark 1.
Ω
∥v∥ δ = sup |v(x, t)| . It is known that the solution of the initial-
V
Ω
(x,t)∈Ω boundary value problem (1)–(3) depending on a
δ
δ
It follows that if v ∈ V , then ∥v∥ δ ≤ δ. Simi- heat source v ∈ V admits the following expres-
Ω
V
Ω
Ω
: sion
∞
larly, we introduce the following norm on V δ j
X
= sup |v j (t)| . y(x, t) = sin(β j x)y j (t), (x, t) ∈ Ω, (5)
∥v j ∥ V δ j
t∈[0,T] j=1
≤ δ j .
It follows that if v j ∈ V δ j , then ∥v j ∥ V δ j where y j : [0, T] → R are functions that need to
be determined. It follows from the above that
each of these functions must satisfy the equality
The Cattaneo–Hristov diffusion equation is de- y j (0) = 0 due to the initial condition in (2).
fined by 13,14 :
We note that if on the right-hand side of (1) we
2
2
∂y α ∂ y ∂ y
− κ 2 (1 − α) cf 0 D t = κ 1 , consider the zero heat source ¯v ≡ 0, which is an
∂t ∂x 2 ∂x 2 element of V , then the solution of the initial-
δ
Ω
where κ 1 is the effective heat diffusivity and κ 2 boundary value problem (1)–(3) corresponds to
is the elastic heat diffusivity, which are deter- the trivial solution ¯y ≡ 0. This solution is called
mined by the relations ρC p κ 1 = k 1 and ρC p κ 2 = unperturbed. However, if on the right-hand side of
k 2 , where k 1 is the thermal conductivity, k 2 de- (1) we choose other non-zero heat sources v ∈ V ,
δ
Ω
notes the elastic conductivity, ρ is the density then it is necessary to determine the type of so-
of particles and C p is the specific heat of par- lutions of (1)–(3) that are associated with such
ticles; see. 13 The Cattaneo–Hristov model uses choices. The set of such solutions will be denoted
the Caputo–Fabrizio fractional derivative to ex- by Y Ω . It is clear that ¯y ∈ Y Ω . In the set Y Ω we
press the Cattaneo constitutive equation with the introduce the following norm for its elements:
Jeffrey fading memory, resulting in a heat con-

∂y

∂y
duction equation with a relaxation term, which ∥y∥ Y Ω = max ∥y∥ Ω ,

,

, (6)

∂x

∂t
allows to see an interpretation of the physical Ω Ω
meaning of the Caputo–Fabrizio time-fractional where ∥h∥ Ω = sup (x,t)∈Ω |h(x, t)|. The introduc-
derivative; see. 14–16 In this sense, this equation tion of this norm in Y Ω allows us to compare the
describes transient heat conduction with a damp- unperturbed solution ¯y ≡ 0 with any other solu-
ing term expressed by the Caputo–Fabrizio frac- tion y ∈ Y Ω that is obtained by choosing a heat
tional derivative. The procedure to obtain this source v ∈ V δ Ω using the following real number
model suggests how other constitutive equations ∥y − ¯y∥ Y Ω = ∥y∥ Y Ω . This observation allows us to
could be modified with non-singular fading mem- introduce the following definition of robust stabil-
ories; see. 17–19 It is observed that for α = 1 we ity for the set Y Ω .
obtain the classical diffusion equation.
Definition 1. The unperturbed solution ¯y ≡ 0 of
the Cattaneo–Hristov diffusion model (1)–(3) in
In order to analyze the effect that external sources the absence of heat sources is robustly stable with
δ
have on the Cattaneo–Hristov diffusion model, we respect to heat sources v ∈ V , if for all ε > 0
Ω
consider the following diffusion equation: there exists η(ε) > 0 such that under the con-
2
2
V
∂y α ∂ y ∂ y dition ∥v∥ δ ≤ δ < η(ε), any other solution of
Ω
− κ 2 (1 − α) cf 0 D t = κ 1 + v(x, t), (1)
∂t ∂x 2 ∂x 2 (1)–(3) satisfies ∥y∥ Y Ω < ε.
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