Page 104 - IJOCTA-15-1

´
R. Temoltzi-Avila, J. Temoltzi-Avila / IJOCTA, Vol.15, No.1, pp.92-102 (2025)
(4) on the constants of thermal conductivity and Remark 1. There are some cases that extend our
elastic conductivity. assumption about inequality (4), namely:
This completes the proof. □
(a) one of the following cases is satisfied pro-
Now we can obtain the robust stability criterion vided that R ∈ (0, π):
in the sense of Definition 1. Actually, we observe (a 1 ) 0 < κ 1 + κ 2 ≤ R 2 ,
from Lemma 1 that for every ε > 0, the inequality 2 2π 2
∥y∥ < ε for all y ∈ Y Ω is satisfied whenever R R
Y Ω (a 2 ) 2 < κ 1 + κ 2 < ,
ε 2π 2π
δ < η(ε) = , R
2M (a 3 ) ≤ κ 1 + κ 2 ,
2π
that is, the inequality ∥y∥ < ε is satisfied for (b) equivalently, one of the following cases is
Y Ω
all y ∈ Y Ω , if for all {ε j } j∈N ∈ E ε , we can choose satisfied provided that R ∈ [π, ∞):
{δ j } j∈N ∈ D δ such that 0 < δ j < 1 ε j for each R
2M (b 1 ) 0 < κ 1 + κ 2 ≤ ,
j ∈ N, where M is defined in (17). It follows that 2π
R R 2
¯ y is robustly stable in the sense of Definition 1. (b 2 ) < κ 1 + κ 2 < ,
We get the following result. 2π 2 2π 2
R
(b 3 ) ≤ κ 1 + κ 2 .
Theorem 2. The unperturbed solution ¯y ≡ 0 2π 2
of the Cattaneo–Hristov diffusion equation is ro- Our case study in this paper corresponds to the
bustly stable in the sense of Definition 1, since subcase (b 3 ). The remaining cases can be ana-
for each ε > 0 it suffices to choose the parameter lyzed using the same method, considering the cor-
j=1 j and 0 < δ j <
η(ε) = 1 P ∞ ε 1 ε j , provided
2M 2M responding value of M.
that {ε j } j∈N ∈ E ε .
We show the validity of the results obtained in 6. Conclusion
the following example, which have been obtained
with GNU Octave 6.2.0. We have obtained a robust stability criterion for
the Cattaneo–Hristov diffusion model. The crite-
Example 1. In the Cattaneo–Hristov diffusion
rion has been obtained by generalizing the con-
model we assume κ 1 = 0.35, κ 2 = 0.15, R = 3
cept of stability under constant-acting perturba-
and T = 8. In this case M ≈ 1.42857. We pose
tions. Using this criterion, it has been possible
the problem of finding functions y ∈ Y Ω for which
to establish a priori a bound for the values of the
< ε = 1 is satisfied.
solutions of the Cattaneo–Hristov diffusion model
the inequality ∥y∥ Y Ω
We choose a sequence {ε p,j } j∈N ∈ E 1 defined by and its first partial derivatives with respect to the
ε p,j = ε(1 − p)p j−1 with p = 0.0005. For this longitudinal axis and with respect to time.
choice, we consider the sequence {δ j } j∈N defined
by δ j = 9 ε p,n .
20M Appendix A. Proof of Theorem 1
P N
We employ v k (x, t) = sin(β j x)v j,k (t) to rep-
j=1
resent two approximations of heat sources v 1 and Let {δ j } j∈N be a sequence of positive real num-
P ∞
v 2 in the Cattaneo–Hristov diffusion equation with bers and let v(x, t) = j=1 sin(β j x)v j (t) be a
v j,1 (t) = δ j and v j,2 (t) = δ j (1 − e −µ αt ), N ∈ N. heat source with v j ∈ V δ j . If this heat source
Then the corresponding approximations of the so- is substituted on the right-hand side of (1), then
lutions y 1 and y 2 , as well as their corresponding the solution of (1)–(3) is expressed as
first partial derivatives, are shown in Figure 2 in ∞
X
the case where α = 0.5. On the other hand, if we y(x, t) = sin(β j x)y j (t),
1
set x 0 = R, then we obtain the approximations j=1
3
of the functions y 1 (x 0 , ·) and y 2 (x 0 , ·), as well where the Fourier coefficients y j are described by
as their corresponding first partial derivatives,
Z t
as shown in Figure 3 assuming that α = 0.10, y j (t) = φ j (η)v j (t − η)dη,
α = 0.25 and α = 1.00. 0
We observe that with
q
≲ 0.762810. 2 −(µ α+a j )η
∥y 1 ∥ Y Ω ≲ 0.792972, ∥y 2 ∥ Y Ω φ j (η) = 1 − d e ·
j
We conclude this section with the following obser- · cosh(c j η − arctanh(d j )),
vation that justifies our choice of the inequality 1
2
2
where a j = ((κ 1 + κ 2 )β − µ α ), b j = κ 2 µ α β ,
satisfied by the coefficients of thermal conductiv- 2 j j
2
ity and elastic conductivity. c j = (a + b j ) 1/2 , and d j = a j . We observe that
j
c j
98