Page 101 - IJOCTA-15-1

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

This definition of robust stability constitutes a must satisfy:
generalization of the concept of stability under α
κ 2 (1 − α)β 2 cf D y j + ˙y j
constant-action perturbations used for systems j 0 t
2
of ordinary differential equations, and which was + κ 1 β y j = v j (t), y j (0) = 0. (8)
j
first proposed by Duboshin and Malkin; see e.g. 26
Here, v j ∈ V δ j is an external perturbation.
This concept of robust stability has been es-
The set of solutions y j obtained by choosing an
tablished in another partial differential equation
model; see. 34 external perturbation v j ∈ V δ j on the right-hand
. This set is non-
side of (8) is denoted by Y δ j
4. Application of the Fourier method empty, since if we choose the null external pertur-
bation ¯v j ≡ 0, then we obtain the trivial solution
If we use the method of separation of variables ¯ y j ≡ 0. The solution ¯y j is called unperturbed. It
to find the solution of (1)–(3), then we can de- follows that Y δ j ⊂ Y.
termine a sequence of eigenvalues {λ j } j∈N such
We now determine other elements of the set Y δ j
2
π
that λ j = ( j) and a sequence of eigenfunctions 31
R using the method described in.
{X j } j∈N such that X j (x) = sin(β j x), x ∈ [0, R],
1/2 It is well known that, according to the concept
where β j = λ j . Since {X j } j∈N is a basis of of the Caputo–Fabrizio fractional derivative, af-
2
L (0, R), it follows that the solutions of (1)–(3) ter integrating by parts we obtain the following
admit the representation given in (5). equivalent representation
It is well known that the method to justify the ex- Z t µ 2
cf α µ α α −µ α(t−η)
istence of solutions for the initial-boundary value 0 D y j (t) = α y j (t) − α e y j (η)dη.
t
problem (1)–(3) that depends on a heat source 0
R t µ αη
δ
v ∈ V , consists in verifying the uniform conver- If we choose z j (t) = 0 e y j (η)dη, then we ob-
Ω
gence on Ω of the series on the right-hand side tain ˙z j (t) = e µ αt y j (t), since y j is continuous be-
1
of (5), and the uniform convergence on Ω of four cause y j ∈ H (0, T). This allows us to obtain the
other series defining the functions following expression
2
2
∂y ∂y ∂ y α ∂ y cf α 1 −µ αt
, , , cf 0 D t . (7) 0 D y j (t) = e ( ˙z j (t) − µ α z j (t)) .
t
∂t ∂x ∂x 2 ∂x 2 1 − α
In the following result we justify the application We observe that ˙y j (t) = e −µ αt (¨z j (t) − µ α ˙z j (t)).
of the method of separation of variables. By substituting in equation (8) we get
P ∞
Theorem 1. Let v(x, t) = j=1 sin(β j x)v j (t) ¨ z j (t) + 2a j ˙z j (t) − b j z j (t)
for µ αt
a heat source for (1)–(3), where v j ∈ V δ j = e v j (t), z j (0) = ˙z j (0) = 0,
all j ∈ N. Then the function y expressed in
2
2
1
where a j = ((κ 1 +κ 2 )β −µ α ) and b j = κ 2 µ α β .
the form (5) is a unique solution to the initial- 2 j j
Since β j → ∞ as j → ∞, then we can assume
boundary value problem (1)–(3) provided that the
P ∞ that a j > 0 for all j ∈ N. Therefore, if we denote
series j=1 j is convergent, that is, y ∈ Y Ω .
δ
2
by c j = (a + b j ) 1/2 , then the solution of (10) is
j
We present the proof of Theorem 1 in Appen- expressed as
dix A. Z t 1
z j (t) = e µ αη−a j (t−η) sinh(c j (t − η))v j (η)dη.
We now obtain the main properties of the Fourier
0 c j
coefficients. We choose an arbitrary external per- −µ αt
δ
turbation v ∈ V and assume that y is the solu- In this case, since y j (t) = ˙z j (t)e , then the
Ω
tion admitting the representation given in (5). If solution of the (8) can be expressed as
we substitute the functions y and v in (1) and Z t
y j (t) = φ j (η)v j (t − η)dη, (9)
equate the Fourier coefficients with respect to
0
each eigenfunction, then we obtain the following
where
equality:
e
∞ φ j (η) = 1 − d 2 1/2 −(µ α+a j )η ·
j

X α
sin(β j x) κ 2 (1 − α)β 2 cf 0 D y j (t)+ · cosh (c j η − arctanh(d j )) ,
t
j
j=1 a j
∞ with d j = . We observe that φ j (η) > 0 for
X c j
2
˙ y j (t) + κ 1 β y j (t) = sin(β j x)v j (t).
j η ∈ (0, t) since d j ∈ (−1, 1). We conclude that
j=1 on the right-hand side of
for every choice v j ∈ V δ j
If in this equality we compare the terms of each (8), the solution y j defined in (9) associated with
series, we conclude that the functions y j and v j this choice is an element of Y δ j .
95