Environmental Engineering Reference
In-Depth Information
6.8 Cauchy Problems
In this section we develop the solution structure theorem for Cauchy problems of
dual-phase-lagging heat-conduction equations. We also briefly discuss the methods
of solving Cauchy problems without going into the details.
In this section we let
be
R
1
,
R
2
or
R
3
.
stands for one-, two- or three-
dimensional Laplace operator.
M
represents a point in
R
1
,
R
2
or
R
3
Ω
Δ
and
u
(
M
,
t
)
is the temperature at point
M
and time instant
t
.
Theorem 1
.Let
u
=
W
ψ
(
M
,
t
)
be the solution of
⎧
⎨
u
t
τ
0
+
B
2
∂
∂
A
2
u
tt
=
Δ
u
+
t
Δ
u
,
Ω
×
(
0
,
+
∞
)
,
(6.156)
⎩
u
(
M
,
0
)=
0
,
u
t
(
M
,
0
)=
ψ
(
M
)
.
The solution of
⎧
⎨
⎩
u
t
τ
0
+
B
2
∂
∂
A
2
u
tt
=
Δ
u
+
t
Δ
u
,
Ω
×
(
0
,
+
∞
)
,
(6.157)
u
(
M
,
0
)=
ϕ
(
M
)
,
u
t
(
M
,
0
)=
0
is
1
W
ϕ
(
τ
0
+
∂
B
2
u
=
t
−
Δ
M
,
t
)
.
(6.158)
∂
Proof.
By its definition, the
W
ϕ
(
M
,
t
)
satisfies
⎧
⎨
2
W
ϕ
∂
τ
0
∂
W
ϕ
∂
+
∂
1
B
2
∂
∂
A
2
−
W
ϕ
−
W
ϕ
=
,
Δ
t
Δ
0
(6.159a)
t
t
2
t
=
0
=
ϕ
(
⎩
∂
W
ϕ
∂
W
ϕ
(
,
)=
,
)
.
M
0
0
M
(6.159b)
t
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