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In-Depth Information
t
t 2 t
2
1
τ 0
∂
∂
τ + ∂
=
W f τ (
M
,
t
− τ )
d
W f τ (
M
,
t
− τ )
d
τ
t
∂
0
0
t
t
B 2 ∂
∂
A 2
−
Δ
W f τ (
M
,
t
− τ )
d
τ −
t Δ
W f τ (
M
,
t
− τ )
d
τ
0
0
t
t
− τ ) τ = t
2 W f τ
∂
∂
W f τ
∂
∂
1
τ 0
=
d
τ +
W f τ (
M
,
t
+
d
τ
t
t 2
0
0
τ = t −
A 2 t
t
0 Δ
+ ∂
W f τ
∂
B 2 ∂
∂
0 Δ
W f τ d
τ −
W f τ d
τ
t
t
t
t
A 2 t
2 W f τ
∂
1
τ
∂
W f τ
∂
∂
=
d
τ +
d
τ +
f
(
M
,
t
) −
0 Δ
W f τ d
τ
t 2
t
0
0
0
B 2 t
0
W f τ τ = t
∂
∂
−
t Δ
W f τ d
τ + Δ
1
d
t
2 W f τ
∂
τ 0 ∂
W f τ
∂
+ ∂
B 2 ∂
∂
A 2
=
−
W f τ −
τ +
(
,
)
Δ
t Δ
W f τ
f
M
t
t
t 2
0
t
=
τ +
(
,
)=
(
,
) .
0d
f
M
t
f
M
t
0
Satisfaction of Boundary Conditions
By substituting Eq. (6.10) into the boundary conditions of PDS (6.9) and applying
the boundary conditions of PDS (6.11), we have
L u
∂Ω =
∂Ω
L t
0
t
, ∂
u
∂
∂
W f τ (
M
,
t
− τ )
d
τ ,
W f τ (
M
,
t
− τ )
d
τ
∂
n
n
0
L t
0
∂Ω
t
∂
∂
=
W f τ d
τ ,
n W f τ d
τ
0
L W f τ ,
∂Ω
t
∂
∂
=
n W f τ
d
τ =
0
.
0
Satisfaction of Initial Conditions
It is straightforward to show that the u in Eq. (6.10) satisfies the initial condition
u
(
M
,
0
)=
0. Also,
t
)= ∂
∂
u t (
M
,
t
W f τ (
M
,
t
− τ )
d
τ
t
0
t
− τ ) τ = t .
∂
W f τ
∂
=
d
τ +
W f τ (
M
,
t
t
0
 
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