Biomedical Engineering Reference
In-Depth Information
lines of constant ϕ
lines of constant F
(characteristics)
F
0
initial conditions
Γ
Γ
solution
(A, B)
(A, B)
Figure 4.14:
Illustration of a sample initial condition and the corresponding
solution.
with
φ
(
x
,
y
)
=
0
,F
(
x
,
y
)
=
F
0
(
x
,
y
)
, have a solution of the form
+
b
2
X
(
x
,
y
)
2
d
b
−
ea
√
a
2
F
(
x
,
y
)
=
+
Y
(
x
,
y
)
2
(4.46)
,
√
a
2
tan
−
1
Y
(
x
,
y
)
X
(
x
,
y
)
+
b
2
db
−
ea
φ
(
x
,
y
)
=
.
(4.47)
If db
−
ea
=
0
, where
b
a
X
(
x
,
y
)
=
√
a
2
+
b
2
(
x
−
A
)
−
√
a
2
+
b
2
(
y
−
B
)
,
(4.48)
a
b
Y
(
x
,
y
)
=
√
a
2
+
b
2
(
x
−
A
)
+
√
a
2
+
b
2
(
y
−
B
)
(4.49)
ec
+
fb
ae
−
bd
,B
=
af
+
cd
bd
−
ae
. The solution is valid in the set
R
2
and where A
=
\
L, where
L is an arbitrary line passing through the point
(
A
,
B
)
.
If db
−
ea
=
0
, then F
0
(
x
,
y
)
=
F
0
is constant on
, and the solution be-
comes
F
(
x
,
y
)
=
F
0
,
(4.50)
±
1
F
0
√
a
2
φ
(
x
,
y
)
=
+
b
2
(
ax
+
by
−
c
)
,
(4.51)
valid on all
R
2
.
Given an initial piece of the interface, the interface is approximated us-
ing a linear function, and also the speed,
F
, along the interface uses a linear
