Biomedical Engineering Reference
In-Depth Information
x
t
head given by the characteristic
c
which coalesces with the blood/no
blood front. To determine
S
∗
L
we select a point
P
on the characteristic line
=
S
∗
L
=
u
−
x
t
=
S
∗
L
=
c
on which the particle velocity and wave speed take on respectively the
values
u
and
c
. Then we connect this unknown point to the left data state via the left
generalized Riemnan invariant
u
u
−
+
4
c
=
constant
to obtain
u
L
+
4
c
L
=
u
+
4
c
.But
since
A
=
0 right at the blood/no blood front,
c
=
0 and therefore
S
∗
L
=
u
L
+
4
c
L
follows.
To find the solution inside the fan
W
Lfan
it is sufficient to select a point
P
=(
,
)
x
t
x
t
=
−
inside
W
Lfan
. A characteristic through the origin and
P
has slope
c
, with
u
and
c
two unknowns. Then, connecting the left data state to the unknown point
P
via
the left generalized Riemann invariant gives
u
L
u
4
c
. Solving these two
linear equations for the two unknowns
u
and
c
leads to the claimed solution (2.54)
and the proposition is thus proved.
+
4
c
L
=
u
+
Now we consider the mirror-image problem case.
Proposition 6.2.
The solution of the Riemann problem (2.24) with data
A
L
=
0,
u
L
=
0and
u
R
arbitrary has exact solution consisting of a single rarefaction
wave associated with the right eigenvalue
0;
A
R
>
λ
3
=
u
+
c
given as
⎧
⎨
/
≤
,
W
0
if
x
t
S
∗
R
W
RO
(
,
)=
W
Rfan
if
≤
S
∗
R
≤
x
/
tu
R
+
c
R
,
x
t
(2.55)
⎩
W
R
if
u
R
+
c
R
≤
x
/
t
,
where
W
R
is the right data state,
W
L
=
W
0
is the collapsed left data state and
S
∗
R
=
u
R
−
4
c
R
(2.56)
is the speed of the blood/no blood front separating the collapsed state (left) from the
uncollapsed new state (right)
W
Rfan
, with
⎧
⎨
⎩
u
R
−
1
5
4
x
t
u
=
4
c
R
+
,
W
Lfan
≡
(2.57)
5
t
1
x
=
−
+
+
.
c
u
R
4
c
R
Proof.
The proof follows analogous steps to those of Proposition 6.1 and details are
therefore omitted.
Now we consider the most interesting case in which no collapsed state is present
at the initial time
t
0 but a collapsed state arises as the result of the interaction of
the data states via the differential equations.
=
