Biomedical Engineering Reference
In-Depth Information
14.2 Consider the differential equation
c
du
dx
du
dx
+
d
dx
u
+
+
f
=
0,
b
.
Derive the weak form of this differential equation, and explain what steps
are taken.
14.3 Let
f
(
x
) be a function on the domain 0
on the domain
a
≤
x
≤
1. Let
f
(
x
) be known at
n
points, denoted by
x
i
, homogeneously distributed on the above domain.
Hence the distance
≤
x
≤
x
between two subsequent points equals
1
n
−
1
.
A polynomial
f
h
(
x
)oforder
n
−
1 can be constructed through these points,
which generally will form an approximation of
f
(
x
):
x
=
a
n
−
1
x
n
−
1
.
f
h
(
x
)
=
a
0
+
a
1
x
+···+
(a)
Show that the coefficients of
a
i
can be found by solving
⎡
⎤
⎡
⎤
⎡
⎤
x
1
x
n
−
1
1
1
x
1
···
a
0
a
1
a
2
.
a
n
−
1
f
1
f
2
f
3
.
f
n
⎣
⎦
⎣
⎦
⎣
⎦
x
2
x
n
−
1
2
1
x
2
···
x
3
x
n
−
1
3
1
x
3
···
=
,
.
.
.
.
.
x
n
x
n
−
1
1
x
n
···
n
f
(
x
i
).
(b) Use this to find a polynomial approximation for different values of
n
to the function:
where
f
i
=
f
(
x
)
=
1for0
≤
x
≤
0.5,
f
(
x
)
=
0 for 0.5
<
x
≤
1.
Compare the results to those obtained using the weighted residuals
formulation in Exercise 14.
1
(c). Explain the differences.
14.4 Consider the domain
−
1
≤
x
≤
1. Assume that the function
u
is known
at
x
1
1, say
u
1
,
u
2
and
u
3
respectively. The
polynomial approximation of
u
, denoted by
u
h
is written as
=−
1,
x
2
=
0 and
x
3
=
u
h
=
a
0
+
a
1
x
+
a
2
x
2
.
(a) Determine the coefficients
a
0
,
a
1
and
a
2
to be expressed as a function
of
u
1
,
u
2
and
u
3
.
