Biomedical Engineering Reference
In-Depth Information
b
g
2
(
x
)
dx
=
0
⇒
g
(
x
)
=
0on
a
≤
x
≤
b
.
(14.7)
a
This follows immediately from the observation that the square of a function
g
(
x
)
is always greater than or equal to zero for any value of
x
,i.e.
g
2
(
x
)
≥
0, such that
the integral of
g
2
(
x
) over the domain
a
≤
x
≤
b
must be greater than or equal to
zero, i.e.
b
g
2
(
x
)
dx
≥
0,
(14.8)
a
and can only be
equal
to zero if
g
(
x
)
b
.
Effectively, the method of weighted residuals transforms the requirement that a
function, say
g
(
x
), must be equal to zero on a given domain at an
infinite
number
of points into a
single evaluation
of the integral, that must be equal to zero.
Using the method of weighted residuals, the differential equation, Eq. (
14.1
), is
transformed into an integral equation:
b
=
0 for all
a
≤
x
≤
w
d
dx
c
du
dx
+
f
dx
=
0,
(14.9)
a
which should hold for all weighting functions
w
(
x
). The term between the square
brackets contains second order derivatives
d
2
u
dx
2
of the function
u
. As has been
outlined in the introduction, approximate solutions of
u
are sought by defining an
interpolation of
u
on the domain of interest and transforming the integral equation
into a discrete set of linear equations. Defining interpolation functions that are
both second-order differentiable and still integrable is far from straightforward,
in particular in the multi-dimensional case on arbitrarily shaped domains. Fortu-
nately, the second-order derivatives can be removed by means of an integration by
parts:
/
w
c
du
dx
b
b
b
a
−
dw
dx
c
du
dx
dx
+
wf dx
=
0.
(14.10)
a
a
This introduces the boundary terms:
w
c
du
dx
a
=−
w
(
a
)
c
du
x
=
a
+
w
(
b
)
c
du
x
=
b
b
.
(14.11)
dx
dx
At the boundary either
u
is prescribed (i.e. the essential boundary condition at
x
=
a
) or the derivative
cdu
/
dx
is prescribed (i.e. the natural boundary condition
at
x
b
). Along the boundary where
u
is prescribed the corresponding flux, say
p
u
, with
=
x
=
a
p
u
=
c
du
dx
,
(14.12)
