Civil Engineering Reference
In-Depth Information
x
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
0.2
0.4
y
(
x
)
0.6
0.8
Exact
1 term
2 terms
1.0
1.2
Figure 5.2
Solutions to Example 5.2.
so the two-term approximate solution is
19
6
5
3
x
2
(
x
−
1)
=
5
3
x
3
3
2
x
2
19
6
y
*
=
x
(
x
−
1)
+
+
−
x
For comparison, the exact, one-term and two-term solutions are plotted in Figure 5.2. The
differences in the exact and two-term solutions are barely discernible.
EXAMPLE 5.3
Use Galerkin's method of weighted residuals to obtain a one-term approximation to the
solution of the differential equation
d
2
y
d
x
2
+
y
=
4
x
0
≤
x
≤
1
with boundary conditions
y
(0)
=
0,
y
(1)
=
1
.
■
Solution
Here the boundary conditions are not homogeneous, so a modification is required. Unlike
the case of homogeneous boundary conditions, it is not possible to construct a trial solu-
tion of the form
c
1
N
1
(
x
)
that satisfies both stated boundary conditions. Instead, we as-
sume a trial solution as
+
f
(
x
)
where
N
1
(
x
)
satisfies the homogeneous boundary conditions and
f
(
x
) is chosen to
satisfy the nonhomogeneous condition. (Note that, if both boundary conditions were
nonhomogeneous, two such functions would be included.) One such solution is
y
*
y
*
=
c
1
N
1
(
x
)
=
c
1
x
(
x
−
1)
+
x
which satisfies
y
(0)
=
0 and
y
(1)
=
1 identically.
