Civil Engineering Reference
In-Depth Information
PROBLEMS
5.1
Verify the integration and subsequent determination of
c
1
in Example 5.1.
5.2
Using the procedure discussed in Example 5.4, determine three trial functions for
the problem of Example 5.1.
5.3
It has been stated that the trial functions used in the method of weighted residuals
generally satisfy the physics of the problem described by the differential equation
to be solved. Why does the trial function assumed in Example 5.3 not satisfy the
physics of the problem?
5.4
For each of the following differential equations and stated boundary conditions,
obtain a one-term solution using Galerkin's method of weighted residuals and the
specified trial function. In each case, compare the one-term solution to the exact
solution.
a.
d
2
y
d
x
2
+
y
=
2
x
0
≤
x
≤
1
y
(0)
=
0
y
(1)
=
0
N
1
(
x
)
=
x
(1
−
x
2
)
d
2
y
d
x
2
b.
+
y
=
2 sin
x
0
≤
x
≤
1
y
(0)
=
0
y
(1)
=
0
N
1
(
x
)
=
sin
x
d
y
d
x
+
y
2
c.
=
4
x
0
≤
x
≤
1
y
(0)
=
0
y
(1)
=
0
N
1
(
x
)
=
x
2
(1
−
x
)
d
y
d
x
−
y
=
2
d.
0
≤
x
≤
10
0
y
(10)
=
0
N
1
(
x
)
=
x
2
(10
−
x
)
2
y
(0)
=
e.
d
2
y
d
x
2
−
3
d
y
d
x
+
y
=
x
0
≤
x
≤
1
y
(0)
=
0
y
(1)
=
0
1)
2
N
1
(
x
)
=
x
(
x
−
