Civil Engineering Reference
In-Depth Information
5.15
Consider a two-dimensional problem governed by the differential equation
2
2
∂
∂
x
2
∂
∂
y
2
+
=
0
(this is Laplace's equation) in a specified two-dimensional domain with specified
boundary conditions. How would you apply the Galerkin finite element method
to this problem?
5.16
Reconsider Equation 5.24. If we do
not
integrate by parts and simply substitute
the discretized solution form, what is the result? Explain.
5.17
Given the differential equation
d
2
y
d
x
2
+
4
y
=
x
Assume the solution as a power series
n
a
i
x
i
=
a
0
+
a
1
x
+
a
2
x
2
y
(
x
)
=
+···
i
=
0
and obtain the relations governing the coefficients of the power series solution.
How does this procedure compare to the Galerkin method?
5.18
The differential equation
d
y
d
x
+
y
=
3
0
≤
x
≤
1
has the exact solution
+
Ce
−
x
y
(
x
)
=
3
where
C
=
constant. Assume that the domain is
0
≤
x
≤
1
and the specified
boundary condition is
y
(0)
=
0. Show that, if the procedure of Example 5.4 is
followed, the exact solution is obtained.
