Other numerical methods 261
From Equation 9.31 it can be seen that the approximate equation is
again the same as in Equation 9.23. Therefore, the four methods (collocation, sub-domain, moments and Galerkin) result in the same approximated solution (Equation 9.23), while the method of least squares
gives another approximating solution (Equation 9.28). Comparison of
the results shows overall that the methods are not very accurate and
this is particularly true for the least squares method (Figure 9.1). This
inaccuracy occurs because the selected weight functions are incompatible with the physical problem.
PROBLEM 9.1
Solve Equation 9.13 for the same constants m, c and k and the same boundary conditions by using:
1. The moments method and the approximated function

f
x
=
+
+
α α
0
1
α
α
2
2
3
3
x
x
+
2. The least squares method and the approximated function

f
x
=
+
+
α α
0
1
α
α
2
2
3
3
x
x
+
1.2
0.8
0.6
0.4
0.2
–0.2
0
1
0
100
200
300
400
500
600
700
800
900 1000
Solution domain × 10 –3
Function f(x)
Exact solution
WRM
Least squares
Figure 9.1 Comparison of the weighted residual methods.