8.3 The Ritz Method
465
Table 8.1 The values and errors of the approximate solution and analytical solution
x
y(x)
y 1 (x)
y 2 (x)
Errors of y 1 %
Errors of y 2 %
0.25
0.0440137
0.0520833
0.0440803
18.334490
0.1513848
0.5
0.0697470
0.0694444
0.0694444
−0.8232490
−0.8232490
0.75
0.0600562
0.0520833
0.0600864
−13.275628
0.0503131
Thus, we get
y 2 (x) = x(1 − x)
71
369
+
7
41
x
The analytical solution of this problem is
y(x) =
sin x
sin 1
− x
The values and errors of the approximate solution and analytical solution are
shown in Table 8.1.
It can be seen from the table that error of the quadratic approximation solution
generally decreases.
Example 8.3.3 Let a rocket of mass m make level flight, the flight distance is the
following functional
J [v(m)] =
m 0
m 1
v
av 2 + bm 2 (c + mv
)dm
where, a, b and c are all constants greater than zero, try to use the Ritz method to
solve the above problem.
Solution Let the approximate solution be
v = αm
(1)
where, α is an undetermined constant.
Derive the expression (1) with respect to m, we get
v
= α
(2)
Substituting the expression (1) and expression (2) into the expression of the
functional, simplifying and integrating, we obtain