Unformatted text preview: olution of Example 9.1.1 using the midpoint rule and collocation at two Gauss-Legendre points.
The error formula (9.1.31b) can be used to obtain a more precise estimate of the
collocation error than given by (9.1.28).
Let us conclude this section with two examples.
Example 9.1.1 (cf. 1], Chapter 5). Consider the problem
y + y = ( 8 ; x2 )2
0 00 which has the exact solution 0<x<1 y (0) = y(1) = 0
y(x) = 2 ln( 8 ; x2 ): The solution is smooth on 0 x 1, but the coe cient p(x) = 1=x is unbounded
at x = 0. This would not be a problem when using collocation at the Gauss-Legendre
points or, e.g., the midpoint rule since functions need not be evaluated at the endpoints.
Formulas, such as the trapezoidal rule, would have to be modi ed to account for the
singularity in p(x). We'll avoid this and present results using both the midpoint rule and
collocation (9.1.3) at the two Gauss-Legendre points (9.1.32). The maximum pointwise
errors are presented in Table 9.1.1.
A simple calculation will verify that the midpoint rule and collocation solutions are
converging at their expected rates of O(N 2) and O(N 4), respectively. The advantages
of the higher-o...
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