130 is a cubic polynomial so e 0 we see that we can

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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 8 y + y = ( 8 ; x2 )2 x 0 00 which has the exact solution 0<x<1 y (0) = y(1) = 0 0 7 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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