# This would not be a problem when using collocation at

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Unformatted text preview: rder collocation method on this problem are apparent. Example 9.1.2 4]. Consider ; ; 2 y + (1 + x)2 y] = e 2 (1 + x)(3 ; x) + 2 ] y(0) = 0 y(1) = e 1 2 ; e 7 3 ;x= 00 0 ;= 12 ;= 0<x<1 N Error 4 0:44 10 4 8 0:37 10 4 16 0:58 10 8 32 0:60 10 9 64 0:19 10 10 128 0:64 10 12 Table 9.1.2: Maximum pointwise errors in the solution of Example 9.1.2 using collocation at Flaherty and Mathon's 4] points. ; ; ; ; ; ; which has the exact solution y(x) = e 2 ; e ( 2 +3 +3) 3 ;x= ;x x x = : For 0 < 1 there is a boundary layer of width O( ) at x = 0. Collocation using cubic polynomials was performed at the Gauss-Legendre points and at the points x 1=x 12; h x 2=x 12+ h i where i; = i 1 =2; 2 i = p(x ) ; q(x ) p(x ) k i k k i i; = ( p1!; 4=2) =2)= 1+ !( i i i i )+ i f ( i;12 ( i ) k = i ; 1 iotherwise < 0 1 !(z) = coth z ; z : px px With = 10 4 and N = 8, collocation at the Gauss-Legendre points produced a solution with a maximum pointwise error of approximately 15. The results using collocation at Flaherty and Mathon's 4] points are shown in T...
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