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# which has the exact solution yx e 2 e

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Unformatted text preview: able 9.1.2. Following the logic introduced in (9.1.30), Flaherty and Mathon selected their collocation points to re ect the boundary layer in the Green's function. Speci cally, they chose points 1=x 12; h 2=x 12+ h ; i i; = i i i; = i symmetrically disposed with respect to the center of each subinterval so that Z1 ; 1 e ; (1 ) 2 2( ; 4 2 )P ( )d = 0 13 for polynomials P ( ) of as high a degree as possible. 9.2 Collocation for First-Order Systems Let us extend the collocation methods to rst-order vector BVPs of the usual form y = f (x y) a<x<b (9.2.1a) g (y(a)) = g (y(b)) = 0: (9.2.1b) 0 L R We'll follow a di erent approach than the one used in Section 9.1 and integrate (9.2.1a) on a subinterval (x 1 x ) to obtain i; i y(x) = y(x 1 ) + Z x Z y (x)dx = y(x 1 ) + 0 i; xi ;1 x i; xi ;1 f (x y)dx: (9.2.2) As with initial value methods, we'll construct a numerical method by approximating y (or f ) by a polynomial and integrating the result. Any polynomial basis may be used, but let us concentrate on the Lagrange form of the interpolating polynomial 0 X J y (x) = 0 k where Y J L ( )=...
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