{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

which has the exact solution yx e 2 e

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ( )=...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online