113c ex yx y x 9114a finally letting denote

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Unformatted text preview: mark 1. Each result (9.1.13b), (9.1.13c), or (9.1.14b) relates a global quantity (y Y e) to its local counterpart (Ly LY Le) through the Green's function. Let us write (9.1.14b) in the more explicit form e( ) = XZ N i xi ;1 =1 xi G( x)Le(x)dx = X N =1 e ( ): (9.1.15) i i Suppose that 2 (x 1 x ) so that G( x) is smooth for x 2 (x 1 x ). Write = i; i i; Le = L(y ; Y ) = Ly ; Pr + Pr ; LY i (9.1.16a) where Pr is a linear polynomial for x 2 (x 1 x ) that interpolates both Ly and LY at the two collocation points 1 and 2 on this subinterval. Thus, i; i i i Pr = r( 1) x ; 1; i i i 2 i + r( 2) x ; 1 : i 2 5 i i 2; 1 i (9.1.16b) LY Pr Ly x i-1 ξ i,1 ξ i,2 x i Figure 9.1.2: Functions Ly, LY , and Pr on a subinterval (x 1 x ) not containing the point x = . i; i The functions Ly, LY , and Pr are illustrated in Figure 9.1.2. The di erences Ly ; Pr and LY ; Pr can be estimated using formulas for the error in linear interpolation 2] as Ly ; Pr = 1 (x ; 1)(x ; 2)(Ly) ( ) (9.1.17a) 2 00 i i i LY ; Pr = 1 (x ; 1)(x ;...
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