11 geometry for the collocation solution of 911

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Unformatted text preview: ion solution of (9.1.1) showing the restriction of the cubic Hermite polynomial basis to the subinterval x 1 x ]. i; i In order to obtain approximate solutions of (9.1.1) using (9.1.2), we collocate at two points 1, 2 per subinterval and satisfy the boundary conditions. Thus, the unknown coe cients c , d , i = 0 1 ::: N , are determined as the solution of i i i i LY ( ) = r( ) ij j=1 2 ij Y (a) = A i = 1 2 ::: N (9.1.3a) Y (b) = B: (9.1.3b) The restriction of (9.1.2a) to the subinterval x 1 x ) is i; Y (x) = c i; 1 3 i; i 3 3 3 1 (x) + d 1 ! 1 (x) + c (x) + d ! (x): i; i i; Substituting (9.1.4) into (9.1.3a) LY ( 1) = L c 1 + R c = r( 1) LY ( 2) d1 d r( 2) where 3 3 3 L = L 3 1( 1) L!3 1( 1) R = L 3( L 1( 2) L! 1( 2) L( With (9.1.4), the boundary conditions (9.1.3b) become i i; i i i i; i i; i i i i; i; i i; i i c L0 d00 = A R 2 N +1 c d N N i i i i (9.1.4) i = 1 2 ::: N i i i i i i (9.1.5a) L!3( 1) : L!3( 2) (9.1.5b) 1) 2) =B i i i i (9.1.6a) where L0 = R N 1 0]: +1 = Combining (9.1.5) and (...
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This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.

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