92 collocation for first order systems let us extend

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Unformatted text preview: k j =1 = j6 y ( )L ( ) + R( ) ik =1 k ; = ( ; 1 )( ; 2) : : : ( ; ; ( ; 1)( ; 2 ) : : : ( ; j k k j k R=y k 0 i 1 i k 2 ::: iJ Y J j x=x 1+ h i; i ik (9.2.3a) 0 1 )( 1 )( k; k; ; +1) : : : ( ; ) ; +1) : : : ( ; ) k k J k k 0 k (9.2.3b) (; ) (9.2.3c) 1 =1 (9.2.3d) j =x 1+ h: i; J (9.2.3e) i The image of the collocation points , k = 1 2 : : : J , are ordered such that k 0 1 < 2< 14 < J 1: (9.2.3f) The divided di erence y 1 2: : : : ] will be de ned shortly. Substituting (9.2.3a) into (9.2.2) yields 0 i i iJ X J y(x) = y(x 1) + h i; i y( ) ik =1 k Z 0 X J y( ) = y(x 1 ) + h ij i; i k =1 y( ) ik a= Z jk and j 0 J i =1 i i i; i k =1 y( ) b= k and E= ik Z1 0 R( )d : 0 (9.2.5c) 0 ik jk i L ( )d + h k i j Z1 0 R( )d : (9.2.6a) R( )d : (9.2.6b) k Z1 0 y(x ) = y(x 1) + h i; j L ( )d 0 we have i i y ( )a + h E : Z1 0 Letting k (9.2.5b) Evaluating (9.2.4) at x = x yields y(x ) = y(x 1) + h L ( )d + h (9.2.4) R( )d : k J Z R( )d : (9.2.5a) X y( ) = y(x 1) + h X 0 k j i; i L ( )d 0 Then ij j...
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