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Unformatted text preview: 0 j Z E= k Z 0 Let L ( )d + h 0 Evaluating (9.2.4) at the collocation points Z X J i k 15 =1 y ( )b + h E: 0 ik k i (9.2.6c) If the errors E , j = 1 2 : : : J , and E are neglected, we see that the collocation solution is obtained from a J stage implicit Runge-Kutta method. Thus, letting Y(x) denote the approximate solution, we have j Y(x ) = Y(x 1) + h i i; X J bk (9.2.7a) ak) k = 1 2 : : : J: (9.2.7b) i =1 k where k = Y ( ) = f (x 1 + h Y(x 1) + h 0 ik ik i; k i i; X J i j =1 kj k ik ij Solution continuity, required for rst-order BVPs, y(x ) = y(x+) i = 1 2 ::: N ;1 ; i i is automatically satis ed. Solutions at any point x 2 x 1 x ] are de ned by the interpolation polynomial (9.2.4) and (9.2.3a) as i; i X J Y(x) = Y(x 1) + h i; i k =1 Y( ) 0 Z ik 0 L ( )d : (9.2.7c) k Some common choices of the collocation points are: 1. Gauss-Legendre points. The points , k = 1 2 : : : J , are the roots of the Legendre polynomial of degree J mapped to the interval (0 1). The roots of the Legendre polynomial are normally prescribed on (;1 1). This may be done by the linear transformation = (1 + )=2, 2 ;1 1]. For all J , 1 > 0 and < 1 thus, s...
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