21 multiple shooting procedures that are based on the

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Unformatted text preview: mpler of the two, so let's begin there. 1. On each subinterval (xj ; 1 xj ], j = 1 2 : : : N , solve the IVPs vj = Avj + b 0 9 vj (a) = 0 (7.2.2a) Yj = AYj 0 where j, Yj (a) = I xj 1 < x xj j = 1 2 ::: N ; (7.2.2b) j = 1 2 : : : N , are chosen and I is the m m identity matrix. 2. As with simple shooting, consider the solution of the BVP in the form (7.1.2a), which now becomes y(x) = Yj (x)cj + vj (x) xj 1 < x xj j = 1 2 : : : N: ; (7.2.3) The solution must be continuous at the interior shooting points xj , j = 1 2 : : : N ; 1 thus, with the initial conditions speci ed by (7.2.2) Yj (xj )cj + vj (xj ) = cj+1 j = 1 2 : : : N ; 1: The boundary conditions (7.2.1b,c) must also be satis ed and, with (7.2.3) and (7.2.2), this implies Ly(a) = LY1c1 + Lv(a) = Lc1 = l Ry(b) = RYN (b)cN + RvN (b) = r: Writing this system in matrix form Ac = g where (7.2.4a) 2 L 6 ;Y1 (x1 ) 6 6 A=6 6 6 3 I ;Y2(x2 ) I ... 6 4 ... ;YN 1(xN 1 ) ; 2 6 c=6 6 4 c1 c2 ... cN 3 7 7 7 5 ; 2 I RYN (b) (7.2.4b) 3 l 6 v1(x1 ) 6 6 ... g=6 6 4 vN 1 (xN 1 )...
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