12 713 is the reduced number of odes that have to be

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Unformatted text preview: ing a basis U, v to satisfy the initial conditions (7.1.1b). Example 7.1.2. Conte 5] constructed the constant-coe cient fourth-order BVP 2 0<x<1 yiv ; (1 + R)y + Ry = ;1 + Rx 2 3 y(0) = y (0) = 1 y(1) = 2 + sinh 1 y (1) = 1 + cosh 1: to try to explain di culties that researchers were having when solving certain owinstability problems. We'll rewrite the problem as the rst-order system 2 3 2 3 2 3 01 0 0 y1 0 6 7 6 7 6 7 0 7: A=6 0 0 1 0 7 y = 6 y2 7 b=6 40 0 4 y3 5 4 5 0 15 0 2 ;R 0 1 + R 0 y4 ;1 + Rx =2 00 0 L=R= 1 0 0 0 0100 0 l= 1 1 2 + sinh r = 1 + cosh 1 : 1 3 Let's go through the steps in solving this problem by the shooting procedure that we just described. We note that v(0) = 1 1 0 0]T satis es the initial conditions and, thus, the particular-solution problem (7.1.7) has been speci ed. Similarly, 23 23 0 0 607 607 u(2) = 6 0 7 : u(1) = 6 1 7 45 45 1 0 7 satisfy trivial versions of the initial conditions hence, the homogeneous problems (7.1.9) are speci ed. Conte 5] solved the IVPs (7.1.7, 7.1.9) using a xed-step Runge-Kutta method. (It was done in the 1960s.) His...
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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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