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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 constantcoe cient fourthorder 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 rstorder 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 particularsolution 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 xedstep RungeKutta 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.
 Spring '14
 JosephE.Flaherty

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