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Unformatted text preview: results for the error ke1 k in the rst component y1(x) of
the solution are reported in Table 7.1.2 for R = 400 3600. The exact solution of this
problem is
2
y1(x) = 1 + x + sinh x:
2
1 R jje1jj
400 0.00173
3600 4000
Table 7.1.2: Maximum errors of Example 7.1.2 using Conte's 5] linear shooting procedure.
1 The di culties with this problem are similar to those of Example 7.1.1. The fundamental solutions of the ODE are
sinh x cosh x p sinh Rx p cosh Rx: p Although the exact solution doesn't depend on the rapidly growing components (sinh Rx
p
and cosh Rx), the IVPs (7.1.7) and (7.1.9) do. Thus, the matrix RU(b) appearing in
(7.1.10) will be illconditioned when the parameter R is large. The two components of
p
p
the fundamental solution sinh Rx and cosh Rx are numerically linearly dependent for
p
large values of Rx. Although the exact solution of the BVP is independent of these
rapidly growing components, small round o errors introduce them and they eventually
dominate the exact solution. Problems 1. Show that the solution representation (7.1.4 satis es the linear BVP (7.1.1).
2. Find the Green's function for the BVP of E...
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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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