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19 are speci ed conte 5 solved the ivps 717 719 using

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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 ill-conditioned 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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