# chapter11 - i book 2005/4/7 22:31 page S-93 #114 i i i...

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± ± “book” — 2005/4/7 — 22:31 — page S-93 — #114 ± ± ± ± Answers to Selected Exercises for Chapter 11 Section 11.1 (page 899) 1. The approximate solutions in the plot in Figure S.16(a) were obtained with x =∆ t =1 / 20. The numerical boundary conditions u (0 ,t ) = 0 was applied. Note that no numerical boundary condition needs to be specifed at the right end o± the domain. 0 0.5 1 1.5 2 2.5 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t = 1 t = 5 approximate exact 0 0.5 1 1.5 2 2.5 3 3.5 4 0 5 10 15 20 25 30 approximate solution exact solution Figure S.16 (a, left) Figure for solution to Exercise 1 (Section 11.1). (b, right) Figure for solution to Exercise 3 (Section 11.1). 3. The approximate solution in the plot in Figure S.16(b) was obtained with x / 20, ∆ t / 320 and t f = 2. From the characteristics, we fnd that the appropriate numerical boundary condition at x =0is u (0 )=0 . 9. The approximate solutions in the plot in Figure S.17(a) were obtained with x / 25 and ∆ t / 4. Note that no numerical boundary condition needs to be specifed at the right end o± the domain. 13. The approximate solutions in the plot Figure S.17(b) were obtained with x t / 20. From the characteristics, we fnd that the appropriate numerical boundary conditions at x = ± 1are u ( ± 1 . S-93

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± ± “book” — 2005/4/7 — 22:31 — page S-94 — #115 ± ± ± ± S-94 Answers to Selected Exercises for Chapter 11 0 0.5 1 1.5 2 2.5 3 3.5 4 35 40 45 50 55 60 65 10 seconds 20 seconds 30 seconds 40 seconds 0 0.1 0.2 0.3
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## This note was uploaded on 10/13/2009 for the course MATH 471 taught by Professor Anna during the Spring '09 term at University of Michigan-Dearborn.

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chapter11 - i book 2005/4/7 22:31 page S-93 #114 i i i...

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