AEM_3e_Chapter_06

AEM_3e_Chapter_06 - 6 3. h=0.1 Numerical Solutions of...

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x n y n 0.00 0.0000 0.10 0.1005 0.20 0.2030 0.30 0.3098 0.40 0.4234 0.50 0.5470 x n y n 0.00 0.0000 0.05 0.0501 0.10 0.1004 0.15 0.1512 0.20 0.2028 0.25 0.2554 0.30 0.3095 0.35 0.3652 0.40 0.4230 0.45 0.4832 0.50 0.5465 x n y n 0.00 0.0000 0.10 0.0050 0.20 0.0200 0.30 0.0451 0.40 0.0805 0.50 0.1266 x n y n 0.00 0.0000 0.05 0.0013 0.10 0.0050 0.15 0.0113 0.20 0.0200 0.25 0.0313 0.30 0.0451 0.35 0.0615 0.40 0.0805 0.45 0.1022 0.50 0.1266 x n y n 1.00 1.0000 1.10 1.0095 1.20 1.0404 1.30 1.0967 1.40 1.1866 1.50 1.3260 x n y n 1.00 1.0000 1.05 1.0024 1.10 1.0100 1.15 1.0228 1.20 1.0414 1.25 1.0663 1.30 1.0984 1.35 1.1389 1.40 1.1895 1.45 1.2526 1.50 1.3315 1.1 1.2 1.3 1.4 x 5 10 15 20 y x n Euler Imp. Euler 1.00 1.0000 1.0000 1.10 1.2000 1.2469 1.20 1.4938 1.6430 1.30 1.9711 2.4042 1.40 2.9060 4.5085 6 6 Numerical Solutions of Ordinary Differential Equations EXERCISES 6.1 Euler Methods and Error Analysis 3. h =0 . 1 h =0 . 05 6. h =0 . 1 h =0 . 05 9. h =0 . 1 h =0 . 05 12. (a) (b) 95
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x n y n 1.00 5.0000 1.10 3.9724 1.20 3.2284 1.30 2.6945 1.40 2.3163 1.50 2.0533 x n y n 0.00 1.0000 0.10 1.1115 0.20 1.2530 0.30 1.4397 0.40 1.6961 0.50 2.0670 6.1 Euler Methods and Error Analysis 15. (a) Using Euler’s method we obtain y (0 . 1) y 1 =0 . 8. (b) Using y 0 =5 e 2 x we see that the local truncation error is 5 e 2 c (0 . 1) 2 2 . 025 e 2 c . Since e 2 x is a decreasing function, e 2 c e 0 = 1 for 0 c 0 . 1. Thus an upper bound for the local truncation error is 0 . 025(1) = 0 . 025. (c) Since y (0 . 1) = 0 . 8234, the actual error is y (0 . 1) y 1 . 0234, which is less than 0 . 025. (d) Using Euler’s method with h . 05 we obtain y (0 . 1) y 2 . 8125. (e) The error in (d) is 0 . 8234 0 . 8125 = 0 . 0109. With global truncation error O ( h ), when the step size is halved we expect the error for h . 05 to be one-half the error when h . 1. Comparing 0 . 0109 with 0 . 0234 we see that this is the case.
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This note was uploaded on 01/03/2011 for the course BIS 511 at Yale.

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AEM_3e_Chapter_06 - 6 3. h=0.1 Numerical Solutions of...

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