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HW8 Solutions

HW8 Solutions - Problem 3 a= b= N= 0 1.57 10000 Exact...

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Problem 3 a = 0 Exact analytical result b = 1.57 12.42 N = 10000 Step size (h) 0 Trapezoidal rule result Summary of results 12.42 h result error 0.16 12.42 4.97E-04 Error 0.02 12.42 4.96E-06 0 0 12.42 4.96E-08 0 12.42 4.96E-10 which is in agreement with the theoretical prediction. As step size (h) decreases, the error decreases as h 2 ,

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Problem 4 Initial conditions x0 = 0 xf = 2 y(0) = 1 Steps = 10000 Step size = 0 N = 1000 Global error = 0 (print every Nth point) x y ε 0.2 0.8 0.8 0 0.4 0.66 0.66 0 0.6 0.56 0.56 0 0.8 0.49 0.49 0 Summary of results 1 0.46 0.46 0 h result error 1.2 0.48 0.48 0 0.2 1.52 4.52E-02 1.4 0.54 0.54 0 0.02 1.59 6.77E-04 1.6 0.67 0.67 0 0 1.59 7.03E-06 1.8 0.96 0.96 0 0 1.59 0 2 1.59 1.59 0 which is in agreement with the theoretical pr y a As step size (h) decreases, the error decrea

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rediction. ases as h 2 ,
Problem 5b Initial conditions t y t0 = 0 0.25 4.63 tf = 25 0.5 3.18 Equations = 2 0.75 1.03 Steps = 10000 1 -1.25 N = 100 1.25 -3.13 (print every Nth point) 1.5 -4.16 Step size = 0 1.75 -4.16 2 -3.15 y(0) [1] = 5 2.25 -1.45 y(0) [2] = 1 2.5 0.52 2.75 2.27 3 3.39 3.25 3.65 3.5 3.02 3.75 1.7 4 0.04 4.25 -1.54 4.5 -2.69 4.75 -3.13 5 -2.81 5.25 -1.83 5.5 -0.46 5.75 0.95 6 2.07 6.25 2.63 6.5 2.55 6.75 1.85 7 0.75 7.25 -0.48 7.5 -1.53 7.75 -2.17 8 -2.25 8.25 -1.8 8.5 -0.94 8.75 0.11 9 1.07 9.25 1.74 9.5 1.95 9.75 1.69 10 1.04 10.25 0.17 10.5 -0.7 10.75 -1.36 11 -1.66 11.25 -1.55 11.5 -1.08 11.75 -0.37 12 0.39 0 5 10 -6 -4 -2 0 2 4 6

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12.25 1.02 12.5 1.38 12.75 1.39 13 1.07 13.25 0.51 13.5 -0.15 13.75 -0.74 14 -1.12 14.25 -1.21 14.5 -1.02 14.75 -0.59 15 -0.04 15.25 0.5 15.5 0.88 15.75 1.04 16 0.94 16.25 0.62 16.5 0.17 16.75 -0.3 17 -0.68 17.25 -0.87 17.5 -0.85 17.75 -0.63 18 -0.26 18.25 0.14 18.5 0.5 18.75 0.72 19 0.75 19.25 0.61 19.5 0.32 19.75 -0.02 20 -0.35 20.25 -0.57 20.5 -0.65 20.75 -0.57 21 -0.36 21.25 -0.07 21.5 0.22 21.75 0.44 22 0.55 22.25 0.52 22.5 0.37 22.75 0.13

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