Читать книгу Solutions Manual to Accompany An Introduction to Numerical Methods and Analysis - James F. Epperson - Страница 13

1 Use Euler's method with to compute approximate solution values for the initial value problemYou should get (be sure your calculator is set in radians).Solution:

2 Repeat the above with . What value do you now get for ?

3 Repeat the above with . What value do you now get for ?Solution:

4 Use Euler's method with to compute approximate solution values forWhat approximate value do you get for ?

5 Repeat the above with . What value do you now get for ?Solution: We have the computationwhere , , and . Hence

6 Repeat the above with . What value do you now get for ?

7 Use Euler's method with to compute approximate solution values over the interval for the initial value problemwhich has exact solution . Plot your approximate solution as a function of , and plot the error as a function of .Solution: The plots are given in Figures 2.1 and 2.2.

8 Repeat the above for the equationwhich has exact solution .

9 Repeat the above for the equationwhich has exact solution .Solution: The plots are given in Figures 2.3 and 2.4.Figure 2.1 Exact solution for Exercise 2.3.7.Figure 2.2 Error plot for Exercise 2.3.7.Figure 2.3 Exact solution for Exercise 2.3.9.Figure 2.4 Error plot for Exercise 2.3.9.

10 Use Euler's method to compute approximate solutions to each of the initial value problems below, using . Compute the maximum error over the interval for each value of . Plot your approximate solutions for the case. Hint: Verify that your code works by using it to reproduce the results given for the examples in the text., ; ;, ; ;, ; .

11 Consider the approximate values in Tables 2.5 and 2.6 in the text. Let denote the approximate values for , and denote the approximate values for . Note thatandthus and are both approximations to the same value. Compute the set of new approximationsand compare these to the corresponding exact solution values. Are they better or worse as an approximation?Solution: Table 2.4 shows the new solution and the error, which is much smaller than that obtained using Euler's method directly.Table 2.4 Solution values for Exercise 2.3.11.0.00000.1250.2500.3750.5000.6250.7500.8751.000

12 Apply the basic idea from the previous problem to the approximation of solutions towhich has exact solution .

13 Assume that the function satisfiesfor some constant . Use this and (2.9)‐(2.10) to show that the error satisfies the recursion,whereSolution: If we take and subtract (2.10) from (2.9) we getfrom which we haveand the desired result follows immediately.