Topic 14.2: Heun's Method (Questions)

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Question 1

Given the IVP

y(1)(t) = 1 - 0.25 y(t) + 0.2 t
y(0) = 1

approximate y(0.5), y(1), and y(1.5) using Heun's method.

Answer: 1.3765625, 1.75625, 2.1390625.

Question 2

Given the same ODE as in Question 1, but with the initial condition y(1) = 2, approximate y(1.5) and y(2.0).

Answer: 2.353125 and 2.7125

Question 3

Given an IVP with an initial condition y(0) = y0, if the second derivative is bounded by -8 < y(3)(t) < 8, on how large an interval can we estimate y(t) if we want to ensure that the error is less than 0.0001? Compare this with the range for Question 3 of Euler's method.

Answer: (-0.0531, 0.0531).

Question 4

Consider the following modification to Huen's method:

Instead of defining K1 = f(t1, y0 + hK0), use this as an initial value and iterate:

K1 = f(t1, y0 + ½h(K0 + K1)).

For Question 1, use this modification, iterating the previous step 10 times.

Copyright ©2005 by Douglas Wilhelm Harder. All rights reserved.