Download Numerical Analysis, Exercises for lecture 13 [§10.1–4, 10.7] Model

Numerical Analysis, Exercises for lecture 13 [§10.1–4, 10.7]
1. Consider the Bernoulli differential equation initial value problem
y 0 + xy = xy 3 ,
y(0) = 0.5
(a) Find the value of y(0.5) using Heun’s method (by hand) with step size h =
0.25.
(b) Repeat using your Matlab/Octave implementation of the classical RungeKutta method.
(c) Repeat using Matlab’s ode45 or Octave’s lsode.
(d) Repeat using the exact solution formula y(x) = √
1
.
1 + 3ex2
2. Solve the initial value problem
u00 + 2u0 + 100u = 0,
u(0) = 1.1,
u0 (0) = −11
using Matlab’s ode45 or Octave’s lsode. Plot u over the interval [0, 6].
3. Prove: the local truncation error of the following method is at least O(h3 ):
k1 = f (xn , yn ), k2 = f (xn + 21 h, yn + 21 hk1 ), yn+1 = yn + hk2
Model quiz questions
(A) Compute the solution of the initial value problem y 0 = x + y, y(0) = 10 at the
point x = 1 using Euler’s method with step h = 0.5.
Answer: y(1) ≈
(B) Write the differential equation u000 + 3xu = 4 as a set of first-order differential
equations.
1
0.5
Exercise Answers
1. (a) 0.45388, (b) 0.45398
0
2.
−0.5
−1
Quiz Solutions
0
2
4
6
(A) Euler’s method is yn+1 = yn + hf (xn , yn ). With y 0 = f (x, y) = x + y, h = 0.5,
x0 = 0 and y0 = 10, Euler’s method gives
y(0.5) ≈ y1 = y0 + h · (x0 + y0 ) = 10 + 0.5 · (0 + 10) = 15
x1 = x0 + h = 0 + 0.5 = 0.5
y(1) ≈ y2 = y1 + h · (x1 + y1 ) = 15 + 0.5 · (0.5 + 15) = 22.75
(B) Introducing the variables y1 = u, y2 = u0 and y3 = u00 , we have y10 = u0 = y2 , y20 =
u00 = y3 , and y30 = u000 = 4−3xu = 4−3xy1 . That is, y10 = y2 , y20 = y3 , y30 = 4 − 3xy1