Math 450 Homework 4 - Spring 2009

HOMEWORK #4 due Wednesday April 8

Text references are to
Introduction to partial differential equations with MATLAB
Jeffrey Cooper (1998) (Birkhauser)

NOTE: In the following, ^ means superscript, _ (underscore) means subscript, and Sum(i=1,9) means the sum for i=1 to 9.

1. Problem 1 page 124 in the text

2. Problem 3 page 124 in the text.
(Hint: The general solution of phi''(x)=w phi(x) is phi(x)=C_1 exp(xz) + C_2 exp(-xz) where z^2=w, whether w is real or complex, by the same argument as in the case w>0.)

3. (i) Do Problem 4 page 124 in the text EXCEPT plot values at the four times t=0, t1, 2*t1, and 3*t1 for t1=0.050 INSTEAD OF AT the five t-values given, and answer the question in part (b) for time t=3*t1 instead of time t=5. As part of the same plot, add a fifth line u4=zeros(size(x)) where x is the X-grid array in order to highlight the X-axis so as to make the other plotted curves clearer. Hint: Colors appear in the plots in the order blue, green, red, bluegreen (or cyan), purple (or magenta), and a sickly yellow-green (yellow), in that order. Enter, for example, ``help plot'' at the MATLAB command line.

(ii) Find the maximum of the absolute values of the differences between the first term and the solution u(x,t) (call this maximum emax) at each of the time points. At which time is the maximum relative error emax/tmax the smallest, where tmax is the maximum of the first term? (Hint For arrays x and y, consider max(abs(x-y)). Note that tmax is the value of the first term at pi/4.)

(Hint: Instead of including an m-file for a function u=u(x,t), you can include the following command in your MATLAB program (for example)

u = @(x,t) 5*sin(2*x)*A_1(t) - 2*sin(3*x)*A_2(t) + 3*sin(5*x)*A_3(t)

for appropriate expressions A_1(t),A_2(t),A_3(t), and define similar functions for the first term and the maximum of the first term.)

4. Problem 5 (page 125) in the text.

5. Problem 14a (that is, only part (a)) (page 129) in the text. Use k=1. Note that heat3.m solves the PDE using the Crank-Nicolson method, not an eigenfunction expansion. In part (a), give an upper bound for the first value of t such that |u(x,t)-U(x)| <= 0.10 for 0 < x < 10, not necessarily the first such value. (Hint: Note that lim_{t to infinity} u(0,t)=1 while u(10,t)=0 for all t.)

6. Problem 1 (page 138) in the text.

7. Problem 4 (page 138) in the text.

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