612 AMSC Numerical Methods for Partial Dierential Equations Spring Term 2004 Instructor: Georg Dolzmann Homework set #1
Problem 1: [Morton&Meyers 2.1] (i) The function u0 (x) is dened on [0, 1] by u0 (x) = 2x 2 2x
1 if 0 x 2 , 1 if 2 x 1.
Show that the Fourier sine series of u0 is given by
u0 (x) =
m=1
am sin mx
where am = (ii) Show that
2p+2 2p
m 8 sin . m2 2 2
1 2 dx 2 x (2p + 1)2 1 1 < . 2 (2p + 1) 4p0
and hence that
p=p0 0
(iii) Deduce that u (x) is approximated on the interval [0, 1] to within 0.001 by the sine series in part (i) truncated after m = 405. Problem 2: Write a MATLAB program to solve the equation heat ut = uxx u(0, t) = 0 u(1, t) = 0
0
in (0, 1) (0, ), for t > 0, for t > 0,
u(x, 0) = u (x) for x (0, 1), where u0 (x) = 2x 2 2x
1 if 0 x 2 , 1 if 2 x 1.
Implement the explicit scheme that leads to
n+1 n n n n Uj = Uj + Uj+1 2Uj + Uj1
2
for j = 1, . . . , J 1, and n 1. Here = t/(x)2 and
0 Uj = u0 (xj ),
j = 1, 2, . . . , J 1,
n n U0 = UJ = 0...