University The of Sydney Applied Mathematics 2
MATH2965 Lecturers:
Introduction to PDEs (Advanced) D. J. Galloway & R. Thompson
2008
Tutorial 7 For the week beginning Monday 15th September 1. Define the Laplace transforms I = L t-3/2 e-a (a)
2 2
/4t
and J = L t-1/2 e-a
/4t
.
(b) (c) 2.
Determine a relationship between I and J by substituting the expression 2 u = s1/2 t1/2 - (a/2)t-1/2 into the well-known integral - e-u du = . (If you have not seen this integral derived before, ask your tutor to show you; it is a marvellous piece of lateral thought.) Determine another relationship between I and J by introducing the change of variables sw = (a2 /4)t-1 into the definition of I. Hence show that I = (2 /a)e-a s and J = ( /s) e-a s .
(Haberman Exercise 13.2.9) Solve the two-dimensional heat equation u 2u 2u = + 2 t x2 y in a rectangle with sides L and H, subject to the boundary conditions u = 0 on all four boundaries. The initial condition is u = u0 (x, y) at t = 0. Proceed as follows. Write u = h(t)(x, y) and show that 2 2 dh = -h; + 2 = -, dt x2 y where is the separation constant. Now separate variables again, letting (x, y) = f (x)g(y) and show that d2 f = -f dx2 d2 g = -( - )g. dy 2
with corresponding boundary f conditions (0) = f (L) = 0 and g(0) = g(H) = 0. The eigenfunctions for f are then fn = sin nx and the corresponding L eigenvalues are n = (n/L)2 , n = 1, 2, 3, . . . . Each of these eigenvalues in turn gives rise to an eigenvalue problem for g with corresponding eigenvalues nm . Show that m 2 n 2 m 2 nm = n + = + H L H and find the corresponding eigenfunctions gnm (y).
Finally demonstrate that u(x, y, t) = where Anm = 4 LH
0
Anm sin
m=1 n=1
my -[(n/L)2 +(m/H)2 ]t nx sin e L H
H 0
L
u0 (x, y) sin
nx my sin dxdy L H
Note that the corresponding steady-state solution is u(x, y) = 0 everywhere, as expected. 3. Solve the 2-D Laplace equation for u(x, y) in a square of side L, subject to the boundary conditions u(x, 0) = 0, u(x,...