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The University of Sydney

Applied Mathematics 2

MATH2965 Introduction to PDEs (Advanced) 2009

Lecturer: F. Viera

Tutorial 7

For the week beginning Monday 14th September

1. Define the Laplace transforms

I= Lt3/2ea2/4t and J= Lt1/2ea2/4t .

(a) Determine a relationship betweenIand Jby substituting the expressionu= s1/2t1/2(a/2)t1/2 into the well-known integral eu2du= .(If you have not seen this integral derived before, ask your tutor to showyou; it is a marvellous piece of lateral thought.)

(b) Determine another relationship betweenIand Jby introducing the changeof variables sw = (a2/4)t1 into the definition ofI.

(c) Hence show thatI= (2/a)ea

s and J= (

/s) ea

s.

(Haberman Exercise 13.2.9)

2. Solve the two-dimensional heat equation

ut

=2ux2

+ 2u

y2

in a rectangle with sides L and H, subject to the boundary conditions u = 0on 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

dh

dt = h;

2

x2+

2

y2 = ,

where is the separation constant. Now separate variables again, letting(x, y) =f(x)g(y) and show that

d2f

dx2 = f d

2g

dy2 = ( )g.

with corresponding boundary conditions f(0) =f(L) = 0 and g(0) =g(H) =0. The eigenfunctions for f are then fn = sin

nxL and the corresponding

eigenvalues are n = (n/L)2, n = 1, 2, 3, . . . . Eachof these eigenvalues in

turn gives rise to an eigenvalue problem for g with corresponding eigenvaluesnm. Show that

nm= n+mH

2=nL

2+mH

2

and find the corresponding eigenfunctions gnm(y).

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Finally demonstrate that

u(x,y,t) =

m=1

n=1

Anmsinnx

L sin

my

H e[(n/L)

2+(m/H)2]t

where

Anm= 4LH

H

0

L

0

u0(x, y)sinnxL

sinmyH

dxdy

Note that the corresponding steady-state solution is u(x, y) = 0 everywhere, asexpected.

3. Solve the 2-D Laplace equation foru(x, y) in a square of side L, subject to theboundary conditions u(x, 0) = 0, u(x,H) = 0, u(0, y) = 0, u(L, y) = y .

4. Consider the heat equation in three dimensions with no sources but with non-constant thermal properties

cu

t = (K0u) ,

where c and K0 are functions of x, y and z. Assume that u = 0 on theboundary. Show that the time variable can be separated by assuming that

u(x,y,z,t) = (x,y,z)h(t).

Show that (x,y,z) satisfies the eigenvalue problem

(p) +(x,y,z)= 0

with = 0 on the boundary. What are (x,y,z) and p(x,y,z)?

(Haberman exercise 7.5.4) .