APM3701 Assignments 2 Semester 1 2026 – Due 17 August 2026

Description

APM3701 Assignment 2 2026
Assignment Unique Number: 192483
Closing Date: 17 August 2026
QUESTION 1
Consider the heat flow in a homogeneous rod of length   with heat conductivity   and
constant source of energy  . We assume that initially the rod was submerged in a medium
where the temperature at each point   of the rod is described by the function
L
k
. We
A
x
1 − sin x
also suppose that the heat flux is
−t
e
units at the left end and
cos(t − π)
units at the right end.
(a) Write down the initial-boundary problem satisfied by the
temperature distribution
u(x, t)
in the rod at any point   and time  .
x
t
(Explain all the meaning of the variables and parameters used). [10
Marks]
The one-dimensional heat equation with a constant heat source   (energy generated per unit
volume per unit time) is given by (see Section 4.4 of the Study Guide):
A
2

=
∂u
k
∂ u

+
A,
0 < x < L, t > 0,
∂t
∂x
2
u(x, t)
= temperature at position   and time  ,
x
t
k
= thermal diffusivity (heat conductivity divided by heat capacity per unit volume),
A
= constant rate of internal heat generation per unit volume.
Initial condition: