APM3701 Assignment 02 2025 (700123) Closing Date 8 August 2025.

APM3701
Assignment 02
Unique No: 700123
Closing Date: 8 August 2025

,APM3701 Assignment 02 Page 1



Question 1: Heat Flow in a Rod
Consider a horizontal rod of length L units with heat conductivity 1. The initial temper-
ature at each point x is given by f (x). The temperatures at the left and right ends are
described by time-dependent functions g1 (t) and g2 (t), respectively.

(a) Write the initial-boundary value problem satisfied by the temperature distribution
u(x, t) at point x and time t. Define all variables and parameters used. (5 Marks)
Variables and Parameters:

• u(x, t): Temperature at position x and time t.

• x: Position along the rod, 0 ≤ x ≤ L.

• t: Time, t ≥ 0.

• L: Length of the rod.

• f (x): Initial temperature distribution at t = 0.

• g1 (t): Temperature at the left end (x = 0).

• g2 (t): Temperature at the right end (x = L).

• Heat conductivity: 1.

Solution:

The temperature distribution u(x, t) satisfies the heat equation:

∂u ∂ 2u
= , 0 < x < L, t>0
∂t ∂x2

Initial condition:
u(x, 0) = f (x), 0≤x≤L

Boundary conditions:


u(0, t) = g1 (t), u(L, t) = g2 (t), t≥0

, APM3701 Assignment 02 Page 2


(b) Suppose f , g1 , and g2 are bounded, such that there exist constants m and M
satisfying:
m ≤ f (x) ≤ M, m ≤ g1 (t) ≤ M, m ≤ g2 (t) ≤ M,

for all x in the domain of f and all t ≥ 0, and the solution u(x, t) satisfies:


m ≤ u(x, t) ≤ M,


for all x and t ≥ 0. Show that the solution u(x, t) is unique. (10 Marks)

Solution:

Assume two solutions u1 (x, t) and u2 (x, t) satisfy the same initial-boundary value
problem. Define:
w(x, t) = u1 (x, t) − u2 (x, t).

Then w(x, t) satisfies:

∂w ∂ 2w
= , 0 < x < L, t > 0,
∂t ∂x2

with initial condition:


w(x, 0) = u1 (x, 0) − u2 (x, 0) = 0, 0 ≤ x ≤ L,


and boundary conditions:


w(0, t) = 0, w(L, t) = 0, t ≥ 0.


Define the energy functional:
Z L
1
E(t) = w2 (x, t) dx.
2 0


Compute:
L L
∂ 2w
Z Z
dE ∂w
= w dx = w 2 dx.
dt 0 ∂t 0 ∂x