APM3701 Assignment 2 2025 (700123) Due 8 August 2025

APM3701
Assignment 2
Unique No: 700123
Due 8 August 2025

,Student Name: APM3701 Assignment 2


APM3701 Assignment 2
Due Date: 8 August 2025



Heat Flow Problem

Question 1(a)

Formulate the initial-boundary value problem (IBVP) governing the temperature u(x, t)
in a rod of length L, with heat conductivity 1, initial temperature f (x), and time-
dependent boundary temperatures g1 (t) and g2 (t).

IBVP: 
∂u ∂ 2u
= , 0 < x < L, t > 0



 ∂t

 ∂x2
 u(x, 0) = f (x), 0≤x≤L



u(0, t) = g1 (t), u(L, t) = g2 (t), t ≥ 0



Variables and Parameters:

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

• x: Spatial coordinate, 0 ≤ x ≤ L

• t: Time, t ≥ 0

• L: Length of the rod

• f (x): Initial temperature distribution

• g1 (t), g2 (t): Boundary temperatures at x = 0 and x = L




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, Student Name: APM3701 Assignment 2


Question 1(b)

Assume f (x), g1 (t), and g2 (t) are bounded:


m ≤ f (x) ≤ M, m ≤ g1 (t) ≤ M, m ≤ g2 (t) ≤ M


and the solution u(x, t) satisfies:


m ≤ u(x, t) ≤ M, ∀x ∈ [0, L], t ≥ 0


Show that the solution is unique.

Let u1 (x, t) and u2 (x, t) be two solutions. Define:


w(x, t) = u1 (x, t) − u2 (x, t)


Then w satisfies:

∂w ∂ 2w
= , 0 < x < L, t > 0



 ∂t

 ∂x2
 w(x, 0) = 0, 0≤x≤L



w(0, t) = 0, w(L, t) = 0, t ≥ 0




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


Differentiate:
L L
∂ 2w
Z Z
dE ∂w
=2 w dx = 2 w 2 dx
dt 0 ∂t 0 ∂x
Integrating by parts and using boundary conditions:
Z L  2
dE ∂w
= −2 dx ≤ 0
dt 0 ∂x

Since E(0) = 0, we conclude E(t) = 0 ⇒ w(x, t) = 0 for all x, t.




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