APM3701 Assignment 2 (700123) 2025 (Correct Solutions) Due 8 August 2025

APM3701
Assignment 2
Unique No: 700123
Due 8 August 2025

, APM3701 Assignment 2 — Due: Friday, 8 August 2025




APM3701 Assignment 2
Due Date: Friday, 8 August 2025


Heat Flow Problem

Question 1(a)

Formulate the initial–boundary value problem (IBVP) for the temperature distribution
u(x, t) along a homogeneous rod of length L with thermal conductivity equal to 1, initial
temperature f (x), and prescribed time-dependent boundary temperatures g1 (t) and g2 (t).

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


Variables and Parameters:

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

• x: Spatial coordinate along the rod, 0 ≤ x ≤ L.

• t: Time variable, t ≥ 0.

• L: Length of the rod (constant).

• f (x): Prescribed initial temperature distribution.

• g1 (t), g2 (t): Prescribed boundary temperatures at x = 0 and x = L respectively.




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, APM3701 Assignment 2 — Due: Friday, 8 August 2025


Question 1(b)

Assume f , g1 , and g2 are uniformly bounded:


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


and that the solution satisfies


m ≤ u(x, t) ≤ M, ∀x ∈ [0, L], ∀t ≥ 0.


We prove uniqueness of the solution.

Let u1 and u2 be two solutions to the IBVP, and define


w(x, t) = u1 (x, t) − u2 (x, t).


Then w satisfies the homogeneous problem

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



Consider the energy functional
Z L
E(t) = w2 (x, t) dx.
0


Differentiating with respect to t:

L L
∂ 2w
Z Z
dE ∂w
=2 w dx = 2 w dx.
dt 0 ∂t 0 ∂x2

Integrating by parts and using the homogeneous Dirichlet conditions yields:
Z L  2
dE ∂w
= −2 dx ≤ 0.
dt 0 ∂x




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