APM3701 Assignment 2 2026 (Answer Guide) 192483 – Due 17 August 2026

APM3701 Assignment 2 2026 (Answer Guide) 192483 – Due
17 August 2026

APM3701 – Assignment 02 (2026)
Question 1



Question 1(a)

Formulation of the initial–boundary value problem
We consider heat flow in a homogeneous rod of length L, with:

heat conductivity k,
a constant internal heat source of strength A.

Let
u(x, t) = temperature at position x ∈ (0, L) at time t > 0.



Governing equation
For a rod with an internal heat source, the temperature satisfies the non-homogeneous
heat equation:


k
∂t ∂x2

where:

k is the thermal diffusivity,
A represents the rate of heat generation per unit length.



Initial condition
Initially, the rod is submerged in a medium such that the temperature is given by:

u(x, 0) = 1 − sin x, 0≤x≤L

,Boundary conditions (given heat fluxes)
Heat flux at a boundary is proportional to the spatial temperature gradient.
Using Fourier’s law of heat conduction:
∂u
Heat flux = −k
∂x


Left end x = 0
The heat flux is e−t , hence:

, Right end x = L
The heat flux is cos(t − π). Since cos(t − π) = − cos t, we write:




Complete initial–boundary value problem

⎧ ∂u ∂2 u
= + , 0 < x < L, t > 0,
k A
∂t ∂x2
u(x, 0) = 1 − sin x, 0 ≤ x ≤ L,
⎨
∂u −t
−k (0, t) = e , t > 0,
∂x
⎩−k ∂u (L, t) = cos(t − π), t > 0.
∂x

This completes the formulation of the problem.




Question 1(b)

Uniqueness of the solution using the energy method



Step 1: Assume two solutions
Let u1(x, t) and u2(x, t) be two solutions of the problem.
Define their difference:

w(x, t) = u1(x, t) − u2(x, t)



Step 2: PDE satisfied by w
Since both solutions satisfy the same PDE and boundary conditions, w satisfies:

∂w
k
∂t ∂x2