APM3701 Assignment 2 Exceptional Solutions Due 8 August 2025

APM3701
Assignment 2
Due 8 August 2025

, APM3701

Assignment 2: Step-by-Step Calculations Provided.

Due 8 August 2025.

Question 1: Heat Flow in a Horizontal Rod

(a) Initial-Boundary Value Problem (IBVP) for Temperature Distribution

The heat flow in a horizontal rod of length L with heat conductivity 1 is modeled by the
one-dimensional heat equation, describing the temperature distribution u(x,t) at position x
∈ [0,L] and time t ≥ 0. The rod’s initial temperature is given by f(x), and the temperatures at
the left (x = 0) and right (x = L) ends vary with time as g1(t) and g2(t), respectively.
The heat equation, derived from Fourier’s law and energy conservation, is:

0 (1)
Initial Condition: At t = 0, the temperature across the rod is:

u(x,0) = f(x), 0≤x≤L (2)
Boundary Conditions: The temperatures at the ends are prescribed as:
u(0,t) = g1(t), u(L,t) = g2(t), t ≥ 0 (3)


Variables and Parameters:

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

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

• t: Time, t ≥ 0.

• L: Length of the rod.

• f(x): Initial temperature distribution, a bounded function.

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

• Heat conductivity is 1, simplifying the diffusion coefficient.