UNIVERSITY OF SOUTH AFRICA
College of Science, Engineering and Technology
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MAT4847: Partial Differential Equations I
Assignment 02 — Question 3 — Year Module 2026
⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄⋄
MAT4847
Module Code:
Partial Differential Equations I
Module Name:
Heat Flow in a Rod (Question 3)
Topic:
02
Assignment Number:
196817
Unique Number:
07 September 2026
Due Date:
25
Total Marks (Q3):
Submitted in partial fulfilment of the requirements for MAT4847 — UNISA 2026
, UNISA | MAT4847 Assignment 02 — Question 3
Question 3(a): Initial Boundary Value Problem for Heat Flow in a Rod
Question: Consider the heat flow in a horizontal rod of length p units and heat conductivity
k. If initially, the left half of the rod is in contact with ice at 0◦ C, and the right half of the rod
is at the air temperature A◦ C, write down the initial boundary value problem that is satisfied
by the rod, if both ends are isolated. Explain the meaning of every constant and variable. [10
Marks]
3.1 The Heat Equation (Governing PDE)
The temperature distribution in a thin homogeneous rod of length p is governed by the one-
dimensional heat equation. This equation models how heat diffuses through the rod over time
in the absence of internal heat sources (Haberman, 2013:43). Let u(x, t) denote the temper-
ature at position x along the rod at time t > 0. The governing partial differential equation
is:
∂u ∂2u
= k 2, 0 < x < p, t > 0, (1)
∂t ∂x
where k > 0 is the thermal diffusivity (heat conductivity constant) of the rod.
3.2 Boundary Conditions
Since both ends of the rod are thermally insulated (isolated), no heat flows through either end-
point. Insulation means the temperature gradient (rate of change of temperature with respect
to position) is zero at each end. This yields the two Neumann boundary conditions:
∂u
(0, t) = 0, t > 0, (2)
∂x
∂u
(p, t) = 0, t > 0. (3)
∂x
Page 2 of 11
College of Science, Engineering and Technology
⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄⋄
MAT4847: Partial Differential Equations I
Assignment 02 — Question 3 — Year Module 2026
⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄⋄
MAT4847
Module Code:
Partial Differential Equations I
Module Name:
Heat Flow in a Rod (Question 3)
Topic:
02
Assignment Number:
196817
Unique Number:
07 September 2026
Due Date:
25
Total Marks (Q3):
Submitted in partial fulfilment of the requirements for MAT4847 — UNISA 2026
, UNISA | MAT4847 Assignment 02 — Question 3
Question 3(a): Initial Boundary Value Problem for Heat Flow in a Rod
Question: Consider the heat flow in a horizontal rod of length p units and heat conductivity
k. If initially, the left half of the rod is in contact with ice at 0◦ C, and the right half of the rod
is at the air temperature A◦ C, write down the initial boundary value problem that is satisfied
by the rod, if both ends are isolated. Explain the meaning of every constant and variable. [10
Marks]
3.1 The Heat Equation (Governing PDE)
The temperature distribution in a thin homogeneous rod of length p is governed by the one-
dimensional heat equation. This equation models how heat diffuses through the rod over time
in the absence of internal heat sources (Haberman, 2013:43). Let u(x, t) denote the temper-
ature at position x along the rod at time t > 0. The governing partial differential equation
is:
∂u ∂2u
= k 2, 0 < x < p, t > 0, (1)
∂t ∂x
where k > 0 is the thermal diffusivity (heat conductivity constant) of the rod.
3.2 Boundary Conditions
Since both ends of the rod are thermally insulated (isolated), no heat flows through either end-
point. Insulation means the temperature gradient (rate of change of temperature with respect
to position) is zero at each end. This yields the two Neumann boundary conditions:
∂u
(0, t) = 0, t > 0, (2)
∂x
∂u
(p, t) = 0, t > 0. (3)
∂x
Page 2 of 11
