UNIVERSITY OF SOUTH AFRICA (UNISA)
Department of Mathematical Sciences
⋄
ASSIGNMENT 02
Year Module — 2026
⋄
Module Code: MAT4847
Module Name: Partial Differential Equations I
Assignment No.: 02
Unique Number: 196817
Due Date: 07 September 2026
Year: 2026
Submitted in partial fulfilment of the requirements for MAT4847
at the University of South Africa.
, UNISA | MAT4847 Assignment 02 — Heat Equation
Question 3: Heat Flow in a Rod with Insulated Ends
Full Question: Consider the heat flow in a horizontal rod of length p units and heat conduc-
tivity k.
(a) 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]
(b) Determine the temperature of the rod at any point x of the rod at time t > 0.
(Explain all the steps). [15 Marks]
3(a) Formulation of the Initial Boundary Value Problem
The temperature distribution in a uniform rod is governed by the one-dimensional heat (diffu-
sion) equation. Let u(x, t) denote the temperature at position x along the rod at time t. The
governing partial differential equation is:
∂u ∂2u
= k 2, 0 < x < p, t > 0. (1)
∂t ∂x
Boundary Conditions
Since both ends of the rod are thermally insulated (isolated), there is no heat flux
through either endpoint. By Fourier’s law of heat conduction, zero heat flux means zero tem-
perature gradient at each end. This gives the Neumann (insulated) boundary conditions:
∂u
(0, t) = 0, t > 0, (2)
∂x
∂u
(p, t) = 0, t > 0. (3)
∂x
Page 1 of 9
Department of Mathematical Sciences
⋄
ASSIGNMENT 02
Year Module — 2026
⋄
Module Code: MAT4847
Module Name: Partial Differential Equations I
Assignment No.: 02
Unique Number: 196817
Due Date: 07 September 2026
Year: 2026
Submitted in partial fulfilment of the requirements for MAT4847
at the University of South Africa.
, UNISA | MAT4847 Assignment 02 — Heat Equation
Question 3: Heat Flow in a Rod with Insulated Ends
Full Question: Consider the heat flow in a horizontal rod of length p units and heat conduc-
tivity k.
(a) 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]
(b) Determine the temperature of the rod at any point x of the rod at time t > 0.
(Explain all the steps). [15 Marks]
3(a) Formulation of the Initial Boundary Value Problem
The temperature distribution in a uniform rod is governed by the one-dimensional heat (diffu-
sion) equation. Let u(x, t) denote the temperature at position x along the rod at time t. The
governing partial differential equation is:
∂u ∂2u
= k 2, 0 < x < p, t > 0. (1)
∂t ∂x
Boundary Conditions
Since both ends of the rod are thermally insulated (isolated), there is no heat flux
through either endpoint. By Fourier’s law of heat conduction, zero heat flux means zero tem-
perature gradient at each end. This gives the Neumann (insulated) boundary conditions:
∂u
(0, t) = 0, t > 0, (2)
∂x
∂u
(p, t) = 0, t > 0. (3)
∂x
Page 1 of 9
