APM3701 ASSIGNMENT 02 SOLUTIONS 2026
UNISA
Module: Partial Differential Equations
TAKE NOTE PLEASE
, APM3701 – ASSIGNMENT 02 SOLUTIONS
Module: Partial Differential Equations
Assignment Unique Number: 192483
QUESTION 1
1(a) Initial–Boundary Value Problem
Let:
• 𝑢(𝑥, 𝑡)= temperature at position 𝑥and time 𝑡
• 𝐿= length of the rod
• 𝑘= thermal diffusivity (heat conductivity constant)
• 𝐴= constant internal heat source
• 0 < 𝑥 < 𝐿, 𝑡 > 0
The heat equation with a constant heat source is
∂𝑢 ∂2 𝑢
=𝑘 2+𝐴
∂𝑡 ∂𝑥
Thus,
𝑢𝑡 = 𝑘𝑢𝑥𝑥 + 𝐴, 0 < 𝑥 < 𝐿, 𝑡 > 0
Initial condition
Initially,
𝑢(𝑥, 0) = 1 − sin 𝑥
UNISA
Module: Partial Differential Equations
TAKE NOTE PLEASE
, APM3701 – ASSIGNMENT 02 SOLUTIONS
Module: Partial Differential Equations
Assignment Unique Number: 192483
QUESTION 1
1(a) Initial–Boundary Value Problem
Let:
• 𝑢(𝑥, 𝑡)= temperature at position 𝑥and time 𝑡
• 𝐿= length of the rod
• 𝑘= thermal diffusivity (heat conductivity constant)
• 𝐴= constant internal heat source
• 0 < 𝑥 < 𝐿, 𝑡 > 0
The heat equation with a constant heat source is
∂𝑢 ∂2 𝑢
=𝑘 2+𝐴
∂𝑡 ∂𝑥
Thus,
𝑢𝑡 = 𝑘𝑢𝑥𝑥 + 𝐴, 0 < 𝑥 < 𝐿, 𝑡 > 0
Initial condition
Initially,
𝑢(𝑥, 0) = 1 − sin 𝑥
