APM3701
Assignment 2
Unique No: 192483
Due 17 August 2026
,Assignment 02
Question 1(a)
Q
Consider heat flow in a homogeneous rod of length 𝐿 with heat conductivity 𝑘 and
constant energy source 𝐴.
Initially the temperature is
𝑢(𝑥, 0) = 1 − sin𝑥
Heat flux at the ends:
• Left end: 𝑒 −𝑡
• Right end: cos(𝑡 − 𝜋)
Write down the initial–boundary value problem and explain all variables and
parameters.
A
This is a 1D heat equation with a source term and nonhomogeneous Neumann
boundary conditions.
Governing equation
Since there is a constant internal energy source 𝐴,
𝑢𝑡 = 𝑘𝑢 𝑥𝑥 + 𝐴, 0 < 𝑥 < 𝐿, 𝑡 > 0
, Meaning of symbols
• 𝑢(𝑥, 𝑡) temperature at position 𝑥 and time 𝑡
• 𝑥 ∈ [0, 𝐿] position along rod
• 𝑡 > 0 time
• 𝑘 thermal diffusivity
• 𝐴 constant internal heat source
• 𝐿 rod length
Initial condition
𝑢(𝑥, 0) = 1 − sin𝑥
Boundary conditions
Heat flux is proportional to negative gradient:
Flux = −𝑘𝑢 𝑥
Left end 𝑥 = 0
𝑒 −𝑡
−𝑘𝑢 𝑥 (0, 𝑡) = 𝑒 −𝑡 𝑢𝑥 (0, 𝑡) = − 𝑘
Right end 𝑥 = 𝐿
Given flux cos(𝑡 − 𝜋).
Note:
Assignment 2
Unique No: 192483
Due 17 August 2026
,Assignment 02
Question 1(a)
Q
Consider heat flow in a homogeneous rod of length 𝐿 with heat conductivity 𝑘 and
constant energy source 𝐴.
Initially the temperature is
𝑢(𝑥, 0) = 1 − sin𝑥
Heat flux at the ends:
• Left end: 𝑒 −𝑡
• Right end: cos(𝑡 − 𝜋)
Write down the initial–boundary value problem and explain all variables and
parameters.
A
This is a 1D heat equation with a source term and nonhomogeneous Neumann
boundary conditions.
Governing equation
Since there is a constant internal energy source 𝐴,
𝑢𝑡 = 𝑘𝑢 𝑥𝑥 + 𝐴, 0 < 𝑥 < 𝐿, 𝑡 > 0
, Meaning of symbols
• 𝑢(𝑥, 𝑡) temperature at position 𝑥 and time 𝑡
• 𝑥 ∈ [0, 𝐿] position along rod
• 𝑡 > 0 time
• 𝑘 thermal diffusivity
• 𝐴 constant internal heat source
• 𝐿 rod length
Initial condition
𝑢(𝑥, 0) = 1 − sin𝑥
Boundary conditions
Heat flux is proportional to negative gradient:
Flux = −𝑘𝑢 𝑥
Left end 𝑥 = 0
𝑒 −𝑡
−𝑘𝑢 𝑥 (0, 𝑡) = 𝑒 −𝑡 𝑢𝑥 (0, 𝑡) = − 𝑘
Right end 𝑥 = 𝐿
Given flux cos(𝑡 − 𝜋).
Note:
