APM3701
Assignment 2
Unique No: 192483
Due 17 August 2026
,Assignment 02 – Question 1(a)
Q: Write down the initial–boundary value problem satisfied by the
temperature 𝑢(𝑥, 𝑡) in a homogeneous rod of length 𝐿 with heat conductivity
𝑘, constant heat source 𝐴, initial temperature 1 − sin𝑥, heat flux 𝑒 −𝑡 at the
left end and cos(𝑡 − 𝜋) at the right end.
A:
Let 𝑢(𝑥, 𝑡) be the temperature at position 𝑥 and time 𝑡, where
0 < 𝑥 < 𝐿, 𝑡 > 0.
Since the rod has heat conductivity 𝑘 and a constant internal heat source 𝐴, the heat
equation becomes
𝑢𝑡 = 𝑘𝑢 𝑥𝑥 + 𝐴, 0 < 𝑥 < 𝐿, 𝑡 > 0.
Initial condition:
The rod is initially in a medium with temperature distribution
𝑢(𝑥, 0) = 1 − sin𝑥, 0 < 𝑥 < 𝐿.
Boundary conditions:
Heat flux at the ends is prescribed. Using Fourier’s law, heat flux is proportional to −𝑢𝑥 .
Left end 𝑥 = 0 : flux = 𝑒 −𝑡 ,
−𝑘𝑢 𝑥 (0, 𝑡) = 𝑒 −𝑡 .
Right end 𝑥 = 𝐿 : flux = cos(𝑡 − 𝜋),
−𝑘𝑢 𝑥 (𝐿, 𝑡) = cos(𝑡 − 𝜋).
Meaning of variables and parameters:
, 𝑢(𝑥, 𝑡) = temperature in the rod
𝑥 = position along the rod 0 ≤ 𝑥 ≤ 𝐿
𝑡 = time
𝑘 = thermal conductivity constant
𝐴 = constant internal heat source
𝑒 −𝑡 , cos(𝑡 − 𝜋) = prescribed heat flux at the ends
Thus the required initial–boundary value problem is
𝑢𝑡 = 𝑘𝑢 𝑥𝑥 + 𝐴, 0 < 𝑥 < 𝐿, 𝑡 > 0, 𝑢(𝑥, 0) = 1 − sin𝑥, −𝑘𝑢𝑥 (0, 𝑡) = 𝑒 −𝑡 , −𝑘𝑢 𝑥 (𝐿, 𝑡) =
cos(𝑡 − 𝜋).
Assignment 2
Unique No: 192483
Due 17 August 2026
,Assignment 02 – Question 1(a)
Q: Write down the initial–boundary value problem satisfied by the
temperature 𝑢(𝑥, 𝑡) in a homogeneous rod of length 𝐿 with heat conductivity
𝑘, constant heat source 𝐴, initial temperature 1 − sin𝑥, heat flux 𝑒 −𝑡 at the
left end and cos(𝑡 − 𝜋) at the right end.
A:
Let 𝑢(𝑥, 𝑡) be the temperature at position 𝑥 and time 𝑡, where
0 < 𝑥 < 𝐿, 𝑡 > 0.
Since the rod has heat conductivity 𝑘 and a constant internal heat source 𝐴, the heat
equation becomes
𝑢𝑡 = 𝑘𝑢 𝑥𝑥 + 𝐴, 0 < 𝑥 < 𝐿, 𝑡 > 0.
Initial condition:
The rod is initially in a medium with temperature distribution
𝑢(𝑥, 0) = 1 − sin𝑥, 0 < 𝑥 < 𝐿.
Boundary conditions:
Heat flux at the ends is prescribed. Using Fourier’s law, heat flux is proportional to −𝑢𝑥 .
Left end 𝑥 = 0 : flux = 𝑒 −𝑡 ,
−𝑘𝑢 𝑥 (0, 𝑡) = 𝑒 −𝑡 .
Right end 𝑥 = 𝐿 : flux = cos(𝑡 − 𝜋),
−𝑘𝑢 𝑥 (𝐿, 𝑡) = cos(𝑡 − 𝜋).
Meaning of variables and parameters:
, 𝑢(𝑥, 𝑡) = temperature in the rod
𝑥 = position along the rod 0 ≤ 𝑥 ≤ 𝐿
𝑡 = time
𝑘 = thermal conductivity constant
𝐴 = constant internal heat source
𝑒 −𝑡 , cos(𝑡 − 𝜋) = prescribed heat flux at the ends
Thus the required initial–boundary value problem is
𝑢𝑡 = 𝑘𝑢 𝑥𝑥 + 𝐴, 0 < 𝑥 < 𝐿, 𝑡 > 0, 𝑢(𝑥, 0) = 1 − sin𝑥, −𝑘𝑢𝑥 (0, 𝑡) = 𝑒 −𝑡 , −𝑘𝑢 𝑥 (𝐿, 𝑡) =
cos(𝑡 − 𝜋).
