APM3701
ASSIGNMENT 1 2025
UNIQUE NO. 608471
DUE DATE: 29 MAY 2025
, lOMoARcPSD|18222662
APM3701/102
QUESTION 1
Solve the following (initial)-boundary value problem,
a.
uxy (x, y) = xy 3 , x, y ≥ 0.
u (x, 0) = f (x) , and uy (0, y) = g (y) .
Determine u (x, y) , if f (x) = cosx and g (y) = y +sin y. (Check your answer by substituting,
and explain all the steps clearly) (15 Marks)
b.
xu x + yu y = yu
(1)
u (2x2, x) = x 2 − 1.
(Check your answer by substituting before applying the boundary condition, and ex-
plain all the steps clearly) (15
Marks)
[30 Marks]
QUESTION 2
Consider the heat flow in a rod of length a unit and heat conductivity c.
a. If initially the rod was submerged in a fluid which is kept at a temperature K ◦
C, write down the
initial boundary value problem that is satisfied by the rod, if the heat flux is of k unit at the left
end and A units at the right end. (Explain the meaning of every constant and variable).
(5 Marks)
b. Determine the temperature of the rod at any point x of the rod at time t > 0. (Explain all the
steps). (20 Marks)
[25 Marks]
QUESTION 3
Consider the temperature distribution in a rectangular plate as described in the figure below. De-
termine the steady-state temperature distribution u (x, y) in the plate. Calculate the steady-sate
temperature at the centre of the plate. [25 Marks]
Ice at 00
W
Ice at 00 Ice at 00
L
Ambiant temperature
5
ASSIGNMENT 1 2025
UNIQUE NO. 608471
DUE DATE: 29 MAY 2025
, lOMoARcPSD|18222662
APM3701/102
QUESTION 1
Solve the following (initial)-boundary value problem,
a.
uxy (x, y) = xy 3 , x, y ≥ 0.
u (x, 0) = f (x) , and uy (0, y) = g (y) .
Determine u (x, y) , if f (x) = cosx and g (y) = y +sin y. (Check your answer by substituting,
and explain all the steps clearly) (15 Marks)
b.
xu x + yu y = yu
(1)
u (2x2, x) = x 2 − 1.
(Check your answer by substituting before applying the boundary condition, and ex-
plain all the steps clearly) (15
Marks)
[30 Marks]
QUESTION 2
Consider the heat flow in a rod of length a unit and heat conductivity c.
a. If initially the rod was submerged in a fluid which is kept at a temperature K ◦
C, write down the
initial boundary value problem that is satisfied by the rod, if the heat flux is of k unit at the left
end and A units at the right end. (Explain the meaning of every constant and variable).
(5 Marks)
b. Determine the temperature of the rod at any point x of the rod at time t > 0. (Explain all the
steps). (20 Marks)
[25 Marks]
QUESTION 3
Consider the temperature distribution in a rectangular plate as described in the figure below. De-
termine the steady-state temperature distribution u (x, y) in the plate. Calculate the steady-sate
temperature at the centre of the plate. [25 Marks]
Ice at 00
W
Ice at 00 Ice at 00
L
Ambiant temperature
5
