APM3711
ASSIGNMENT 04
Due date: 11 September 2025
, 1 Question 1
Problem Statement
Consider the partial differential equation:
yuxx − 2∇2 u = 12, 0 < x < 4, 0<y<3
with boundary conditions:
• At x = 0 and x = 4: u = 60
• At y = 0 and y = 3: ∂u
∂y
=5
Tasks:
(a) Taking h = 1, sketch the region and the grid points. Use symmetry to minimize
the number of unknowns ui and indicate them in the sketch.
(b) Use the 5-point difference formula for the Laplacian to derive a system of equations
for the ui .
Part (a): Grid and Symmetry Analysis
Domain and Grid Construction
The domain is a rectangle: 0 < x < 4, 0 < y < 3. With h = 1:
• In x-direction: x = 0, 1, 2, 3, 4 (5 grid points)
• In y-direction: y = 0, 1, 2, 3 (4 grid points)
This results in a 5 × 4 grid.
1
ASSIGNMENT 04
Due date: 11 September 2025
, 1 Question 1
Problem Statement
Consider the partial differential equation:
yuxx − 2∇2 u = 12, 0 < x < 4, 0<y<3
with boundary conditions:
• At x = 0 and x = 4: u = 60
• At y = 0 and y = 3: ∂u
∂y
=5
Tasks:
(a) Taking h = 1, sketch the region and the grid points. Use symmetry to minimize
the number of unknowns ui and indicate them in the sketch.
(b) Use the 5-point difference formula for the Laplacian to derive a system of equations
for the ui .
Part (a): Grid and Symmetry Analysis
Domain and Grid Construction
The domain is a rectangle: 0 < x < 4, 0 < y < 3. With h = 1:
• In x-direction: x = 0, 1, 2, 3, 4 (5 grid points)
• In y-direction: y = 0, 1, 2, 3 (4 grid points)
This results in a 5 × 4 grid.
1
