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
∂u
with boundary conditions: - At x = 0 and x = 4: u = 60 - At y = 0 and y = 3: ∂y =5
(a) Taking h = 1, sketch the region and the grid points. Use symmetry to minimize the number of
unknowns ui that have to be calculated and indicate the ui in the sketch.
(b) Use the 5-point difference formula for the Laplace operator to derive a system of equations for
the ui .
Part (a): Sketch the Region and Grid Points, Use Symmetry
Step 1: Define the Domain and Grid
The domain is a rectangle defined by 0 < x < 4 and 0 < y < 3. With a step size h = 1: - For x-direction:
x = 0, 1, 2, 3, 4 (5 points, since 4/1 + 1 = 5). - For y-direction: y = 0, 1, 2, 3 (4 points, since 3/1 + 1 = 4).
This forms a 5 × 4 grid with points at (xi , yj ) = (i, j), where i = 0, 1, 2, 3, 4 and j = 0, 1, 2, 3.
Step 2: Apply Boundary Conditions
- At x = 0 and x = 4: u = 60. This applies to points (0, j) and (4, j) for j = 0, 1, 2, 3. - At y = 0 and
∂u
y = 3: ∂y = 5 (Neumann condition), which will be approximated later.
Step 3: Identify Symmetry
The boundary conditions at x = 0 and x = 4 are identical (u = 60), suggesting symmetry about x = 2.
The PDE yuxx − 2∇2 u = 12 has a symmetric structure in x if u(x, y) = u(4 − x, y). The term yuxx
depends on x-derivatives, and ∇2 u = uxx + uyy is symmetric about x = 2 given the boundary conditions.
With symmetry, u(1, y) = u(3, y). Interior points are at x = 1, 2, 3 and y = 1, 2. Unknowns are: -
u1,1 = u(1, 1), u1,2 = u(1, 2) - u2,1 = u(2, 1), u2,2 = u(2, 2) - u3,1 = u1,1 , u3,2 = u1,2
Thus, 4 unknowns: u1,1 , u1,2 , u2,1 , u2,2 .
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
∂u
with boundary conditions: - At x = 0 and x = 4: u = 60 - At y = 0 and y = 3: ∂y =5
(a) Taking h = 1, sketch the region and the grid points. Use symmetry to minimize the number of
unknowns ui that have to be calculated and indicate the ui in the sketch.
(b) Use the 5-point difference formula for the Laplace operator to derive a system of equations for
the ui .
Part (a): Sketch the Region and Grid Points, Use Symmetry
Step 1: Define the Domain and Grid
The domain is a rectangle defined by 0 < x < 4 and 0 < y < 3. With a step size h = 1: - For x-direction:
x = 0, 1, 2, 3, 4 (5 points, since 4/1 + 1 = 5). - For y-direction: y = 0, 1, 2, 3 (4 points, since 3/1 + 1 = 4).
This forms a 5 × 4 grid with points at (xi , yj ) = (i, j), where i = 0, 1, 2, 3, 4 and j = 0, 1, 2, 3.
Step 2: Apply Boundary Conditions
- At x = 0 and x = 4: u = 60. This applies to points (0, j) and (4, j) for j = 0, 1, 2, 3. - At y = 0 and
∂u
y = 3: ∂y = 5 (Neumann condition), which will be approximated later.
Step 3: Identify Symmetry
The boundary conditions at x = 0 and x = 4 are identical (u = 60), suggesting symmetry about x = 2.
The PDE yuxx − 2∇2 u = 12 has a symmetric structure in x if u(x, y) = u(4 − x, y). The term yuxx
depends on x-derivatives, and ∇2 u = uxx + uyy is symmetric about x = 2 given the boundary conditions.
With symmetry, u(1, y) = u(3, y). Interior points are at x = 1, 2, 3 and y = 1, 2. Unknowns are: -
u1,1 = u(1, 1), u1,2 = u(1, 2) - u2,1 = u(2, 1), u2,2 = u(2, 2) - u3,1 = u1,1 , u3,2 = u1,2
Thus, 4 unknowns: u1,1 , u1,2 , u2,1 , u2,2 .
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