2-D poison equation

1 Problem

I need a tutorial on solving the following 2D Poisson equation on the triangle seen

in Figure 1.

∂

2w

∂x2

+

∂

2w

∂y2

= a (1)

The boundary conditions are:

On OA (y = 0) we have ∂w

∂y = 0.

On OB (y = bx) we have c

∂w

∂x + d

∂w

∂y = 0.

On AB (y = ex + f) we have w + g

∂w

∂x + h

∂w

∂y = 0.

Note: a, b, c, d, e, f, g, h are constants.

The solution should be submitted in Latex format.

All necessary steps should be explained briefly (details appropriate for first year

undergraduate level).

Any solution method is accepted as long as it is well known (e.g. separation of

variable with Fourier series).

O

A

B

x

y

Figure 1: Schematic.

1