Numerical Method - Old Questions

12. Solve the Poisson’s equation ∇ ² f=2x² y ² over the square domain 0 <= x <= 3 and 0 <= y <= 3 with f=0 on the boundary and h=1.

Given Poisson’s eqn. is

h = 1

Let’s divide the domain into grids of 3 x 3 with f = 0 at the boundary as below

Now, from the difference equation for the Poisson’s equation,

Solving these equations,

Using eq.(ii) in (iii)

Using eq.(ii) in (iv)

& we have eq. (i)

Solving equation (a), (b) & (i) we get,

Using these values in eq. (ii)