Write a python code for both the problems. The Poisson Equation

The Poisson equation in two dimensions is written as follows:

$gra{d}^2(x,y)=\frac{de{l}^2}{del{x}^2}+\frac{de{l}^2}{del{y}^2}=f(x,y)$

where $f(x,y)$ is a known function. If $f(x,y)=0,$ then we have the Laplace

equation:

$gra{d}^2(x,y)=\frac{de{l}^2}{del{x}^2}+\frac{de{l}^2}{del{y}^2}=0$

To solve Equations $(1)$ and $(2),$ we need to know the boundary conditions

at the border of the domain. There are three types of boundary conditions:

Dirichlet: We know the value of $$ at the border.

Neumann: We know the normal derivative of $$ at the border.

Mixed Dirichlet and Neumann.

If only Neumann boundary conditions are imposed, the problem is ill$-$posed.

Find a problem that involves the Poisson and Laplace equations in a rect$-$

angular domain and solve it using the Finite Difference Method.

$($b$)$ Laplace Equation

$($a$)$ Square domain with a hole inside

$($b$)$ Laplace Equation

$($c$)$ Mesh representation

Figure $2$: Problem $2$

Figure $1$: Problem $1$