I have written the following Mathematica codes to solve the LAPLACE equation. using the finite difference method.

```
In(1):= Remove(a, b, Nx, Ny, h, xgrid, ygrid, u, i, j)
a = 0; b = 0.5; n = 4;
h = (b - a)/n;
xgrid = Table(x(i) -> a + i h, {i, 1, n});
ygrid = Table(y(j) -> a + j h, {j, 1, n});
eqnstemplate = {-4 u(i, j) + u(i + 1, j) + u(i - 1, j) + u(i, j - 1) +
u(i, j + 1) == 0};
BC1 = Table(u(i, 0) == 0, {i, 1, n - 1});
BC2 = Table(u(i, 4) == 200 x(i), {i, 1, n - 1});
BC3 = Table(u(4, j) == 200 y(j), {j, 2, n - 1});
BC4 = Table(u(0, j) == 0, {j, 2, n - 1});
Eqns = Table(eqnstemplate, {i, 1, n - 1}, {j, 1, n - 1}) /. xgrid /.
ygrid // Flatten;
systemEqns = Join(Eqns, BC1, BC2, BC3, BC4) /. xgrid /. ygrid
Out(12)= {u(0, 1) + u(1, 0) - 4 u(1, 1) + u(1, 2) + u(2, 1) == 0,
u(0, 2) + u(1, 1) - 4 u(1, 2) + u(1, 3) + u(2, 2) == 0,
u(0, 3) + u(1, 2) - 4 u(1, 3) + u(1, 4) + u(2, 3) == 0,
u(1, 1) + u(2, 0) - 4 u(2, 1) + u(2, 2) + u(3, 1) == 0,
u(1, 2) + u(2, 1) - 4 u(2, 2) + u(2, 3) + u(3, 2) == 0,
u(1, 3) + u(2, 2) - 4 u(2, 3) + u(2, 4) + u(3, 3) == 0,
u(2, 1) + u(3, 0) - 4 u(3, 1) + u(3, 2) + u(4, 1) == 0,
u(2, 2) + u(3, 1) - 4 u(3, 2) + u(3, 3) + u(4, 2) == 0,
u(2, 3) + u(3, 2) - 4 u(3, 3) + u(3, 4) + u(4, 3) == 0, u(1, 0) == 0,
u(2, 0) == 0, u(3, 0) == 0, u(1, 4) == 25., u(2, 4) == 50.,
u(3, 4) == 75., u(4, 2) == 50., u(4, 3) == 75., u(0, 2) == 0,
u(0, 3) == 0}
```

I need this to be in the form of a matrix only for the unknown variable with the substitution of the known value to obtain the linear system that arises from the resolution of the equation of laplacses.