1) Derive the following fourth order central finite difference schemes:
$\left(\frac{\partial \phi}{\partial x}\right)_{i,j} \approx \frac{-\phi_{i+2,j} + 8\phi_{i+1,j} - 8\phi_{i-1,j} + \phi_{i-2,j}}{12\Delta x}$
$\left(\frac{\partial^2 \phi}{\partial x^2}\right)_{i,j} \approx \frac{-\phi_{i+2,j} + 16\phi_{i+1,j} - 30\phi_{i,j} + 16\phi_{i-1,j} - \phi_{i-2,j}}{12\Delta x^2}$