Consider the generic transport equation \(\phi_{t}+u \phi_{x}=\mu \phi_{x x}\), where \(u\) is a known function of \(x\) and \(t\) and \(\mu\) is a constant coefficient. Assume that the computational grid is uniform with steps \(\Delta x\) and \(\Delta t\). Write the finite difference schemes satisfying the following requirements.

a) Explicit scheme of the first order in time and second order in space. Use central differences for the space derivatives.

b) The same requirements as in (a), but the scheme is implicit.

c) Scheme of the first order in time. Use implicit central difference approximation for the diffusion term \(\mu \phi_{x x}\) and explicit backward difference approximation for the convection term \(u \phi_{x}\).