While the answers above are great, I Felt it was lacking what The question asked regarding an equation analogous to the Schrodinger (or Dirac) equation.
There is a quantity called the Riemann-Silberstein vector ( https://en.wikipedia.org/wiki/Riemann%E2%80%93Silberstein_vector#Photon_wave_function ), First used by infamous Bernhardt Riemann to demonstrate a concise formulation of Maxwell's equations.
This “vector” has the form:
$$\vec{F}=\vec{E}+ic\vec{B}$$
A quick search online, demonstrates that classical electrodynamics written in this form can be quite useful in solving problems.
In the quantum realm, a quantity analogous to the wavefunction can be written for a single photon. Such a quantity has the form:
$$i\hbar\partial_{t}\vec{F}=c\left(\vec{S}\cdot\frac{\hbar}{i}\vec{\nabla}\right)\vec{F}$$ Which can be written simply in the form:
$$i\hbar\partial_{t}\vec{F}=c\left(\vec{S}\cdot\hat{P}\right)\vec{F}$$
This can be a useful quantity for examining the properties of a single photon. Start with the Wikipedia page, it's actually quite an interesting and useful quantity.