...to your question about an analytical solution to the equations of airgun interior ballistics. Unfortunately, I don't.
However, if "(un)reasonable approximations," interest you, here're a couple.
The "average" driving pressure at the breech required to achieve a given muzzle velocity is determined by...
2. Barrel length.
3. Pellet weight.
4. Propellant density.
Here's the semi-empirical formula I've had some predictive success with...
Feet_per_second = 172 x sqrt(Q / (grains + Z x Q))
Where: Q = Pavg x barrel_inches x caliber^2
Pavg = average driving pressure in psi
Z = 0.011 for air, 0.02 for CO2, 0.002 for helium.
In a PCP, the generation of the driving pressure occurs in two phases...
1. "Isobaric" phase = the interval while the firing valve is open and connecting the breech to the reservoir, maintaining (approximate) constant pressure.
2. "Adiabatic" phase = after the valve closes and the dispensed charge expands (and cools) behind the pellet, causing an exponential drop in pressure proportional (for a - mostly - diatomic gas like air) to volume1.4
If there's a close-form solution to this mish-mash, I have no idea what it would look like. But simple iterative solutions converge rapidly and seem to lead to fairly realistic results.