Here it has been discussed why the moon is moving away from earth at a rate of 1.5 inches per year. Is there any way we can estimate the value (e.g, 1.5 inches per year) mathematically?
This is such a complex problem that I think your chances of getting an accurate answer are almost nil.
First, the equations are likely to be highly complex if you want to be accurate down to the tenth of an inch. For example, you'd need to model the exact shape of the Earth due to the Moon's tides, most likely accurate down to the meter or so. Scientists have done this on average, but finding how these types of models minutely change on an hourly basis is near impossible. The data just doesn't exist.
Second, filling in the variables you'll have in your equations is likely going to be impossible because many won't be known. For example, do you know the flexure/elasticity/tensile strength/etc. of the entire Earth's crust at every point on Earth over time? My guess is you don't. You'd need to know that though to do really accurate calculations.
The only way we know the recession rate of the Moon is through measurements. Crude models might get you crude order of magnitude numbers, but to match the precision of observations would require models that are far more informed than we currently are.
Easy. Bounce a laser off the moon at a specific time of day and do this everyday for years and years. You could in fact get the distance it moves away from earth down to the day just by the time it takes for the laser light to come back to your device.
Multiply the time it takes for the light to come back with the speed of light and divide by 2. Note down the time each day and check the differences over time. You'll be able to get the change of rate down to the day.
Heck, we could even fly back to the moon, house a computer with some batteries and a few solar panels to charge them and ping it. On the trip back just drop off a few satellites 22,000 miles apart back to earth to boost signal so it could connect to the internet lol. Fun aside.