How does tyre pressure affect the frictional force on a tyre even though friction is independent of area of contact? And how does one analyze or calculate this friction?
As the other answer says, tires are complicated by having dynamical changes in surface that depend on the pressure, which change the effective pressure coefficient.
What happens when the tire is flattened a larger area takes up the normal force , so normal per unit area goes down. The fact is that a flat tire does not roll, so kinetic friction is important .
New models are beginning to show how kinetic friction can be greater than static friction. Kinetic friction is now understood, in many cases, to be primarily caused by chemical bonding between the surfaces, rather than interlocking asperities; however, in many other cases roughness effects are dominant, for example in rubber to road friction.
Here is an engineering table with the caveats:
The coefficient of friction between two materials in relative sliding may depend on contact pressure, surface roughness of the relative harder contact surface, temperature, sliding velocity and the type of lubricant whether the level of contamination. It's the reason that the data found in the many reference tables available may show a large variation.
The rolling coefficient for cars depends additively and inversely on pressure:
c = 0.005 + 1/p (0.01 + 0.0095(v/100)^2)
v velocity km/h , p tire pressure in bars
So it is an empirical formula.I would guess that the higher the pressure, the smaller the effective area of contact, which makes the friction coefficient pressure dependent due to the changes on the type of contact, i.e. the type of contact surface-surface changes with the change in pressure and effectively gives the formula above.
For a general discussion, from CarTalk, see http://www.cartalk.com/content/service-your-car-13
They repeat the "low tire pressure increases surface contact", as stated in the answer by kpv. Rolling resistance does increase with low tire pressure.
Tires are quite complicated, and involve many engineering trade-offs; you cannot use rigid body notions in order to evaluate a tire!
For the case at hand, low tire pressure, the immediate problem is excess flexing, especially in the sidewalls, which results in accelerated wear. Heating is increased by at least two of the new conditions: the flexing, and the reduction in the exposed surface area due to the surface contact.