American options are much harder to deal with than European ones.
This optimal exercise policy will affect the value of the option, and the exercise policy needs to be known when solving the pde. There is therefore no general analytical solutions for American call and put options. There is some special cases. For American Call options on assets that do not have any payouts, the American call price is the same as the European one, since the optimal exercise policy is to not exercise. For American Put is this not the case, it may pay to exercise them early.