There have been many successful models using fractional differential equations with order between 0 and 1. Is there any physical meaning to a model when its order is between 1 and 2?
A fractional differential equation with an order between 1 to 2 can be stable if you can ensure that the poles are in the left-half plane of the complex plane. There is a paper (I forgot who the author is) showing that stable systems of orders between 1 and 2 exhibit overshooting property in its transient response, which is desirable in many engineering applications.