Rigorous time domain responses of polarizable media

The scope of this note is to study a model of induced polarization which fits the usually accepted frequency dependent formula of Cole and Cole, but is more general and allows the time domain observations to retrieve the parameters describing the induced polarization phenomena of the medium. By introducing the memory mechanisms, represented by derivatives of fractional order, in the relation between the electric flux density and the electric field and considering the fractional order differential equation which follows, I solve it with mathematically rigorous and closed formulae and compute the responses to a step function, a box, a set of positive boxes and a set of alternating positive and negative boxes. I also introduce a method which retrieves the parameters describing the medium when comparing the theoretical curves with the observed ones. The responses to these signals also allow to estimate the temporary alteration of the medium when repeated positive (negative) signals are input; the response increases (decreases) in amplitude when the signals are all positive (negative), it decreases when the signals are alternatively positive and negative in agreement with the known attitude of the medium to induced polarization