Self-similarity and probability density function of the transient response of fractional compound motion

In recent years, there has been an increasing use of fractional differential equations due to their ability to effectively represent various engineering phenomena, including viscoelasticity, heat transport, non-local continuum, and others. These equations take into account certain effects that cannot be accurately predicted using classical differential equations. This paper provides a comprehensive analysis of the fractional compound motion, specifically focusing on the response of a one-term fractional differential equation that is excited by a Poissonian white noise process. The present study introduces a straightforward equation for the probability density function of fractional compound motion. The validity of this equation is subsequently confirmed by the execution of various numerical simulations. Furthermore, a comprehensive analysis is conducted on the self-similarity of fractional compound motion, demonstrating that the phenomenon can be regarded as self-similar in weak sense. This characteristic can be effectively employed to mitigate the loss of Markovianity in fractional differential equations. • The PDF of the fCM tends to be Gaussian for instants sufficiently distant from zero. • For instants distant from zero the PDF of the fCM tends to the PDF of a fBM. • FCM tends to be self-similar for instants sufficiently distant from zero. • Wake self-similarity of fCM allows to overcome the loss of Markovianity.