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

A C TIn 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.