Langevin equation with two fractional orders

Abstract

A new type of fractional Langevin equation of two different orders is introduced. The solutions for this equation, known as the fractional Ornstein–Uhlenbeck processes, based on Weyl and Riemann–Liouville fractional derivatives are obtained. The basic properties of these processes are studied. An example of the spectral density of ocean wind speed which has similar spectral density as that of Weyl fractional Ornstein–Uhlenbeck process is given.