On the relation between non-homogeneous fractional Burgers equations and time-dependent harmonic oscillator

In this paper we discuss the relation between non-homogeneous nonlinear fractional diffusive equations and the Schrodinger equation with time-dependent harmonic potential. It is well known that the Cole-Hopf transformation allows to linearize non-homogeneous nonlinear diffusive equations (NHNDEs) into a Schrodinger-type equation with time-dependent potential. We first discuss the utility of the results about time-dependent harmonic oscillator to build explicit solution of such non-homogeneous nonlinear partial differential equations. In particular, we recall that starting from a trial polynomial solution of the NHNDE, it is possible to construct other solutions by using linear invariants of the Schrodinger equation with time-dependent potential. Finally we apply these results to find explicit solutions to a novel non-homogeneous fractional Burgers-type equation.