A full probabilistic solution of the random linear fractional differential equation via the random variable transformation technique

Abstract

This paper provides a full probabilistic solution of the randomized fractionallinear nonhomogeneous differential equation with a random initial conditionvia the computation of the first probability density function of the solution sto-chastic process. To account for most generality in our analysis, we assume thatuncertainty appears in all input parameters (diffusion coefficient, source term,and initial condition) and that a wide range of probabilistic distributions can beassigned to these parameters. Throughout our study, we will consider that thefractional order of Caputo derivative lies in ]0,1], that corresponds to the mainstandard case. To conduct our analysis, we take advantage of the random var-iable transformation technique to construct approximations of the first proba-bility density function of the solution process from a suitable infinite seriesrepresentation. We then prove these approximations do converge to the exactdensity assuming mild conditions on random input parameters. Our theoreti-cal findings are illustrated through 2 numerical examples.