Solving random mean square fractional linear differential equations by generalized power series: Analysis and computing

This paper deals with solving the general random (Caputo) fractional linear differential equation under general assumptions on random input data (initial condition, forcing term and diffusion coefficient). Our contribution extends, in two directions, the results presented in a recent contribution by the authors. In that paper, a mean square random generalized power series solution has been constructed in the case that the fractional order, say α, of the Caputo derivative lies on the interval ]0,1] and assuming that the diffusion coefficient belongs to a class, ℭ, of random variables that contains all bounded random variables. However, significant families of unbounded random variables, such as Gaussian and Exponential, for example, do not fall into class ℭ. Now, in this contribution we first enlarge the class of random variables to which the diffusion coefficient belongs and we prove that the constructed random generalized power series solution is mean square convergent too. We show that any bounded random variable and important unbounded random variables, including Gaussian and Exponential ones, are allowed to play the role of the diffusion coefficient as well. Secondly, we construct a mean square random generalized power series solution in the case that α parameter lies on the larger interval ]0,2]. As a consequence, the results established in our previous contribution are fairly generalized. It is particularly enlightening, the numerical study of the convergence of the approximations to the mean and the standard deviation of the solution stochastic process in terms of α parameter and on the type of the probability distribution chosen for the diffusion coefficient.