Solution processes for second-order linear fractional differential equations with random inhomogeneous parts

The novelty of this paper is to derive a mean square solution of a second-order (Caputo) fractional linear differential equation in which the coefficients and initial conditions are random variables, and the forcing term is a second order stochastic process. Using the so-called mean square calculus and assuming mild conditions on the random variables of the equation together with an exponential growth condition on the forcing term, a mean square convergent generalized power series solution is constructed. As a result of this convergence, the sequences of the mean and correlation obtained from the truncated power series solution are convergent as well. The theory is illustrated with several examples in which different kind of distributions on the input parameters are assumed.