Generalized polynomial chaos expansions for the random fractional Bateman equations

Bateman equations model mass balance in a linear radioactive decay chain of isotopes. A generalization of this model may be based on the introduction of a fractional derivative, to include memory effects, and on the incorporation of randomness in the input parameters (decay rate and initial concentrations), since it is not possible to predict when a particular nuclide will decay from a quantum-mechanical point of view. In this new context, a recent contribution studied the stochastic solution in the pathwise and the mean-square senses and, for a chain of length three, determined analytical forms for the probability density functions, with several numerical examples. In this paper, we focus on the computation of statistical moments of the solution, instead, in the setting of forward uncertainty quantification. We investigate the use of stochastic polynomial approximations to efficiently address that matter: for independent random inputs, generalized polynomial chaos expansions and the Galerkin projection technique are employed; and for non-independent random inputs, the Galerkin method is mimicked from the canonical polynomial basis. The deterministic system for the expansion's coefficients can be explicitly solved in terms of the matrix Mittag-Leffler function. Albeit these polynomial expansions are theoretically optimal for estimation of the expectation and the variance, we conduct numerical experiments to compare with Monte Carlo simulation and derive interesting conclusions that depend on the specific problem analyzed.