  We introduce and study a new class of partial differential equations (PDEs)
with hybrid fuzzy-stochastic parameters, coined fuzzy-stochastic PDEs. Compared
to purely stochastic PDEs or purely fuzzy PDEs, fuzzy-stochastic PDEs offer
powerful models for accurate representation and propagation of hybrid
aleatoric-epistemic uncertainties inevitable in many real-world problems. We
will use the level-set representation of fuzzy functions and define the
solution to fuzzy-stochastic PDE problems through a corresponding parametric
problem, and further present theoretical results on the well-posedness and
regularity of such problems. We also propose a numerical strategy for computing
output fuzzy-stochastic quantities, such as fuzzy failure probabilities and
fuzzy probability distributions. We present two numerical examples to compute
various fuzzy-stochastic quantities and to demonstrate the applicability of
fuzzy-stochastic PDEs to complex engineering problems.
