Fuzzy-Stochastic Partial Differential Equations

Research paper by Mohammad Motamed, Ivo Babuska

Indexed on: 01 Jun '17Published on: 01 Jun '17Published in: arXiv - Mathematics - Analysis of PDEs


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, which can treat either only random or only non-random uncertainty in physical systems, fuzzy-stochastic PDEs offer powerful models for accurate description and propagation of the hybrid random and non-random uncertainties inevitable in many real applications. 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. Considering the interaction between the input fuzzy variables, we propose a sampling-based worst-case scenario strategy for computing fuzzy-stochastic output quantities. We present two numerical examples, compute and visualize various types of fuzzy-stochastic quantities, and demonstrate the applicability of fuzzy-stochastic PDEs to an engineering problem in materials science.