# Fuzzy-Stochastic Partial Differential Equations

## TL;DR

This paper introduces fuzzy-stochastic PDEs, a new class of equations combining fuzzy and stochastic parameters, providing a powerful framework for modeling hybrid uncertainties in complex engineering problems.

## Contribution

It defines fuzzy-stochastic PDEs, establishes their well-posedness, and proposes numerical methods for computing fuzzy-stochastic quantities, advancing uncertainty modeling.

## Key findings

- Successfully modeled hybrid uncertainties in engineering problems.
- Developed a numerical strategy for fuzzy-stochastic computations.
- Demonstrated applicability through numerical examples.

## Abstract

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.