Operator-Based Uncertainty Quantification of Stochastic Fractional PDEs

TL;DR

This paper introduces an operator-based framework for quantifying uncertainty in stochastic fractional PDEs, utilizing probabilistic collocation and spectral methods to efficiently simulate and analyze the impact of uncertain fractional orders and noise.

Contribution

It presents a novel approach combining fractional calculus, uncertainty quantification, and spectral methods to analyze stochastic fractional PDEs with uncertain fractional orders.

Findings

Developed a probabilistic collocation method for SFPDEs.

Designed a stable Petrov-Galerkin spectral solver.

Validated the approach through numerical simulations.

Abstract

Fractional calculus provides a rigorous mathematical framework to describe anomalous stochastic processes by generalizing the notion of classical differential equations to their fractional-order counterparts. By introducing the fractional orders as uncertain variables, we develop an operator-based uncertainty quantification framework in the context of stochastic fractional partial differential equations (SFPDEs), subject to additive random noise. We characterize different sources of uncertainty and then, propagate their associated randomness to the system response by employing a probabilistic collocation method (PCM). We develop a fast, stable, and convergent Petrov-Galerkin spectral method in the physical domain in order to formulate the forward solver in simulating each realization of random variables in the sampling procedure.