Operator-Based Uncertainty Quantification of Stochastic Fractional PDEs
This work addresses the need for efficient uncertainty quantification in fractional PDEs, which are used to model anomalous stochastic processes in physics and engineering.
The paper develops an operator-based uncertainty quantification framework for stochastic fractional partial differential equations (SFPDEs) with additive random noise, using a probabilistic collocation method and a fast Petrov-Galerkin spectral solver. The method propagates uncertainties from fractional orders and noise to the system response.
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.