On the Solution Variability of Random-Order Fractional Differential Equations with Orders on Bounded Supports

In this study, random-order FDEs are studied with a focus on the variability of their solutions and random orders when the order of differentiation is governed by a probability distribution. Fractional differential equations (FDEs) offer a generalized modeling approach that enables the analysis of nonlocality and memory effects. However, the deterministic framework for studying FDEs neglects the random nature of real-life events. In this regard, four continuous probability distributions with bounded support (uniform, Beta, triangle and Bates) are used to analyze the variability of the solutions depending on the random order, which is defined to vary between [0.65, 0.95] according to these probability distributions with identical expected values. Monte-Carlo simulations with N=215 repetitions are performed to investigate the random-order FDEs using a predictor-corrector approach. Results show that the decrease in the deviation from the uniform distribution to the Bates distribution is reflected in the random characteristics of the solutions and the order of differentiation to almost the same extent. The findings indicate that the average solutions obtained with random orders from each distribution show almost identical results, whereas the variability changes significantly based on the distribution of the order of differentiation. This information provides useful guidance in working with probabilistic models instead of deterministic systems for uncertainty quantification or sensitivity analysis.