Dealing with Dependent Uncertainty in Modelling: A Comparative Study Case through the Airy Equation

Figure 2

Comparison of the approximations for the standard deviation 𝜎[𝑋(𝑡)] on the time-interval 𝑡∈[0,5] for different correlation values 𝜌=𝜌_𝐴,𝑌0∈{−0.5,0,0.5,0.9} in Example 2 by using Fröbenius, Polynomial Chaos, and Monte Carlo simulations. Since the numerical values are indistinguishable graphically, for the sake of clarity in the presentation, nonlabel has been introduced. We assume that 𝑌₁=1 and 𝜂=(𝐴,𝑌₀)^⊤ follows a bivariate Gaussian distribution: 𝜂∼𝑁(𝝁_𝜂;𝚺_𝜂) where, according to (34), 𝜇_𝐴=1, 𝜇_𝑌0=1, 𝜎_𝐴=0.2, 𝜎_𝑌0=0.2.