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

Table 1

Comparison of the approximations for the expectations at different time instants 𝑡 and correlation values 𝜌_𝑌0,𝑌1 in Example 1 by using Fröbenius method with a truncated series with 𝑁+1 terms (𝐸[𝑋_𝑁(𝑡)]), Polynomial Chaos of order 𝑃(𝜇^𝑃_𝑋(𝑡)), and Monte Carlo with 𝑚 simulations (𝜇^𝑚_𝑋(𝑡)). We assume that 𝐴=1 and the initial condition: 𝜂=(𝑌₀,𝑌₁)^⊤ follows a bivariate Gaussian distribution: 𝜂∼𝑁(𝝁_𝜂;𝚺_𝜂) where, according to (25), 𝜇_𝑌0=1, 𝜇_𝑌1=1, 𝜎_𝑌0=0.2, 𝜎_𝑌1=0.2.

𝑡	𝜌𝑌0,𝑌1=−0.5			𝜌𝑌0,𝑌1=0			𝜌𝑌0,𝑌1=0.5			𝜌𝑌0,𝑌1=0.9
	𝐸[𝑋𝑁(𝑡)]𝑁=14	𝜇^𝑃𝑋(𝑡)𝑃=1	^𝑚𝑋𝜇(𝑡)𝑚=500000	𝐸[𝑋𝑁(𝑡)]𝑁=14	𝜇^𝑃𝑋(𝑡)𝑃=1	^𝑚𝑋𝜇(𝑡)𝑚=500000	𝐸[𝑋𝑁(𝑡)]𝑁=14	𝜇^𝑃𝑋(𝑡)𝑃=1	^𝑚𝑋𝜇(𝑡)𝑚=500000	𝐸[𝑋𝑁(𝑡)]𝑁=14	𝜇^𝑃𝑋(𝑡)𝑃=1	^𝑚𝑋𝜇(𝑡)𝑚=500000
0.00	1.	1.	0.999945	1.	1.	0.999723	1.	1.	0.999838	1.	1.	0.999615
0.25	1.24707	1.24707	1.24705	1.24707	1.24707	1.24689	1.24707	1.24707	1.24681	1.24707	1.24707	1.24661
0.50	1.47406	1.47406	1.47408	1.47406	1.47406	1.47397	1.47406	1.47406	1.47371	1.47406	1.47406	1.47353
0.75	1.65457	1.65457	1.65462	1.65457	1.65457	1.65457	1.65457	1.65457	1.65413	1.65457	1.65457	1.65398
1.00	1.75744	1.75744	1.75752	1.75744	1.75744	1.75754	1.75744	1.75744	1.75694	1.75744	1.75744	1.75683
2.00	0.884201	0.884201	0.884327	0.884201	0.884201	0.884535	0.884201	0.884201	0.88385	0.884201	0.884201	0.883924
3.00	−1.20538	−1.20538	−1.20541	−1.20538	−1.20538	−1.20537	−1.20538	−1.20538	−1.20506	−1.20538	−1.20538	−1.20495
4.00	−0.353252	−0.353252	−0.353344	−0.353252	−0.353252	−0.353523	−0.353252	−0.353252	−0.353062	−0.353252	−0.353252	−0.353156
5.00	1.21343	1.21343	1.21352	1.21343	1.21343	1.21363	1.21343	1.21343	1.21304	1.21343	1.21343	1.21302