Computing the two first probability density functions of the random Cauchy-Euler differential equation: Study about regular-singular points Publié en ligne: 23 juin 2017 Pages: 213 - 224 Reçu: 22 févr. 2017 Accepté: 23 juin 2017 © 2017 J.-C. Cortés, A. Navarro-Quiles, J.-V. Romero, M.-D. Roselló, published by Sciendo This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.
Fig. 1 Top: Plots of the 1-PDF of the solution SP, W(u), to the random IVP (4) with u0 = 1 given in (8), (11), (13) in Cases I–III at different values of u ∊ {2,3,...,10}. Bottom: Plots of the 1-PDF of the solution SP, V(s), to the random IVP (17) with s0 = 0.5 given in (19), (20), (22) in Cases I–III at different values of s ∊ {0.05,0.1,...,0.5}. Fig. 2 Top: Plots of the mean, μW(u), and plus/minus the standard deviation, σW(u), of the solution SP, W (u), to IVP (4) in Cases I–III at different values of u ε [1,10]. Bottom: Plots of the mean, μV(s), and plus/minus the standard deviation, σV(s), of the solution SP, V (s), to IVP (17) in Cases I–III at different values of s ε [0.05,0.5]. Fig. 3 Covariance function given by (3) in the Case I to both problems, IVP (4) (left) and IVP (17) (right) for the values of u1,u2 ε [1,3] and s1,s2 ε [0.05,0.5]. Columns p1 and p2 = 1 − p1 collect the values of the probabilities given by (25) corresponding to Cases I–III, when Ji ~ N(μi,Σ), being μi and Σ specified in (24). Values of ps represent the probabilities associated with asymptotic stability according to (26). Cases p 1 p 2 ps I 0.978524 0.021476 0.966055 II 0.530394 0.469606 0.966054 III 0.045171 0.954829 0.999866