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Computing the two first probability density functions of the random Cauchy-Euler differential equation: Study about regular-singular points

J.-C. Cortés

J.-C. Cortés

Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de ValènciaValencia, Spain
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Cortés, J.-C.
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A. Navarro-Quiles

A. Navarro-Quiles

Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de ValènciaValencia, Spain
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Navarro-Quiles, A.
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J.-V. Romero

J.-V. Romero

Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de ValènciaValencia, Spain
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Romero, J.-V.
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M.-D. Roselló

M.-D. Roselló

Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de ValènciaValencia, Spain
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Roselló, M.-D.
  
23 jun 2017
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Publicado en línea: 23 jun 2017
Páginas: 213 - 224
Recibido: 22 feb 2017
Aceptado: 23 jun 2017
DOI: https://doi.org/10.21042/AMNS.2017.1.00018
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© 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}.
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].
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].
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)_

Casesp1p2ps
I0.9785240.0214760.966055
II0.5303940.4696060.966054
III0.0451710.9548290.999866
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Ciencias de la vida, Ciencias de la vida, otros, Matemáticas, Matemáticas aplicadas, Matemáticas generales, Física, Física, otros
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