Beyond the hypothesis of boundedness for the random coefficient of Airy, Hermite and Laguerre differential equations with uncertainties

Abstract

In this work, we study the full randomized versions of Airy, Hermite and Laguerre differential equations, which depend on a random variable appearing as an equation coefficient as well as two random initial conditions. In previous contributions, the mean square stochastic solutions to the aforementioned random differential equations were constructed via the Fröbenius method, under the assumption of exponential growth of the absolute moments of the equation coefficient, which is equivalent to its essential boundedness. In this paper we aim at relaxing the boundedness hypothesis to allow more general probability distributions for the equation coefficient. We prove that the equations are solvable in the mean square sense when the equation coefficient has finite moment-generating function in a neighborhood of the origin. A thorough discussion of the new hypotheses is included.

On the mean‐square solution to the Legendre differential equation with random input data

“…Likewise, this research may be applied to other linear random ordinary or fractional differential equations, such as Calatayud Gregori et al [19, 20], by employing (). From Calatayud Gregori et al [19], we notice that Airy, Hermite, and Laguerre random differential equations have a m.s. solution on $$$ \mathrm{\mathbb{R}} $$$ if the m.g.f.…”