Abstract
In this contribution, we construct approximations for the density associated with the solution of second-order linear differential equations whose coefficients are analytic stochastic processes about regular-singular points. Our analysis is based on the combination of a random Fröbenius technique together with the random variable transformation technique assuming mild probabilistic conditions on the initial conditions and coefficients. The new results complete the ones recently established by the authors for the same class of stochastic differential equations, but about regular points. In this way, this new contribution allows us to study, for example, the important randomized Bessel differential equation.
Highlights
Introduction and MotivationAs a main difference with respect to deterministic differential equations, solving a random differential equation (RDE) does consist of determining, exactly or approximately, its solution stochastic process (SP), say X (t), and of computing its main statistical properties such as the mean and the variance
For the sake of completeness, below, we summarize the main results about the deterministic theory of second-order linear differential equations about ordinary and regular-singular points
We presented a methodology to construct reliable approximations to the first probability density function of the solution of second-order linear differential equations about a regular-singular point with a finite degree of randomness in the coefficients and assuming that both initial conditions were random variables
Summary
As a main difference with respect to deterministic (or classical) differential equations, solving a random differential equation (RDE) does consist of determining, exactly or approximately, its solution stochastic process (SP), say X (t), and of computing its main statistical properties such as the mean and the variance. We address the problem of constructing reliable approximations for the 1-PDF of the solution SP to second-order linear differential equations whose coefficients are analytic SPs depending on a random variable (RV) about a regular-singular point and whose initial conditions (ICs) are RVs. we are dealing with what is often called RDEs having a finite degree of randomness [5] According to the Fröbenius method (see Theorem 1) and under the analyticity condition assumed in Hypothesis H2, the solution SP of RDE (10) about a regular-singular point, t0 , can be written as a linear combination of two uniformly convergent independent random series, X1 (t; A) and X2 (t; A),.
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