Solving Second-Order Linear Differential Equations with Random Analytic Coefficients about Regular-Singular Points

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.

Keywords

• random variable transformation technique
• second-order random linear differential equation
• regular-singular point
• first probability density function