Asymptotics of Solutions of Linear Differential Equations with Holomorphic Coefficients in the Neighborhood of an Infinitely Distant Point

This study is devoted to the description of the asymptotic expansions of solutions of linear ordinary differential equations with holomorphic coefficients in the neighborhood of an infinitely distant singular point. This is a classical problem of analytical theory of differential equations and an important particular case of the general Poincare problem on constructing the asymptotics of solutions of linear ordinary differential equations with holomorphic coefficients in the neighborhoods of irregular singular points. In this study we consider such equations for which the principal symbol of the differential operator has multiple roots. The asymptotics of a solution for the case of equations with simple roots of the principal symbol were constructed earlier.

The Radiation-Dominated Universe in Nonstandard Cosmology

We continue the recent study of our model theory of low-density cosmology without dark matter. We assume a purely radiative spherically symmetric background and treat matter as anisotropic perturbations. Einstein’s equations for the background are solved numerically. We find two irregular singular points, one is the Big Bang and the other a Big Crunch. The radiation temperature continues to decrease for another 0.21 Hubble times and then starts to increase towards infinity. Then we derive the four evolution equations for the anisotropic perturbations. In the Regge- Wheeler gauge there are three metric perturbations and a radial velocity perturbation. The solution of these equations allow a detailed discussion of the cosmic evolution of the model universe under study.

A System Associated with the Confluent Hypergeometric Function Φ3 and a Certain Linear Ordinary Differential Equation with Two Irregular Singular Points

The confluent hypergeometric function Φ3 satisfies a system of partial differential equations on P1(C) × P1(C) with the singular loci x = 0, x = ∞, y = ∞ of irregular type and y = 0 of regular type. We obtain asymptotic expansions and Stokes multipliers of linearly independent solutions near the singular loci x = 0 and x = ∞. Applying the results we also clarify the global behaviour of the solutions of a third order linear ordinary differential equation with two irregular singular points.