Seminár z kvalitatívnej teórie diferenciálnych rovníc

Seminar on Qualitative Theory of Differential Equations

Thursday 22.10.2015 at 14:00 Lecture room M-223

Pavol Quittner (KAMŠ FMFI UK):

Asymptotika riešení rovnice špekulanta

Abstract: We consider an initial value problem for a singular second-order ODE stemming from an optimal control problem

in financial economics and possessing infinitely many solutions. If one looks for these solutions in the form of a power series

then a formal approach leads to a series whose radius of convergence is zero (for some values of the parameters of the problem).

We prove that each partial sum of this series is a good approximation of all the solutions close to the singularity:

This solves an open question posed by P. Brunovský, A. Černý and M. Winkler. We also obtain uniqueness and comparison results

for the solutions satisfying an additional boundary condition and use these results in the study of the asymptotics

of the difference of two solutions of the initial value problem. Our results are in sharp contrast to known results for

a less singular ODE problem stemming from the study of self-similar solutions of a semilinear wave equation.