Higher order asymptotics of solutions of a singular ODE

Article type: Research Article

Authors: Quittner, Pavol

Affiliations: Department of Applied Mathematics and Statistics, Comenius University, Mlynská dolina, 84248 Bratislava, Slovakia. Tel.: +421 2 60295670; Fax: +421 2 65412305; E-mail: quittner@fmph.uniba.sk

Abstract: We consider an initial value problem for a singular second-order differential equation 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.

Keywords: singular differential equation, asymptotic expansion

DOI: 10.3233/ASY-151314

Journal: Asymptotic Analysis, vol. 94, no. 3-4, pp. 293-308, 2015