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Computational Mathematics and Mathematical Physics

, Volume 58, Issue 12, pp 1948–1966 | Cite as

The Behavior of Solutions to a Special Abel Equation of the Second Kind near a Nodal Singular Point

  • S. V. PikulinEmail author
Article
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Abstract

The propagation of a diffusion–reaction plane traveling wave (for example, a flame front), the charge distribution inside a heavy atom in the Thomas–Fermi model, and some other models in natural sciences lead to bounded solutions of a certain autonomous nonlinear second-order ordinary differential equation reducible to an Abel equation of the second kind. In this study, a sufficient condition is obtained under which all solutions to a special second-kind Abel equation that pass through a nodal singular point of the equation can be represented by a convergent power series (in terms of fractional powers of the variable) in a neighborhood of this point. Under this condition, new parametric representations of bounded solutions to the corresponding autonomous nonlinear equation are derived. These representations are efficient for numerical implementation.

Keywords:

Kolmogorov–Petrovskii–Piskunov equation Abel equation of the second kind Thomas–Fermi model autonomous nonlinear equation Fuchs index parametric representation 

Notes

ACKNOWLEDGMENTS

This study was supported by the Russian Foundation for Basic Research, project no. 16-01-00781.

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Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Dorodnicyn Computing Center, Federal Research Center “Computer Science and Control”, Russian Academy of SciencesMoscowRussia

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