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.
Key words:
Kolmogorov–Petrovskii–Piskunov equation, Abel equation of the second kind, Thomas–Fermi model, autonomous nonlinear equation, Fuchs index, parametric representation.
Citation:
S. V. Pikulin, “The behavior of solutions to a special Abel equation of the second kind near a nodal singular point”, Zh. Vychisl. Mat. Mat. Fiz., 58:12 (2018), 2026–2047; Comput. Math. Math. Phys., 58:12 (2018), 1948–1966
\Bibitem{Pik18}
\by S.~V.~Pikulin
\paper The behavior of solutions to a special Abel equation of the second kind near a nodal singular point
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2018
\vol 58
\issue 12
\pages 2026--2047
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\crossref{https://doi.org/10.31857/S004446690003550-6}
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\transl
\jour Comput. Math. Math. Phys.
\yr 2018
\vol 58
\issue 12
\pages 1948--1966
\crossref{https://doi.org/10.1134/S0965542518120151}
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Linking options:
https://www.mathnet.ru/eng/zvmmf10803
https://www.mathnet.ru/eng/zvmmf/v58/i12/p2026
This publication is cited in the following 3 articles:
S. V. Pikulin, “Parametrization of solutions to the Emden–Fowler equation and the Thomas–Fermi model of compressed atoms”, Comput. Math. Math. Phys., 60:8 (2020), 1271–1283
V S. Pikulin, “Analytical-numerical method for calculating the Thomas-Fermi potential”, Russ. J. Math. Phys., 26:4 (2019), 544–552
S. V. Pikulin, “The Thomas–Fermi problem and solutions of the Emden–Fowler equation”, Comput. Math. Math. Phys., 59:8 (2019), 1292–1313
Citation in format AMSBIB
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