The Behavior of Solutions to a Special Abel Equation of the Second Kind near a Nodal Singular Point
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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
This study was supported by the Russian Foundation for Basic Research, project no. 16-01-00781.
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