Analytical-numerical investigation of bubble-type solutions of nonlinear singular problems
type: Item
instance of: scholarly article
title: Analytical-numerical investigation of bubble-type solutions of nonlinear singular problems
zbMATH Open document ID: 1100.65066
author: N. B. Konyukhova
author: A. I. Sukov
author: Pedro M. Lima
author: Nikolai Chemetov
published in: Journal of Computational and Applied Mathematics
publication date: 2006-03-24T00:00:00Z
review text: The paper represents a continuation of the previous work of \textit{P. M. Lima, N. V. Chemetov, N. B. Konyukhova}, and \textit{A. I. Sukov} [Analytical-numerical approach to a singular boundary value problem, Proceedings of CILAMCE XXIV, Ouro Preto, Brasil, (2003), ISBN 85-288-0040-7]. A sophisticated analysis shall improve the construction of numerical methods tailored to the considered problem. The underlying physical problem means the determination of the density in non-homogeneous fluids. Simplifying assumptions on corresponding partial differential equations yield a time-independent problem in one or more space dimensions. In the paper at hand, bubble-type solutions shall be determined, where according spatial symmetries arise. Consequently, the solution depends only on the radial variable \(r\) in the polar system. Thus a scalar ordinary differential equation (ODE) of second order arises for the unknown density function. Since bubble-type functions are considered, the solution shall increase monotonically. The physical problem causes boundary conditions for \(r=0\) and \(r \rightarrow \infty\). Thereby, a parameter \(\xi\) has to be chosen appropriately in the right-hand boundary condition. However, the ODE exhibits a singularity in each of the two cases. The authors analyse the two boundary conditions separately. In both cases, a one-parameter family of solutions arises, which satisfies one of the boundary constraints but not the other condition. Furthermore, the solution can be expanded in a Taylor series at \(r=0\) and in an exponential Lyapunov series for \(r \rightarrow \infty\). The authors prove that \(0 < \xi < 1\) represents a necessary condition for the existence of non-constant solutions satisfying the boundary value problem. From previous results, it follows that this condition is sufficient, too. Using the analytical properties, the authors construct a numerical method for solving the boundary value problem of the ODE. The idea consists in splitti…
reviewed by: Roland Pulch
Mathematics Subject Classification ID: 65L10
Mathematics Subject Classification ID: 34B16
zbMATH DE Number: 5015133
zbMATH Keywords: singular boundary value problem
zbMATH Keywords: singular Cauchy problem
zbMATH Keywords: bubble-type solutions
zbMATH Keywords: shooting method
zbMATH Keywords: numerical examples
Wikidata QID: Q57622122
describes a project that uses: Mathematica
MaRDI profile type: Publication
full work available at URL: j.cam.2005.05.004
OpenAlex ID: W2056520979
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DOI: 10.1016/J.CAM.2005.05.004
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description: scientific article; zbMATH DE number 5015133