Analytical-numerical investigation of bubble-type solutions of nonlinear singular problems
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 splitting the domain \([0,\infty)\) into three parts. In the two outer parts, the respective expansions of the solution into series are applied. In addition, the middle part is divided at a unique zero of the solution again, where a shooting method yields a corresponding approximation. Numerical simulations of the boundary value problem demonstrate that the constructed technique produces results, which agree with the expectations according to the underlying physical problem with bubble-type solutions. The paper is carefully written and well comprehensible.
scientific article; zbMATH DE number 5015133