The paper represents a continuation of the previous work of P. M. Lima, N. V. Chemetov, N. B. Konyukhova, and 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 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 and . Thereby, a parameter 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 and in an exponential Lyapunov series for . The authors prove that 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 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.