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Analytical-numerical investigation of bubble-type solutions of nonlinear singular problems. (English) Zbl 1100.65066

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 $r$ 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 $r\to \infty$. 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 $r\to \infty$. 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 $\left[0,\infty \right)$ 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.

##### MSC:
 65L10 Boundary value problems for ODE (numerical methods) 34B16 Singular nonlinear boundary value problems for ODE