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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 {\it P. M. Lima, N. V. Chemetov, N. B. Konyukhova}, and {\it 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.

65L10Boundary value problems for ODE (numerical methods)
34B16Singular nonlinear boundary value problems for ODE
Full Text: DOI
[1] Abramowitz, M.; Stegun, I.: Handbook of mathematical functions. (1965) · Zbl 0171.38503
[2] Baxley, J. V.: Boundary value problems on infinite intervals. Boundary value problems for functional differential equations, 49-62 (1995) · Zbl 0845.34032
[3] Berestycki, H.; Lions, P. -L.: Nonlinear scalar field equations, I, existence of a ground state. Arch. rational mech. Anal. 82, 313-345 (1983) · Zbl 0533.35029
[4] Dell’isola, F.; Gouin, H.; Rotoli, G.: Nucleation of spherical shell-like interfaces by second gradient theory: numerical simulations. European J. Mech. B/fluids 15, 545-568 (1996) · Zbl 0887.76008
[5] Derrick, G. H.: Comments on nonlinear wave equations as models for elementary particles. J. math. Phys. 5, 1252-1254 (1965)
[6] Gavrilyuk, S. L.; Shugrin, S. M.: Media with equations of state that depend on derivatives. J. appl. Mech. tech. Phys. 37, 177-189 (1996) · Zbl 1031.76501
[7] Gazzola, F.; Serrin, J.; Tang, M.: Existence of ground states and free boundary problems for quasilinear elliptic operators. Add. differential equations 5, 1-30 (2000) · Zbl 0987.35064
[8] Gouin, H.; Rotoli, G.: An analytical approximation of density profile and surface tension of microscopic bubbles for van der Waals fluids. Mech. res. Comm. 24, 255-260 (1997) · Zbl 0899.76064
[9] Konyukhova, N. B.: Singular Cauchy problems for systems of ordinary differential equations. USSR comput. Math. math. Phys. 23, 72-82 (1983) · Zbl 0555.34002
[10] P.M. Lima, N.V. Chemetov, N.B. Konyukhova, A.I. Sukov, Analytical -- numerical approach to a singular boundary value problem, in: Proceedings of CILAMCE XXIV, Ouro Preto, Brasil (CD-ROM).
[11] A.M. Lyapunov, Problème Général de la Stabilité du Mouvement, Annals of Mathematics Studies, vol. 17, Princeton University Press, Princeton, NJ, 1947.
[12] Rubakov, V. A.: Classical gauge fields. (1999) · Zbl 1036.81002
[13] Rybakov, Yu.P.; Sanyuk, V. I.: Many-dimensional solitons. (2001)
[14] Wasov, W.: Asymptotic expansions for ordinary differential equations. (1965)
[15] Wolfram, S.: The Mathematica book. (1996) · Zbl 0878.65001