SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Invited Talk

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Analysis and Numerical Approximation of the Generalized Density Profile Equation

P. Lima, G.Y. Kulikov and M.L. Morgado

Abstract
We are concerned with a generalization of the Cahn-Hilliard continuum model for multiphase fluids [1] where the classical Laplacian has been replaced by a degenerate one (so-called p-Laplacian). Using spherical symmetry, the model can be reduced to a boundary value problem for a second order nonlinear ordinary differential equation. One searches for a monotone solution of this equation which satisfies certain boundary conditions. The case of the classical Laplacian was studied in detail in [2] and [3]. In the present work, it is proved that the arising boundary value problem possesses a unique strictly monotone solution The asymptotic behavior of the solution is also analyzed at two singular points; namely, at the origin and at infinity. An efficient numerical technique for treating such singular boundary value problems is presented, based on the shooting method and on nested implicit Runge-Kutta formulas with global error control.

Bibliography
[1] J.W. Cahn and J.E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy, J. Chem Phys., 28 (1958) 258-267.
[2] P.M. Lima, N.B. Konyukhova, N.V. Chemetov, and A.I. Sukov, Analytical-numerical investigation of bubble-type solutions of nonlinear singular problems, J. Comput. Appl. Math., 189 (2006) 260-273.
[3] G. Kitzhofer, O. Koch, P.M. Lima, and E. Weinmuller, Efficient numerical solution of profile equation in hydrodynamics, J. Sci. Comput., 32, (2007) 411-424.

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