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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for multiphase fluids [1] where the classical Laplacian has been
replaced by a degenerate one (so-called p-Laplacian).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.