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A NOTE ON THE NUMERICAL APPROACH FOR THE REACTION–DIFFUSION PROBLEM WITH A FREE BOUNDARY CONDITION

Part of: Parabolic equations and systems Partial differential equations, initial value and time-dependent initial-boundary value problems Incompressible viscous fluids Incompressible inviscid fluids

Published online by Cambridge University Press:  13 October 2010

E. ÖZUĞURLU
E. ÖZUĞURLU*
Affiliation:
Bahçeşehir University, Faculty of Arts and Sciences, Department of Mathematics and Computer Sciences, 34353 Beşiktaş, Istanbul, Turkey (email: ersin.ozugurlu@bahcesehir.edu.tr)
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Abstract

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The equation modelling the evolution of a foam (a complex porous medium consisting of a set of gas bubbles surrounded by liquid films) is solved numerically. This model is described by the reaction–diffusion differential equation with a free boundary. Two numerical methods, namely the fixed-point and the averaging in time and forward differences in space (the Crank–Nicolson scheme), both in combination with Newton’s method, are proposed for solving the governing equations. The solution of Burgers’ equation is considered as a special case. We present the Crank–Nicolson scheme combined with Newton’s method for the reaction–diffusion differential equation appearing in a foam breaking phenomenon.

Keywords

foam drainagenonlinear partial differential equationCrank–Nicolson methodbreaking frontPlateau borderreaction–diffusion problemfree boundary problem

MSC classification

Secondary: 35K55: Nonlinear parabolic equations 65M06: Finite difference methods 65M12: Stability and convergence of numerical methods 76B45: Capillarity (surface tension) 76D99: None of the above, but in this section
Type
Research Article
Information
The ANZIAM Journal , Volume 51 , Issue 3 , January 2010 , pp. 317 - 330
Copyright
Copyright © Australian Mathematical Society 2010

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