scholarly journals Optimal Control of a Free Boundary Problem with Surface Tension Effects: A Priori Error Analysis

2015 ◽  
Vol 53 (5) ◽  
pp. 2279-2306 ◽  
Author(s):  
Harbir Antil ◽  
Ricardo H. Nochetto ◽  
Patrick Sodré
Keyword(s):  
Surface Tension ◽  
Optimal Control ◽  
Free Boundary ◽  
Error Analysis ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
A Priori ◽  
A Priori Error Analysis ◽  
A Free Boundary Problem

Stability and bifurcation analysis of a free boundary problem modelling multi-layer tumours with Gibbs–Thomson relation

2015 ◽  
Vol 26 (4) ◽  
pp. 401-425 ◽  
Author(s):  
FUJUN ZHOU ◽  
JUNDE WU
Keyword(s):  
Surface Tension ◽  
Free Boundary ◽  
Boundary Problem ◽  
Stationary Solution ◽  
Free Boundary Problem ◽  
Bifurcation Analysis ◽  
Surface Tension Coefficient ◽  
Stability And Bifurcation ◽  
Stability And Bifurcation Analysis ◽  
A Free Boundary Problem

Of concern is the stability and bifurcation analysis of a free boundary problem modelling the growth of multi-layer tumours. A remarkable feature of this problem lies in that the free boundary is imposed with nonlinear boundary conditions, where a Gibbs–Thomson relation is taken into account. By employing a functional approach, analytic semigroup theory and bifurcation theory, we prove that there exists a positive threshold value γ* of surface tension coefficient γ such that if γ > γ* then the unique flat stationary solution is asymptotically stable under non-flat perturbations, while for γ < γ* this unique flat stationary solution is unstable and there exists a series of non-flat stationary solutions bifurcating from it. The result indicates a significant phenomenon that a smaller value of surface tension coefficient γ may make tumours more aggressive.

Remarks concerning a free boundary problem arising in the theory of liquid drops and in plasma physics

1989 ◽  
Vol 111 (1-2) ◽  
pp. 169-181 ◽  
Author(s):  
John W. Barrett ◽  
Charles M. Elliott
Keyword(s):  
Surface Tension ◽  
Free Boundary ◽  
Plasma Physics ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Liquid Drop ◽  
Liquid Drops ◽  
Plasma Problem ◽  
A Free Boundary Problem ◽  
Surface Tension Coefficients

SynopsisWe consider a generalisation of the liquid drop problem, introduced in [1, Part II], by allowing the upper and lower surfaces to have different surface tension coefficients γv and γu. We study the existence, uniqueness and regularity of this problem. In addition, we show that as γv/γu →0, the solution of this problem converges to the solution of the “plasma problem”.

scholarly journals Modelo matemático y simulaciones numéricas para un problema de frontera libre ecológico-Mathematical model and numerical simulations for a free boundary problem of ecological

2014 ◽  
Vol 3 (20) ◽  
pp. 128 ◽  
Author(s):  
Deccy Y. Trejos-Angel ◽  
Oscar A. Ramírez-Céspedes
Keyword(s):  
Mathematical Model ◽  
Free Boundary ◽  
Numerical Simulations ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
A Priori ◽  
A Free Boundary Problem

En este artículo se estudia una aproximación numérica del Problema de Frontera Libre (PFL) de un sistema de ecuaciones diferenciales de tipo parabólico unidimensional, asociado con la evolución de la interface, que describe la partición regional de dos grupos de individuos de una misma especie que interactúan en un límite espacial para obtener sus propios hábitats y que es a priori totalmente desconocido. Considerando la dinámica local del sistema, el esquema implícito de diferencias finitas es utilizado, obteniendo así un sistema algebraico no lineal de ecuaciones en cada paso de tiempo. Finalmente, algunas simulaciones de la distribución de densidad poblacional y de la evolución de la frontera libre conforme al tiempo son exhibidas en diferentes escenarios, en base a un algoritmo propuesto e implementado en MATLAB, esto permite validar el modelo matemático PFL.

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