A note on the numerical approach for the reaction-diffusion problem with a free boundary condition

2011 ◽  
Vol 51 ◽  
pp. 317
Author(s):  
Ersin Özuğurlu
Keyword(s):  
Boundary Condition ◽  
Free Boundary ◽  
Reaction Diffusion ◽  
Numerical Approach ◽  
Diffusion Problem ◽  
Free Boundary Condition

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

2010 ◽  
Vol 51 (3) ◽  
pp. 317-330
Author(s):  
E. ÖZUĞURLU
Keyword(s):  
Differential Equation ◽  
Free Boundary ◽  
Newton’S Method ◽  
Newton's Method ◽  
Reaction Diffusion ◽  
Numerical Approach ◽  
Liquid Films ◽  
Diffusion Problem ◽  
Nicolson Scheme ◽  
Crank Nicolson

AbstractThe 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.

scholarly journals SUPERSYMMETRIC WZW σ MODEL ON INFINITE AND HALF-PLANE

2005 ◽  
Vol 20 (13) ◽  
pp. 2763-2772
Author(s):  
R. A. ZAIT ◽  
M. F. MOURAD
Keyword(s):  
Boundary Condition ◽  
Free Boundary ◽  
Half Plane ◽  
Infinite Number ◽  
Equations Of Motion ◽  
Free Boundary Condition ◽  
Infinite Plane ◽  
Recursion Relations ◽  
Conserved Currents

We study classical integrability of the supersymmetric U(N) σ model with the Wess–Zumino–Witten term on infinite and half-plane. We demonstrate the existence of nonlocal conserved currents of the model and derive general recursion relations for the infinite number of the corresponding charges in a superfield framework. The explicit forms of the first few supersymmetric charges are constructed. We show that the considered model is integrable on infinite plane as a consequence of the conservation of the supersymmetric charges. Also, we study the model on half-plane with free boundary, and examine the conservation of the supersymmetric charges on half-plane and find that they are conserved as a result of the equations of motion and the free boundary condition. As a result, the model on half-plane with free boundary is integrable. Finally, we conclude the paper and some features and comments are presented.

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