Abstract

This paper is concerned with the study of the numerical approximation for  the following nonlinear diffusion equation $ \dfrac{\partial u}{\partial t}=u_{xx} + (1-u)^{-p}, \quad 0<x<1, \quad 0< t< T, $ with a nonlinear singular boundary flux $ u_{x}(0,t)= 0, \quad u_{x}(1,t)=(1-u(1,t))^{-q}, \quad 0< t< T $ and an initial solution $ u(x,0)=u_{0}(x), \quad 0\leq x \leq 1. $ We use the finite differences method to obtain the discrete scheme. Under some conditions, we show that the discrete solution obtained quenches in a finite time and we estimate this quenching time. We also prove that the discrete quenching time converges to the theoretical one when the mesh size tends to zero. Finally, we provide numerical results in tables and present curves and graphs to demonstrate the relevance of our analyses.

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