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Quenching for discretization of a semilinear heat equation with singular boundary outflux
Anoh Assiedou Rodrigue,N'Guessan Koffi,Coulibaly Adama,Toure Kidjegbo Augustin 원광대학교 기초자연과학연구소 2021 ANNALS OF FUZZY MATHEMATICS AND INFORMATICS Vol.22 No.2
This paper concerns the study of the discret approximation for the following semilinear heat equation with a singular boundary outflux $$\\ \left\{% \begin{array}{ll} \hbox{$\dfrac{\partial u}{\partial t}=u_{xx} + (1-u)^{-p}, \quad 0<x<1,\; t>0$,} \\ \hbox{$u_{x}(0,t)= 0, \quad u_{x}(1,t)=-u(1,t)^{-q},\quad t>0$,} \\ \hbox{$u(x,0)=u_{0}(x),\quad 0\leq x \leq 1$,} \\ \end{array}% \right.$$ where $ p>0, $ $ q > 0. $ We find some conditions under which the solution of a discrete form of above problem quenches in a finite time and estimate its discrete quenching time. We also establish the convergence of the discrete quenching time to the theoretical one when the mesh size tends to zero. Finally, we give some numerical experiments for a best illustration of our analysis.
Quenching for discretization of a nonlinear diffusion equation with singular boundary flux
NGuessan Koffi,Anoh Assiedou Rodrigue,Coulibaly Adama,Toure Kidjegbo Augustin 원광대학교 기초자연과학연구소 2022 ANNALS OF FUZZY MATHEMATICS AND INFORMATICS Vol.23 No.1
In this paper, we study the discrete approximation for the following nonlinear diffusion equation with nonlinear source and singular boundary flux $$\\ \left\{% \begin{array}{ll} \hbox{$\dfrac{\partial A(u)}{\partial t}=u_{xx} + (1-u)^{-\alpha}, \quad 0<x<1,\; t>0$,} \\ \hbox{$u_{x}(0,t)= 0, \quad u_{x}(1,t)=-B(u(1,t)),\quad t>0$,} \\ \hbox{$u(x,0)=u_{0}(x),\quad 0\leq x \leq 1$,} \\ \end{array}% \right. $$ with $ \alpha > 0. $ We find some conditions under which the solution of a discrete form of above problem quenches in a finite time and estimate its discrete quenching time. We also establish the convergence of the discrete quenching time to the theoretical one when the mesh size tends to zero. Finally, we give some numerical experiments for a best illustration of our analysis.