NUMERICAL BLOW-UP TIME FOR NONLINEAR PARABOLIC PROBLEMS

Adou KoffiAchille; Diop FatouN.; N’Guessan Koffi; Touré KidjégboAugustin

Adou KoffiAchille Direction Pédagogique, Ecole Supérieure Africaine des TIC (ESATIC), UP Mathématiques, 18 BP 1501 Abidjan 18, Côte D’Ivoire
Diop FatouN. Direction Pédagogique, Ecole Supérieure Africaine des TIC (ESATIC), UP Mathématiques, 18 BP 1501 Abidjan 18, Côte D’Ivoire
N’Guessan Koffi Département de Mathématiques et Informatique, Université Alassane Ouattara de Bouaké, UFR-SED, 01 BP V 18 Bouaké 01, Côte D’Ivoire
Touré KidjégboAugustin Département de Mathématiques et Informatique, Institut National Polytechnique Houphouët-Boigny, UMRI Mathématiques et Nouvelles Technologies de L'Information, Yamoussoukro, BP 2444, Côte D'Ivoire

Article ID: 2574

Keywords: blow-up; nonlinear parabolic equation; finite difference scheme; numerical blow-up time

Abstract

In this paper, we analyze numerically some of the features of theblow-up phenomena arising from a nonlinear parabolic equationsubject to nonlinear boundary conditions. More precisely, we studynumerical approximations of solutions of the problem ${\begin{cases} (\log u (x, t))_{t} = u_{x x} (x, t) + u^{β - 1} (x, t), (x, t) \in (0, 1) \times (0, T), \\ - u_{x} (0, t) + u^{α} (0, t) = 0, t > 0, \\ u_{x} (1, t) + u^{α} (1, t) = 0, t > 0, \\ u (x, 0) = u_{0} (x) \geq γ > 0, 0 \leq x \leq 1, \end{cases}$ , where $β \geq α > 1$ . We obtain some:conditions under which thesolution of the semidiscrete form blows up in a finite time. Weestimate its semidiscrete blow-up time and also establish theconvergence of the semidiscrete blow-up time to the real one. Finally.we give some numerical experiments to illustrate our analysis.

References

[1]In this paper, we analyze numerically some of the features of theblow-up phenomena arising from a nonlinear parabolic equationsubject to nonlinear boundary conditions. More precisely, we studynumerical approximations of solutions of the problem

[2]http://www.w3.org/1998/Math/MathML" display="block"> {(log⁡u(x,t))t=uxx(x,t)+uβ−1(x,t),(x,t)∈(0,1)×(0,T),−ux(0,t)+uα(0,t)=0,t>0,ux(1,t)+uα(1,t)=0,t>0,u(x,0)=u0(x)≥γ>0,0≤x≤1,

[3]where http://www.w3.org/1998/Math/MathML" display="block"> β≥α>1. We obtain some:conditions under which thesolution of the semidiscrete form blows up in a finite time. Weestimate its semidiscrete blow-up time and also establish theconvergence of the semidiscrete blow-up time to the real one. Finally.we give some numerical experiments to illustrate our analysis.