Quenching for semidiscretizations of a semilinear heat equation with Dirichlet and Neumann boundary conditions

Diabate Nabongo; Théodore K. Boni

Commentationes Mathematicae Universitatis Carolinae (2008)

  • Volume: 49, Issue: 3, page 463-475
  • ISSN: 0010-2628

Abstract

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This paper concerns the study of the numerical approximation for the following boundary value problem: u t ( x , t ) - u x x ( x , t ) = - u - p ( x , t ) , 0 < x < 1 , t > 0 , u x ( 0 , t ) = 0 , u ( 1 , t ) = 1 , t > 0 , u ( x , 0 ) = u 0 ( x ) > 0 , 0 ≤ x ≤ 1 , where p > 0 . We obtain some conditions under which the solution of a semidiscrete form of the above problem quenches in a finite time and estimate its semidiscrete quenching time. We also establish the convergence of the semidiscrete quenching time. Finally, we give some numerical experiments to illustrate our analysis.

How to cite

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Nabongo, Diabate, and Boni, Théodore K.. "Quenching for semidiscretizations of a semilinear heat equation with Dirichlet and Neumann boundary conditions." Commentationes Mathematicae Universitatis Carolinae 49.3 (2008): 463-475. <http://eudml.org/doc/250301>.

@article{Nabongo2008,
abstract = {This paper concerns the study of the numerical approximation for the following boundary value problem: \[ \left\lbrace \begin\{array\}\{ll\}u\_t(x,t)-u\_\{xx\}(x,t) = -u^\{-p\}(x,t), & 0<x<1, t>0, \ u\_\{x\}(0,t)=0, & u(1,t)=1, t>0, \ u(x,0)=u\_\{0\}(x)>0, & 0\le x \le 1, \end\{array\}\right.\] where $p>0$. We obtain some conditions under which the solution of a semidiscrete form of the above problem quenches in a finite time and estimate its semidiscrete quenching time. We also establish the convergence of the semidiscrete quenching time. Finally, we give some numerical experiments to illustrate our analysis.},
author = {Nabongo, Diabate, Boni, Théodore K.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {semidiscretizations; discretizations; heat equations; quenching; semidiscrete quenching time; convergence; semidiscretizations; discretizations; heat equations; quenching; semidiscrete quenching time; convergence},
language = {eng},
number = {3},
pages = {463-475},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Quenching for semidiscretizations of a semilinear heat equation with Dirichlet and Neumann boundary conditions},
url = {http://eudml.org/doc/250301},
volume = {49},
year = {2008},
}

TY - JOUR
AU - Nabongo, Diabate
AU - Boni, Théodore K.
TI - Quenching for semidiscretizations of a semilinear heat equation with Dirichlet and Neumann boundary conditions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2008
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 49
IS - 3
SP - 463
EP - 475
AB - This paper concerns the study of the numerical approximation for the following boundary value problem: \[ \left\lbrace \begin{array}{ll}u_t(x,t)-u_{xx}(x,t) = -u^{-p}(x,t), & 0<x<1, t>0, \ u_{x}(0,t)=0, & u(1,t)=1, t>0, \ u(x,0)=u_{0}(x)>0, & 0\le x \le 1, \end{array}\right.\] where $p>0$. We obtain some conditions under which the solution of a semidiscrete form of the above problem quenches in a finite time and estimate its semidiscrete quenching time. We also establish the convergence of the semidiscrete quenching time. Finally, we give some numerical experiments to illustrate our analysis.
LA - eng
KW - semidiscretizations; discretizations; heat equations; quenching; semidiscrete quenching time; convergence; semidiscretizations; discretizations; heat equations; quenching; semidiscrete quenching time; convergence
UR - http://eudml.org/doc/250301
ER -

References

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  2. Acker A., Kawohl B., 10.1016/0362-546X(89)90034-5, Nonlinear Anal. 13 (1989), 53-61. (1989) Zbl0676.35021MR0973368DOI10.1016/0362-546X(89)90034-5
  3. Boni T.K., 10.1016/S0764-4442(01)02078-X, C.R. Acad. Sci. Paris Sér. I Math. 333 (2001), 795-800. (2001) Zbl0999.35004MR1868956DOI10.1016/S0764-4442(01)02078-X
  4. Boni T.K., On quenching of solutions for some semilinear parabolic equations of second order, Bull. Belg. Math. Soc. Simon Stevin 7 (2000), 73-95. (2000) Zbl0969.35077MR1741748
  5. Fila M., Kawohl B., Levine H.A., Quenching for quasilinear equations, Comm. Partial Differential Equations 17 (1992), 593-614. (1992) Zbl0801.35057MR1163438
  6. Guo J.S., Hu B., The profile near quenching time for the solution of a singular semilinear heat equation, Proc. Edinburgh Math. Soc. 40 (1997), 437-456. (1997) Zbl0903.35007MR1475908
  7. Guo J., 10.1016/0362-546X(91)90154-S, Nonlinear Anal. 17 (1991), 803-809. (1991) MR1131490DOI10.1016/0362-546X(91)90154-S
  8. Levine H.A., Quenching, nonquenching and beyond quenching for solutions of some parabolic equations, Annali Mat. Pura Appl. 155 (1990), 243-260. (1990) MR1042837

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