Numerical quenching of a heat equation with nonlinear boundary conditions

Volume 13, Issue 1, pp 65--74 http://dx.doi.org/10.22436/jnsa.013.01.06

Publication Date: September 11, 2019 Submission Date: April 10, 2019 Revision Date: June 15, 2019 Accteptance Date: June 16, 2019

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1625 Downloads
• 2960 Views

Authors

Kouame Beranger Edja - Institut National Polytechnique Felix Houphouet-Boigny Yamoussoukro, BP 2444, Cote d'Ivoire. Kidjegbo Augustin Toure - Institut National Polytechnique Felix Houphouet-Boigny Yamoussoukro, BP 2444, Cote d'Ivoire. Brou Jean-Claude Koua - UFR Mathematique et Informatique, Universite Felix Houphouet Boigny, Cote d'Ivoire.

Abstract

In this paper, we study the quenching behavior of semidiscretizations of the heat equation with nonlinear boundary conditions. We obtain some conditions under which the positive solution of the semidiscrete problem quenches in a finite time and estimate its semidiscrete quenching time. We also establish the convergence of the semidiscrete quenching time and obtain some results on numerical quenching rate. Finally we give some numerical results to illustrate our analysis.

Share and Cite

Keywords

• Numerical quenching
• heat equation
• nonlinear boundary

MSC

•  35K05
•  34B15
•  74S20

References

• [1] K. A. Adou, K. A. Touré, A. Coulibaly, Numerical study of the blow-up time of positive solutions of semilinear heat equations, Far East J. Appl. Math., 4 (2018), 291--308
  • View Article


• [2] T. K. Boni, Extinction for discretizations of some semilinear parabolic equations, C. R. Acad. Sci. Paris Sér. I Math., 333 (2001), 795--800
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] T. K. Boni, B. Y. Diby, Quenching time of solutions for some nonlinear parabolic equations with Dirichlet boundary condition and a potential, Ann. Math. Inform., 35 (2008), 31--42
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [4] C. Y. Chan, S. I. Yuen, Parabolic problems with nonlinear absorptions and releases at the boundaries, Appl. Math. Comput., 121 (2001), 203--209
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] K. B. Edja, K. A. Touré, B. J.-C. Koua, Numerical Blow-up for a Heat Equation with Nonlinear Boundary Conditions, J. Math. Res., 10 (2018), 119--128
  • View Article
  • Google Scholar


• [6] H. Kawarada, On Solutions of Initial-Boundary Problem for $u_t=u_{xx}+\frac{1}{1-u}$, Publ. RIMS, Kyoto Univ., 10 (1975), 729--736
  • View Article
  • Google Scholar


• [7] C. M. Krirk, C. A. Roberts, A quenching problem for the heat equation, J. Integral Equations Appl., 14 (2002), 53--72
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [8] H. A. Levine, The Quenching of Solutions of Linear Parabolic and Hyperbolic Equations with Nonlinear Boundary Conditions, SIAM J. Math. Anal., 4 (1983), 1139--1153
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] H. A. Levine, J. T. Montgomery, The quenching of solutions of some nonlinear parabolic equations, SIAM J. Math. Anal., 11 (1980), 842--847
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] K. W. Liang, P. Lin, R. C. E. Tan, Numerical Solution of Quenching Problems Using Mesh-Dependent Variable Temporal Steps, Appl. Numer. Math., 57 (2007), 791--800
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] T. M. Mathurin, T. K. Augustin, M. E. Patrice, Numerical approximation of the blow-up time for a semilinear parabolic equation with nonlinear boundary equation, Far East J. Math. Sci. (FJMS), 60 (2012), 125--167
  • View Article
  • MathSciNet
  • MATH


• [12] D. Nabongo, T. K. Boni, Quenching for semidiscretizations of a heat equation with a singular boundary condition, Asymptot. Anal., 59 (2008), 27--38
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [13] K. C. N'dri, K. A. Touré, G. Yoro, Numerical blow-up time for a parabolic equation with nonlinear boundary conditions, Int. J. Numer. Methods Appl., 17 (2018), 141--160
  • View Article


• [14] B. Selçuk, Quenching behavior of a semilinear reaction-diffusion system with singular boundary condition, Turkish J. Math., 40 (2016), 166--180
  • View Article
  • MathSciNet
  • Google Scholar


• [15] B. Selçuk, N. Ozalp, The quenching behavior of a semilinear heat equation with a singular boundary outflux, Quart. Appl. Math., 72 (2014), 747--752
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] M. G. Teixeira, M. A. Rincon, I.-S. Liu, Numerical analysis of quenching-Heat conduction in metallic materials, Appl. Math. Model., 33 (2009), 2464--2473
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] Y. H. Zhi, C. L. Mu, The quenching behavior of a nonlinear parabolic equation with nonlinear boundary outflux, Appl. Math. Comput., 184 (2007), 624--630
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH