Quenching for discretizations of a semilinear parabolic equation with nonlinear boundary outflux

Authors

  • Kouakou Cyrille N'Dri Institut National Polytechnique Félix Houphouët-Boigny, Côte d'Ivoire https://orcid.org/0000-0001-6421-8672
  • Ardjouma Ganon Institut National Polytechnique Félix Houphouët-Boigny, Côte d'Ivoire
  • Gozo Yoro Département Mathématiques et Informatique, Université Nangui Abrogoua d'Abobo Adjamé, Abidjan, Côte d'Ivoire
  • Kidjegbo Augustin Touré Institut National Polytechnique Félix Houphouët-Boigny, Côte d'Ivoire

DOI:

https://doi.org/10.33993/jnaat522-1325

Keywords:

Nonlinear parabolic equations, Nonlinear boundary outflux, Semidiscretizations, Numerical quenching, Quenching rate
Abstract views: 180

Abstract

In this paper, we study numerical approximations of a semilinear parabolic problem in one-dimension, of which the nonlinearity appears both in source term and in Neumann boundary condition. By a semidiscretization using finite difference method, we obtain a system of ordinary differential equations which is an approximation of the original problem. We obtain some conditions under which the positive solution of our system quenches in a finite time and estimate its semidiscrete quenching time. Convergence of the numerical quenching time to the theoretical one is established. Next, we show that the quenching rate of the numerical scheme is different from the continuous one. Finally, we give some numerical results to illustrate our analysis.

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References

L. M. Abia, J. C. Lopez-Marcos and J. Martınez , Blow-up for semidiscretizations of reaction-diffusion equations, Appl. Numer. Math., 20 (1996) no. 1, pp. 145-156. https://doi.org/10.1016/0168-9274(95)00122-0 DOI: https://doi.org/10.1016/0168-9274(95)00122-0

K. A. Adou, K. A. Toure and A. Coulibaly, On the numerical quenching time at blow-up, Adv. Math., Sci. J., 8 (2019) no. 2, pp. 71-85, from http://research-publication.com/wp-content/uploads/2019/09/AMSJ-2019-N2-3.pdf

A. C. Aitken, On Bernoulli’s numerical solution of algebraic equations, Proc. R. Soc. Edinburgh, 46 (1926), pp. 289-305. https://doi.org/10.1017/S0370164600022070 DOI: https://doi.org/10.1017/S0370164600022070

L. A. Assale, T. K. Boni and D. Nabongo , Numerical blow-up time for a semilinear parabolic equation with nonlinear boundary conditions, J. Appl. Math., 2008 (2008), pp. 29. https://doi.org/10.1155/2008/753518, id/No 753518 DOI: https://doi.org/10.1155/2008/753518

T. K. Boni and D. Nabongo, Numerical quenching for a nonlinear diffusion equation with a singular boundary condition., Bull. Belg. Math. Soc. - Simon Stevin, 16 (2009) no. 2, pp. 289-303. https://doi.org/10.36045/bbms/1244038140 DOI: https://doi.org/10.36045/bbms/1244038140

K. Deng and C. A. Roberts , Quenching for a diffusive equation with a concentrated singularity, Differ. Integral Equ., 10 (1997) no. 2, pp. 369-379. https://doi.org/10.57262/die/1367526343 DOI: https://doi.org/10.57262/die/1367526343

K. Deng and C-L. Zhao, Blow-up versus quenching, Commun. Appl. Anal., 7 (2003) no. 1, pp. 87-100, from https://www.researchgate.net/publication/266607632

J. R. Dormand and P. J. Prince, A family of embedded Runge-Kutta formulae, J. Comput. Appl. Math., 6 (1980), pp. 19-26. https://doi.org/10.1016/0771-050X(80)90013-3 DOI: https://doi.org/10.1016/0771-050X(80)90013-3

K. B. Edja, K. A. Toure and B. J-C. Koua, Numerical quenching of a heat equation with nonlinear boundary conditions, J. Nonlinear Sci. Appl., 13 (2020), pp. 65-74. https://doi.org/10.22436/jnsa.013.01.06 DOI: https://doi.org/10.22436/jnsa.013.01.06

A. Ganon, M. M. Taha and A. K. Toure, Numerical blow-up on whole domain for a quasilinear parabolic equation with nonlinear boundary condition, Adv. Math., Sci. J., 9 (2020) no. 1, pp. 49-58. https://doi.org/10.37418/amsj.9.1.5 DOI: https://doi.org/10.37418/amsj.9.1.5

E. Hairer, S. P. Nørsett and G. Wanner, Solving ordinary differential equations. I: Nonstiff problems, Springer Ser. Comput. Math., vol 8, 2nd ed. Berlin: Springer-Verlag, 1993. https://doi.org/10.1007/978-3-540-78862-1 DOI: https://doi.org/10.1007/978-3-540-78862-1

C. Hirota and K. Ozawa, Numerical method of estimating the blow-up time and rate of the solution of ordinary differential equations - an application to the blow-up problems of partial differential equations, J. Comput. Appl. Math., 193 (2006) no. 2, pp. 614-637. https://doi.org/10.1016/j.cam.2005.04.069 DOI: https://doi.org/10.1016/j.cam.2005.04.069

H. Kawarada, On solutions of initial-boundary problem for ut = uxx + 1/(1−u), Publ. Res. Inst. Math. Sci., 10 (1975), pp. 729-736. https://doi.org/10.2977/prims/1195191889 DOI: https://doi.org/10.2977/prims/1195191889

D. Nabongo and T. K. Boni, Numerical quenching solutions of localized semi-linear parabolic equation, Bol. Mat. (N.S.), 14 (2007) no. 2, pp. 92-109, from https://revistas.unal.edu.co/index.php/bolma/article/view/40463

D. Nabongo and T. K. Boni, Numerical quenching for a semilinear parabolic equation, Math. Model. Anal., 13 (2008) no. 4, pp. 521-538. https://doi.org/10.3846/1392-6292.2008.13. DOI: https://doi.org/10.3846/1392-6292.2008.13.521-538

K. C. N’Dri, K. A. Tour e and G. Yoro, Numerical blow-up time for a parabolic equation with nonlinear boundary conditions, Int. J. Numer. Methods Appl., 17 (2018) no. 3-4, pp. 141-160. https://doi.org/10.17654/NM017340141 DOI: https://doi.org/10.17654/NM017340141

K. C. N’Dri, K. A. Toure and G. Yoro, Numerical quenching versus blow-up for a nonlinear parabolic equation with nonlinear boundary outflux, Adv. Math., Sci. J., 9 (2020) no. 1, pp. 151-171. https://doi.org/10.37418/amsj.9.1.14 DOI: https://doi.org/10.37418/amsj.9.1.14

C. V. Pao, Quasilinear parabolic and elliptic equations with nonlinear boundary conditions, Nonlinear Anal., Theory Methods Appl., 66 (2007) no. 9, pp. 639-662. https://doi.org/10.1016/j.na.2005.12.007 DOI: https://doi.org/10.1016/j.na.2005.12.007

T. Salin, On quenching with logarithmic singularity, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, 52 (2003) no. 1, pp. 261-289. https://doi.org/10.1016/S0362-546X(02)00110-4 DOI: https://doi.org/10.1016/S0362-546X(02)00110-4

B. Selcuk and N. Ozalp, The quenching behavior of a semilinear heat equation with a singular boundary outflux, Q. Appl. Math., 72 (2014) no. 4, pp. 747-752. https://doi.org/10.1090/S0033-569X-2014-01367-9 DOI: https://doi.org/10.1090/S0033-569X-2014-01367-9

J. Wu, W. H. Hui and H. Ding, Arc-length method for differential equations, Appl. Math. Mech., Engl. Ed., 20 (1999) no. 8, pp. 936-942. https://doi.org/10.1007/BF02452494 DOI: https://doi.org/10.1007/BF02452494

Y. Zhi, The boundary quenching behavior of a semilinear parabolic equation, Appl. Math. Comput., 218 (2011) no. 2, pp. 233-238. https://doi.org/10.1016/j.amc.2011.05.002 DOI: https://doi.org/10.1016/j.amc.2011.05.002

Y. Zhi and C. Mu, The quenching behavior of a nonlinear parabolic equation with nonlinear boundary outflux, Appl. Math. Comput., 184 (2007) no. 2, pp. 624-630. https://doi.org/10.1016/j.amc.2006.06.061 DOI: https://doi.org/10.1016/j.amc.2006.06.061

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Published

2023-12-28

How to Cite

N’Dri, K. C., Ganon, A., Yoro, G., & Touré, K. A. . (2023). Quenching for discretizations of a semilinear parabolic equation with nonlinear boundary outflux. J. Numer. Anal. Approx. Theory, 52(2), 241–252. https://doi.org/10.33993/jnaat522-1325

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Vol. 52 No. 2 (2023)

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Copyright (c) 2023 Kouakou Cyrille N'Dri

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