Geodesic
Numerical blow-up for nonlinear diffusion equation with neumann boundary conditions
Ganon, Ardjouma 1 ; Taha, Manin Mathurin 1 ; Koffi, N'guessan 2 ; Toure, Augustin Kidjegbo 2
1 Institut National Polytechnique Felix Houphouet-Boigny Yamoussoukro, BP 2444, Cote d'Ivoire
2 UFR SED, Universite Alassane Ouattara de Bouake, 01 BP V 18 Bouake 01, Cote d'Ivoire
Journal of nonlinear sciences and its applications, Tome 14 (2021) no. 2, p. 80-88.

Voir la notice de l'article provenant de la source International Scientific Research Publications

This work is concerned with the study of the numerical approximation for the nonlinear diffusion equation $ (u^{m})_t= u_{xx}, \ 0$, under Neumann boundary conditions $ u_x(0,t)=0, \ u_x(1,t)=u^{\alpha}(1,t), \ t>0 $. First, we obtain a semidiscrete scheme by the finite differences method and prove the convergence of its solution to the continuous one. Then, we establish the numerical blow-up and the convergence of the numerical blow-up time to the theoretical one when the mesh size goes to zero. Finally, we illustrate our analysis with some numerical experiments.
DOI : 10.22436/jnsa.014.02.03
Classification : 35B44, 35B51, 35K20, 65M06
Keywords: Nonlinear diffusion equation, numerical blow-up, arc length transformation, Aitken \( \Delta^{2} \) method

Ganon, Ardjouma  1 ; Taha, Manin Mathurin  1 ; Koffi, N'guessan  2 ; Toure, Augustin Kidjegbo  2

1 Institut National Polytechnique Felix Houphouet-Boigny Yamoussoukro, BP 2444, Cote d'Ivoire
2 UFR SED, Universite Alassane Ouattara de Bouake, 01 BP V 18 Bouake 01, Cote d'Ivoire 
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Ganon, Ardjouma ; Taha, Manin Mathurin ; Koffi, N'guessan ; Toure, Augustin Kidjegbo . Numerical blow-up for nonlinear diffusion equation with neumann boundary conditions. Journal of nonlinear sciences and its applications, Tome 14 (2021) no. 2, p. 80-88. doi : 10.22436/jnsa.014.02.03. http://geodesic.mathdoc.fr/articles/10.22436/jnsa.014.02.03/

[1] Adou, K. A.; Touré, K. A.; Coulibaly, A. On the numerical quenching time at blow-up, Adv. Math. Sci. J., Volume 8 (2019), pp. 71-85 | Zbl

[2] Deng, K.; Xu, M. X. Remarks on blow-up behavior for a nonlinear diffusion equation with neumann boundary conditions, Proc. Amer. Math. Soc., Volume 127 (1999), pp. 167-172 | Zbl | DOI

[3] Edja, K. B.; N'guessan, K.; Koua, B. J.-C.; Touré, K. A. Numerical quenching for heat equations with coupled nonlinearboundary flux, Int. J. Anal. Appl., Volume 17 (2019), pp. 1034-1051

[4] Ferreira, R.; Groisman, P.; Rossi, J. D. Numerical Blow-up for the Porous Medium Equation with a Source, Numer. Methods Partial Differential Equations, Volume 20 (2004), pp. 552-575 | Zbl | DOI

[5] Ferreira, R.; Quirós, F. The Balance bet we en Nonlinear Inwards and Outwards Boundary Flux for a Nonlinear Heat Equation, J. Differential Equations, Volume 184 (2002), pp. 259-282

[6] Filo, J. Diffusivity versus Absorption through the Boundary, J. Differential Equations, Volume 99 (1992), pp. 281-305 | DOI | Zbl

[7] Ganon, A.; Taha, M. M.; Touré, A. K. Blow-up for Semidiscretization of Semilinear Parabolic Equation With Nonlinear Boundary Condition, J. Math. Res., Volume 11 (2019), pp. 1-10

[8] Ganon, A.; Taha, M. M.; Touré, A. K. Numerical blow-up on whole domain for a quasilinear parabolic equation with nonlinear boundary condition, Adv. Math. Sci. J., Volume 9 (2020), pp. 49-58

[9] Hirota, C.; Ozawa, K. 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., Volume 193 (2006), pp. 614-637 | DOI | Zbl

[10] Jiang, Z. X.; Zheng, S.; Song, X. F. Blow-Up Analysis for a Nonlinear Diffusion Equation with Nonlinear Boundary Conditions, Appl. Math. Lett., Volume 17 (2004), pp. 193-199 | DOI | Zbl

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

[12] Ushijima, T. K. On the Approximation of Blow-up Time for Solutions of Nonlinear Parabolic Equations, Publ. Res. Inst. Math. Sci., Volume 36 (2000), pp. 613-640 | DOI | Zbl

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