International Journal of Numerical Methods and Applications

The International Journal of Numerical Methods and Applications publishes research articles on numerical methods and their applications in various fields, including differential equations, fluid dynamics, and bioinformatics. It also welcomes survey articles on new methods in numerical analysis.

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ON THE NUMERICAL APPROXIMATIONS OF BLOW-UP TIME IN SEMILINEAR PARABOLIC EQUATIONS

Authors

  • ADOU Koffi Achille
  • ABRO Goh André Pascal
  • GANON Ardjouma

Keywords:

blow-up time, blow-up problems, nonlinear parabolic equation, semidiscrete solution, arc length transformation, Aitken method

DOI:

https://doi.org/10.17654/0975045226005

Abstract

This paper investigates the numerical estimation of the blow-up time for solutions of semilinear parabolic problems defined on a bounded domain. We study the behavior of semidiscrete approximations applied to reaction-diffusion equations and establish both necessary and sufficient conditions for blow-up to occur in the discrete setting. For the numerical computations, the problem is transformed into a more tractable form using the arc-length transformation technique, which enables to generate a linearly convergent sequence to the blow-up time. This sequence is then accelerated using the Aitken  method. Several numerical experiments are presented to illustrate the proposed approach.

Received: August 11, 2025
Revised: September 12, 2025
Accepted: September 25, 2025

References

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[8] A. Friedman and B. Mcleod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34 (1985), 425-447.

[9] Ardjouma Ganon, Manin Mathurin Tahaa, N’Guessan N’Guessan Koffi and Augustin Kidjegbo Touré, Numerical blow-up for nonlinear diffusion equation with Neumann boundary conditions, J. Nonlinear Sci. Appl. 14 (2021), 80-88.

[10] A. Ganon, M. M. Taha and K. A. Touré, Numerical blow-up for a quasilinear parabolic equation with nonlinear boundary condition, Far East J. Math. Sci. (FJMS) 114 (2019), 19-38.

[11] 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), 614-637.

[12] B. Perthame, Parabolic equations in biology, Lecture Notes on Mathematical Modelling in the Life Sciences, Springer Switzerland, 2015.

DOI 10.1007/978-3-319-19500-1.

[13] P. Quittner and Ph. Souplet, Superlinear parabolic problems, blow-up, global existence and steady states, Birkhäuser Advanced Texts, Birkhäuser, Basel, 2007.

[14] M. A. Rasheed, Blow-up properties of a coupled system of reaction-diffusion equations, Iraqi Journal of Science 62(9) (2021), 3052-3060.

[15] T. K. Ushijima, On the approximation of blow-up time for solutions of nonlinear parabolic equations, Publ. RIMS Inst. Math. Sci. 36 (2000), 613-640.

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Published

2025-11-18

Issue

Vol. 26 No. 1 (2026)

Section

Articles

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How to Cite

ON THE NUMERICAL APPROXIMATIONS OF BLOW-UP TIME IN SEMILINEAR PARABOLIC EQUATIONS. (2025). International Journal of Numerical Methods and Applications, 26(1), 95-113. https://doi.org/10.17654/0975045226005

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