An Ushijima-type analysis for the numerical blow-up of a fractional reaction-diffusion equation
DOI:
https://doi.org/10.33993/jnaat551-1661Keywords:
Fractional reaction-diffusion equation, Numerical blow-up, L1 scheme, Ushijima framework, Blow-up time convergence, Caputo derivative, Adaptive time-steppingAbstract
This paper is devoted to the rigorous analysis of the blow-up phenomenon for a nonlinear reaction-diffusion equation with a Caputo fractional time derivative. Our approach is twofold. First, we prove that the solution of a semi-discretized system
blows up in a finite time Th under a suitable condition on the initial data. Second, we study a full discretization based on the explicit L1 scheme. By using a discrete weighted functional and comparing it to an auxiliary sequence, we prove that the numerical
solution also blows up in finite time, provided an adaptive time-stepping strategy is chosen. The main contribution of this work is to demonstrate that the established lemmas lay the groundwork for applying the theoretical framework of Ushijima to this class of fractional problems. This theoretical bridge allows us to conclude on the convergence of the numerical blow-up time towards its semi-discrete counterpart, thus validating the scheme’s ability to faithfully capture the dynamics of the singularity.
Downloads
References
[1] P. R. Beesack, Comparison theorems and integral inequalities for Volterra integral equations, Proceedings of the American Mathematical Society, 20(1) (1969), 61–66. DOI: https://doi.org/10.1090/S0002-9939-1969-0234228-3
[2] J. Cao, G. Song, J. Wang, Q. Shi, and S. Sun, Blow-up and global solutions for a class of time fractional nonlinear reaction–diffusion equation with weakly spatial source, Applied Mathematics Letters, 91 (2019), 201–206. DOI: https://doi.org/10.1016/j.aml.2018.12.020
[3] H. Chen and M. Stynes, A discrete comparison principle for the time-fractional diffusion equation, Computers & Mathematics with Applications, 80(5) (2020), 917–922. DOI: https://doi.org/10.1016/j.camwa.2020.04.018
[4] K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Lecture Notes in Mathematics, Vol. 2004, Springer, Berlin, Heidelberg, 2010. DOI: https://doi.org/10.1007/978-3-642-14574-2
[5] X. Huang, Y. Liu, and M. Yamamoto, Blow-up for time-fractional diffusion equations with superlinear convex semilinear terms, arXiv preprint arXiv:2310.14295, 2023.
[6] B. Jin, R. Lazarov and Z. Zhou, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA Journal of Numerical Analysis, 36(1) (2016), 197–221. DOI: https://doi.org/10.1093/imanum/dru063
[7] M. D. Kassim, K. M. Furati, and N.-E. Tatar, Asymptotic behavior of solutions to nonlinear fractional differential equations, Mathematical Modelling and Analysis, 21(5) (2016), 610–629. DOI: https://doi.org/10.3846/13926292.2016.1198279
[8] A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
[9] M. Kirane, M. Medved, and N.-E. Tatar, On the nonexistence of blowing-up solutions to a fractional functional-differential equation, Georgian Mathematical Journal, 19(1) (2012), 127–144. DOI: https://doi.org/10.1515/gmj-2012-0006
[10] C. M. Kirk, W. E. Olmstead, and C. A. Roberts, A system of nonlinear Volterra equations with blow-up solutions, Journal of Integral Equations and Applications, 25(3) (2013), 377–394. DOI: https://doi.org/10.1216/JIE-2013-25-3-377
[11] W. Li, S. Wang, and V. Rehbock, A 2nd-order one-point numerical integration scheme for fractional ordinary differential equations, Numerical Algebra, Control and Optimization, 7(3) (2017), 273–287. DOI: https://doi.org/10.3934/naco.2017018
[12] B. Li, X. Xie, and S. Zhang, A new smoothness result for Caputo-type fractional ordinary differential equations, Applied Mathematics and Computation, 349 (2019), 408–420. DOI: https://doi.org/10.1016/j.amc.2018.12.052
[13] Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, Journal of Computational Physics, 225(2) (2007), 1533–1552. DOI: https://doi.org/10.1016/j.jcp.2007.02.001
[14] Y. Luchko, Maximum principle for the generalized time-fractional diffusion equation, Journal of Mathematical Analysis and Applications, 351(1) (2009), 218–223. DOI: https://doi.org/10.1016/j.jmaa.2008.10.018
[15] Y. Luchko and M. Yamamoto, On the maximum principle for a time-fractional diffusion equation, Fractional Calculus and Applied Analysis, 20(5) (2017), 1131–1145. DOI: https://doi.org/10.1515/fca-2017-0060
[16] H. Nachid, B. Yekre, and Y. Gozo, Simulation of the blow-up and the quenching time for a positive solutions of singular boundary value problems for a nonlinear parabolic systems, Annales Mathématiques Africaines, 8 (2020), 53–70.
[17] C. A. Roberts and W. E. Olmstead, Growth rates for blow-up solutions of nonlinear Volterra equations, Quarterly of Applied Mathematics, 54(1) (1996), 153–159. DOI: https://doi.org/10.1090/qam/1373844
[18] A. Safsaf, S. Alfalqi, A. Bchatnia, and A. Beniani, Blow-up dynamics in nonlinear coupled wave equations with fractional damping and external source, Electronic Research Archive, 32(10) (2024), 5738–5751. DOI: https://doi.org/10.3934/era.2024265
[19] Z. Z. Sun and X. Wu, A fully discrete difference scheme for a diffusion-wave system, Applied Numerical Mathematics, 56(2) (2006), 193–209. DOI: https://doi.org/10.1016/j.apnum.2005.03.003
[20] V. E. Tarasov, On history of mathematical economics: Application of fractional calculus, Mathematics, 7(6) (2019), 509. DOI: https://doi.org/10.3390/math7060509
[21] V. E. Tarasov, Exact solutions of Bernoulli and Logistic fractional differential equations with power law coefficients, Mathematics, 8(12) (2020), 2231. DOI: https://doi.org/10.3390/math8122231
[22] T. K. Ushijima, On the Approximation of Blow-up Time for Solutions of Nonlinear Parabolic Equations, Publications of the Research Institute for Mathematical Sciences, 36(5) (2000), 613–640. DOI: https://doi.org/10.2977/prims/1195142812
[23] J. Villa-Morales, Upper bounds for the blow-up time of a system of fractional differential equations with Caputo derivatives and a numerical scheme for the solution of the system, arXiv preprint arXiv:2310.13584, 2023. DOI: https://doi.org/10.1016/j.rinam.2023.100408
[24] Q. Wang, Z. Yang, and C. Zhao, Numerical blow-up analysis of the explicit L1-scheme for fractional ordinary differential equations, Numerical Algorithms, 89 (2022), 819–846. DOI: https://doi.org/10.1007/s11075-021-01121-w
[25] S. Zeng, S. Migórski, V. T. Nguyen, and Y. R. Bai, Maximum Principles for a Class of Generalized Time-Fractional Diffusion Equations, Fractional Calculus and Applied Analysis, 23(3) (2020), 822–836. DOI: https://doi.org/10.1515/fca-2020-0041
Downloads
Published
Issue
Section
License
Copyright (c) 2026 Benjamin Yekre, Kra Amani N'da Rodrigue, Halima Nachid
This work is licensed under a Creative Commons Attribution 4.0 International License.
Open Access. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
