Finite Difference Scheme for One-dimensional Coupled Parabolic System with Blow-up

Authors

  • Manar Khalil Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysiay, 43600 UKM Bangi Selangor, Malaysia
  • Ishak Hashim Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi Selangor, Malaysia https://orcid.org/0000-0003-4237-7140
  • Maan A. Rasheed Department of Mathematics, College of Basic Education, Mustansiriyah University, Baghdad Iraq. https://orcid.org/0000-0002-7955-1424
  • Shaher Momani Nonlinear Dynamics Research Center (NDRC),Ajman University,Ajman PO Box 346, United Arab Emirates 2-Department of Mathematics, Faculty of Science, University of Jordan, https://orcid.org/0000-0002-6326-8456

DOI:

https://doi.org/10.29020/nybg.ejpam.v18i3.6140

Keywords:

fully discrete, Crank-Nicolson formula, numerical blow-up times, convergence, blow-up time, consistency, stability, convergence (CSC)

Abstract

This study aims to find an efficient technique to estimate the blow-up time (BUT) of one-dimensional semilinear coupled parabolic systems. Firstly, a fully discrete finite difference formula is derived with a nonfixed time-stepping formula, based on the Crank-Nicolson method. In addition, the consistency, stability, and convergence of the proposed scheme are considered. Secondly, two numerical experiments are presented. For each experiment, we apply the proposed scheme to calculate the numerical blow-up time, error bounds, and the numerical order of convergence for blow-up times.  The results obtained show that the proposed C.N scheme is consistent with the system considered. However, it is conditionally stable. In addition, the Crank-Nicolson scheme converges in the stability region and achieves first- and second-order accuracy in temporal and spatial dimensions, respectively. Moreover, the suggested nonfixed-stepping formula plays an important role in avoiding any possible instability that may appear near the blow-up point. In addition, it helps to increase the order of numerical convergence. 
Furthermore, the numerical experiments demonstrate that the numerical blow-up simultaneously occurs at only the center point. Finally, the numerical blow-up time sequence is convergent, as the space step is small enough. In addition, the numerical order of convergence for the blow-up time agrees well with the theoretical order of convergence of the proposed scheme. 

Author Biographies

  • Ishak Hashim, Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi Selangor, Malaysia

    Ishak Hashim was born in Melaka, Malaysia, in 1969. He obtained the degree of Bachelor of Science (Mathematics) from The Ohio State University, US, in June 1992. He joined Universiti Kebangsaan Malaysia (UKM) as a Mathematics Tutor in March 1993. He obtained the degree of Master of Science in the Mathematics of Nonlinear Models from Heriot-Watt University, UK, in November 1994. His degree of Doctor of Philosophy (Mathematics) was obtained in March 1998 from the University of Strathclyde, UK, under the supervision of Professor S.K. Wilson in Fluid Mechanics. Ishak has been a Professor at the School of Mathematical Sciences, Faculty of Science and Technology, UKM, since August 2008. His research interests in applied mathematics include convective instabilities in fluid flows, boundary layers, computational fluid dynamics and analytical/numeric-analytic methods for nonlinear equations.

  • Maan A. Rasheed, Department of Mathematics, College of Basic Education, Mustansiriyah University, Baghdad Iraq.

    B.SC., in Mathematics, University of Baghdad, College of Science, (Ranked (1) amongst (86) graduates), 1998-2002. M.Sc., in Applied Mathematics, College of Science, University of Baghdad, 2002-2005. Thesis title: “On Numerical solutions of second order parabolic partial differential equations”. Ph.D., in Applied Mathematics, School of Physical and Mathematical Sciences, University of Sussex, England, UK, 2009-2012. Thesis title: “On blow-up solutions of parabolic problems”.

    Fulbright Visiting Scholar (FVS), University of Central Oklahoma (UCO), USA, 2017. University teaching: Since 2006, he has taught many subjects for BSc and MSc students at Mustansiriyah University, and Al-Nahrain University such as: • Numerical Analysis • Differential Equations • Graph Theory  • Complex Analysis • Probability Theory • Integration Theory • General Mathematics.Research interests:

     Partial (ordinary) differential equations: theory and applications

     Numerical solutions of differential equations.

  • Shaher Momani, Nonlinear Dynamics Research Center (NDRC),Ajman University,Ajman PO Box 346, United Arab Emirates 2-Department of Mathematics, Faculty of Science, University of Jordan,

    B.Sc. in mathematics in 1984 from Yarmouk University.

    Ph.D. in mathematics in 1991 from University of Wales.

    Islamic World Academy of Sciences fellowship.

    Dean of the Faculty of Science at university of Jordan between 2014-2016.

    Dean of Academic Research at university of Jordan between 2016-2018.

    Member of Princess Sumaya University for Technology board of trustees.

    Jordan Scientific Research Support Fund member between 2010-2012.

    Member at the editorial board of The Jordanian Journal of Mathematics and Statistics.

    Editor in chief of University of Jordan Dirasat journal.

    Chairman of the Department of Mathematics at the University of Jordan between 2012 and 2014.

    Chairman of the Department of Mathematics at Mutah University between 1994 and 1995.

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Published

2025-08-01

Issue

Vol. 18 No. 3: (July 2025)

Section

Mathematical Modeling and Numerical Analysis

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Copyright (c) 2025 Manar Khalil, Ishak Hashim, Maan A. Rasheed, Shaher Momani

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How to Cite

Finite Difference Scheme for One-dimensional Coupled Parabolic System with Blow-up. (2025). European Journal of Pure and Applied Mathematics, 18(3), 6140. https://doi.org/10.29020/nybg.ejpam.v18i3.6140