Numerical Finite-Difference Approximations of a Coupled Reaction-Diffusion System with Gradient Terms

Authors

DOI:

https://doi.org/10.29020/nybg.ejpam.v17i3.5246

Keywords:

finite difference schemes, blow-up solutions, Euler explicit (implicit), blow-up time, gradient terms

Abstract

This study focuses on the derivation of explicit and implicit finite difference formulas.The objective of this study is to derive an estimation of the blow-up time for a coupled reaction-diffusion system incorporating gradient terms, employing numerical finite difference approximations. Furthermore, an examination is conducted on the consistency, stability, and convergence of the proposed schemes. Additionally, the study presents two numerical experiments. In each instance, the numerical blow-up time is calculated benefit the suggested methodologies, employing varying space steps and non-fixed time-stepping. The numerical findings obtained demonstrate that the blow-up time sequence exhibits convergence as the space step decreases. Moreover, the
numerical orders of convergence for the blow-up time goes well with the theoretical orders observed in the numerical solutions.

Author Biographies

  • Manar I. Khalil, Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysiay, 43600 UKM Bangi Selangor, Malaysia

    Manar Ismail Khalil completed her bachelor’s degree in mathematics at the College of Education, Tikrit University, Iraq (2002), and obtained a master’s degree in mathematics, specializing in mathematics teaching methods, Tikrit University, Iraq (2013). Currently studying for a PhD. At Universiti Kebangsaan Malaysia, in mathematics. Numerical analysis.

  • 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.

  • Faieza Samat, Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi Selangor, Malaysia.

    Dr. Faieza Samat obtained her B.Sc and M.Sc degrees in Mathematics from Universiti Kebangsaan Malaysia (UKM) in 1999 and 2000 respectively. She completed her Ph.D in Numerical Analysis from Universiti Putra Malaysia (UPM) in 2012. Prior to working with UKM, she served as a lecturer in Universiti Pendidikan Sultan Idris (UPSI) from 2000 until 2014. Currently, Dr Faieza Samat is a mathematics lecturer for ASASIpintar pre-university program. Her main interest is on development of numerical methods for ordinary differential equations.

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

    Shaher Momani is Distinguished Professor, Department of Mathematics, School of Science, The University of Jordan.  Momani also works as the Dean of College of Humanities and Science at Ajman University in UAE.   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 the 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. Momani has published more than 300 articles in ISI international journals of high quality. He was selected by Clarivate Analytics in its prestigious list of Highly Cited Researchers in Mathematics in 2014, 2015, 2016, and 2017, and in 2018, he was selected by Clarivate Analytics as a researcher with substantial influence over the last decade in the Cross-Field category.  He was also selected by Clarivate Analytics in its prestigious list of The World's Most Influential Scientific Minds from 2014 to 2018. Prof. Shaher Momani is an active member of the Royal Scientific Society of Jordan (RSS)’s distinguished science community and was selected as one of the 1st patches of Jordan’s Stars of Science in 2017 (honored by His Majesty the King during the 2017 World Science Forum). 

Downloads

Published

2024-07-31

Issue

Vol. 17 No. 3: (July 2024)

Section

Nonlinear Analysis

License

Copyright (c) 2024 European Journal of Pure and Applied Mathematics

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

Upon acceptance of an article by the European Journal of Pure and Applied Mathematics, the author(s) retain the copyright to the article. However, by submitting your work, you agree that the article will be published under the Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0). This license allows others to copy, distribute, and adapt your work, provided proper attribution is given to the original author(s) and source. However, the work cannot be used for commercial purposes.

By agreeing to this statement, you acknowledge that:

  • You retain full copyright over your work.
  • The European Journal of Pure and Applied Mathematics will publish your work under the Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0).
  • This license allows others to use and share your work for non-commercial purposes, provided they give appropriate credit to the original author(s) and source.

How to Cite

Numerical Finite-Difference Approximations of a Coupled Reaction-Diffusion System with Gradient Terms. (2024). European Journal of Pure and Applied Mathematics, 17(3), 1516-1538. https://doi.org/10.29020/nybg.ejpam.v17i3.5246