Strong Solutions of the Neuronal Responses System

Authors

  • Raad Abd Faculty of Computer Science and Mathematics/university of kufa
  • Ghassan A. Al-Juaifri Faculty of Computer Science and Mathematics University of Kufa

DOI:

https://doi.org/10.31642/JoKMC/2018/110208

Keywords:

Continuous dependence, Existence, Neumann boundary, Strong solution, uniqueness

Abstract

We explore a connected reaction-diffusion system involving neural responses within limited convex open domains . Utilizing the Classical Faedo-Galerkin technique and employing compactness arguments, we establish the existence, uniqueness, and continuous dependence on initial data for strong solutions.

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References

Jacques Louis Lions. Quelques m´ethodes de r´esolution des problemes aux limites non lin´eaires. 1969.

Nicholas F Britton et al. Reaction-diffusion equations and their applications to biology. Academic Press, 1986.

P Fife. Mathematical approach of reacting and diffusing systems, 1979. DOI: https://doi.org/10.1007/978-3-642-93111-6

James D Murray. Mathematical biology springer-verlag berlin heidelberg new york, 1993. Second Corrected Edition.

Joel Smoller. Shock waves and reactiondiffusion equations, volume 258. Springer Science & Business Media, 2012.

Al-Juaifri, Ghassan A., and Akil J. Harfash. Finite element analysis of nonlinear reactiondiffusion system of FitzhughNagumo type with Robin boundary conditions. Mathematics and Computers in Simulation 203 : 486-517, Elsevier, 2023. DOI: https://doi.org/10.1016/j.matcom.2022.07.005

A I Vol’pert, V A Volpert, and V A Volpert. Traveling wave solutions of parabolic systems, volume 140. American Mathematical Soc., 1994.

James P Keener and James Sneyd. Mathematical physiology, volume 1. Springer, 1998. DOI: https://doi.org/10.1007/b98841

Richard FitzHugh. Impulses and physiological states in theoretical models of nerve membrane. Biophysical journal, 1(6):445–466, 1961. DOI: https://doi.org/10.1016/S0006-3495(61)86902-6

Jinichi Nagumo, Suguru Arimoto, and Shuji Yoshizawa. An active pulse transmission line simulating nerve axon. Proceedings of the IRE, 50(10):2061–2070, 1962. DOI: https://doi.org/10.1109/JRPROC.1962.288235

James D Murray. Mathematical biology: I and ii, interdisciplinary applied mathematics. Mathematical Biology, Springer, 2002. DOI: https://doi.org/10.1007/b98868

Markus B¨ar, N Gottschalk, Markus Eiswirth, and Gerhard Ertl. Spiral waves in a surface reaction: model calculations. The Journal of chemical physics, 100(2):1202–1214, 1994. DOI: https://doi.org/10.1063/1.466650

Dwight Barkley. A model for fast computer simulation of waves in excitable media. Physica D: Nonlinear Phenomena, 49(1-2):61–70, 1991. DOI: https://doi.org/10.1016/0167-2789(91)90194-E

Alain Karma. Meandering transition in two-dimensional excitable media. Physical review letters, 65(22):2824, 1990. DOI: https://doi.org/10.1103/PhysRevLett.65.2824

John J Tyson. What everyone should know about the belousov-zhabotinsky reaction. In Frontiers in mathematical biology, pages 569–587. Springer, 1994. DOI: https://doi.org/10.1007/978-3-642-50124-1_33

Al-Juaifri, G.A., Harfash, A.J. Analysis of a nonlinear reaction-diffusion system of the Fitzhugh-Nagumo type with Robin boundary conditions. In Ricerche di Matematica,72(1) pages 335–357. Springer 2023. DOI: https://doi.org/10.1007/s11587-022-00711-7

Ghassan A. Al-Juaifri and Akil J. Harfash. Existence and uniqueness of solution for the nonlinear Brusselator system with Robin boundary conditions In Georgian Mathematical Journal,2023 https://doi.org/10.1515/gmj-2023-2091. DOI: https://doi.org/10.1515/gmj-2023-2091

V Krinsky and A Pumir. Models of defibrillation of cardiac tissue. Chaos: An Interdisciplinary Journal of Nonlinear Science, 8(1):188–203, 1998. DOI: https://doi.org/10.1063/1.166297

Houria Triki and Abdul-Majid Wazwaz. On soliton solutions for the fitzhugh– nagumo equation with time-dependent coefficients. Applied Mathematical Modelling, 37(6):3821–3828, 2013. DOI: https://doi.org/10.1016/j.apm.2012.07.031

Zhang, Chunrui and Ke, Ai and Zheng, Baodong. Patterns of interaction of coupled reaction–diffusion systems of the FitzHugh–Nagumo type In Nonlinear Dynamics, pages 1451–1476. Springer, 2019. DOI: https://doi.org/10.1007/s11071-019-05065-8

Abdul-Majid Wazwaz. Partial differential equations and solitary waves theory. Springer Science & Business Media, 2010. DOI: https://doi.org/10.1007/978-3-642-00251-9

Daniel Daners. Robin boundary value problems on arbitrary domains. Transactions of the American Mathematical Society, 352(9):4207–4236, 2000. DOI: https://doi.org/10.1090/S0002-9947-00-02444-2

VG Maz’ya. Zur Theorie Sobolewscher R¨aume, volume 38. TeubnerTexte zur Mathematik,Teubner, Leipzig, 1981.

RE Showalter. Variational theory and approximation of boundary value problems. In Numerical Analysis Lancaster 1984, pages 140–179. Springer, 1985. DOI: https://doi.org/10.1007/BFb0075159

Jonathan A Sherratt. A comparison of periodic travelling wave generation by robin and dirichlet boundary conditions in oscillatory reaction–diffusion equations. IMA journal of applied mathematics, 73(5):759–781, 2008. DOI: https://doi.org/10.1093/imamat/hxn015

Abdalaziz Saleem Al-Ofl. Analysis of complex nonlinear reaction-diffusion equations. PhD thesis, Durham University, 2008.

Roger Temam. Infinite-dimensional dynamical systems in mechanics and physics, volume 68. Springer Science & Business Media, 2012.

Thierry Cazenave. Semilinear Schrodinger Equations, volume 10. American Mathematical Soc., 2003. DOI: https://doi.org/10.1090/cln/010

Philippe G Ciarlet. The finite element method for elliptic problems. SIAM, 2002. DOI: https://doi.org/10.1137/1.9780898719208

Philip Hartman. Ordinary Differential Equations. John Wiley & Sons, 1973.

Robert Dautray and Jacques-Louis Lions. Mathematical Analysis and Numerical Methods for Science and Technology: Volume 2 Functional and Variational Methods, volume 2. Springer Science & Business Media, 1999.

J. C. Robinson. Infinite-dimensional dynamical systems. Cambridge Texts in Applied Mathematics. Cambridge University Press., 2001.

Alexander Zen´ıˇsek and John Robert Whiteman.ˇ Nonlinear elliptic and evolution problems and their finite element approximations. Academic Press,

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Published

2025-05-19

How to Cite

Abd, R., & Al-Juaifri , G. A. (2025). Strong Solutions of the Neuronal Responses System. Journal of Kufa for Mathematics and Computer, 11(2), 62-69. https://doi.org/10.31642/JoKMC/2018/110208

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