Existence and Uniqueness of a Weak Solution of a Time Fractional Reaction-Diffusion System of Fitzhugh-Nagumo Type

Yong-Dok Han; Kang-Song Yun; Chol-Gwang Kim

doi:doi:10.11648/j.engmath.20261001.11

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Existence and Uniqueness of a Weak Solution of a Time Fractional Reaction-Diffusion System of Fitzhugh-Nagumo Type

Yong-Dok Han^*, Kang-Song Yun, Chol-Gwang Kim

Published in Engineering Mathematics (Volume 10, Issue 1)

Received: 28 January 2025 Accepted: 12 August 2025 Published: 23 January 2026

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Abstract

In the mathematical theory of nerve impulse propagation, the Fitzhugh-Nagumo Reaction-Diffusion System has attracted a great deal of attention. The Fitzhugh-Nagumo Reaction-Diffusion System provides a prototype for chemical and other nerve conduction and biological systems. In this paper, we define two types of weak solutions of the time fractional Fitzhugh-Nagumo Reaction-Diffusion System, namely (1) -weak solutions and (2) -weak solutions, and demonstrate the existence and uniqueness of these weak solutions. First, we have obtained a generalization of [1, Lemma 1] in Lemma 2.1 and using Lemma 2.1 and Galerkin’s approximation sequence, we have found the existence of (1)-weak solutions and (2)-weak solutions. We also obtained a generalization of the result of [10, Lemma 6] to Hilbert spaces in Lemma 2.2, and using this result we proved the uniqueness of the (2)-weak solution. Lemma 2.1 and Lemma 2.2 of this paper are results that can be effectively used to show the existence and uniqueness of weak solutions of time fractional partial differential equations. And the existence and uniqueness results of the weak solution of the time fractional Fitzhugh-Nagumo Reaction-Diffusion System can be used in the numerical solutions of this reaction-diffusion system. Also, we can be used in the optimal control problems described in this system.

Published in	Engineering Mathematics (Volume 10, Issue 1)
DOI	10.11648/j.engmath.20261001.11
Page(s)	1-13
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Time-Fractional Reaction-Diffusion System, Existence, Uniqueness

References

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[2]	Armanyos, M., Radwan, A. G., Fractional-Order Fitzhugh-Nagumo and Izhikevich Neuron Models: 978-1-4673-9749-0/16/$31.00 (2016) IEEE.
[3]	Ford, N. J., Xiao, J. Y., Yan, Y. B., A Finite Element Method for Time Fractional Partial Differential Equations: Fract. Calc. Appl. Anal., 14, 454-474(2011).
[4]	Kilbas, A. A., Srivastava, H. M., Trujllo, J. J., Theory and Applications of Fractional Differential Equations: ELSEVIER, North-Holland, 2006.ISBN-13: 978-0-444-51832-3
[5]	Lions, J. L., Quelques méthodes de r ésolution des probl émes aux limites non linéaire: DUNOD GAUTHIER-VILLARS, Paris, 1969.
[6]	Lions, J. L., Magenes, E., Problémes aux limites non homogénes et applications: DUNOD, Paris, 1968.
[7]	Li, X. J., Xu, C. J., A space-time spectral method for the time fractional diffusion equation: SIAM J. NUMER. ANAL., 47(3), 2108-2131(2009). https://doi.org/10.1137/080718942
[8]	Plotnikov, S. A., Alexander, L. F., Controlled synchronization in two hybrid FitzHugh-Nagumo systems: IFAC-PapersOnLine, 49-14, 137-141(2016). https://doi.org/10.1016/j.ifacol.2016.07.998
[9]	Qiao, Q., Zhang, X., Spectrum and stability of travelling pulses in a coupled Fitzhugh-Nagumo equation: arXiv: 2302.06110V1 [math.DS] 13 Feb 2023, 1-44.
[10]	Taki-Eddine, O., Abdelfatah, B., A priori estimates for weak solution for a time-fractional nonlinear reaction-diffusion equations with an integral condition: Chaos, Solitons and Fractals, 103, 79-89(2017). https://doi.org/10.1016/j.chaos.2017.05.035
[11]	Ucar, A., Lonngren, K. E., Bai, E. W., Synchronization of the coupled FitzHugh-Nagumo systems: Chaos, Solitons and Fractals, 20, 1085-1090(2004). https://doi.org/10.1016/j.chaos.2003.09.039
[12]	Webb, J. R. L., Weakly singular Gronwall inequalities and applications to fractional differential equations: J. Math. Anal. Appl., 2018: 30939-9. https://doi.org/10.1016/j.jmaa.2018.11.004
[13]	Wei, L., Global existence theory and chaos control of fractional differential equations: J.Math.Anal.Appl., 332,709-726(2007). https://doi.org/10.1016/j.jmaa.2006.10.040
[14]	Zhang, C., Ke, A., Zheng, B. D., Patterns of interaction of coupled reaction-diffusion systems of the FitzHugh-Nagumo type: Nonlinear Dyn., https://doi.org/10.1007/s11071-2019-05065-8
[15]	Zhou, Y., Peng, L., Weak solutions of the time-fractional Navier- Stokes equations and optimal control: Computers and Mathematics with Applications, 2016: 0898-1221. https://doi.org/10.1016/j.camwa.2016.7.007

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Han, Y., Yun, K., Kim, C. (2026). Existence and Uniqueness of a Weak Solution of a Time Fractional Reaction-Diffusion System of Fitzhugh-Nagumo Type. Engineering Mathematics, 10(1), 1-13. https://doi.org/10.11648/j.engmath.20261001.11

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Han, Y.; Yun, K.; Kim, C. Existence and Uniqueness of a Weak Solution of a Time Fractional Reaction-Diffusion System of Fitzhugh-Nagumo Type. Eng. Math. 2026, 10(1), 1-13. doi: 10.11648/j.engmath.20261001.11

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AMA Style

Han Y, Yun K, Kim C. Existence and Uniqueness of a Weak Solution of a Time Fractional Reaction-Diffusion System of Fitzhugh-Nagumo Type. Eng Math. 2026;10(1):1-13. doi: 10.11648/j.engmath.20261001.11

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@article{10.11648/j.engmath.20261001.11,
  author = {Yong-Dok Han and Kang-Song Yun and Chol-Gwang Kim},
  title = {Existence and Uniqueness of a Weak Solution of a Time Fractional Reaction-Diffusion System of Fitzhugh-Nagumo Type
},
  journal = {Engineering Mathematics},
  volume = {10},
  number = {1},
  pages = {1-13},
  doi = {10.11648/j.engmath.20261001.11},
  url = {https://doi.org/10.11648/j.engmath.20261001.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.engmath.20261001.11},
  abstract = {In the mathematical theory of nerve impulse propagation, the Fitzhugh-Nagumo Reaction-Diffusion System has attracted a great deal of attention. The Fitzhugh-Nagumo Reaction-Diffusion System provides a prototype for chemical and other nerve conduction and biological systems. In this paper, we define two types of weak solutions of the time fractional Fitzhugh-Nagumo Reaction-Diffusion System, namely (1) -weak solutions and (2) -weak solutions, and demonstrate the existence and uniqueness of these weak solutions. First, we have obtained a generalization of [1, Lemma 1] in Lemma 2.1 and using Lemma 2.1 and Galerkin’s approximation sequence, we have found the existence of (1)-weak solutions and (2)-weak solutions. We also obtained a generalization of the result of [10, Lemma 6] to Hilbert spaces in Lemma 2.2, and using this result we proved the uniqueness of the (2)-weak solution. Lemma 2.1 and Lemma 2.2 of this paper are results that can be effectively used to show the existence and uniqueness of weak solutions of time fractional partial differential equations. And the existence and uniqueness results of the weak solution of the time fractional Fitzhugh-Nagumo Reaction-Diffusion System can be used in the numerical solutions of this reaction-diffusion system. Also, we can be used in the optimal control problems described in this system.
},
 year = {2026}
}

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TY  - JOUR
T1  - Existence and Uniqueness of a Weak Solution of a Time Fractional Reaction-Diffusion System of Fitzhugh-Nagumo Type

AU  - Yong-Dok Han
AU  - Kang-Song Yun
AU  - Chol-Gwang Kim
Y1  - 2026/01/23
PY  - 2026
N1  - https://doi.org/10.11648/j.engmath.20261001.11
DO  - 10.11648/j.engmath.20261001.11
T2  - Engineering Mathematics
JF  - Engineering Mathematics
JO  - Engineering Mathematics
SP  - 1
EP  - 13
PB  - Science Publishing Group
SN  - 2640-088X
UR  - https://doi.org/10.11648/j.engmath.20261001.11
AB  - In the mathematical theory of nerve impulse propagation, the Fitzhugh-Nagumo Reaction-Diffusion System has attracted a great deal of attention. The Fitzhugh-Nagumo Reaction-Diffusion System provides a prototype for chemical and other nerve conduction and biological systems. In this paper, we define two types of weak solutions of the time fractional Fitzhugh-Nagumo Reaction-Diffusion System, namely (1) -weak solutions and (2) -weak solutions, and demonstrate the existence and uniqueness of these weak solutions. First, we have obtained a generalization of [1, Lemma 1] in Lemma 2.1 and using Lemma 2.1 and Galerkin’s approximation sequence, we have found the existence of (1)-weak solutions and (2)-weak solutions. We also obtained a generalization of the result of [10, Lemma 6] to Hilbert spaces in Lemma 2.2, and using this result we proved the uniqueness of the (2)-weak solution. Lemma 2.1 and Lemma 2.2 of this paper are results that can be effectively used to show the existence and uniqueness of weak solutions of time fractional partial differential equations. And the existence and uniqueness results of the weak solution of the time fractional Fitzhugh-Nagumo Reaction-Diffusion System can be used in the numerical solutions of this reaction-diffusion system. Also, we can be used in the optimal control problems described in this system.

VL  - 10
IS  - 1
ER  -

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Yong-Dok Han

Faculty of Mathematics, University of Sciences, Pyongyang, DPR Korea

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Kang-Song Yun

Faculty of Mathematics, University of Sciences, Pyongyang, DPR Korea

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Chol-Gwang Kim

Faculty of Mathematics, University of Sciences, Pyongyang, DPR Korea

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APA Style

Han, Y., Yun, K., Kim, C. (2026). Existence and Uniqueness of a Weak Solution of a Time Fractional Reaction-Diffusion System of Fitzhugh-Nagumo Type. Engineering Mathematics, 10(1), 1-13. https://doi.org/10.11648/j.engmath.20261001.11

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ACS Style

Han, Y.; Yun, K.; Kim, C. Existence and Uniqueness of a Weak Solution of a Time Fractional Reaction-Diffusion System of Fitzhugh-Nagumo Type. Eng. Math. 2026, 10(1), 1-13. doi: 10.11648/j.engmath.20261001.11

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AMA Style

Han Y, Yun K, Kim C. Existence and Uniqueness of a Weak Solution of a Time Fractional Reaction-Diffusion System of Fitzhugh-Nagumo Type. Eng Math. 2026;10(1):1-13. doi: 10.11648/j.engmath.20261001.11

Copy | Download

@article{10.11648/j.engmath.20261001.11,
  author = {Yong-Dok Han and Kang-Song Yun and Chol-Gwang Kim},
  title = {Existence and Uniqueness of a Weak Solution of a Time Fractional Reaction-Diffusion System of Fitzhugh-Nagumo Type
},
  journal = {Engineering Mathematics},
  volume = {10},
  number = {1},
  pages = {1-13},
  doi = {10.11648/j.engmath.20261001.11},
  url = {https://doi.org/10.11648/j.engmath.20261001.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.engmath.20261001.11},
  abstract = {In the mathematical theory of nerve impulse propagation, the Fitzhugh-Nagumo Reaction-Diffusion System has attracted a great deal of attention. The Fitzhugh-Nagumo Reaction-Diffusion System provides a prototype for chemical and other nerve conduction and biological systems. In this paper, we define two types of weak solutions of the time fractional Fitzhugh-Nagumo Reaction-Diffusion System, namely (1) -weak solutions and (2) -weak solutions, and demonstrate the existence and uniqueness of these weak solutions. First, we have obtained a generalization of [1, Lemma 1] in Lemma 2.1 and using Lemma 2.1 and Galerkin’s approximation sequence, we have found the existence of (1)-weak solutions and (2)-weak solutions. We also obtained a generalization of the result of [10, Lemma 6] to Hilbert spaces in Lemma 2.2, and using this result we proved the uniqueness of the (2)-weak solution. Lemma 2.1 and Lemma 2.2 of this paper are results that can be effectively used to show the existence and uniqueness of weak solutions of time fractional partial differential equations. And the existence and uniqueness results of the weak solution of the time fractional Fitzhugh-Nagumo Reaction-Diffusion System can be used in the numerical solutions of this reaction-diffusion system. Also, we can be used in the optimal control problems described in this system.
},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - Existence and Uniqueness of a Weak Solution of a Time Fractional Reaction-Diffusion System of Fitzhugh-Nagumo Type

AU  - Yong-Dok Han
AU  - Kang-Song Yun
AU  - Chol-Gwang Kim
Y1  - 2026/01/23
PY  - 2026
N1  - https://doi.org/10.11648/j.engmath.20261001.11
DO  - 10.11648/j.engmath.20261001.11
T2  - Engineering Mathematics
JF  - Engineering Mathematics
JO  - Engineering Mathematics
SP  - 1
EP  - 13
PB  - Science Publishing Group
SN  - 2640-088X
UR  - https://doi.org/10.11648/j.engmath.20261001.11
AB  - In the mathematical theory of nerve impulse propagation, the Fitzhugh-Nagumo Reaction-Diffusion System has attracted a great deal of attention. The Fitzhugh-Nagumo Reaction-Diffusion System provides a prototype for chemical and other nerve conduction and biological systems. In this paper, we define two types of weak solutions of the time fractional Fitzhugh-Nagumo Reaction-Diffusion System, namely (1) -weak solutions and (2) -weak solutions, and demonstrate the existence and uniqueness of these weak solutions. First, we have obtained a generalization of [1, Lemma 1] in Lemma 2.1 and using Lemma 2.1 and Galerkin’s approximation sequence, we have found the existence of (1)-weak solutions and (2)-weak solutions. We also obtained a generalization of the result of [10, Lemma 6] to Hilbert spaces in Lemma 2.2, and using this result we proved the uniqueness of the (2)-weak solution. Lemma 2.1 and Lemma 2.2 of this paper are results that can be effectively used to show the existence and uniqueness of weak solutions of time fractional partial differential equations. And the existence and uniqueness results of the weak solution of the time fractional Fitzhugh-Nagumo Reaction-Diffusion System can be used in the numerical solutions of this reaction-diffusion system. Also, we can be used in the optimal control problems described in this system.

VL  - 10
IS  - 1
ER  -

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