Numerical experiments demonstrate that our scheme provides highly accurate solutions for the F-N equation and shows highly compatible with other numerical techniques in terms of accuracy and efficiency.
📌 3 Key Takeaways
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Objective & Hypothesis:In the present paper, an accurate and reliable numerical scheme has been created for solving FitzHugh-Nagumo (F-N) equation which is a mathematical model that describes the spiking behaviorof neurons.
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Key Findings & Data:For this, the scheme is constructed on the collocation finite element method using the septic B-spline technique. $L_{2}$ and $L_{\infty }$ error norms are computed to check the validity and reliability of the current method.
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Conclusion & Impact:Numerical experiments demonstrate that our scheme provides highly accurate solutions for the F-N equation and shows highly compatible with other numerical techniques in terms of accuracy and efficiency.
🔬Design:Modeling & Simulation
📍Scope:Comprehensive Sample
⚙️Variables:Fitzhugh-Nagumo equation · Finite element method · Collocation
Abstract
In the present paper, an accurate and reliable numerical scheme has been created for solving FitzHugh-Nagumo (F-N) equation which is a mathematical model that describes the spiking behaviorof neurons. For this, the scheme is constructed on the collocation finite element method using the septic B-spline technique. $L_{2}$ and $L_{\infty }$ error norms are computed to check the validity and reliability of the current method. Tables and graphs are used to demonstrate the effectiveness of the solutions produced. The obtained numerical results are compared with other existing numerical methods to validate the accuracy and efficiency of the proposed scheme. Numerical experiments demonstrate that our scheme provides highly accurate solutions for the F-N equation and shows highly compatible with other numerical techniques in terms of accuracy and efficiency.
Sucu, D. Y., Karakoç, S. B. G., Güngör, M. (2025). Investigation on the New Numerical Soliton Solutions of FitzHugh-Nagumo Equation with Collocation Method. Fundamental Journal of Mathematics and Applications, Volume: 8 Issue: 3. https://doi.org/10.33401/fujma.1617563
T. Li, D.A. Soba Fuentes, A. Columbu and G. Viglialoro, Dissipative gradient nonlinearities prevent d-formations in local and nonlocal attraction–repulsion chemotaxis models, Stud. Appl. Math., 154(2) (2025), e70018. $ \href{https://doi.org/10.1111/sapm.70018}{\mbox{[CrossRef]}} $
D. Y. Sucu and S.B.G. Karakoç, A novel numerical approach for solving the Newell-Whitehead equation, Afyon Kocatepe Üniv. Fen ve Müh. Bil. Derg., 23(6) (2023), 1428–1433. $ \href{https://doi.org/10.35414/akufemubid.1323548}{\mbox{[CrossRef]}} $
H. Ali, Md. Kamrujjaman and Md. Shafiqul Islam, Numerical computation of FitzHugh-Nagumo equation: a novel Galerkin finite element approach, Int. J. Math. Res., 9(1) (2020), 20–27. $ \href{https://doi.org/10.18488/journal.24.2020.91.20.27}{\mbox{[CrossRef]}} $
A.K.A. Khalifa, Theory and application of the collocation method with splines for ordinary and partial differential equations [Ph.D. thesis], Heriot-Watt Univ., (1979). $ \href{http://hdl.handle.net/10399/1529}{\mbox{[Web]}} $
A.R. Gourlay, Some recent methods for the numerical solution of time-dependent partial differential equations, Proc. R. Soc., 323 (1971), 219–235. $ \href{https://doi.org/10.1098/rspa.1971.0099}{\mbox{[CrossRef]}} $
A. Devi and O.P. Yadav, Higher Order Galerkin Finite Element Method for Generalized FitzHugh-Nagumo Reaction Diffusion Equation, Numer. Heat Transf., Part B: Fundam., (2024), 1–25.
S.B.G. Karakoç and D.Y. Sucu, New exact and numerical experiments for the Caudrey-Dodd-Gibbon equation, Fundam. J. Math. Appl., 7(1) (2024), 26-34. $ \href{https://doi.org/10.33401/fujma.1389595}{\mbox{[CrossRef]}} $
H. Ali, M. Kamrujjaman and M.S. Islam, Numerical computation of FitzHugh-Nagumo equation: a novel Galerkin finite element approach, Int. J. Math. Res., 9(1) (2020), 20–27. $ \href{https://doi.org/10.18488/journal.24.2020.91.20.27}{\mbox{[CrossRef]}} $