Traveling pulse solutions to FitzHugh–Nagumo equations

Chen, Chao-Nien; Choi, Y. S.

doi:10.1007/s00526-014-0776-z

Cited by 40 publications

(52 citation statements)

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“…In Section 5 we present, compare and perform some checks on numerical results from our algorithm in one dimension of space. The fastest traveling pulse speed has an asymptotic limit as d → 0 , see Chen and Choi (2015); we check our numerical wave speed against this theoretical result and find excellent agreement. In Section 6 we present numerical results for two-dimensional domains and find as many as four traveling pulse solutions (with different wave speeds) for Fig.…”

Section: An Illustration Of Robustnessmentioning

confidence: 60%

“…We will give a heuristic argument in subsection 2.1 on why J (c) = 0 determines the wave speed. The rigorous proof is in Chen and Choi (2015).…”

Section: A Variational Formulation For Traveling Pulsementioning

confidence: 99%

“…Remark 2 The admissible set A defined above differs slightly from that in (2.9) of Chen and Choi (2015), because 1. the weighted H 1 norm w H 1 ex employed here is equivalent to the norm R e x w 2…”

Section: A Variational Formulation For Traveling Pulsementioning

confidence: 99%

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A steepest descent algorithm for the computation of traveling dissipative solitons

Choi

Connors

2019

Japan J. Indust. Appl. Math.

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An algorithm is proposed to calculate traveling dissipative solitons for the FitzHugh-Nagumo equations. It is based on the application of the steepest descent method to a certain functional. This approach can be used to find solitons whenever the problem has a variational structure. Since the method seeks the lowest energy configuration, it has robust performance qualities. It is global in nature, so that initial guesses for both the pulse profile and the wave speed can be quite different from the correct solution. Also, bifurcations have a minimal effect on the performance. With an appropriate set of physical parameters in two dimensional domains, we observe the co-existence of single-soliton and 2-soliton solutions together with additional unstable traveling pulses. The algorithm automatically calculates these various pulses as the energy minimizers at different wave speeds. In addition to finding individual solutions, this approach could be used to augment or initiate continuation algorithms.

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Section: An Illustration Of Robustnessmentioning

confidence: 60%

“…We will give a heuristic argument in subsection 2.1 on why J (c) = 0 determines the wave speed. The rigorous proof is in Chen and Choi (2015).…”

Section: A Variational Formulation For Traveling Pulsementioning

confidence: 99%

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A steepest descent algorithm for the computation of traveling dissipative solitons

Choi

Connors

2019

Japan J. Indust. Appl. Math.

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“…This may not be true if γ 2 δ ≤ 1. An existence result for traveling pulse solution can be found in [6]. The situation for N > 1 seems to be more delicate.…”

Section: Resultsmentioning

confidence: 96%

“…These patterns have been found [5,[8][9][10]25,26,33,44] not only in a neighborhood induced by Turing's instability but also observed in the places which are far from equilibrium. In addition, a very important feature of these models is the appearance of wave propagation [6,12,18,22]. The aim of this paper is to investigate the existence of traveling waves of (1.1)-(1.2) on under the Dirichlet boundary condition…”

Section: Introductionmentioning

confidence: 99%

Traveling waves for the FitzHugh–Nagumo system on an infinite channel

Chen

Huang

2016

Journal of Differential Equations

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We are concerned with the traveling wave solutions for the FitzHugh-Nagumo system on an infinite channel. Based on a variational formulation in which a non-local term depends on a parameter, the speed of a traveling wave can be selected out. Furthermore, to show the existence of a traveling wave solution with such a speed, we seek a minimizer subject to a constraint. In the way of solving the variational problem, we apply a truncation technique to the nonlocal term to obtain a minimizer located in a bounded invariant region.

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A comparative investigation of a time‐dependent mesh method and physics‐informed neural networks to analyze the generalized Kolmogorov–Petrovsky–Piskunov equation

Sultan,

Zhang

2024

Numerical Methods in Fluids

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The Kolmogorov–Petrovsky–Piskunov (KPP) partial differential equation (PDE) is solved in this article using the moving mesh finite difference technique (MMFDM) in conjunction with physics‐informed neural networks (PINNs). We construct a time‐dependent mesh to obtain approximate solutions for the KPP problem. The temporal derivative is discretized using a backward Euler, while the spatial derivatives are discretized using a central implicit difference scheme. Depending on the error measure, several moving mesh partial differential equations (MMPDEs) are employed along the arc‐length and curvature mesh density functions (MDF). The proposed strategy has been suggested to yield remarkably precise and consistent results. To find the approximate solution, we additionally employ physics‐informed neural networks (PINNs) to compare the outcomes of the adaptive moving mesh approach. It has been observed that solutions obtained using the moving mesh method (MMM) are sufficiently accurate, and the absolute error is also much lower than the PINNs.

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Traveling pulse solutions to FitzHugh–Nagumo equations

Cited by 40 publications

References 50 publications

A steepest descent algorithm for the computation of traveling dissipative solitons

A steepest descent algorithm for the computation of traveling dissipative solitons

Traveling waves for the FitzHugh–Nagumo system on an infinite channel

A comparative investigation of a time‐dependent mesh method and physics‐informed neural networks to analyze the generalized Kolmogorov–Petrovsky–Piskunov equation

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