A reliable numerical algorithm for a fractional model of Fitzhugh-Nagumo equation arising in the transmission of nerve impulses

Prakash, Amit; Kaur, Hardish

doi:10.1515/nleng-2018-0057

Cited by 13 publications

(6 citation statements)

References 27 publications

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“…A widely studied reaction–diffusion system used to model wave propagation and pattern formation in excitable media is the Fitzhugh–Nagumo system of equations [ 40 , 41 ]

named after Fizthugh [ 42 ] and Nagumo [ 43 ]. In recent years, the single component fractional equation

has been studied as a test equation for various methods of solution of time fractional reaction–diffusion equations (see, for example, [ 27 , 30 ] and references there-in).…”

Section: Examplesmentioning

confidence: 99%

“…has been studied as a test equation for various methods of solution of time fractional reaction-diffusion equations (see, for example, [27,30] and references there-in).…”

Section: Fractional Fitzhugh-nagumo Equationmentioning

confidence: 99%

“…These examples have been selected to emphasize the importance of considering the details of the reaction kinetics when dealing with reaction-subdiffusion problems. Whilst there have been many papers published on various methods of solution for variants of Equation (10) (see, for example, [27][28][29][30][31]), there have been very few papers published considering algebraic or numerical solution methods for variants of Equation (11). We hope that the examples below will stimulate further activity in this area, where the physical motivation for the modelling equation is stronger.…”

Section: Introductionmentioning

confidence: 99%

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Time Fractional Fisher–KPP and Fitzhugh–Nagumo Equations

Angstmann

Henry

2020

Entropy

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A standard reaction–diffusion equation consists of two additive terms, a diffusion term and a reaction rate term. The latter term is obtained directly from a reaction rate equation which is itself derived from known reaction kinetics, together with modelling assumptions such as the law of mass action for well-mixed systems. In formulating a reaction–subdiffusion equation, it is not sufficient to know the reaction rate equation. It is also necessary to know details of the reaction kinetics, even in well-mixed systems where reactions are not diffusion limited. This is because, at a fundamental level, birth and death processes need to be dealt with differently in subdiffusive environments. While there has been some discussion of this in the published literature, few examples have been provided, and there are still very many papers being published with Caputo fractional time derivatives simply replacing first order time derivatives in reaction–diffusion equations. In this paper, we formulate clear examples of reaction–subdiffusion systems, based on; equal birth and death rate dynamics, Fisher–Kolmogorov, Petrovsky and Piskunov (Fisher–KPP) equation dynamics, and Fitzhugh–Nagumo equation dynamics. These examples illustrate how to incorporate considerations of reaction kinetics into fractional reaction–diffusion equations. We also show how the dynamics of a system with birth rates and death rates cancelling, in an otherwise subdiffusive environment, are governed by a mass-conserving tempered time fractional diffusion equation that is subdiffusive for short times but standard diffusion for long times.

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“…A widely studied reaction–diffusion system used to model wave propagation and pattern formation in excitable media is the Fitzhugh–Nagumo system of equations [ 40 , 41 ]

named after Fizthugh [ 42 ] and Nagumo [ 43 ]. In recent years, the single component fractional equation

has been studied as a test equation for various methods of solution of time fractional reaction–diffusion equations (see, for example, [ 27 , 30 ] and references there-in).…”

Section: Examplesmentioning

confidence: 99%

“…has been studied as a test equation for various methods of solution of time fractional reaction-diffusion equations (see, for example, [27,30] and references there-in).…”

Section: Fractional Fitzhugh-nagumo Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Time Fractional Fisher–KPP and Fitzhugh–Nagumo Equations

Angstmann

Henry

2020

Entropy

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“…Abdel-Aty et al [26] studied the time-fractional FitzHugh-Nagumo equation, both computationally and numerically, employing the im-proved Riccati expansion method and the B-spline method with a focus on the Atangana-Baleanu derivative. Additionally, Prakash and Kaur [27] explored the fractional model of the FitzHugh-Nagumo equation, which is relevant to the transmission of nerve impulses. They developed a reliable and computationally effective numerical scheme that combines the homotopy perturbation method with the Laplace transform approach.…”

Section: Introductionmentioning

confidence: 99%

Radial Basis Functions Approximation Method for Time-Fractional FitzHugh–Nagumo Equation

Alam,

Haq,

Ali

et al. 2023

Fractal Fract

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In this paper, a numerical approach employing radial basis functions has been applied to solve time-fractional FitzHugh–Nagumo equation. Spatial approximation is achieved by combining radial basis functions with the collocation method, while temporal discretization is accomplished using a finite difference scheme. To evaluate the effectiveness of this method, we first conduct an eigenvalue stability analysis and then validate the results with numerical examples, varying the shape parameter c of the radial basis functions. Notably, this method offers the advantage of being mesh-free, which reduces computational overhead and eliminates the need for complex mesh generation processes. To assess the method’s performance, we subject it to examples. The simulated results demonstrate a high level of agreement with exact solutions and previous research. The accuracy and efficiency of this method are evaluated using discrete error norms, including L2, L∞, and Lrms.

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“…The FDE is also applicable in various physical and engineering fields models as physics, mathematical biology, fluid mechanics, optical fibers, quantum field theory, neural physics, and many more. [10][11][12][13][14][15][16][17] There are several definitions of fractional derivatives like Riemann-Liouville and Caputo but have some limitations as the Caputo derivative is impotent to explain the singular kernel of the phenomena. But Caputo and Fabrizio 18 defeated the above obliges in 2015, by introducing CF derivative operator.…”

Section: Introductionmentioning

confidence: 99%

Numerical simulation of coupled fractional‐order Whitham‐Broer‐Kaup equations arising in shallow water with Atangana‐Baleanu derivative

Prakash

Kaur

2022

Math Methods in App Sciences

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In this manuscript, fractional nonlinear coupled Whitham‐Broer‐Kaup equation associated with Atangana‐Baleanu fractional derivative is considered. General conditions under which a system solution exists and is unique are presented using the fixed‐point theorem method. For numerical simulations, coupled fractional modified Boussinesq equations and coupled fractional approximate long wave equations are investigated using the homotopy perturbation transform technique (HPTT). The suggested technique is an elegant compilation of the Laplace transform technique with the homotopy perturbation approach. The physical behavior of the obtained solutions has been presented graphically as well as in tables for diverse fractional order. Comparative simulations have been performed to validate the efficiency and accuracy of the suggested technique.

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A reliable numerical algorithm for a fractional model of Fitzhugh-Nagumo equation arising in the transmission of nerve impulses

Cited by 13 publications

References 27 publications

Time Fractional Fisher–KPP and Fitzhugh–Nagumo Equations

Time Fractional Fisher–KPP and Fitzhugh–Nagumo Equations

Radial Basis Functions Approximation Method for Time-Fractional FitzHugh–Nagumo Equation

Numerical simulation of coupled fractional‐order Whitham‐Broer‐Kaup equations arising in shallow water with Atangana‐Baleanu derivative

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