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

Alam, Mehboob; Haq, Sirajul; Ali, Ihteram; Ebadi, M. J.; Salahshour, Soheil

doi:10.3390/fractalfract7120882

Cited by 3 publications

(2 citation statements)

References 38 publications

(45 reference statements)

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“…Additionally, INan B., Ali K.K., Saha A., and Ak T applied the exponential finite difference method (10) , and Berat Karaagac discussed the finite element method (11) . Alam, M., Haq S., Ali I., and Ebadi M.J. employed radial basis functions (12) . Bhausaheb Sontakke and Rajashri Pandit utilized the Adomian Decomposition method to obtain numerical solutions for the time-fractional FN equation (13) .…”

Section: Introductionmentioning

confidence: 99%

Solving Time-Fractional Fitzhugh–Nagumo Equation using Homotopy Perturbation Method

Dhumal,

Sontakke

2024

IJST

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Objectives: This study aims to explore solutions to the time-fractional Fitzhugh-Nagumo equation, a nonlinear reaction-diffusion equation. Method: We utilize the Homotopy Perturbation Method (HPM) as a proficient analytical approach for addressing the time-fractional Fitzhugh-Nagumo equation. The HPM offers a structured method for deriving approximate solutions in the shape of convergent series, enabling accurate solutions even for intricate nonlinear fractional equations. Finding: The series solution obtained is validated by comparing it with numerical methods, showcasing its precision and effectiveness. Additionally, we assessed the error across various time and space values. Our analysis and computations reveal that the Homotopy Perturbation Method (HPM) stands out for providing precise approximations while maintaining computational efficiency. It's clear that this method presents a robust alternative to conventional numerical techniques, particularly in situations where analytical solutions are difficult to obtain. Novelty: The application of the Homotopy Perturbation Method to the Time-fractional Fitzhugh-Nagumo Equation has been effectively explored, with specific examples showing a strong agreement between the exact solution and the obtained solution. Keywords: Time-Fractional Fitzhugh–Nagumo Equation, Homotopy Perturbation Method, Riemann-Liouville fractional integral, Caputo fractional derivative, Fractional Homotopy Perturbation Method

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Section: Introductionmentioning

confidence: 99%

Solving Time-Fractional Fitzhugh–Nagumo Equation using Homotopy Perturbation Method

Dhumal,

Sontakke

2024

IJST

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“…For the solution of fifth-order boundary value problems, Kasi and Ballem [41] employed the finite element approach incorporating the Galerkin method with the quartic B spline as the basis function. Alam et al [42] studied the RBF approximation method for the time-fractional FitzHugh-Nagumo equation. Radmanesh and Ebadi [43] used the local RBF method for solving fractional integral equations.…”

Section: Introductionmentioning

confidence: 99%

Approximate Solution of PHI-Four and Allen–Cahn Equations Using Non-Polynomial Spline Technique

Haq,

Ali

et al. 2024

Mathematics

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The aim of this work is to use an efficient and accurate numerical technique based on non-polynomial spline for the solution of the PHI-Four and Allen–Cahn equations. A recent discovery suggests that the PHI-Four equation focuses on its implications for particle physics and the behavior of scalar fields in the quantum realm. In materials science, ongoing research involves using the Allen–Cahn equation to understand and predict the evolution of microstructures in various materials as well as in biophysics. It depicts pattern formation in biological systems and the dynamics of spatial organization in tissues. To obtain an approximate solution of both equations, this technique uses forward differences for the time and cubic non-polynomial spline function for spatial descretization. The stability of the suggested technique is addressed using the von Neumann technique. Convergence test is carried out theoretically to show the order of convergence of the scheme. Some numerical tests are carried out to confirm accuracy and efficiency in terms of absolute error LR. Convergence rates for different test problems are also computed numerically. Numerical results and simulations obtained are compared with the existing methods.

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A Collocation Procedure for Treating the Time-Fractional FitzHugh–Nagumo Differential Equation Using Shifted Lucas Polynomials

Abd-Elhameed,

Alqubori,

Atta

2024

Mathematics

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This work employs newly shifted Lucas polynomials to approximate solutions to the time-fractional Fitzhugh–Nagumo differential equation (TFFNDE) relevant to neuroscience. Novel essential formulae for the shifted Lucas polynomials are crucial for developing our suggested numerical approach. The analytic and inversion formulas are introduced, and after that, new formulas that express these polynomials’ integer and fractional derivatives are derived to facilitate the construction of integer and fractional operational matrices for the derivatives. Employing these operational matrices with the typical collocation method converts the TFFNDE into a system of algebraic equations that can be addressed with standard numerical solvers. The convergence analysis of the shifted Lucas expansion is carefully investigated. Certain inequalities involving the golden ratio are established in this context. The suggested numerical method is evaluated using several numerical examples to verify its applicability and efficiency.

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

Cited by 3 publications

References 38 publications

Solving Time-Fractional Fitzhugh–Nagumo Equation using Homotopy Perturbation Method

Solving Time-Fractional Fitzhugh–Nagumo Equation using Homotopy Perturbation Method

Approximate Solution of PHI-Four and Allen–Cahn Equations Using Non-Polynomial Spline Technique

A Collocation Procedure for Treating the Time-Fractional FitzHugh–Nagumo Differential Equation Using Shifted Lucas Polynomials

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