Search Results (28)

The generalized Burgers–Huxley (gBH) equation arises as a canonical model in nerve-pulse propagation (generalizing the Hodgkin–Huxley/FitzHugh–Nagumo excitable-media framework), in population dynamics with Allee-threshold reaction kinetics, and in nonlinear wave propagation in dispersive media; accurate benchmark solutions are essential for quantitative predictions in these domains. We couple the fourth-order exponential time differencing scheme ETDRK4 with a Chebyshev collocation spatial discretization and a linear boundary-lifting procedure to solve the gBH equation on a bounded interval with non-homogeneous Dirichlet data. On the canonical Ismail–Raslan–Rabboh travelling-wave benchmark the scheme attains $L^{\infty }$ errors at the level of floating-point round-off (∼10⁻¹⁹ absolute, ∼10⁻¹⁵ relative) with as few as $N=2$ collocation points and a single time step of size $\Delta t=1.0$—that is, three total nodes and one ETDRK4 advance. In strongly nonlinear regimes ($\gamma =0.1, 0.3, 0.5, 0.9$) the scheme exhibits approximately $\mathcal{O}(\Delta t^{2.45})$ temporal convergence across all four parameter values, consistent with the classical Hochbruck–Ostermann order reduction for exponential integrators on parabolic PDEs with non-homogeneous Dirichlet data. Used as a high-accuracy probe, the scheme provides a diagnostic of independent interest: the wave-speed formula of Wang, Zhu and Lu, still appearing as the exact-solution benchmark in numerical studies as recently as 2020, does not satisfy the partial differential equation. The corrected formula stated by Deng and verified symbolically by Appadu and Tijani is the unique value that makes the travelling-wave ansatz a genuine solution. We derive the residual associated with Wang’s formula in closed form, $R=\frac{\gamma A_1}{2}(A_2-A_2^{\mathrm{W}})(1-v^2)$, and show both analytically and numerically that reported errors for schemes benchmarked against Wang’s formula coincide with the analytical wave-profile gap $\frac{\gamma A_1}{2}|A_2-A_2^{\mathrm{W}}|$ rather than with true scheme accuracy. At the Ismail benchmark this gap equals $3.748\times {10}^{-7}$, which matches the N- and $\Delta t$-independent plateau observed when the scheme is measured against Wang’s profile. In the nerve-pulse and excitable-media interpretation, the two formulas correspond to action-potential propagation speeds of opposite sign at the Ismail benchmark, underscoring that the correction is not a mere algebraic curiosity but changes the qualitative physical prediction of the model. Full article

Singular behavior near the initial times, arising from the presence of fractional-order derivatives in fractional differential equations, often leads to the deterioration of solutions obtained using spectral methods when applied based on classical orthogonal polynomials. Consequently, instead of relying on smooth polynomials as the basis for spectral approaches, significant attention has been devoted to developing appropriate nonsmooth functions that can form the basis for various spectral methods to address the limitation imposed by the fractional-order derivatives. In this study, we develop a numerical framework for solving one- and two-dimensional time-fractional coupled FitzHugh–Nagumo (FHN) models. We construct a novel, time-nonsmooth but spatially smooth function, called the orthogonal shifted Chebyshev function, in both one- and two-dimensional dimensions, that serves as the basis of the spectral collocation approach. Furthermore, we derive novel operational matrices of second-order and fractional-order derivatives, based on the new basis function in the spatial and time directions, respectively. These matrices are then used in conjunction with the spectral collocation technique in both spatial and time directions to reduce the problem to a system of algebraic equations. The numerical results demonstrate the accuracy of the presented numerical scheme and confirm the superiority of the new basis over the classical shifted Chebyshev polynomials when applied to time-fractional models. Full article

In this paper, we investigate the (1 + 1)-dimensional nonlinear truncated M-fractional FitzHugh–Nagumo model. The main objective is to analyze the dynamical behavior and obtain exact solutions for the model. First, a fractional transformation is applied to convert the governing partial differential equation into an ordinary differential equation. Subsequently, a Galilean transformation is employed to reduce the resulting equation to a dynamical system. The bifurcation structure and chaotic dynamics of the model are then examined. The presence of chaos is further confirmed through the phase portrait, basin of attraction, return map, Lyapunov exponent, permutation entropy, Poincaré map, power spectrum, attractor, fractal dimension, multistability, time analysis, and recurrence plot. In addition, the sensitivity of the system to the initial conditions is analyzed. Finally, exact solutions for the model are constructed using the unified Riccati equation expansion method. The obtained results are illustrated using two-dimensional, three-dimensional, and contour plots. Full article

Neural field dynamics in the cerebral cortex exhibit complex spatiotemporal patterns inadequately captured by classical integer-order diffusion models that assume exponentially decaying spatial interactions. This study establishes a stochastic fractional FitzHugh–Nagumo framework incorporating power-law spatial correlations through fractional Laplacian operators, providing explicit parameterization of non-local cortical connectivity characteristics. The inverse problem of estimating fractional orders and model parameters from electroencephalographic data is addressed through multi-objective optimization with rigorous train–test validation. Systematic sensitivity analysis across the parameter space $({\alpha }_u,{\alpha }_v)\in [1.0,2.0]\times [1.0,2.0]$ identifies optimal subdiffusive characteristics at ${\alpha }_u={\alpha }_v=1.5$, corresponding to power-law spatial kernels $C\left(x\right)\sim {\left|x\right|}^{-1.5}$ consistent with anatomical connectivity measurements. The optimized model achieves out-of-sample performance $R^2=0.973$ on held-out test data, approaching the measurement noise ceiling. While classical FitzHugh–Nagumo models achieve comparable test accuracy, the fractional framework provides enhanced interpretability through explicit spatial interaction parameterization. The fractional orders serve as quantitative biomarkers of cortical network organization, enabling data-driven characterization across brain states and neurological conditions. The methodology establishes computational foundations for clinical applications in epilepsy monitoring, neurodegenerative disease detection, and brain–computer interfaces. Full article

This study presents the mathematical formulation of a symmetry-compact three-step algorithm (TSA) for the numerical computation of the spatio-temporal generalized FitzHugh–Nagumo equation (FHNE), a class of one-dimensional time-dependent initial-boundary value partial differential equations. The proposed symmetry-compact TSA is constructed using the Lagrange polynomial as the basis function, yielding a structurally balanced and computationally compact formulation with an inherent symmetry that facilitates automatic step-size adaptation over the integration interval. The symmetry-compact nature of the formulation enhances numerical stability while maintaining a reduced computational footprint, thereby improving both accuracy and efficiency when compared with existing numerical schemes. Prior to the application of the TSA, the FHNE is discretized in space, resulting in a system of ordinary differential equations suitable for time integration. Rigorous analyses of the stability and convergence properties of the symmetry-compact TSA are carried out to establish the reliability and robustness of the method. The performance of the proposed algorithm is quantitatively assessed using absolute error, maximum error, root mean square error, and central processing unit time for selected spatio-temporal test cases of the FHNE. The numerical results and corresponding solution profiles clearly demonstrate that the symmetry-compact TSA delivers superior accuracy, enhanced computational efficiency, and improved stability characteristics relative to existing methods, particularly in the presence of stiffness and chaotic dynamics. Full article

This work develops and analyzes an incommensurate fractional FitzHugh–Nagumo (FHN) reaction–diffusion system in which each state variable evolves with a distinct fractional order. The formulation extends the classical and commensurate fractional models by incorporating heterogeneous memory effects that break temporal symmetry between the activator and inhibitor variables. After establishing the mathematical framework, the equilibrium states of the system are derived and subjected to a detailed local stability analysis in both diffusion-free and diffusion-driven regimes. Explicit stability criteria are obtained by examining the spectral properties of the linearized operator under incommensurate fractional dynamics. Numerical simulations based on a Caputo L1 discretization scheme corroborate the theoretical results and demonstrate how asymmetric memory orders influence transient behavior, convergence rates, and the qualitative structure of the solutions. The study provides the first systematic stability characterization of an incommensurate fractional FitzHugh–Nagumo reaction–diffusion model, highlighting the role of fractional-order asymmetry in shaping the system’s dynamical response. Full article

This work focuses on solving the singularly perturbed generalized Hodgkin-Huxley (HH) problem. The HH equation is numerically solved by a collocation approach using third-degree splines. The forward difference technique is utilized for time discretization, while $\theta$-weighted schemes are employed for space discretization. Solving non-linear models using discretization and quasi-linearization results in a set of linear algebraic equations, which are solved using matrices. Furthermore, Von Neumann’s (VN) stability and Spectral Radius (S.R) reveal that the suggested technique is unconditionally stable. To assess the performance and accuracy of this method, absolute error (AE), $L_2$, and $L_{\infty }$ norms are offered. The results align with the literature. Simulation results show that the proposed strategy produces accurate results. Full article

The FitzHugh–Nagumo (FHN) equation in one dimension is solved in this paper using an improved physics-informed neural network (PINN) approach. Examining test problems with known analytical solutions and the explicit finite difference method (EFDM) allowed for the demonstration of the PINN’s effectiveness. Our study presents an improved PINN formulation tailored to the FitzHugh–Nagumo reaction–diffusion system. The proposed framework is efficiently designed, validated, and systematically optimized, demonstrating that a careful balance among model complexity, collocation density, and training strategy enables high accuracy within limited computational time. Despite the very strong agreement that both methods provide, we have demonstrated that the PINN results exhibit a closer agreement with the analytical solutions for Test Problem 1, whereas the EFDM yielded more accurate results for Test Problem 2. This study is crucial for evaluating the PINN’s performance in solving the FHN equation and its application to nonlinear processes like pulse propagation in optical fibers, drug delivery, neural behavior, geophysical fluid dynamics, and long-wave propagation in oceans, highlighting the potential of PINNs for complex systems. Numerical models for this class of nonlinear partial differential equations (PDEs) may be developed by existing and future model creators of a wide range of various nonlinear physical processes in the physical and engineering sectors using the concepts of the solution methods employed in this study. Full article

This study is devoted to solving the nonlinear inhomogeneous time-fractional FitzHugh–Nagumo differential problem ($\mathsf{TFFNDP}$) using the spectral modified collocation method. The proposed algorithm uses non-symmetric polynomials, namely shifted Legendre polynomials ($\mathsf{SLPs}$), which are orthogonal. The orthogonality property of $\mathsf{SLPs}$ and certain relations facilitate the acquisition of accurate spectral approximations. Comprehensive convergence and error studies are conducted to validate the accuracy of the suggested shifted Legendre expansion. Several numerical examples are presented to demonstrate the method’s effectiveness and accuracy. The proposed scheme is benchmarked against known analytical solutions and compared with other algorithms to ensure the applicability and efficiency of the proposed algorithm. Full article

This paper introduces new formulas for non-symmetric Jacobi polynomials of specific parameters, focusing specifically on the subclasses where the difference between the two parameters of Jacobi polynomials is two or three. First, several key expressions of these polynomials are established, such as the power form expression and its inverse expression. After that, further essential formulas such as the derivatives of moments, linearization and connection formulas, and a formula for the repeated integrals are developed. Symbolic algebra is pivotal for summing some sums in closed forms. An application of some of the introduced formulas is included. The FitzHugh–Nagumo equation—a nonlinear differential equation arising in neuroscience—is solved using the collocation method. The presented numerical examples demonstrate the accuracy and efficiency of the proposed algorithm. Full article

Traditional brain emulation approaches often rely on classical computational models that inadequately capture the stochastic, nonlinear, and potentially coherent features of biological neural systems. In this position paper, we introduce NeuroQ a quantum-inspired framework grounded in stochastic mechanics, particularly Nelson’s formulation. By reformulating the FitzHugh–Nagumo neuron model with structured noise, we derive a Schrödinger-like equation that encodes membrane dynamics in a quantum-like formalism. This formulation enables the use of quantum simulation strategies—including Hamiltonian encoding, variational eigensolvers, and continuous-variable models—for neural emulation. We outline a conceptual roadmap for implementing NeuroQ on near-term quantum platforms and discuss its broader implications for neuromorphic quantum hardware, artificial consciousness, and time-symmetric cognitive architectures. Rather than demonstrating a working prototype, this work aims to establish a coherent theoretical foundation for future research in quantum brain emulation. Full article

This paper consists of various kinds of wave solitons to the mathematical model known as the truncated M-fractional FitzHugh–Nagumo model. This model explains the transmission of the electromechanical pulses in nerves. Through the application of the modified extended tanh function technique and the modified $(G^{\prime }/G^2)$-expansion technique, we are able to achieve the series of exact solitons. The results differ from the current solutions because of the fractional derivative. These solutions could be helpful in the telecommunication and bioscience domains. Contour plots, in two and three dimensions, are used to describe the results. Stability analysis is used to check the stability of the obtained solutions. Moreover, the stationary solutions of the focusing equation are studied through modulation instability. Future research on the focused model in question will benefit from the findings. The techniques used are simple and effective. Full article

Understanding how past factors influence ion channel kinetics is essential for understanding complex phenomena in cardiac electrophysiology, such as early afterdepolarizations (EADs), which are abnormal depolarizations during the action potential plateau associated with life-threatening arrhythmias. We developed a mathematical framework that extends Hodgkin-Huxley type equations with gamma Mittag-Leffler distributed delays, using tools from Fractional Calculus. Traditional memoryless two-variable models fail to reproduce EADs. Our approach modifies FitzHugh-Nagumo, Mitchell-Schaeffer, and Karma cardiac models, enabling the generation of EADs in each of them. We analyze the emergence of these oscillations by discussing the fractional parameters and the mean and variance of the memory kernels. Stability observations are also presented. Full article

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. Full article

The FitzHugh–Nagumo model has been used empirically to model certain types of neuronal activities. It is also a non-linear dynamical system applicable to chemical kinetics, population dynamics, epidemiology and pattern formation. In the literature, many approaches have been proposed to study its dynamics. In this paper, initially, we have employed cutting-edge tools from discrete dynamics for discretization and fixed points. It has been proven that an exact discrete scheme exists for this paradigm. This project also considers the phase space and integral surfaces of these evolutionary equations. In addition, it carries out a thorough symmetry analysis of this reaction diffusion system to find equivalent systems. Moreover, steady-state solutions are obtained using ansatzes for traveling wave solutions. The existence of infinite traveling wave solutions has also been proven. Yet again, this investigation establishes the potential of symmetry methods to unravel non-linearity. Finally, singular perturbation theory has been employed to obtain analytical approximations and to study stability in different parameter regimes. Full article