This research explores the higher-order nonlinear fractional Huxley equation formulated with the $\beta$, and $M$-truncated fractional derivatives to account for memory and hereditary effects present in nonlinear diffusion and excitation wave dynamics. The fractional formulation expands on the standard Huxley model by incorporating nonlocal temporal and spatial correlations, providing a more realistic description of finite-amplitude wave propagation and spectral energy transfer in complex media. Two advanced analytical techniques are used to derive exact solutions: the enhanced modified extended tanh expansion method (EMETEM) and the improved $F$-expansion technique. Compared to classic perturbation and variational techniques, these methods offer greater algebraic freedom and faster convergence, resulting in a diverse family of closed-form traveling-wave solutions expressed in trigonometric, hyperbolic, exponential, and rational forms. The analytical results are further validated, and the spatiotemporal evolution of the resulting wave structures is investigated using a finite-difference numerical scheme. The accuracy and robustness of the suggested framework are confirmed by the numerical findings, which show good agreement with the analytical results. Phase-plane and bifurcation analyses demonstrate transitions between periodic, quasi-periodic, and chaotic regimes by revealing both stable and unstable spiral formations. The findings show that fractional derivatives significantly improve the dynamical characteristics of the Huxley system by allowing for finer control of diffusion, dispersion, and localized energy concentration, advancing our understanding of nonlinear fractional wave behavior in excitable and dispersive media.

This study investigated the influence of Brownian motion and noise effects on the dynamics of the stochastic Poisson-Nernst-Planck system with M-truncated fractional derivative. To explore exact analytical representations of the soliton solutions, the study employed the modified extended direct algebraic method. The method successfully produces closed-form exact soliton solutions that capture the stochastic behavior of the system in the presence of random perturbations. The addition of the M-truncated fractional derivative provides a more flexible structure to describe anomalous transport, offering an advanced mathematical representation of electro-diffusion processes. The obtained results highlight the combined role of noise, fractional dynamics, and stochastic fluctuations in shaping the system's evolution, thereby deepening the theoretical understanding of nonlinear stochastic transport models and opening potential avenues for applications in complex biological and physical systems. Moreover, the study presented graphical demonstrations that illustrate the effect of noise and the fractional order of derivation in 3D, 2D, and contour surfaces.

The stochastic FitzHugh-Nagumo (FHN) equation is a fundamental model for excitable media, widely used to describe neuronal signal transmission and population dynamics under random perturbations. However, analytical studies that simultaneously incorporate stochastic effects, nonlinear wave propagation, and bifurcation behavior are still limited. In this work, we investigated a stochastic FHN model with multiplicative Gaussian noise and developed a unified analytical framework by integrating the unified method with a novel auxiliary equation technique. Using a stochastic transformation and expectation operator, the governing stochastic partial differential equation was reduced to a deterministic traveling-wave ordinary differential equation. This framework enables the systematic derivation of multiple classes of exact solutions, including rational, trigonometric, and hyperbolic soliton structures such as kink, anti-kink, periodic, and singular waves. Compared with existing methods, the proposed approach provides a more general mechanism for constructing diverse solution families within a single analytical structure. Furthermore, the reduced equation was reformulated into a planar dynamical system via a Galilean transformation, allowing bifurcation and phase-portrait analysis. Graphical results clearly illustrate the influence of noise intensity on wave morphology and temporal dynamics. The findings provide new analytical insights into noise-modulated excitation phenomena in stochastic nonlinear systems.

In this paper, various types of exact soliton solutions of the truncated M-fractional Kudryashov–Sinelshchikov equation, a significant fluid surge model, were obtained. This model accounted for density and heat transfer effects while describing the propagation of pressure waves in mixtures of liquid–gas bubbles. By applying the modified $(G^{\prime }/G^2)$-expansion method and the extended Sinh–Gordon equation expansion method, we derived new solutions in the forms of trigonometric, hyperbolic, and rational functions. The obtained solutions were illustrated dynamically using 2D, 3D, and contour plots. These results were novel due to the use of a new definition of fractional derivatives. The effect of the fractional derivative on the solutions was demonstrated through 2D plots. To examine the stability of the obtained solutions, stability analysis was performed. Steady-state solutions were derived using modulation instability analysis. The obtained solutions may be useful in various fields of science and engineering.