Fractional-order calculus has become a useful mathematical framework to describe the complex super-diffusive process; however, numerical solutions of the two-sided space-fractional super-diffusive model with variable coefficients are difficult to obtain, and almost no method can obtain an analytical solution. In this paper, a class of new fractional dimensional reproducing kernel spaces (RKS) based on Caputo fractional derivatives is given, and we give analytical and numerical solutions of the two-sided space-fractional super-diffusive model based on the class of new RKS. The analytical solution is represented in the form of series in the reproducing kernel space. Numerical experiments indicate that the piecewise reproducing kernel method is more accurate than the traditional reproducing kernel method (RKM), and these new fractional reproducing kernel spaces are efficient for the two-sided space-fractional super-diffusive model.

In this paper, the Fourier spectral method is used to solve the fractional-in-space nonlinear coupled FitzHugh–Nagumo model.Numerical simulation is carried out to elucidate the diffusion behavior of patterns for the fractional 2D and 3D FitzHugh–Nagumo model. The results of numerical experiments are consistent with the theoretical results of other scholars, which verifies the accuracy of the method. We show that stable spatio-temporal patterns can be sustained for a long time; these patterns are different from any previously obtained in numerical studies. Here, we show that behavior patterns can be described well by the fractional FitzHugh–Nagumo and Gray–Scott models, which have unique properties that integer models do not have. Results show that the Fourier spectral method has strong competitiveness, reliability, and solving ability for solving 2D and 3D fractional-in-space nonlinear reaction-diffusion models.

This paper investigates a three-species reaction-diffusion model with fractional-order derivatives and proposes an innovative numerical method for its simulation. The method integrates an optimized Grünwald-Letnikov discretization scheme, enhanced by a short-memory principle, with a high-order accurate nine-point compact difference scheme, enabling an efficient and stable solution of fractional operators. Rigorous convergence and stability analyses confirm the theoretical reliability of the algorithm. Through stability and Turing bifurcation analyses, the study systematically reveals the regulatory mechanism of the fractional-order exponent on the dynamic behavior of the system. The numerical results demonstrate that the present method accurately captures the effect of fractional derivatives on the formation process of spatial patterns.

This paper investigates the chaotic dynamics in a fractional-order vocal fold vibration (VCV) model based on the Grünwald-Letnikov fractional derivative (GLFD). Studying the characteristics of vocal fold vibration is of great significance for revealing its vibration mechanism, the etiology of abnormal vibrations, and natural speech synthesis. Traditional vocal fold vibration models are based on integer-order systems and are unable to describe the memory effects present in real physical systems. To overcome this limitation, this paper introduces fractional derivatives and develops a high-precision numerical method to simulate the fractional-order VCV model. By incorporating nonlinear elastic and damping forces, the model can more accurately describe the complex dynamic characteristics of vocal fold vibrations, including memory effects and non-locality. The numerical simulation results reveal novel chaotic behaviors in the fractional-order VCV model, which have not been observed in integer-order models. These findings provide new insights into the possible dynamic states of vocal fold vibrations and lay the foundation for further theoretical and experimental studies on the vocal cord vibration mechanism.

This paper investigates influence of fractional order on the dynamic behavior of a cross-reaction diffusion model. Stability analysis determines the existence conditions of equilibrium points, and bifurcation analyses are performed. A numerical method combining high-order time-precision generating functions and a nine-point difference spatial discretization scheme is proposed, incorporating the short memory principle to improve computational efficiency. Numerical simulations demonstrate the spatial pattern formation of the system at different parameters, validating the effectiveness of the theoretical results. Some novel patterns are shown.