In this work, a novel approach based on a single-layer machine learning Legendre spectral neural network (LSNN) method is used to solve an elliptic partial differential equation. A Legendre polynomial based approach is utilized to generate neurons that fulfill the boundary conditions. The loss function is computed by using the error back-propagation principles and a feed-forward neural network model combined with automatic differentiation. The main advantage of using this methodology is that it does not need to solve a system of nonlinear and nonsparse equations compared with other traditional numerical schemes, which makes this algorithm more convenient for solving higher-dimensional equations. Further, the hidden layer is eliminated with the help of a Legendre polynomial to enlarge the input pattern. The neural network's training accuracy and efficiency were significantly enhanced by the innovative sampling technique and neuron architecture. Moreover, the Legendre spectral approach can handle equations on more complex domains because of numerous networks. Several test problems were used to validate the proposed scheme, and a comparison was made with other neural network schemes consisting of the physics-informed neural network (PINN) scheme. We found that our proposed scheme has a very good agreement with PINN, which further enhances the reliability and efficiency of our proposed method. The absolute and relative error in both $L_2$ and $L_{\mathrm{\infty }}$ between exact and numerical solutions are provided, which shows that our numerical method converges exponentially.

The aim of this study is to investigate the dynamics of epidemic transmission of COVID-19 SEIR stochastic model with generalized saturated incidence rate. We assume that the random perturbations depends on white noises, which implies that it is directly proportional to the steady states. The existence and uniqueness of the positive solution along with the stability analysis is provided under disease-free and endemic equilibrium conditions for asymptotically stable transmission dynamics of the model. An epidemiological metric based on the ratio of basic reproduction is used to describe the transmission of an infectious disease using different parameters values involve in the proposed model. A higher order scheme based on Legendre spectral collocation method is used for the numerical simulations. For the better understanding of the proposed scheme, a comparison is made with the deterministic counterpart. In order to confirm the theoretical analysis, we provide a number of numerical examples.

We study the stability of a two-dimensional fractional reaction-diffusion system under the Caputo differential operator in time. Our model is based on the Grey-Scott model, a well-known coupled reaction-diffusion system that describes the interaction between two chemical species. The diffusion term captures the species special spread, while the nonlinear term in the system describes the chemical reaction, resulting in a wide range of difficult, self-organizing patterns, including spots, stripes, or spirals, depending on the parameter values. We derive conditions for local stability of the homogeneous equilibrium by linearizing the system and analyzing the eigenvalues of the Jacobian. Furthermore, we construct appropriate Lyapunov functionals to establish global asymptotic stability of the discrete model under suitable conditions. This approach seeks to provide a robust framework for analyzing complex dynamical behaviors in systems governed by fractional-order in-time reaction-diffusion systems. The numerical simulations employ the Chebyshev spectral method for spatial discretization and the $L1$ scheme for fractional time derivatives. These simulations validate the theoretical findings, demonstrating the model's ability to replicate intricate patterns often observed in reaction-diffusion systems. The results suggest that the fractional-order framework enhances the understanding of pattern formation in such systems, making this model a valuable tool for studying anomalous diffusion and non-local dynamics in biological and chemical processes.

The theory of integral inequalities is significantly advanced by the relationship between fractional calculus and convexity. The current study used extended convex functions and Katugampola fractional operators to derive Hermite-Hadamard, Fejér-Hermite-Hadamard-type inequalities as well as a few other fractional integral inequalities. We used two extended-type convex functions: log-convex and exponentially trigonometric convex functions. Our conclusions were supported by tabular data and graphical representations, which offer numerical and visual validation of the explored results. This study improves mathematical analysis and expands the scope of these inequalities by emphasizing their applicability across different forms of convexity. The knowledge acquired is highly valuable for theoretical investigation as well as real-world applications in a variety of scientific fields.

Nonlinear stochastic Itô–Volterra integral equations (NSIVIEs) represent systems whose current state is influenced by random fluctuations and is dependent on previous information. These equations appear in many real-world scenarios, including engineering systems, biological processes, financial markets, heterogeneous media, complex transport phenomena, and viscoelastic materials. Strong numerical frameworks are required because analytical solutions for these equations are rarely available, particularly when nonlinearities and random fluctuations are present. To effectively solve NSIVIEs, in this study we propose a new hybrid numerical framework that combines Monte Carlo simulation and Legendre spectral collocation. By using orthogonal polynomial basis functions to approximate the solution, this method provides spectral accuracy while handling the hereditary memory component of the Volterra equation through a high-order Legendre spectral collocation method. A precise statistical treatment of the random fluctuations is made possible by simultaneously addressing the stochastic Itô noise through Monte Carlo sampling across numerous independent realizations. We perform a thorough convergence analysis and obtain explicit error bounds that measure the decrease in approximation error with increasing spectral resolution and Monte Carlo sample count. Numerical experiments show that the method can accurately reproduce complex stochastic behaviors and validate theoretical predictions.