Because they are useful for both enabling numerical simulations and containing well-defined physical phenomena, discrete fractional reaction–diffusion models have attracted a great deal of interest from academics. Within the family of fractional reaction–diffusion models, a discrete form is examined in detail in this study. Furthermore, we investigate the complex synchronization dynamics of a suggested discrete master–slave reaction–diffusion system using the accuracy of linear control techniques combined with a fractional discrete Lyapunov approach. This study’s deviation from the behavior of equivalents with integer orders makes it very fascinating. Like the non-local nature inherent in Caputo fractional derivatives, it creates a memory Lyapunov function that is closely linked to the historical background of the system. The investigation provides a strong basis to the theoretical results.

In this article, we focused on the fractional order modeling, simulations and neural networking to observe the correlation between severity of infection in HIV-AIDS patients and the role of treatments and control. The model is structured with eight classes and a modified Atangana–Baleanu derivative in Caputo’s sense. The model has several interlinking parameters which show the rates of transmission between classes. We assumed natural death and death on the disease severity in patients. The model was analyzed mathematically as well as computationally. In the mathematical aspects, ${\mathcal{R}}_0$ was plotted for different cases which play a vital role in the infection spread in the population. The model was passed through qualitative analysis for the existence of solutions and stability results. A computational scheme is developed for the model and is applied for the numerical results to analyze the intricate dynamics of the infection. It has been observed that there is a good resemblance in the results for the correlation between the hospitalization, vaccination and recovery rate of the patients. These are reaffirmed with the neural networking tools for the regression, probability, clustering, mean square error and fitting data.

In this article, we have investigated solutions to a symmetry chaotic system with fractional derivative order using two different methods—the numerical scheme for the ABC fractional derivative, and the Laplace decomposition method, with help from the MATLAB and Mathematica platforms. We have explored progressive and efficient solutions to the chaotic model through the successful implementation of two mathematical methods. For the phase portrait of the model, the profiles of chaos are plotted by assigning values to the attached parameters. Hence, the offered techniques are relevant for advanced studies on other models. We believe that the unique techniques that have been proposed in this study will be applied in the future to build and simulate a wide range of fractional models, which can be used to address more challenging physics and engineering problems.

This study focuses on the solution of the rotationally symmetric Rossler attractor by using the adaptive predictor–corrector algorithm (Apc-ABM-method) and the fractional Laplace decomposition method ($\rho$-Laplace DM). Furthermore, a comparison between the proposed methods and Runge–Kutta Fourth Order (RK4) is made. It is discovered that the proposed methods are effective and yield solutions that are identical to the approximate solutions produced by the other methods. Therefore, we can generalize the approach to other systems and obtain more accurate results. In addition to this, it has been shown to be useful for correctly discovering examples via the demonstration of attractor chaos. In the future, the two methods can be used to find the numerical solution to a variety of models that can be used in science and engineering applications.

In this study, the numerical scheme for the Caputo fractional derivative (NCFD) method and the He–Laplace method (H-LM) are two powerful methods used for analyzing fractional-order systems. These two approaches are used in the study of the complex dynamics of the fractional-order inverted Rössler system, particularly for the detection of chaotic behavior. The enhanced NCFD method is used for reliable and accurate numerical simulations by capturing the intricate dynamics of chaotic systems. Further, analytical solutions are obtained using the H-LM for the fractional-order inverted Rössler system. This method is popular due to its simplicity, numerical stability, and ability to handle most initial values, yielding very accurate results. Combining analytical insights from the H-LM with the robust numerical accuracy of the NCFD approach yields a comprehensive understanding of this system’s dynamics. The advantages of the NCFD method include its high numerical accuracy and ability to capture complex chaotic dynamics. The H-LM offers simplicity and stability. The proposed methods prove to be capable of detecting chaotic attractors, estimating their behavior correctly, and finding accurate solutions. These findings confirm that NCFD- and H-LM-based approaches are promising methods for the modeling and solution of complex systems. Since these results provide improved numerical simulations and solutions for a broad class of fractional-order models, they will thus be of greatest use in forthcoming applications in engineering and science.