The generalized Mycielskian graphs are known for their advantageous properties employed in interconnection networks in parallel computing to provide efficient and optimized network solutions. This paper focuses on investigating the bounds and computation of the harmonic–arithmetic index of the generalized Mycielskian graph of path graph, cycle graph, complete graph, and circulant graph. Furthermore, exploring the harmonic–arithmetic index of graphene provides insights into its structural properties, aiding in material design, predictive modeling, and understanding its behavior in various applications. Additionally, the study delves into analyzing the harmonic–arithmetic index of the curvilinear regression model concerning elucidating specific properties of benzenoid hydrocarbons, offering insights into their structural characteristics.

In the topic of discrete variable‐order systems governed by fractional difference equations, this study makes a significant contribution by introducing two innovative variable‐order versions of the fractional Grassi–Miller system. These new formulations are aimed at deepening our understanding of the complex dynamics that such systems exhibit. The research specifically delves into the chaotic dynamical behaviors manifested by these systems: one version being the fractional Grassi–Miller map with commensurate variable order and the other being the fractional Grassi–Miller map with incommensurate variable order. To provide a comprehensive analysis, this study incorporates a variety of variable orders, encompassing both exponential and sinusoidal functions. These variable orders are crucial in exploring how different functional forms influence the behavior of the system. By varying these orders, the research seeks to uncover the patterns and chaotic dynamics that emerge under different conditions. A suite of advanced numerical methods is employed to rigorously analyze and validate the presence of chaotic attractors in these newly proposed variable fractional versions of the Grassi–Miller system. The methods used include bifurcation diagrams, phase portraits, Lyapunov exponents, approximate entropy, C0 complexity, and 0–1 test for chaos. Through the application of these numerical methods, the study thoroughly validates the existence of chaotic attractors in the proposed variable fractional versions of the Grassi–Miller system. The findings underscore the rich and complex behaviors that arise from different variable orders, offering new insights into the dynamics of fractional‐order systems.

In this study, the Bernoulli subequation method (BS‐EM) is applied to investigate the traveling wave solutions of the (2 + 1)‐dimensional resonant Davey–Stewartson system. By employing a wave transformation, the system’s nonlinear partial differential equation is reduced to a nonlinear ordinary differential equation, which is then solved using the BS‐EM approach. As a result, several new traveling wave solutions, which have not been previously reported in the literature, have been successfully obtained. These solutions provide new insights into the physical dynamics of the system and also satisfy the (2 + 1)‐dimensional time–fractional resonant Davey–Stewartson equation. Furthermore, the analytical and graphical analyses of the obtained solutions have been carried out, and the wave profiles have been examined under various parameter conditions. All computations and graphical visualizations in this study were performed using the Wolfram Mathematica 12 software.

This paper is aimed at studying the dynamics of community transmission of HIV by constructing a fractal fractional mathematical model whose kernel is a generalized Mittag–Leffler type. First, we collect and analyze statistical data for epidemiological surveillance of HIV/AIDS prevalence in Yemen from 2000 to 2022. Then, we employ the statistical analysis software EViews and apply ARIMA models to predict the number of HIV/AIDS cases from 2023 to 2024. The results of the selected model, free of standard problems, predicted a future increase in HIV/AIDS cases in Yemen. Next, relying on the well‐known fixed‐point theorem and a set of other associated results, we prove the existence and uniqueness results of the fractional model. Moreover, we use the Adams–Bashforth method to approximate the solutions of this system numerically. Finally, we plot, tabulate, and simulate our results using the Mathematica software and compare them to the results obtained from the statistical model.

Market fluctuations in the stock sector are common. The possible loss that investors may incur because of their investment activity is referred to as investment risk. Returns on investments may fall short of expectations due to a variety of circumstances. Fit of the model to the data; performance in representing volatility, prediction, stability, and analysis; and interpretation goals are all factors to consider. This study investigates the volatility of the Indonesian composite index (ICI) using the GARCH‐MIDAS model, integrating daily ICI returns with monthly macroeconomic indicators: Indonesian bank interest rates (BIIR), consumer price index (CPI), effective federal fund rate (EFFR), and inflation rate (IR). We begin by graphically analysing the trends in ICI returns and macroeconomic variables to identify potential patterns and shifts. Descriptive statistics offer a detailed numerical summary, setting the stage for in‐depth empirical analysis. The long‐run component of stock market volatility is estimated using the GARCH‐MIDAS model, with macroeconomic variables included to capture their impact on market fluctuations. Maximum likelihood estimation (MLE) is employed to estimate the model parameters, ensuring a robust fit to the observed data. Our findings indicate that the EFFR has the most significant impact on ICI volatility, contrary to previous studies. Forecasting performance is evaluated using mean squared error (MSE) and mean absolute error (MAE), confirming the superior predictive capability of the EFFR variable. The study assesses risk using value at risk (VaR) for the ICI, incorporating the EFFR to account for macroeconomic influences on market volatility. VaR values at 99% and 95% confidence levels provide insights into potential maximum losses, aiding in informed investment decision‐making. This research enhances knowledge of the relationship between macroeconomic variables and stock market volatility, providing investors and policymakers with important information for risk management and investment strategy optimization in the Indonesian equity market.

This paper investigates the finite‐time stability (FTS) of a discrete SIR epidemic reaction–diffusion (R‐D) model. The study begins with discretizing a continuous R‐D system using finite difference methods (FDMs), ensuring that essential characteristics like positivity and consistency are maintained. The resulting discrete model captures the interplay between spatial heterogeneity, diffusion rates, and reaction dynamics, enabling a robust framework for theoretical analysis. Employing Lyapunov‐based techniques and eigenvalue analysis, we derive sufficient conditions for achieving FTS, which is crucial for rapid epidemic containment. The theoretical findings are validated through comprehensive numerical simulations that examine the effects of varying diffusion coefficients, reaction rates, and boundary conditions on system stability. The results highlight the critical role of these factors in achieving FTS of epidemic dynamics. This work contributes to developing efficient computational tools and theoretical insights for modeling and controlling infectious diseases in spatially extended populations, providing a foundation for future research on fractional‐order models and complex boundary conditions.

The goal of survey sampling theory is to produce reliable and precise estimates for population parameters. To achieve this, a new estimator for finite population mean that incorporates dual auxiliary variables in the presence of minimum and maximum values is proposed in this study. Theoretical derivations and empirical evaluations demonstrate the superiority of the proposed estimator over existing alternatives, as it consistently yields lower mean squared errors and biases. While its performance improves with larger sample sizes, it also maintains strong efficiency in small‐sample settings.

In various fields, such as economics, finance, bioinformatics, geology, and medicine, namely, in the cases of electroencephalogram, electrocardiogram, and biotechnology, cluster analysis of time series is necessary. The first step in cluster applications is to establish a similarity/dissimilarity coefficient between time series. This article introduces an extension of the affinity coefficient for the autoregressive expansions of the invertible autoregressive moving average models to measure their similarity between them. An application of the affinity coefficient between time series was developed and implemented in R. Cluster analysis is performed with the corresponding distance for the estimated simulated autoregressive moving average of order one. The primary findings indicate that processes with similar forecast functions are grouped (in the same cluster) as expected concerning the affinity coefficient. It was also possible to conclude that this affinity coefficient is very sensitive to the behavior changes of the forecast functions: processes with small different forecast functions appear to be well separated in different clusters. Moreover, if the two processes have at least an infinite number of π- weights with a symmetric signal, the affinity value is also symmetric.

In this work, we suggest a new numerical scheme called the fractional higher order Taylor method (FHOTM) to solve fractional differential equations (FDEs). Using the generalized Taylor’s theorem is the fundamental concept of this approach. Then, the local truncation error generated by the suggested FHOTM is estimated by proving suitable theoretical results. At last, several numerical applications are given to demonstrate the applicability of the suggested approach in relation to their exact solutions.