Computational and Mathematical Methods最新文献

This research explores the fractional dynamics of two important nonlinear models: the (2 + 1)-dimensional breaking soliton equation, which arises in the description of various physical phenomena such as shallow-water waves, plasma oscillations, and optical solitons, and the (2 + 1)-dimensional Chaffee–Infante equation, which serves as a fundamental model for ion-acoustic wave propagation in plasma physics. By employing the generalized Bernoulli equation method in conjunction with Jumarie′s modified Riemann–Liouville derivative, both equations are transformed into nonlinear ordinary differential equations and solved analytically, facilitating the derivation of 13 unique families of accurate traveling wave solutions articulated in hyperbolic, exponential, and rational forms. The novelty of this work lies in broadening the analytical framework beyond previous methods that were confined to a narrow range of hyperbolic or trigonometric solutions. The present framework reveals a richer solution structure, including kink-type waves, periodic behaviors, rapidly decaying solitary pulses, and algebraically localized profiles. These new classes of solutions not only broaden the mathematical solution space but also provide deeper insights into the physical interpretation of fractional-order models in plasma physics and hydrodynamics. The results demonstrate that the generalized Bernoulli equation method is a versatile and efficient analytical tool that advances the study of fractional nonlinear evolution equations beyond the limitations of previous techniques.

This paper introduces the C-Product Toolbox, a new computational package available for MATLAB and Python, designed to perform operations on third-order tensors using a tensor product known as the reduced c-product. The reduced c-product is a variant of the known c-product, a tensor product based on the discrete cosine transform and belonging to a family of tensor products defined by invertible linear transformations. This work presents the theoretical development of the reduced c-product used and provides a detailed explanation of each tensor operation implemented in the computational package. Additionally, numerical experiments compare the C-Product Toolbox with an existing MATLAB toolbox that employs the t-product, a tensor product based on the discrete Fourier transform. The numerical experiments in this paper demonstrate that the C-Product Toolbox offers superior computational efficiency, achieving faster execution times and lower memory consumption. Furthermore, the practical advantages of the proposed methods are highlighted through an application in video denoising, showcasing the effectiveness of the toolbox in real-world scenarios.

This work emphasizes the investigation of the solution to the nonlinear Volterra–Fredholm integral equation (NV-FIE) and the necessary conditions for a unique solution. The first step is to convert the NV-FIE into a system of nonlinear Fredholm integral equations (NFIEs) using the splitting of the time interval. Analytical and semianalytical approaches are unable to solve this kind of singular integral equation due to the cumulative increase in error. While the Toeplitz matrix method (TMM) is considered one of the best methods to solve singular integral equations, its importance lies in the fact that it addresses singularity and provides simple, direct integrals. Therefore, in this study, the TMM is employed on the MIE to obtain an algebraic system. Finally, a numerical example is discussed as an application, and the error is calculated. One of the most prominent results of this study is the flexibility and efficiency of TMM in solving integral equations when the kernel takes the Hilbert type.

This study presents analytical and numerical-analytical decomposition methods for determining complex one-parameter generalized inverse Moore–Penrose matrices. The analytical approach is based on the third Moore–Penrose condition, offering three solution options. The first option employs complex decompositions of the given matrix and its Moore–Penrose inverse. The second option combines the first and third Moore–Penrose conditions, while the third option integrates the second and third conditions. For the first and third options, if any derived iterative procedure converges, the Moore–Penrose inverse matrix can be constructed using the corresponding matrix blocks. In contrast, the second option provides simplified relations, enabling the direct computation of the Moore–Penrose inverse matrix. Numerical-analytical methods build on the second analytical solution, utilizing differential Pukhov transformations as the primary mathematical tool. A model example featuring a rectangular complex matrix is analyzed. A numerical-analytical solution is derived using three matrix discretes, from which corresponding matrix blocks are reconstructed. The Moore–Penrose inverse matrix is then obtained through its complex decomposition.

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.

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 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.

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.

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.