In this paper, we study the dynamical analysis and solutions of the fractional Benjamin-Bona-Mahony-Burger equation. We demonstrate various derived solutions using different definitions of fractional derivatives, namely the beta-derivative, conformable derivative, and M-truncated derivative, to examine their kinetic characteristics. Firstly, we find the solution of the fractional Benjamin-Bona-Mahony-Burger equation using two different approaches. We then discuss the effects of the fractional derivative on the solutions using 3D graphical discussion. Finally, we discuss the dynamical analysis using sensitivity and chaos analysis. We also discuss the chaos analysis using permutation entropy, 2D and 3D phase portrait, fractal dimension, time analysis, return map, Lyapunov exponent, and multistability through Poincare map and basins of attraction. To explore a diverse range of phenomena across the fields of physical science and engineering, this study highlights the computational strength and flexibility of the proposed method.

This paper introduces a collocation algorithm for numerically solving the third-order Gilson–Pickering equation (GPE) and the classical Rosenau–Hyman equation (RHE). We employ newly developed shifted Pell polynomials as basis functions. Novel formulas for these polynomials are devised and utilized in constructing the proposed algorithm. Specifically, we establish a new power form and its inversion formula, along with an explicit formula for derivatives of the shifted Pell polynomials, from which the operational matrices of derivatives (OMDs) are derived. These matrices facilitate the conversion of nonlinear dispersive models into systems of algebraic equations, efficiently solved using Newton’s iterative technique. The error analysis of the shifted Pell expansion is discussed in depth. Several numerical examples, including the RHE, its fourth-order variant, and the Fornberg–Whitham equation, are provided to demonstrate the method’s performance and accuracy. Comparative results are also reported.

This study uses the spectral tau method to treat the time-fractional cable equation (TFCE). The proposed algorithm uses the shifted Delannoy polynomials, which are non-symmetric orthogonal. The orthogonality property of the non-symmetric shifted Delannoy polynomials and some representations facilitate obtaining accurate spectral approximations for the TFCE. Several numerical examples ensure the efficiency and accuracy of the method. We compare the suggested scheme to other algorithms and benchmark it against existing analytical solutions to demonstrate the high accuracy of our presented algorithm.

In this work, a new spectral Galerkin approach to solving the time-fractional diffusionwave equation (TFDWE) with non-homogeneous initial and boundary conditions is presented. A suitable transformation is used to convert the TFDWE governed by non-homogeneous conditions into a modified one governed by homogeneous conditions. New basis functions in terms of specific shifted Horadam polynomials are used. Some new definite integral formulas that are crucial to the numerical implementation are developed, and the Galerkin scheme is analyzed in detail to obtain the approximate solutions. A thorough convergence and error analysis of the proposed expansion is established. A number of numerical experiments are carried out to show the scheme's applicability and accuracy when compared to other methods.

This paper introduces new formulas for non-symmetric Jacobi polynomials of specific parameters, focusing specifically on the subclasses where the difference between the two parameters of Jacobi polynomials is two or three. First, several key expressions of these polynomials are established, such as the power form expression and its inverse expression. After that, further essential formulas such as the derivatives of moments, linearization and connection formulas, and a formula for the repeated integrals are developed. Symbolic algebra is pivotal for summing some sums in closed forms. An application of some of the introduced formulas is included. The FitzHugh–Nagumo equation—a nonlinear differential equation arising in neuroscience—is solved using the collocation method. The presented numerical examples demonstrate the accuracy and efficiency of the proposed algorithm.

Industrial systems often rely on specialized redundant systems to enhance reliability and prevent unexpected failures. This study introduces a novel three-parameter model, the Poisson–Weibull distribution (PWD), and discovers its various key properties. The primary focus of the study is to develop stress–strength (SS) model based on this newly developed distribution. Parameter estimation for both the PWD and SS models is carried out using maximum likelihood estimation (MLE) and Bayesian estimation techniques. Given the complexity of the proposed distribution, numerical approximation techniques are employed to obtain reliable parameter estimates. A comprehensive simulation study employing the Monte Carlo simulation (MCS) and Markov Chain Monte Carlo (MCMC) examines the behavior of the PWD and SS model parameters under various scenarios. The development of the SS model enhances understanding of the PWD’s dynamics while providing practical insights into its real-life applications and limitations. The effectiveness of the proposed distribution and the SS reliability measure is established through applications to real-life data sets.

The main aims of this study are the introduction of Pell coefficient polynomials , their numerical treatment of the first-order hyperbolic partial differential equations. Our suggested numerical algorithm will be derived from the utilization of some novel formulas of the Pell coefficient polynomials, along with the application of the spectral tau method. For the proposed expansion, we investigate the convergence and error estimations in detail. The presented numerical results indicate that the suggested numerical method is accurate, converges exponentially , is computationally efficient.

In this article, we provide a novel distribution known as the new extended Rayleigh inverted Weibull (NERIW) distribution, which arises from the new extended family of distributions (NE-G). The shapes of the probability density function (PDF) can be decreasing, unimodal, or right-skewed, but the shapes of the hazard rate function (HRF) can be decreasing or upside down. The new model is investigated to identify its various statistical properties, including quantile function, median, moments, moment-generating function, incomplete, and conditional moments, mean deviation, and inequality measures. The model parameters are determined using the maximum likelihood estimation approach. A Monte Carlo simulation analysis is conducted to evaluate the effectiveness of maximum likelihood estimators. This study explores the effectiveness of the NERIW distribution in modeling health and disability statistics, focusing on two real-world datasets from Saudi Arabia. By comparing the NERIW distribution with alternative models and offering a more accurate representation of disability prevalence between age groups. The findings provide valuable information for policy makers and researchers in understanding disability trends and improving data-driven decision making in health planning.

This study presents an in-depth analysis of the distribution of disability in Saudi Arabia according to age groups, as documented in the latest KSA Census of 2022. "4.2%" of the population of the Kingdom experiences difficulties or disabilities. The prevalence of disabilities increases significantly with age, where the percentage of people with disabilities is very low in the younger age groups and increases dramatically among the older individuals. These findings provide valuable information for policy makers in developing targeted intervention programs for people with disabilities based on age groups. Traditional probability distributions may, on occasion, fail to take into account this complexity, which can lead to inaccurate conclusions being drawn instead. So, in this article, we introduce a new three-parameter model called the alpha-power transformed Rayleigh inverted Weibull distribution (APTRIWD) to solve this issue. The probability density curves of APTRIWD provide evidence that it can be used in the analysis of disability data in Saudi Arabia and radiation data, demonstrating its practical applicability. Due to the fact that the hazard rate function (HRF) for APTRIWD can exhibit J-shaped, growing, and declining patterns, researchers have a great deal of flexibility when it comes to constructing statistical models for research on disability concerns. Some important statistical properties of the new suggested model are discussed. The maximum likelihood estimation method is utilized to determine the parameters of the machine learning model. For the purpose of determining the efficiency of maximum likelihood estimators, a Monte Carlo simulation analysis is performed. The proposed distribution was evaluated using three datasets related to disability concerns in Saudi Arabia and radiation data. The APTRIWD exhibited superior goodness of fit compared to several models. The APTRIWD is recommended for data modeling in fields such as disability challenges due to its exceptional fit capabilities and radiation data.

The accurate identification of internal and external pressures in thick-walled hyperelastic vessels is a challenging inverse problem with significant implications for structural health monitoring, biomedical devices, and soft robotics. Conventional analytical and numerical approaches address the forward problem effectively but offer limited means for recovering unknown load conditions from observable deformations. In this study, we introduce a Graph-FEM/ML framework that couples high-fidelity finite element simulations with machine learning models to infer normalized internal and external pressures from measurable boundary deformations. A dataset of 1386 valid samples was generated through Latin Hypercube Sampling of geometric and loading parameters and simulated using finite element analysis with a Neo-Hookean constitutive model. Two complementary neural architectures were explored: graph neural networks (GNNs), which operate directly on resampled and feature-enriched boundary data, and convolutional neural networks (CNNs), which process image-based representations of undeformed and deformed cross-sections. The GNN models consistently achieved low root-mean-square errors (≈0.021) and stable correlations across training, validation, and test sets, particularly when augmented with displacement and directional features. In contrast, CNN models exhibited limited predictive accuracy: quarter-section inputs regressed toward mean values, while full-ring and filled-section inputs improved after Bayesian optimization but remained inferior to GNNs, with higher RMSEs (0.023–0.030) and modest correlations (R2). To the best of our knowledge, this is the first work to combine boundary deformation observations with graph-based learning for inverse load identification in hyperelastic vessels. The results highlight the advantages of boundary-informed GNNs over CNNs and establish a reproducible dataset and methodology for future investigations. This framework represents an initial step toward a new direction in mechanics-informed machine learning, with the expectation that future research will refine and extend the approach to improve accuracy, robustness, and applicability in broader engineering and biomedical contexts.

Ruled surfaces in Minkowski 3-space play a crucial role in differential geometry and have significant applications in physics and engineering. This study explores the fundamental properties of ruled surfaces via orthogonal modified frame in Minkowski space E13, focusing on their minimality, developability, and curvature characteristics. We examine the necessary and sufficient conditions for a ruled surface to be minimal, considering the mean curvature and its implications. Furthermore, we analyze the developability of such surfaces, determining the conditions under which they can be locally unfolded onto a plane without distortion. The Gaussian and mean curvatures of ruled surfaces in Minkowski space are computed and discussed, providing insights into their geometric behavior. Special attention is given to spacelike, timelike, and lightlike rulings, highlighting their unique characteristics. This research contributes to the broader understanding of the geometric properties of ruled surfaces within the framework of Minkowski geometry.

In some instances, the size of the target population might exhibit significant variation. In the medical investigation, the number of individuals troubled with a certain infection and the scale of the medical facilities may differ. Probability proportional to size (PPS) sampling helps to collect data in household income surveys when the number of siblings in houses fluctuates. This work aims to develop an improved estimator for estimating finite population mean using auxiliary information under PPS sampling. Utilizing the Taylor series approach, a novel and enhanced estimator is introduced to determine the expression of the mean square error up to the first degree of approximation. This estimator performs better as compared to some current existing estimators using theoretical efficiency constraints. The performance of the existing and newly designed estimators was evaluated by analyzing two actual data sets. The performance was assessed based on maximizing the percentage relative efficiency and minimizing the marginal mean square error. Compared to other estimator which is examined in this work, we found that the proposed technique exhibited superior performance and increased efficiency.

This study explores the application of Romanovski–Jacobi polynomials (RJPs) in spectral Galerkin methods (SGMs) for solving differential equations (DEs). It uses a suitable class of modified RJPs as basis functions that meet the homogeneous initial conditions (ICs) given. We derive spectral Galerkin schemes based on modified RJP expansions to solve three models of high-order ordinary differential equations (ODEs) and partial differential equations (PDEs) of first and second orders with ICs. We provide theoretical assurances of the treatment’s efficacy by validating its convergent and error investigations. The method achieves enhanced accuracy, spectral convergence, and computational efficiency. Numerical experiments demonstrate the robustness of this approach in addressing complex physical and engineering problems, highlighting its potential as a powerful tool to obtain accurate numerical solutions for various types of DEs. The findings are compared to those of preceding studies, verifying that our treatment is more effective and precise than that of its competitors.

We introduce a new flexible statistical model for disability and reliability case studies, designed to enhance modeling capabilities in reliability engineering and disability prevalence analysis. This model extends the classical Pareto Type II distribution by incorporating additional shape parameters, allowing for greater flexibility in modeling various hazard rate shapes and tail behaviors, particularly light tails. Essential properties of the model are explored. The model's applicability and risk assessment potential are demonstrated through case studies: one focusing on reliability engineering (aircraft windshield failure and service times) and another analyzing disability prevalence rates in Saudi Arabia for 2016. Advanced tail analysis tools such as the Hill estimator, Value-at-Risk (VaR), and Peaks Over a Random Threshold (PORT) were employed. The analyses reveal that the proposed OGPTII model offers superior goodness-of-fit compared to several existing models for both reliability and Saudi Arabia disability data, accurately captures the light-tailed nature of the datasets, and provides crucial insights into potential risk factors. This makes it a valuable tool for predictive maintenance strategies in engineering and for evidence-based policy planning and resource allocation in the context of disability management.

This article provides a detailed baseline description of vision disability in Saudi Arabia, including the prevalence by administrative area, the total population affected by the disability, and the severity (severe or blind) among those 2 years and older, providing essential demographic data that will help shape national policy and planning initiatives. In addition to this comparison, to assess the validity of the performance of the statistical modeling, the analysis is based on a second data set that serves to reflect the true expected survival rates for patients with head and neck cancer (HNC) treated with chemotherapy and radiation therapy. The Harris extended Zeghdoudi distribution (HEZD) can be considered as an improved version of the Zeghdoudi distribution (ZD), based on the Harris extended generated family of distributions. The probability density curves of HEZD show a great importance of HEZD in assessing disability in the Kingdom of Saudi Arabia and radiation. Researchers have a lot of latitude in designing statistical models for studies that involve disability with radiation data, given that the HRF for HEZD can have increasing functions. Some important characteristics of HEZD are obtained, such as quantile function, moments, MGF, incomplete moments, and order statistics. To estimate the parameters of HEZD, we used data correlated with disabilities and radiation and applied various estimation techniques. To test the efficiency of the proposed distribution, two data sets for the disability ratio in the Kingdom of Saudi Arabia and radiation fields were used. To address the fit in modeling data in increasingly complex fields that include disabilities, the HEZD has been recommended based on its improved fit capabilities.

In this paper, we present a collocation algorithm for numerically treating the time-fractional Kuramoto–Sivashinsky equation (TFKSE). Certain orthogonal polynomials, which are expressed as combinations of Chebyshev polynomials, and their shifted polynomials are introduced. Some new theoretical formulas regarding these polynomials have been developed, including their operational matrices of both integer and fractional derivatives. The derived formulas will be the foundation for designing the proposed numerical algorithm, which relies on converting the governing problem with its underlying conditions into a nonlinear algebraic system, which can be solved using Newton’s iteration technique. A rigorous error analysis for the proposed combined Chebyshev expansion is presented. Some numerical examples are given to ensure the applicability and efficiency of the presented algorithm. These results demonstrate that the proposed algorithm attains superior accuracy with fewer expansion terms.

In this study, a stochastic computing structure is provided for the numerical solutions of the SIRC epidemic delay differential model, i.e. SIRC-EDDM using the dynamics of the COVID-19. The design of the scale conjugate gradient (CG) neural networks (SCGNNs) is presented for the numerical treatment of SIRC-EDDM. The mathematical model is divided into susceptible S(rho), recovered R(rho), infected I(rho), and cross-immune C(rho), while the numerical performances have been provided into three different cases. The exactitude of the SCGNNs is perceived through the comparison of the accomplished and reference outcomes (Runge-Kutta scheme) and the negligible absolute error (AE) that are performed around 10-06 to 10-08 for each case of the SIRC-EDDM. The obtained results have been presented to reduce the mean square error (MSE) using the performances of train, validation, and test data. The neuron analysis is also performed that shows the AE by taking 14 neurons provide more accurateness as compared to 4 numbers of neurons. To check the proficiency of SCGNNs, the comprehensive studies are accessible using the error histograms (EHs) investigations, state transitions (STs) values, MSE performances, regression measures, and correlation.

The Klein–Gordon equation is a fundamental theoretical physics concept, governing the behavior of relativistic quantum particles with spin-zero. Its numerical solution is crucial in fields like quantum field theory, particle physics, and cosmology. The study explores numerical methodologies for solving this equation, highlighting their significance and challenges. This study uses the collocation method to approximate fractional Klein–Gordon models of distributed order based on Shifted Jacobi orthogonal polynomials and Shifted fractional order Jacobi orthogonal functions. While, the distributed term (integral term) was treat using Legendre–Gauss–Lobatto quadrature. It assesses residuals through finite expansion and yields accurate numerical results. The method is more factual and fair when initial and boundary conditions are enforced. Numerical simulations are presented to demonstrate the method’s accuracy, particularly in fractional Klein–Gordon models of distributed order. Furthermore, we offer a few numerical test scenarios to show that the method is able to maintain the non-smooth solution of the underlying issue.

This paper presents a novel spectral algorithm for the numerical solution of multi-dimensional fractional-order telegraph equations, a critical model used to capture the combined effects of diffusion and wave propagation. The core innovation of this work is the application of Jacobi-Romanovski polynomials as the basis functions for spectral discretization. These polynomials offer unique advantages, including the ability to handle nonstandard domains and boundary conditions, making them particularly suitable for partial differential equation (PDE) applications. A comprehensive error analysis is conducted, providing deep insights into the convergence rates and factors affecting the accuracy of the numerical solutions. Extensive numerical experiments further demonstrate the superior performance of the proposed spectral algorithm in solving a wide range of multi-dimensional fractional-order telegraph equation models. The results show a significant improvement in accuracy and computational efficiency compared to traditional numerical methods, such as finite difference or finite element techniques. This research advances the field of computational science by offering a robust, efficient, and versatile numerical framework for the precise solution of complex multi-dimensional PDEs.

In this study, an improved extension of the Poisson-Ailamujia distribution is introduced. The new distribution was derived using a binomial mixing approach, and the new model is named the “Binomial Poisson-Ailamujia (Bin-PA)” distribution. Some important statistical properties are derived, including mode, quantile function, moments and their associated measures, actuarial (risk) measures, and reliability features such as survival, hazard (failure) rate, and mean residual life function. The parameters of the proposed distribution are estimated using the maximum likelihood estimation method. A comprehensive simulation study is also carried out to access the behavior-derived maximum likelihood estimators. Furthermore, a new count-regression model was also introduced. Two datasets are utilized to demonstrate the applicability and usefulness of the new model. It is concluded that the Binomial Poisson-Ailamujia distribution is more flexible and efficiently analyzed both datasets as compared to competitive discrete distributions.