This study presents an innovative application of Feedforward Neural Networks ‘FNNs’ to solve Variable-Order Fractional Partial Differential Equations ‘VO-FPDEs’ with time delays. Utilizing the Caputo definition, the variable-order fractional derivatives are approximated in terms of integer-order derivatives. The problem is reformulated as a system of partial differential equations with delay terms, which is then addressed using ‘FNNs’ to achieve explicit approximate solutions. Comprehensive error and convergence analyses validate the method’s precision and reliability. The effectiveness of the proposed approach is highlighted through numerical examples, with graphical and tabular representations showcasing minimal absolute errors and robust convergence. These results demonstrate the proposed method’s efficiency and simplicity, establishing it as a powerful tool for addressing complex fractional delay problems.

The current study presents a novel approach for parameter estimation in LRP based on a linear intensity parameter model of the Non-Homogeneous Poisson Process (NHPP). The Modified Maximum Likelihood Estimation- Particle Swarm Optimization (MMLE-PSO) method enhances prediction accuracy and computational efficiency by integrating the PSO algorithm. Compared to Least Square Estimation (LSE) and PSO, MMLE-PSO achieves superior parameter estimation, reducing errors by 61.2% and 92.6%, respectively. Additionally, it accelerates computational performance by 97% over Maximum Likelihood Estimation (MLE) and PSO due to its faster convergence. The method's effectiveness in event pattern modeling is demonstrated using outage data from the Mosul Dam power facility. In statistical evaluation, MMLE-PSO attains the lowest RMSE value 0.0253, outperforming LSE 0.0652 and PSO 0.3429. With its enhanced estimation precision and operational efficiency, MMLE-PSO proves to be a reliable tool for reliability engineering applications

This paper aims to utilize modified artificial neural networks under stochastic differential equations to improve the estimation of occurrence rates on models towards the growth of software reliability through such models. There has never been a time when reliable assessment methods have been more important as software systems grow ever more sophisticated. This study covers the risks arising from software bugs, security weaknesses, and the need to meet legal requirements. The authors use a new method that combines artificial intelligence techniques with classical reliability models to achieve their goal of a more efficient computational tool for predicting software reliability. It focuses on the assessment of the differentiation of the Goel Okumoto model for the reliability functions and then simulates the event rate to obtain further applications. The implications of this research are vast because it provides a sound method through which time to failure in crucial systems essential in electricity generation, such as reliability engineering, finance, and telecommunication, is forecasted. In the end, this research benefits the existing conversations on enhancing software quality and reliability as organizations strive to provide organizations with robust and secure software solutions to handle today’s advanced technologies.

This analysis investigates a novel two-parameter discrete distribution, namely the Discrete Inverse Rayleigh Exponential (DIRE) distribution, which is derived from the Inverse Rayleigh-G family using a survival discretization method. The DIRE distribution features adaptable probability mass and hazard rate functions, capable of exhibiting symmetric, asymmetric, monotonic, and reversed-J-shaped behaviors, making it highly suitable for modeling a wide range of real-world data. Key statistical properties, such as mean, variance, moment-generating function, and dispersion index, are thoroughly examined. For parameter estimation, both Maximum Likelihood Estimation (MLE) and a machine learning-based K-Nearest Neighbors (K-NN) algorithm are utilized. Extensive simulations and real-world dataset analyses reveal that the DIRE distribution surpasses existing models in goodness-of-fit metrics, with the K-NN estimator demonstrating superior accuracy and robustness compared to MLE. The practical utility of the DIRE distribution is illustrated through two empirical datasets—COVID-19 case counts and failure time data—highlighting its effectiveness in managing complex discrete data. The results indicate that this new model offers improved flexibility and reliability, making it a valuable tool for statistical modeling and machine learning applications.