This paper examines a range of results that can be derived from Einstein’s evolution equation focusing on the effect of introducing a Lévy distribution into the evolution equation. In this context, we examine the derivation (derived exclusively from the evolution equation) of the classical and fractional diffusion equations, the classical and generalised Kolmogorov–Feller equations, the evolution of self-affine stochastic fields through the fractional diffusion equation, the fractional Poisson equation (for the time independent case), and, a derivation of the Lyapunov exponent and volatility. In this way, we provide a collection of results (which includes the derivation of certain fractional partial differential equations) that are fundamental to the stochastic modelling associated with elastic scattering problems obtained under a unifying theme, i.e., Einstein’s evolution equation. This includes an analysis of stochastic fields governed by a symmetric (zero-mean) Gaussian distribution, a Lévy distribution characterised by the Lévy index $\gamma \in [0,2]$ and the derivation of two impulse response functions for each case. The relationship between non-Gaussian distributions and fractional calculus is examined and applications to financial forecasting under the fractal market hypothesis considered, the reader being provided with example software functions (written in MATLAB) so that the results presented may be reproduced and/or further investigated.

Although heart disease stands as a prominent contributor to worldwide deaths, not all individuals affected by it ultimately fall prey to its effects. Timely diagnosis and effective treatment can offer those with heart conditions a high-quality life in their later years. Consequently, early disease detection using accessible medical data has been a central goal for researchers in recent decades. Traditionally, researchers relied on statistical tools for this purpose. However, machine learning algorithms, especially classification models, have gained prominence with the growing accumulated data. These algorithms have shown promise in predicting heart disease based on individual data. Our study employed various classification algorithms to predict heart disease incidence using the available dataset. We prioritized model reliability by incorporating the conformal classifier. Our results have shown that boosting algorithms, such as XGBoost and CatBoost, demonstrated exceptional performance with promising metrics. These models identified chest pain type and ST segment slope as crucial indicators of heart disease. Boosting algorithms exhibited a compelling combination of broad coverage and a small prediction set size, making them well-suited for heart disease prediction. Furthermore, we employed explainable artificial intelligence-boosting algorithms to enhance the interpretability of our predictions.

In the current study, a numerical model has been developed to simulate the blood flow characteristics in the human carotid artery. The data thus generated is analyzed to understand the blood flow variations and predict the flow characteristics using Machine Learning techniques. In developing the numerical model, the key features of the system, namely, the blood, is modeled as an incompressible Newtonian fluid, and the artery is an elastic pipe. This model is simulated using COMSOL software by varying the material properties of the artery. Univariate analysis was performed to gain insight into the features' behaviour and target variables. Subsequently, machine-learning regression models were trained using the data generated from the idealized human carotid artery. Furthermore, the validity of the data was ensured by comparing it with flow division ratios available in the literature. The evaluation of these models was conducted by calculating the Mean Absolute Error values for the test dataset, resulting in the following values: polynomial regressor (0.0106), hyper-tuned support vector regressor (0.0487), decision tree regressor (0.000), random forest regressor (0.0156), Adaboost (0.0508), gradient-boosting (0.0044), and XGboost (0.0043). A quantile loss function was employed to assess the prediction uncertainty. According to the theory of loss function, models with low loss values are considered good predictors. The prediction uncertainty was measured by applying quantile loss function, and it identified that the random forest regressor as the best predictor model for the data, followed by the polynomial regression of degree 3. Prediction intervals for the target variable were computed by leveraging the random forest quantile regressor model. Moreover, the developed polynomial model was utilized to investigate the presence of stenosis in the artery.

The transport of Omani heavy crude oil poses significant thermal–hydraulic challenges due to its high viscosity, yield-stress characteristics, and sensitivity to desert ambient conditions. To address these complexities, a non-Newtonian COMSOL Multiphysics model was developed using the Bingham–Papanastasiou formulation. The system considered three serially connected AISI 1020 steel pipeline segments with fixed diameters of 0.106m, 0.1524m, and 0.2032m under non-isothermal conditions. A total of 647 simulations were conducted, varying geometric, thermal, and rheological parameters to capture velocity, temperature, pressure, head loss, heat loss, and power loss. The outputs were processed through feature engineering to derive normalized indices for heat and power losses, alongside an Efficiency Index and composite performance scores for systematic evaluation. Results revealed distinct trade-offs: some designs achieved minimal heat loss but demanded higher pumping power, while others minimized power loss at the expense of increased thermal dissipation. Designs maintaining stable mean transport temperatures within 296.5–297.1 K were identified as robust performers. Importantly, Design 8 is consistently the best choice when power efficiency is prioritized, while Design 1 proves superior under heat-focused operations. Together, they represent the two most robust and versatile pipeline configurations for sustainable and cost-efficient long-distance transport of Omani heavy crude oil. The optimal rheological window was determined as plastic viscosity ≈ 0.0018–0.0028 Pa·s and yield stress ≈ 10–10.7 Pa, enabling reduced pumping requirements and stable thermal gradients, thereby supporting balanced thermal–hydraulic performance under realistic operating conditions.