This study explores the flow of a non-Newtonian fluid between two rolls that are counter-rotating at the same speed and of equal size. The fluid's viscosity depends on temperature, and we investigate its theoretical impact on the thickness of the sheet and other engineering parameters relevant to the process. We derive non-dimensional mass and momentum balance equations using suitable transformation and the lubrication approximation theory. The expressions for velocity distribution, pressure gradient, flow rate, temperature profile, and pressure fields have been obtained by utilizing the perturbation method. After obtaining these expressions, we compute engineering quantities such as the roll separation force, streamline, Nusselt number, and the power input required to drive both cylinders based on the system's kinematical and geometrical parameters. We also obtain numerical solutions using the finite difference method and built-in (BVP method) in Maple. Further, we use response surface methodology and analysis of variance to determine what the mathematical models mean and whether they are good enough for sensitivity and optimization analysis of the heat transmission and roll separation force. Using statistical tools such as the R2, we determine that our Nusselt number and roll separation force provide the best solution for the considered model. Additionally, it has been observed that as the Weissenberg number increases, velocity tends to rise; conversely, velocity decreases with a higher velocity ratio. Also, the temperature profile is notably influenced by the Brickman number and increases with the increase in the Brickman number. It has also been noted that as the values of velocities ratio increase, the separation points shift toward the nip region, while concurrently, the coating thickness decreases. Furthermore, we also demonstrate that compression between analytical and numerical solutions for the considered problem of fluid flow, which suggests that the results presented here are reasonable. Finally, we compare our work with published studies to validate our findings. Hence, these factors help in an efficient fluid coating process and improve the substrate life.

The roll coating process is broadly employed in the manufacturing of wallpapers, protection of fabrics and metals, wrapping, adhesive tapes, x-ray and photographic films, books and magazines, beautification, magnetic records, film foils, coated paper, etc. This study proposes a new framework for analyzing non-Newtonian fluid flow between co-rotating rolls at identical speed and size. The framework combines analytical, numerical, and computational methods powered by artificial neural networks. A key aspect of the model is the incorporation of temperature-dependent viscosity, allowing us to investigate its theoretical influence on various flow characteristics and relevant engineering parameters. To achieve this, we derive non-dimensionalized mass and momentum balance equations using appropriate transformations and lubrication approximation theory. The analytic expression for velocity distribution, temperature, pressure gradient, pressure fields, and flow rate is achieved by utilizing the perturbation method. The numerical solutions using the collocation method based on Hermite functions and the boundary value problem built-in method are also obtained. After deriving these expressions, we calculate engineering quantities including the Nusselt number, streamline, power input needed to drive both cylinders, and the roll separation force. The impacts of emerging parameters on all quantities of interest are illustrated using graphs and tables. It is interesting to mention that an increase in the non-Newtonian parameter increases the velocity but in the increase in the Vogel viscosity parameter, the velocity decreases. Furthermore, the correctness of the present work is observed by comparing analytic, numeric solutions and previously published work, and observed good agreement. To obtain approximate solutions for various flow scenarios within the proposed model, we employ a supervised neural network solver with Levenberg–Marquardt backpropagation (LMBP-SNNs) for testing, validation, and training. This approach utilizes the mean squared error (MSE) to adjust the network parameters. The efficiency of the proposed LMBP-SNN solver is validated through a combination of comparative analyses, performance studies based on MSE outputs, and visualizations of regression errors. The performance on MSE has been shown for the velocity profiles of the developed model about 9.174 × 10−12, 4.1029 × 10−12, 4.5997 × 10−12, and 2.8300 × 10−13. This study addresses a gap in the existing literature by exploring the rheological properties of the Eyring–Powell fluid model and integrating numerical methods along with machine learning techniques in the forward roll coating process.

This work focuses on valuation scheme of European and American options of single asset with meshless radial basis approximations. The prices are governed by Black – Scholes equations. The option price is approximated with three infinitely smooth positive definite radial basis functions (RBFs), namely, Gaussian (GA), Multiquadrics (MQ), Inverse Multiquadrics (IMQ). The RBFs were used for discretizing the space variables while Runge-Kutta method was used as a time-stepping marching method to integrate the resulting systems of differential equations. Numerical examples are shown to illustrate the strength of the method developed. The findings show that the RBFs has proven to be adaptable interpolation method because it does not depend on the locations of the approximation nodes which have overcome frequently evolving problems in computational finance such as slow convergent numerical solutions. Thus, the results allow concluding that the RBF-FD-GA and RBF-FD-MQ methods are well suited for modeling and analyzing Black and Scholes equation.

This work presents an efficient procedure based on Chebychev spectral collocation method for computing the 2D Laplace’s equation on a rectangular domain. The numerical results and comparison of finite difference and finite element methods are presented. We obtained a satisfactory result when compared with other numerical solutions.

Flooding in Nigeria has evolved into a recurrent humanitarian and economic crisis, with the 2024 floods alone displacing over 641,000 individuals, destroying infrastructure, and severely impacting food security and livelihoods. In regions around the Benue River, where the Lagdo Dam has impacts, there is a need for an accurate flood risk analysis and water levels’ predictive model. The optimal capacity of the Lagdo dam basin is set at 210 m, and the designed level at 216 m, while the overall capacity and area of the submerged zone measure \(7.7 \times 10^9\,\) m³ and \(586\,\) km², respectively. However, over the past few decades, sediment accumulation has significantly affected the effective capacity of the dam and further decreased the remaining capacity to about \(1.6 \times 10^9\,\)m³ in 2021. To address these challenges, this study develops an integrated machine learning–based predictive framework capable of jointly forecasting flood severity, water level variations, and associated economic impacts. The proposed approach employs a multi-task learning (MTL) architecture and benchmarks its performance against single-task machine learning models, including Random Forest, XGBoost, and Logistic Regression. To enhance physical interpretability, we engineered causally informed features, including lagged hydrological drivers, monthly stressor aggregates, and exposure and resilience indices. Time-aware models were trained using historical hydrological (inflow, dam discharge, reservoir level), meteorological (daily rainfall), and socioeconomic (persons affected, agricultural and infrastructure damage) data covering the period 1982–2024. The flood severity classification component achieves a mean cross-validated accuracy of 67.5% (±1.2%) across five folds, with a weighted-average F1-score of 0.662, reflecting balanced performance across severity classes. In water level forecasting, the model achieved a mean \(R2\) score of \(0.90\), signifying strong alignment between predictions and actual water levels, while the cross-validation results show a MAE of 0.15 with minimal variation \((\pm 0.01)\). Similarly, in the economic impact prediction, the model achieved a MAE of 0.16 and a R2 score of 0.88, reflecting its strong predictive accuracy, while the cross-validation results yielded consistent performance, with a mean MAE of 0.17 and a standard deviation of 0.02. The results demonstrate that the MTL framework consistently outperforms single-task models in predicting flood severity and water levels, while providing competitive performance in economic impact estimation. In high-risk situations, the framework might cut emergency decision-making timescales by 25–40%, which would be a paradigm change with important ramifications for climate resilience. This research underscores the transformative potential of AI in disaster management and contributes to Nigeria’s sustainable development and resilience against climate change-induced disasters, while highlighting the need for future work on uncertainty quantification and scenario-based validation.