Mathematical analysis of Prandtl number with Artificial neural network and fractional operator for nanofluid flow

This study investigates the free convection flow of Maxwell fluid using the Prabhakar fractional derivative while accounting for exothermic reactions, chemical processes, heat and concentration gradients, and the Soret effect. The innovation is in achieving unique outcomes by combining the effects of mass transport and heat over a plate. Methods:: The governing partial differential equations are non-dimensionalized and solved using the Laplace transform method. The model is generalized based on Fick’s and Fourier’s laws and fractionalized using the Prabhakar fractional derivative. An artificial neural network (ANN) with 10 hidden layers is employed to capture nonlinear flow dynamics. A supervised learning approach is used, with 70% of data for training, 15% for validation, and 15% for testing to ensure model robustness. Validation is performed through comparison with existing algorithms. Results:: The ANN model predicts flow properties with great accuracy, as evidenced by a mean squared error of less than 10−4. The findings indicate that while Prandtl parameters have a considerable impact on temperature and concentration profiles, fluid velocity decreases as Prandtl number increases. The model’s ability to accurately depict the flow dynamics is confirmed by the numerical and graphical examination. Conclusion:: The study demonstrates the effectiveness of fractional derivatives in modeling complex fluid flow. The Prabhakar fractional operator enhances the accuracy of thermal and concentration predictions, offering deeper insights into Maxwell fluid behavior under convective conditions. Future work may explore non-isothermal boundary conditions and different fractional formulations to further improve predictive performance.