Modelling cross-diffusion in MHD Williamson nanofluid flow over a nonlinear stretching surface via Morlet wavelet neural networks

In this paper, a new numerical technique was developed to investigate magnetohydrodynamic (MHD) flow of Williamson nanofluid past a nonlinear stretching surface imbedded in a porous medium laden with Soret and Dufour effects. The control equations, which are highly nonlinear partial differential equations, are first converted into ordinary differential equation (ODEs) using similarity transformation and then are solved effectively by the hybrid computational method applying Morlet Wavelet Neural Networks (MWNNs) combined with a heuristic optimizers neural network and particle swarm as MWNNs-PSO-NNA. The proposed MWNNs-PSO-NNA shows a very low mean square error and Theil's Inequality Coefficient indicating that the accuracy of the model. To check the convergence and validation of the proposed approach, computing the hundred independent runs for statistical metrics. The fitness function, MSE and TIC values ranging from 10^-07 to 10^-05, 10^-09 to 10^-07 and 10^-06 to 10^-04 respectively. It is found that increasing the effects of the Williamson number, magnetic parameter, porosity and stretching index inhibit the velocity field while Brownian motion as well as the Williamson number enhances the temperature profile. The concentration rises with Soret and Brownian motion parameters but diminishes with intensified thermophoresis and magnetic influences. These findings confirm that the proposed hybrid model is not only computationally robust but also highly effective for solving complex fluid flow problems in engineering and applied sciences.