Arif Khalid, Saeed Syed Tauseef, Aslam Muhammad Naeem, Younis Jihad, Riaz Arshad, Saleem Salman
Department of Mathematics and Statistics, The University of Lahore, Lahore, 54770, Pakistan.
Department of Mathematics, Lahore Garrison University, Lahore, 54770, Pakistan.
Sci Rep. 2025 Jul 26;15(1):27287. doi: 10.1038/s41598-025-06912-y.
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 to 10, 10 to 10 and 10 to 10 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.
本文开发了一种新的数值技术，用于研究威廉姆森纳米流体在嵌入充满索雷特和杜福尔效应的多孔介质中的非线性拉伸表面上的磁流体动力学（MHD）流动。控制方程是高度非线性偏微分方程，首先使用相似变换将其转换为常微分方程（ODEs），然后通过应用莫雷小波神经网络（MWNNs）与启发式优化器神经网络和粒子群相结合的混合计算方法（即MWNNs - PSO - NNA）有效地求解。所提出的MWNNs - PSO - NNA显示出非常低的均方误差和泰尔不等式系数，表明了模型的准确性。为了检验所提出方法的收敛性和有效性，计算了用于统计指标的一百次独立运行。适应度函数、MSE和TIC值分别在10⁻¹⁰到10⁻¹⁰、10⁻¹⁰到10⁻¹⁰和10⁻¹⁰到10⁻¹⁰范围内。结果发现，增加威廉姆森数、磁参数、孔隙率和拉伸指数的影响会抑制速度场，而布朗运动以及威廉姆森数会增强温度分布。浓度随索雷特和布朗运动参数增加而上升，但随强化热泳和磁影响而降低。这些发现证实，所提出的混合模型不仅在计算上稳健，而且对于解决工程和应用科学中的复杂流体流动问题非常有效。
