MHD Boundary Layer Flow over a Stretching Sheet: A New Stochastic Method

Department of Mathematics, Abdul Wali Khan University, Mardan 23200, KP, Pakistan Department of Basic Sciences, College of Science and 'eoretical Studies, Saudi Electronic University, Madinah 11673, Saudi Arabia Department of Mathematics, Bangladesh University of Engineering and Technology, Dhaka 1000, Bangladesh Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Majmaah 11952, Saudi Arabia Future Technology Research Center, National Yunlin University of Science and Technology, 123 University Road, Section 3, Douliou, Yunlin 64002, Taiwan Department of Mathematics, COMSATS University Islamabad, Attock Campus, Attock 43600, Pakistan


Introduction
e boundary layer flow of an incompressible liquid over a stretching sheet is common in many engineering and industrial processes. e field has attracted researchers in the last few decades. e boundary layer flow has major applications in industries such as in the aerodynamic extrusion of a polymer sheet from a die, hot rolling, the cooling of an infinite metallic plate in a cooling bath, the boundary layer along a liquid film in condensation process, and glass-fiber production [1][2][3]. Many metallurgical processes contain the cooling of continuous filaments by drawing them through a quiescent fluid. e mechanical properties of the product depend on the rate of cooling and the stretching. Sakiadis [4,5] pioneered the study on the stretching sheet and boundary layer flow. e boundary layer flow over a continuous stretching sheet with constant speed was studied with various conditions. Cran [6] studied the closed form solution of the boundary layer flow with the stretching sheet. Gupta and Gupta [7] investigated the magnetic field effects on the boundary layer flow with the stretching sheet. Anderson [8] studied the porosity effects on the stretching sheet with the conducting particles of fluid. Ariel [9] investigated the combined effect of viscoelasticity and magnetic field on the Cranes problem. Since stretching sheet can occur in a variety of ways, the flow through the stretched sheet does not always need to be of two sizes. If the extension is radial, it can be three. A flat surface of three stretches and the same width was examined by Wang [10]. Brady and Acrivos [11] monitor the flow inside the channel or tube and the Wang flow outside the performing tube [12]. Wang [13] and Usha and Sidharan [14] tested the unstable stretching sheet. Ariel et al. [15] used HPM and expanded HPM to obtain a solution for analysis in the axisymmetric flow across the flat layer of the boundary layer flow. A noniterative solution for MHD flow of the boundary layer flow over a stretching sheet was provided by Ariel [16]. Magnetohydrodynamics (MHD) is the study of the interaction of electromagnetic conditions with the transfer of liquid heat. e flow of conducting fluids is important in many areas of science and engineering, such as MHD power generation, MHD generators, and MHD pumps. Recently, many researchers worked on boundary layer flow [17][18][19][20][21][22][23][24][25]. ey considered the MHD effects of different boundary layer flow with the stretching sheet. Recently, many scientists studied the effects of MHD and heat transfer on various boundary layer flows for different parameters [26][27][28][29][30][31]. All these numerical methods are applied to solve the problem in different scenarios, and each has advantages and disadvantages. Although stochastic numerical computing based on artificial intelligence has been developed to solve stiff nonlinear problems, these solvers are not yet used to analyze this boundary layer flow model's governing system. Stochastic numerical computing solvers are generated basically by taking advantage of computing based on artificial neural networks (ANN) modeling and its optimization of the process to solve different problem systems of ordinary or partial differential equations. ere are many modern applications of stochastic numerical computing solvers in various branch of sciences such as thermodynamics, astrophysics, offline circuits, atomic physics, MHD, plasma physics, fluid dynamics, electromagnetics, nanotechnology, bioinformatics, electricity, energy and finance, and random matrix theory [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48]. Inspired from these facts, the authors study to explore and incorporate the soft computing architectures as an alternative, precise, and feasible computational approaches for solving the fluid mechanics systems associated with the boundary layer flow. e main purpose of this study is to analyze the effect of physical parameters associated with the boundary layer flow system under the influence of the magnetic effect by using an intelligent computing technique based on the Levenberg-Marquard algorithm, whereas, Levenberg-Marquard (LM) inherits accuracy and fastness from the Newton method. Moreover, it also has the steepest descent method convergence capability [42]. Optimization of the data used in the training of deep networks is a very important parameter for the prediction performance of the model [49]. Some of the recent development in AI methods can be seen [50][51][52][53][54][55]. Some structures of our discussion are noted as follows.
e key aspects of the proposed computing paradigm are given as follows: A new application based on artificial intelligence-based computing using neural network backpropagated with Levenberg-Marquard is implemented to study the MHD boundary layer flow with the stretching sheet e dataset for the NN-BLMS is generated for variations of Deborah and magnetic parameters through the OHAM e governing equations are transformed from a set of PDEs into ODE by using similarity transformation e processing of NN-BLMS means training, testing, and validation in employed on the boundary layer flow model for different scenarios to obtain the approximate solution and comparison with reference results e convergence analysis based on mean square, error histogram, and regression plots are employed to ensure the performance of NN-BLMS for the detailed analysis of the boundary layer flow model e mathematical modeling of the boundary layer flow model has been presented in Section 2. e method for the analysis of the MHD boundary layer flow over a stretching sheet has been discussed in Section 3. e numerical and graphical results with discussion and comparison for the MHD boundary layer flow over a stretching sheet through the proposed technique NN-BLMS with numerical reference results are given in Section 4. Finally, concluding remarks for the study on the proposed methodology for the MHD boundary layer flow over a stretching sheet is presented in Section 5.

Mathematical Formulation of the Boundary Layer Flow Model
Consider a viscous fluid on a stretching sheet. Initially, both the sheet and the fluid are at rest. e plate is stretched in the x direction, and fluid starts flowing uniformly. A uniform magnetic field B (x) is applied perpendicular to the flow. e flow is considered steady and incompressible. A boundary layer is originated as shown in Figure 1. e fundamental equations in the form of continuity and momentum equations are given as [17] zu zx + zv zy � 0, (1) Using the boundary conditions (BCs), 2 Mathematical Problems in Engineering (2) In order to reduce the governing equation into boundary value problem using the following similarity transformation [17], We obtain where β � 2n/1 + n and M � 2σβ o 2 /ρc(1 + n).

Solution Procedure
e single neural network model for the proposed NN-BLMS is shown in Figure 2. e general procedure of NN-BLMS presented step by step is given in Figure 3. NN-BLMS is accomplished with the help of nftool in MATLAB by setting for fitting the neural network tools with backpropagation of Levenberg-Marquardt executing the weight of neural networks. It is clear from the analysis that the NN-BLMS is implemented for the fluidic model MHD boundary layer flow over a stretching sheet by setting the values of one parameter β and treating the other physical parameters M as fixed. In the same fashion, the other parameters are changeable, and a total of three scenarios and each scenario have three cases. e proposed NN-BLMS is performed for four scenarios by varying beta and magnetic parameters for f and f ′ with different cases for each scenario as shown in Table 1. For using the NN-BLMS, we used the step size of 0.03 between the intervals of the problem by using the OHAM. We select 80%, 10%, and 10% for testing, training, and validation, respectively, for 301 data input points for the f values randomly. TR is used for the assembly of a result assembled on MSE, and VD data is used for representing NN, while TT data is used to test the PF of random contributions.
e NN-BLMS-based computing paradigm with 1-layer structure of neural networks (hidden and output) is shown in Figure 4. .9537 × 10 -08 , 1.678 × 10 -04 , 2.3718 × 10 -5 , 9.9666 × 10 -8 , 9.9689 × 10 -8 , 9.9879 × 10 -8 , and 9.983 × 10 -08 and 10 -09 , 10 -11 , 10 -08 , 10 -07 , 10 -08 , 10 -09 , 10 -09 , and 10 -08 as shown in Figure 5.  Unit of Neural Network Layer e study of correlation is piloted by regression studies. e results of regression for each scenario are around unity, and the values of correlation R are reliable, which mean that testing, training, and validations are accurately modeled to perform the NN-BLMS. Moreover, for all three cases of each scenario of the MHD boundary layer flow over a stretching sheet, the convergence attains parameter in terms of PF, MSE, performed period, time of performance, and backpropagation measures which are prescribed in Table 2 for all cases of each scenario separately. e PF is approximately 10-09, 10-10 to 10-09, 10-09, 10-08, 10-08, and 10-08 for all scenarios with       Mathematical Problems in Engineering             Targets         In order to validate the efficiency and accuracy of the new proposed method, we compare its results with the methods available in the literature like OHAM and MOHAM as given in Table 3 and Figure 27. All of these numerical and graphical diagrams ensure the precise, flexible, and robust functionality of the NN-BLMS for the MHD boundary layer flow over a stretching sheet.

Conclusions
e computational strength in terms of the supervised learning method NN-BLMS is exploited to obtain a numerical solution for the MHD boundary layer flow over a stretching sheet after the transformation of PDEs based on the flow model into a system of ODEs by using similarity variable conversions. e optimal homotopy asymptotic method is used for the present dataset for the flow model. e data containing training, testing, and validation for NN-BLMS depending on various scenarios are determined by 80%, 10%, and 10%, respectively. e close agreement of both proposed and a reference result is 10 − 2 to 10 − 8 . is means that the proposed model provides highly accurate results for the fluidic system under consideration. e efficacy and performance of the proposed NN-BLMT for the solution of the flow model appears via mean squared error functions, performance measures, regression metrics, and histograms. Some of the key points are given as follows: NN-BLMS contains less computational work and do not required linearization and is fastly convergent. NN-BLMS is simple in applicability. NN-BLMS has better PF as compared to other numerical methods. NN-BLMS minimizes the absolute error. e correctness of NN-BLMS is authenticated by MSC, EH, RG, AE, FT, PF, TS, and TR. NN-BLMS uses 80%, 10% and 10% of the reference data are used as a TR, TS, and VL. Also, the physical variation of the parameters indicates that the boundary layer thickness decreases by increasing the Deborah, porosity, and magnetic parameters.
e boundary layer flows have many applications in engineering and industries such as in the aerodynamic extrusion of a polymer sheet from a die, hot rolling, the cooling of an infinite metallic plate in a cooling bath, the boundary layer along a liquid film in condensation process, and glass-fiber production. e conducting boundary layer flows have industrial and engineering applications such as MHD power generation, MHD generators, and MHD pumps productions.
is procedure will be used for the nanofluid flow problems and nanotechnology. In future, the new types of platforms based on artificial intelligence will be developed for the flow problems.

Data Availability
All the relevant data are available in the manuscript.

Conflicts of Interest
e authors declare that they have no conflicts of interest.

24
Mathematical Problems in Engineering