# Falkner–Skan Equation with Heat Transfer: A New Stochastic Numerical Approach

Authors: Imran Khan; Hakeem Ullah; Hussain AlSalman; Mehreen Fiza; Saeed Islam; Asif Zahoor Raja; Mohammad Shoaib; Abdu H. Gumaei
Journal: Mathematical Problems in Engineering
 (2021)

---

## Abstract

In this study, a new computing model is developed using the strength of feedforward neural networks with the Levenberg–Marquardt method- (NN-BLMM-) based backpropagation technique. It is used to find a solution for the nonlinear system obtained from the governing equations of Falkner–Skan with heat transfer (FSE-HT). Moreover, the partial differential equations (PDEs) for the unsteady squeezing flow of heat and mass transfer of the viscous fluid are converted into ordinary differential equations (ODEs) with the help of similarity transformation. A dataset for the proposed NN-BLMM-based model is generated in different scenarios by a variation of various embedding parameters, Deborah number (β) and Prandtl number (Pr). The training (TR), testing (TS), and validation (VD) of the NN-BLMM model are evaluated in the generated scenarios to compare the obtained results with the reference results. For the fluidic system convergence analysis, a number of metrics such as the mean square error (MSE), error histogram (EH), and regression (RG) plots are utilized for measuring the effectiveness and performance of the NN-BLMM infrastructure model. The experiments showed that comparisons between the results of the proposed model and the reference results match in terms of convergence up to E-05 to E-10. This proves the validity of the NN-BLMM model. Furthermore, the results demonstrated that there is an increase in the velocity profile and a decrease in the thickness of the thermal boundary layer by increasing the Deborah number. Also, the thickness of the thermal boundary layer is decreased by increasing the Prandtl number.


---

## Body

## 1. Introduction

The boundary layer flow of an incompressible liquid through a stretching sheet is commonly used in many industrial and engineering processes. The field has been attracting researchers in the past few years. The boundary layer flow has major applications in industries such as aerodynamic extraction of polymer paper from debris, thermal wrapping, cooling plate with no cooling tuber, boundary layer next to the liquid film in the condensation phase, and glass fiber development [1–3]. By immersing them in quiescent liquids, many metal processes need to cool continuous fibers. The mechanical features of the final product depend only on the drawing costs and the process temperature. Sakiadis [4, 5] experimented with new work in this area, and many researchers in the field have investigated the flow of the boundary layer into the ongoing stretching sheet at an increasing speed. A closed-form solution for the viscous flow of an incompressible fluid was obtained by Crane [6]. Gupta and Gupta [7] studied the same effects as given in [6] by considering the fluid electrically conducting and porous medium with suction/injection by Anderson [8]. Ariel [9] investigated the combined effect of viscoelasticity and magnetic field on the Crane’s problem. Since a stretching sheet can occur in a variety of ways, the flow through the stretching 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] monitored the flow inside the channel or tube and the Wang flow outside the performing tube [12]. Wang [13] and Usha and Sridharan [14] tested the unstable SS. Ariel [15] used HPM and expanded HPM to obtain a solution for analysis in axisymmetric flow across the flat layer. A noniterative solution for MHD flow was provided by Ariel et al. [16]. The problem of a third-order nonlinear two-point boundary value with no exact solutions is commonly known as the Falkner–Skan equation (FSE). Due to the importance of the boundary layer theory, the FSE is considered widely in the last forty years. Due to the nonlinear nature of the FSE having no exact solutions available in the literature, the scientists have tried the analytical and numerical approaches. Hartree [17] obtained a solution of the FSE numerically. Smith and Cebeci and Keller [18, 19] solved the FSE by the shooting method. Meksyn [20] solved the FSE by analytic approximation. Asaithambi [21–23] found the FSE solution by finite differences, and Salama studied the FSE by solving it with a higher-order method [24]. Zhang and Chen used the iterative method for handling the FSE [25]. Rosales-Vera and Valencia used the Fourier transform for the solution of the FSE [26]. Liao [27] applied the homotopy analysis method (HAM) to solve the FSE recently. Recently, Ullah et al. modified the optimal homotopy asymptotic method (OHAM) for the FSE [28]. The important special case of the FSE is known as the Blasius equation. Rosales and Valencia [29] solved this problem using the Fourier series method. Boyd [30] established the solution of the FSE using the numerical method. In the literature survey, the FSE is solved by analytical and numerical methods. Both these methods have some disadvantages as analytical techniques mostly required the initial guess and small parameter assumption and may affect the result. Also, the numerical methods required linearization and discretization techniques which can affect the accuracy. Therefore, a new stochastic method is proposed to solve the FSE-HT, and the results are compared with the literature validating the method.The stochastic methods are relatively less used for the solution of the fluid system. The use of stochastic methods can be seen in thermodynamics, astrophysics, offline circuits, atomic physics, MHD, plasma physics, fluid dynamics, electromagnetics, nanotechnology, bioinformatics, electricity, energy, finance, and random matrix theory [31–36]. The stochastic method would be used for important and reputed fluidic models [37–44].According to the literature survey, no research has applied artificial intelligent techniques through the NN-BLMM to solve the FSE-HT. The purpose of this study is to use the NN-BLMM for the solution of FSE-HT.The critical aspect of the proposed computing paradigm is given as follows:(i)
A new application based on artificial intelligence-based computing using neural network-based backpropagation with Levenberg–Marquardt is implemented for studying and solving the FSE-HT(ii)
The dataset of the NN-BLMM model is generated and produced for variations of Deborah and Prandtl parameters through the OHAM(iii)
The governing equations are transformed from a set of PDEs into ODEs by using a similarity transformation(iv)
The processing of the NN-BLMM means TR, TS, and VD of the FSE-HT model for different scenarios to obtain the approximate solution and comparison with reference results(v)
The convergence analysis based on the MSE, EH, and RG plots is employed to ensure the performance of NN-BLMM for the detailed analysis of the boundary layer flow modelThe mathematical modeling of the FSE-HT is explained in Section2. The analysis of FSE-HT is also given in Section 3. Section 4 presents the numerical results and graphical representations of the NN-BLMM model for the FSE-HT with more discussions and comparisons. Finally, Section 5 gives concluding remarks of the proposed study for the FSE-HT.


## 2. Problem Formulation

We consider two-dimensional laminar viscous flows over a semi-infinite flat plate under the boundary layer approximation as given in Figure1. The governing equations are [28](1)∂u∂x+∂v∂y=0,(2)u∂u∂x+v∂v∂y=UsdUsdx+v∂2u∂2y,(3)u∂T∂x+v∂T∂y=a∂2T∂y2,where u and v are velocity components in x- and y-directions, Usx is the free stream velocity, “a” is the thermal diffusivity, and υ is the kinematic viscosity.Figure 1
The geometry of the problem.In the two-dimensional case, the incompressible boundary layer flow over a wedge of angleβπ is shown in Figure 1. The free stream velocity of the form Uex=cxr can be computed using the similarity transformation as follows:(4)ux,y=Usf′ξ,ξ=yr+1C2xm−1/2,θξ=T−T∞Tw−T∞.Using equation (4) in equations (1)–(3), the Falkner–Skan equation [28] can be calculated as(5)f′′′ξ+fξf″ξ+β1−f′ξ2=0,θ″ξ+Prfξθ′ξ=0,with boundary conditions(6)f0=0,f′0=0,f′∞=1,θ0=1,θ∞=0,where Pr is the Prandtl number.


## 3. Solution Procedure

The solution procedure is performed by implementing the proposed NN-BLMM model using nftool in the MATLAB area. The structure of the NN-BLMM is given in Figure2, and its data process is presented in Figure 3. Essentially, the NN-BLMM is used for three conditions with different parameters as set out in Table 1. The design of the NN-BLMM is based on ten numbers of neurons, as shown in Figure 4. A reference database of 201 input grids has been selected within the interval [0, 5]. 80% of the data is used randomly in training, while 10% is used in the testing and validation process. Training data are used for the construction of a limited solution built on the MSE results. Verification data are used for demonstrating the behavior of the NN, whereas the test data are used for showing the performance of random inputs. Details about the scenarios and cases are given in Table 1.Figure 2
The structure of a single neural network model.Figure 3
Graphical representation of NN-BLMM input, hidden, and output layers.Table 1
Defining the physical parameters of interest for the scenarios and cases of FSE-HT.
ScenariosCasesPhysical quantityβPr1120.5240.5360.52140.5241.0341.5Figure 4
The workflow of the proposed NN-BLMM for processing the FSE-HT. (a) The geometry of the problem. (b) Formulation of the problem. (c) Intelligent computing: neuron model integrated to develop the proposed network. (d) Results. (e) Comparative analysis.
(a)(b)(c)(d)(e)(f)The NN-BLMM-based computing paradigm with one hidden layer structure of neural networks is exposed in Figure2.The input data are operated in the single-layer neural network model with the help of the NN-BLMM to obtain the output result, as represented in Figure3.


## 4. Analysis of Results

The workflow of the proposed NN-BLMM model for processing the FSE-HT is shown in Figure4. Besides, the results of the NN-BLMM for cases 1–3 of all scenarios in working with PF and state are displayed in Figures 5 and 6, separately. The results of FT are shown in Figures 7–12. The EH results are given in Figure 13, whereas RA results are presented in Figures 14–19 on three FSE-HT cases. Figure 5 shows the MSE for scenarios 1 and 2 at the best PF when the TR, VD, and TT data are run. Figure 6 shows the error gradient in the direction of magnitude calculated during the TR of the NN-BLMM for all scenarios at each epoch, and its results are compare with numerical method results for valdiating the proposed method. The absolute errors are further explained with the help of Figures 7–13. The accuracy of the NN-BLMM is verified by checking the results for the regression showing the accuracy of the NN-BLMM as given in Figures 14–19. In addition, the convergence control parameters in terms of executed epochs, MSE, PF, time of execution, and BP measures are presented in Table 2. In Figures 5(a)–5(f), MSE assembly of TR, VD, and TS procedures is presented for cases 1 and 3 of all scenarios (FSE-HT). One can see that the best PF has been achieved at 224, 1000, 366, 72, 143, and 248 epochs with an MSE of about 2.4702 × 10−8, 1.3234 × 10−10, 1.2138 × 10−8, 2.7061 × 10−5, 5.0276 × 10−7, and 1.0658 × 10−9, respectively. Appropriate values for GD and step Mu size of BP are (9.9228 × 10-08, 1.0553 × 10–07, 9.9259 × 10–08, 9.6455 × 10-8, 8.4492 × 10-7, and 2.7359 × 10-8) and (10-08, 10-8, 10-08, 10-08, 10-09) as shown in Figures 6(a)–6(f). The results determine the correct and convergent PF of the NN-BLMM for each case. The maximum error achieved in the TS, PF, and VL by the proposed NN-BLMM is less than 0.1 × 10-03, 0.9 × 10-05, 0.1 × 10-01, 0.5 × 10-03, 3 × 10-03, and 0.3 × 10-03 as given in Figures 7–12. Error variability is also assessed with EH for every input point, and the results are given in Figures 13(a)–13(f). The maximum error achieved is less than −3.5E − 6, 2.37E − 7, 3.64E − 4, −1.1E − 4, −1.2E − 4, and 2.56E − 6 in all cases (FSE-HT). The investigation over RG is performed using the correlation studies. Moreover, the RG results are given in Figures 14–19, and the correlation R values invariably revolve around unity of the NN-BLMM performance results for FSE-HT. Adopted cases are given in Table 2. The NN-BLMM performance is approximately 10−8 to 10−10 and 10−7 to 10−10 in cases 1 and 3 of all scenarios with 1–3 cases of FSE-HT. These results show the stability of PF for the NN-BLMM in each case of FSE-HT equations. The velocity distribution along with the comparison with numerical results is plotted in Figures 20(a) and 21(a). The results of the NN-BLMM have been compared with RK-4 numerical solutions in each case, therefore, to achieve precision gauges, AE is determined from the reference solutions, and the results are shown in Figures 20(b) and 21(b) for case studies 1–3, respectively. One can note that AE is almost 10–02 to 10–04 and 10–03 to 10–06 in all scenarios, respectively. All of these numerical and graphical diagrams ensure the precise, flexible, and robust functionality of the NN-BLMM for FSE-HT. The nondimensional velocity profiles and temperature profiles are given in Figures 20(a) and 21(a), respectively. The variation of physical parameters βandPr is given in Figures 20(a) and 21(a), whereas the absolute error for each variation of βandPr is given in Figures 20(b) and 21(b). The solution obtained by the NN-BLMM is verified by the numerical method. It has been observed that an increase in the Deborah number makes an increase in the velocity profile and a decrease in the thermal boundary layer thickness. Similarly, by increasing the Prandtl number, the thermal boundary layer thickness is decreased. The results obtained by the NN-BLMM are compared with the results available in the literature which proves the accuracy and validation of the method as given in Table 3.Figure 5
The MSE results of the proposed NN-BLMM to solve the FSE-HT. (a) The MSE results in case 1 for scenario 1. (b) The MSE results in case 2 for scenario 1. (c) The MSE results in case 3 for scenario 1. (d) The MSE results in case 1 for scenario 2. (e) The MSE results in case 2 for scenario 2. (f) The MSE results in case 3 for scenario 2.
(a)(b)(c)(d)(e)(f)Figure 6
The dynamic results of transition states for the NN-BLMM to solve the FSE-HT. (a) The transition state results in case 1 for scenario 1. (b) The transition state results in case 2 for scenario 1. (c) The transition state results in case 3 for scenario 1. (d) The transition state results in case 1 for scenario 2. (e) The transition state results in case 2 for scenario 2. (f) The transition state results in case 3 for scenario 2.
(a)(b)(c)(d)(e)(f)Figure 7
The results of the NN-BLMM compared with the reference solution in case 1 of scenario 1 for FSE-HT.Figure 8
The results of the NN-BLMM compared with the reference solution in case 2 of scenario 1 for FSE-HT.Figure 9
The results of the NN-BLMM compared with the reference solution in case 3 of scenario 1 for FSE-HT.Figure 10
The results of the NN-BLMM compared with the reference solution in case 1 of scenario 2 for FSE-HT.Figure 11
The results of the NN-BLMM compared with the reference solution in case 2 of scenario 2 for FSE-HT.Figure 12
The results of the NN-BLMM compared with the reference solution in case 3 of scenario 2 for FSE-HT.Figure 13
Studying the error histograms for the NN-BLMM results in cases 1–3 of scenarios 1 and 2 for FSE-HT. (a) Error histogram for scenario 1 in case 1. (b) Error histogram for scenario 1 in case 2. (c) Error histogram for scenario 1 in case 3. (d) Error histogram for scenario 2 in case 1. (e) Error histogram for scenario 2 in case 2. (f) Error histogram for scenario 2 in case 3.
(a)(b)(c)(d)(e)(f)Figure 14
Regression illustration of NN-BLMM results in case 1 of scenario 1 for FSE-HT.Figure 15
Regression illustration of NN-BLMM results in case 2 of scenario 1 for FSE-HT.Figure 16
Regression illustration of NN-BLMM results in case 3 of scenario 1 for FSE-HT.Figure 17
Regression illustration of NN-BLMM results in case 1 of scenario 2 for FSE-HT.Figure 18
Regression illustration of NN-BLMM results in case 2 of scenario 2 of FSE-HT.Figure 19
Regression illustration of NN-BLMM results in case 3 of scenario 2 for FSE-HT.Table 2
The NN-BLMM results of scenarios 1–3 for FSE-HT.
Scenario 1 casesMean square error (MSE)PerformanceGradientMuEpochTimeTrainingValidationTesting17.77491E−106.78795E−051.51380E−057.70E−107.61E−061.00E−08504<126.74192E−092.10370E−051.52281E−076.74E−091.26E−051.00E−071000<135.21240E−083.20661E−061.71187E−054.85E−081.84E−041.00E−07172<1Scenario 2 casesMean square errorPerformanceGradientMuEpochTimeTrainingValidationTesting11.23663E−073.74133E−076.01470E−059.96E−082.55E−051.00E−0728<123.28408E−103.51972E−081.46030E−093.28E−109.95E−081.00E−09934<132.62694E−104.25158E−083.36674E−072.63E−105.88E−071.00E−091000<1Figure 20
The results of the proposed NN-BLMM compared with reference results regarding scenario 1 of FSE-HT. (a) Variation ofβ. (b) Analysis of AE.
(a)(b)Figure 21
Comparison between the results of the NN-BLMM and reference results for scenario 2 of FSE-HT. (a) Pr variation. (b) Analysis of AE.
(a)(b)Table 3
The results of the initial slopef″0 achieved at different values of β.
βHartree [17]Asaithambi [22]Asaithambi [23]Salama [24]Zhang and Chen [25]Rosales-Vera and Valencia [26]OHAM [28]Stochastic method21.6871.6872221.6872181.6872181.6872181.6872181.6872181.68721811.2331.235891.2325881.2325881.2325871.2325871.2325871.2325870.50.9270.9276820.9276800.9276800.9276800.9276800.9276800.927680−0.10.3190.3192700.3192700.3192700.3192700.3192700.3192700.319270


## 5. Conclusions

In this paper, we conclude that the NN-BLMM contains less computational work and does not require linearization. NN-BLMM is simply applicable. The NN-BLMM has better PF as compared to other numerical methods, and it minimizes the absolute error. Moreover, the NN-BLMM is independent of the assumption of a small parameter, and it does not require the initial guess. The NN-BLMM converges faster when compared to other methods. The correctness of the NN-BLMM is authenticated by MSE, EH, RG, AE, FT, PF, TS, and TR. The NN-BLMM uses 80%, 10%, and 10% of the reference data as TR, TS, and VL. By increasing the Deborah number, the velocity profile increases, whereas the thickness of the thermal boundary layer is decreased by increasing the Deborah number. The thickness of the thermal boundary layer is decreased by increasing the Prandtl number.



---

*Source: 3921481-2021-08-30.xml*