This article discussed the influence of activation energy on MHD flow of third-grade nanofluid model (MHD-TGNFM) along with the convective conditions and used the technique of backpropagation in artificial neural network using Levenberg–Marquardt technique (BANN-LMT). The PDEs representing (MHD-TGNFM) transformed into the system of ODEs. The dataset for BANN-LMT is computed for the six scenarios by using the Adam numerical method by varying the local Hartman number (Ha), Prandtl number (Pr), local chemical reaction parameter (
), Schmidt number (Sc), concentration Biot number (
) and thermal Biot number (
). By testing, validation and training process of (BANN-LMT), the estimated solutions are interpreted for (MHD-TGNFM). The validation of the performance of (BANN-LMT) is done through the MSE, error histogram and regression analysis. The concentration profile increases when there is an increase in Biot number and the local Hartmann number; meanwhile, it decreases for the higher values of Schmidt number and the local chemical reaction parameter.
In this article, we examine the three-dimensional Prandtl nanofluid flow model (TD-PNFM) by utilizing the technique of Levenberg Marquardt with backpropagated artificial neural network (TLM-BANN). The flow is generated by stretched sheet. The electro conductive Prandtl nanofluid is taken through magnetic field. The PDEs representing the TD-PNFM are converted to system of ordinary differential equations, then the obtained ODEs are solved through Adam numerical solver to compute the reference dataset with the variations of Prandtl fluid number, flexible number, ratio parameter, Prandtl number, Biot number and thermophoresis number. The correctness and the validation of the proposed TD-PNFM are examined by training, testing and validation process of TLM-BANN. Regression analysis, error histogram and results of mean square error (MSE), validates the performance analysis of designed TLM-BANN. The performance is ranges 10−10, 10−10, 10−10, 10−11, 10−10 and 10−10 with epochs 204, 192, 143, 20, 183 and 176, as depicted through mean square error. Temperature profile decreases whenever there is an increase in Prandtl fluid number, flexible number, ratio parameter and Prandtl number, but temperature profile shows an increasing behavior with the increase in Biot number and thermophoresis number. The absolute error values by varying the parameters for temperature profile are 10−8 to 10−3, 10−8 to 10−3, 10−7 to 10−3, 10−7 to 10−3, 10−7 to 10−4 and 10−8 to 10−3. Similarly, the increase in Prandtl fluid number, flexible number and ratio parameter leads to a decrease in the concentration profile, whereas the increase in thermophoresis parameter increases the concentration distribution. The absolute error values by varying the parameters for concentration profile are 10−8 to 10−3, 10−7 to 10−3, 10−7 to 10−3 and 10−8 to 10−3. Velocity distribution shows an increasing trend for the upsurge in the values of Prandtl fluid parameter and flexible parameter. Skin friction coefficient declines for the increase in Hartmann number and ratio parameter Nusselt number falls for the rising values of thermophoresis parameter against Nb.
In this research paper, we observed the Prandtl–Eyring magneto hydrodynamic fluid model (PE-MHDFM) by applying the Bayesian regularization scheme as backpropagated artificial neural networks (BRS-BANNs). Effect of suction/injection at the wall is the source of convective steady flow. The nonlinear partial differential equations (PDEs) of PE-MHDFM are converted into ordinary differential equations (ODE) by applying some suitable similarity transformation. These ODEs are solved by utilizing Lobatto IIIA numerical procedure to acquire the reference dataset for different scenarios of BRS-BANN. The reference dataset is used to design the solver BRS-BANN. Further, the performance of BRS-BANN is clarified by MSE results, error analysis plots, regression and error histogram. Moreover, the solution of PE-MHDFM is observed through the validation, training and testing procedures. It is observed that the best correlation between the targeted values outcomes of the study is matched effectively, which definitely authenticates the validity and reliability of the designed solver. Furthermore, the impacts on the velocity profile and temperature profile are examined by the variation of different physical quantities along with their comparison with state-of-the-art Lobatto IIIA numerical approach.
In this paper, the 3D Darcy–Forchheimer flow model (TDDF-FM) is discussed by utilizing the artificial intelligence-based backpropagated neural networks with the algorithm of Levenberg–Marquardt (BNN-ALM) along with the convective conditions. The effects of Brownian diffusion and thermophoresis are also examined. The governing partial differential equations are transformed into a system of ODEs. Lobatto IIIA method is used to interpret the reference dataset of BNN-ALM for different scenarios of TDDF-FM by variation of porosity parameter, Forchheimer parameter, ratio parameter, Prandtl number, Biot number, Brownian motion parameter, Schmidt number, strength parameter of homogeneous reaction and thermophoresis parameter. The solutions are computed for the designed TDDF-FM by testing, training and validation processes of BNN-ALM. The performance analysis of the designed BNN-ALM is validated by histogram analysis, regression studies and the results of mean square error (MSE). Graphs are presented for temperature profile, concentration profile and concentration rate profile. The increasing values of porosity parameter, Forchheimer parameter and Biot number enhance the temperature distribution, whereas the temperature profile shows a decreasing behavior with the increase in Prandtl number and ratio parameter. The concentration profile increases with the increase in porosity parameter, Forchheimer parameter and thermophoresis parameter and it decreases with the increasing behavior of Brownian motion parameter, ratio parameter and Schmidt number. The concentration rate profile decreases with the increase in porosity parameter, Forchheimer parameter and strength parameter of homogeneous reaction and it increases with the ratio parameter. The best performance values for different cases of scenarios are 1.42E−08, 1.70E−08, 1.78E−08, 1.61E−08, 2.43E−08, 1.05E−08, 1.30E−08, 1.59E−08 and 2.31E−08, attained at 215, 186, 178, 99, 125, 213, 187, 190 and 191 epochs. The values for the Gradient factor are 9.93E−08, 9.86E−08, 9.99E−08, 2.97E−07, 1.23E−07, 9.94E−08, 9.94E−08, 9.87E−08 and 9.94E−08, where the value of regression is [Formula: see text] for the training, testing and validation data.
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