Abstract
In current investigation, a novel implementation of intelligent numerical computing solver based on multi-layer perceptron (MLP) feed-forward back-propagation artificial neural networks (ANN) with the Levenberg–Marquard algorithm is provided to interpret heat generation/absorption and radiation phenomenon in unsteady electrically conducting Williamson liquid flow along porous stretching surface. Heat phenomenon is investigated by taking convective boundary condition along with both velocity and thermal slip phenomena. The original nonlinear coupled PDEs representing the fluidic model are transformed to an analogous nonlinear ODEs system via incorporating appropriate transformations. A data set for proposed MLP-ANN is generated for various scenarios of fluidic model by variation of involved pertinent parameters via Galerkin weighted residual method (GWRM). In order to predict the (MLP) values, a multi-layer perceptron (MLP) artificial neural network (ANN) has been developed. There are 10 neurons in hidden layer of feed forward (FF) back propagation (BP) network model. The predictive performance of ANN model has been analyzed by comparing the results obtained from the ANN model using Levenberg-Marquard algorithm as the training algorithm with the target values. When the obtained Mean Square Error (MSE), Coefficient of Determination (R) and error rate values have been analyzed, it has been concluded that the ANN model can predict SFC and NN values with high accuracy. According to the findings of current analysis, ANN approach is accurate, effective and conveniently applicable for simulating the slip flow of Williamson fluid towards the stretching plate with heat generation/absorption. The obtained results showed that ANNs are an ideal tool that can be used to predict Skin Friction Coefficients and Nusselt Number values.
Highlights
In current investigation, a novel implementation of intelligent numerical computing solver based on multi-layer perceptron (MLP) feed-forward back-propagation artificial neural networks (ANN) with the Levenberg–Marquard algorithm is provided to interpret heat generation/absorption and radiation phenomenon in unsteady electrically conducting Williamson liquid flow along porous stretching surface
In the input layer of the ANN model designed for SFC prediction,We, A, S1, M1, E1 and γ1 values have been defined as input parameters, and the skin friction coefficient value has been predicted at output layer
In ANN model developed for prediction of NN; M1, E1, R1, Pr, Ec, Q1, Bi and α values are defined as input parameters and Nusselt Number is predicted at output layer
Summary
Two dimensional incompressible unsteady electrically conducting BLF of Williamson liquid towards a porous stretched surface under velocity as well as thermal slip condition is considered. The related flow equations while obtaining BL approximations takes the following form. GWRM is an effective method for calculating solutions of nonlinear BVP (boundary value problems). It comprises the following main steps: (1) In differential equations, the unknown dependent functions are initially considered to be linear combinations of form or trial functions containing unknown coefficients. GWRM procedures to seek an approximate solution as follows. Gauss–Laguerre formula is employed to integrate every equations in (18) to achieve set of algebraic systems since boundary condition varies from zero to infinity. Using GWRM, the assumed solutions of F(ξ )and θ(ξ )are considered as given below[34]. Together with Eqs. (23–25) produce 2N + 1nonlinear algebraic systems with 2N + 1unknown coefficients (ai, bk) and solved via MATHEMATICA to get ai and bk
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