Levenberg–Marquardt Backpropagation for Numerical Treatment of Micropolar Flow in a Porous Channel with Mass Injection

Sk Md Mizanur Rahman,Muhammad Asif Zahoor Raja,Murari Andrea,Kashif Ullah,Abdu Gumaei,Saeed Islam,Muhammad Ayaz,Mehreen Fiza,Hussain Alsalman,Muhammad Shoaib,Hakeem Ullah,Imran Khan

doi:10.1155/2021/5337589

Abstract

In this research work, an effective Levenberg–Marquardt algorithm-based artificial neural network (LMA-BANN) model is presented to find an accurate series solution for micropolar flow in a porous channel with mass injection (MPFPCMI). The LMA is one of the fastest backpropagation methods used for solving least-squares of nonlinear problems. We create a dataset to train, test, and validate the LMA-BANN model regarding the solution obtained by optimal homotopy asymptotic (OHA) method. The proposed model is evaluated by conducting experiments on a dataset acquired from the OHA method. The experimental results are obtained by using mean square error (MSE) and absolute error (AE) metric functions. The learning process of the adjustable parameters is conducted with efficacy of the LMA-BANN model. The performance of the developed LMA-BANN for the modelled problem is confirmed by achieving the best promise numerical results of performance in the range of E-05 to E-08 and also assessed by error histogram plot (EHP) and regression plot (RP) measures.

Highlights

A short overview of the scheme proposed for finding the proposed LMA-BANN numerical experimentation continuity and momentum equations, i.e., 2–4, based on MPFPCMI is accessible . e proposed structure of stepwise flow is presented in Figure 1 by using “nftool” of the NN tool-box existed in MATLAB
80 percent of data are used for training as 10 percent is for testing and 10 percent is for validation in the event of a 2-layer feed-forward ANN structure fitting tool with LMA backpropagation to solve all problems of MPFPCMI
Conclusion e LMA-BANN is used as an artificial intelligence-based integrated method to find an accurate series solution for the MPFPCMI. e partial differential equations (PDEs) system of the MPFPCMI is converted to the order differential equations (ODEs) system by using the ability of similarity variables. e optimal homotopy asymptotic (OHA) method is used for producing the dataset of the MPFPCMI

Summary

Introduction

A few years ago, Eringen [1, 2] firstly presented the idea of micropolar fluids. eories of non-Newtonian fluid are developed to describe the behavior of the fluid that does not obey Newton’s law, such as micropolar fluids. is fluid summarizes specific non-Newtonian behaviors, such as liquid with polymer additives, liquid crystals, animal blood particles, suspensions, and topographic features. e governing equations of many physical problems are nonlinear in nature and cannot be solved analytically; the scientist developed some approximate and numerical techniques, such as perturbation-based methods [3, 4], homotopy perturbation-based methods [5,6,7], homotopy analysis-based methods [8,9,10], collocation-based method [11,12,13,14,15,16,17,18], and Adomian decomposition-based methods [19, 20]. A few latest research works to solve the problems of Complexity nonlinear systems include the study introduced for local fractional partial differential equations [24], fourth-order nonlinear differential equations [25], Riccati equation control of nonlinear uncertain systems [26], analytic solution of micropolar flow using the homotopy analysis method [27], and pantograph delay differential equation [28]. These numerical-based methods need discretization and improved linearization techniques, which only allow computing the solution for certain standards variables and required huge computer memory and time. There is no study yet has been applied a fast backpropagation method for finding an accurate series solution to micropolar flow in a porous channel with mass injection (MPFPCMI)

Objectives

Results

Conclusion