Design of Morlet Wavelet Neural Network for Solving a Class of Singular Pantograph Nonlinear Differential Models

Kashif Nisar,Fevzi Erdogan,Joel J P C Rodrigues,Muhammad Reazul Haque,Zulqurnain Sabir,Muhammad Asif Zahoor Raja,Danda B Rawat,Ag Asri Ag Ibrahim

doi:10.1109/access.2021.3072952

Abstract

The aim of this study is to design a layer structure of feed-forward artificial neural networks using the Morlet wavelet activation function for solving a class of pantograph differential Lane-Emden models. The Lane-Emden pantograph differential equation is one of the important kind of singular functional differential model. The numerical solutions of the singular pantograph differential model are presented by the approximation capability of the Morlet wavelet neural networks (MWNNs) accomplished with the strength of global and local search terminologies of genetic algorithm (GA) and interior-point algorithm (IPA), i.e., MWNN-GAIPA. Three different problems of the singular pantograph differential models have been numerically solved by using the optimization procedures of MWNN-GAIPA. The correctness of the designed MWNN-GAIPA is observed by comparing the obtained results with the exact solutions. The analysis for 3, 6 and 60 neurons are also presented to check the stability and performance of the designed scheme. Moreover, different statistical analysis using forty number of trials is presented to check the convergence and accuracy of the proposed MWNN-GAIPA scheme.

Highlights

Pantograph equation is one of the specific form of the functional differential system that contain proportional delays
The aim of the present study is to solve the singular pantograph differential model of second kind by designing a layer structure of feed-forward artificial neural networks using the Morlet wavelet activation function, while the optimization task is accomplished with the strength of global and local search terminologies of genetic algorithm (GA) and interior-point algorithm (IPA), i.e., Morlet wavelet neural networks (MWNNs)-GAIPA
The hybridization of the global and local search operators, i.e., GA-IPA is used in the optimization process

Summary

Introduction

Pantograph equation is one of the specific form of the functional differential system that contain proportional delays. The pantograph differential equations have been solved by many techniques, some of them are intelligent networks [5], Chebyshev spectral scheme [6], spectral tau scheme [7], multidimensional homotopy optimal asymptotic scheme [8], Genocchi operational based matrix scheme [9], least-squaresEpsilon-Ritz scheme [10], Taylor operation scheme [11], Galerkin multi-wavelets scheme [12], heuristic computing approach [13], Sinc numerical scheme [14], Laplace. K. Nisar et al.: Design of MWNN for Solving Class of Singular Pantograph Nonlinear Differential Models transform scheme [15], spectral collocation approach [16], multistep block method [17], Legendre Tau computational scheme [18] and Euler–Maruyama scheme [19]

Objectives

Methods

Conclusion