Numeric treatment of nonlinear second order multi-point boundary value problems using ANN, GAs and sequential quadratic programming technique

Zulqurnain Sabir,Muhammad Asif Zahoor Raja

doi:10.5267/j.ijiec.2014.3.004

Abstract

In this paper, computational intelligence technique are presented for solving multi - point nonlinear boundary value problems based on artificial neural networks, evolutionary computing approach, and active-set technique. The neural network is to provide convenient methods for obtaining useful model based on unsupervised error for the differential equations. The motivation for presenting this work comes actually from the aim of introducing a reliable framework that combines the powerful features of ANN optimized with soft computing frameworks to cope with such challenging system. The applicability and reliability of such methods have been monitored thoroughly for various boundary value problems arises in science, engineering and biotechnology as well. Comprehensive numerical experimentations have been performed to validate the accuracy, convergence, and robustness of the designed scheme. Comparative studies have also been made with available standard solution to analyze the correctness of the proposed scheme.

Highlights

Multi-point boundary value problems (BVPs) or non-local boundary value problems were introduced by Ilin and Moiseev (1987)
The proposed methodologies are applied to solve the problem by taking 10 neurons in the hidden layer of neural networks with a total 30 design parameter or weights, W (, w, ) .The fitness functions e and ε is formulated for this case for r ε
A new soft computing approach has been developed effectively for solving multi-point boundary value problems, in- multi-point BVPs and its variants using neural networks optimized with genetic algorithm, sequential quadratic programming technique and their hybrid combination

Summary

Introduction

Multi-point boundary value problems (BVPs) or non-local boundary value problems were introduced by Ilin and Moiseev (1987). These BVPs have become the most important area among researchers These days, because of its widespread applications in engineering as to model the physical problems, including vibration happening in a wire of uniform cross section and combined of material with changed densities, through porous media applications, BVPs are used in fluid flow and in elastic stability. These problems have been normally solved but these techniques have some confines, e.g. the approximate solution consists of a series of very small parameters, which poses difficulty since the popularity of nonlinear problems has no minor parameters. These multi-point BVPs are generally limited to second order equation. The source to apply this theorem can be initiated with semi linear elliptic equations on annuli (Erbe & Wang, 1994)

Methods

Results

Conclusion