1st Edition

Artificial Neural Networks for Engineers and Scientists Solving Ordinary Differential Equations

By S. Chakraverty, Susmita Mall Copyright 2017
    168 Pages 80 B/W Illustrations
    by CRC Press

    Differential equations play a vital role in the fields of engineering and science. Problems in engineering and science can be modeled using ordinary or partial differential equations. Analytical solutions of differential equations may not be obtained easily, so numerical methods have been developed to handle them. Machine intelligence methods, such as Artificial Neural Networks (ANN), are being used to solve differential equations, and these methods are presented in Artificial Neural Networks for Engineers and Scientists: Solving Ordinary Differential Equations. This book shows how computation of differential equation becomes faster once the ANN model is properly developed and applied.

    1. Preliminaries of Artificial Neural Network

    1.1 Introduction

    1.2 Architecture of ANN

    1.2.1 Feed-Forward Neural Network

    1.2.2 Feedback Neural Network

    1.3 Paradigms of Learning

    1.3.1 Supervised Learning or Associative Learning

    1.3.2 Unsupervised or Self-Organization Learning

    1.4 Learning Rules or Learning Processes

    1.4.1 Error Back-Propagation Learning Algorithm or Delta

    Learning Rule

    1.5 Activation Functions

    1.5.1 Sigmoid Function Unipolar Sigmoid Function Bipolar Sigmoid Function

    1.5.2 Tangent Hyperbolic Function


    2. Preliminaries of Ordinary Differential Equations

    2.1 Definitions

    2.1.1 Order and Degree of DEs

    2.1.2 Ordinary Differential Equation

    2.1.3 Partial Differential Equation

    2.1.4 Linear and Nonlinear Differential Equations

    2.1.5 Initial Value Problem

    2.1.6 Boundary Value Problem


    3. Multilayer Artificial Neural Network

    3.1 Structure of Multilayer ANN Model

    3.2 Formulations and Learning Algorithm of Multilayer

    ANN Model

    3.2.1 General Formulation of ODEs Based on ANN Model

    3.2.2 Formulation of nth-Order IVPs Formulation of First-Order IVPs Formulation of Second-Order IVPs

    3.2.3 Formulation of BVPs Formulation of Second-Order BVPs Formulation of Fourth-Order BVPs

    3.2.4 Formulation of a System of First-Order ODEs

    3.2.5 Computation of Gradient of ODEs for Multilayer

    ANN Model

    3.3 First-Order Linear ODEs

    3.4 Higher-Order ODEs

    3.5 System of ODEs


    4. Regression-Based ANN

    4.1 Algorithm of RBNN Model

    4.2 Structure of RBNN Model

    4.3 Formulation and Learning Algorithm of RBNN Model

    4.4 Computation of Gradient for RBNN Model

    4.5 First-Order Linear ODEs

    4.6 Higher-Order Linear ODEs


    5. Single-Layer Functional Link Artificial Neural Network

    5.1 Single-Layer FLANN Models

    5.1.1 ChNN Model Structure of the ChNN Model Formulation of the ChNN Model Gradient Computation of the ChNN Model

    5.1.2 LeNN Model Structure of the LeNN Model Formulation of the LeNN Model Gradient Computation of the LeNN Model

    5.1.3 HeNN Model Architecture of the HeNN Model Formulation of the HeNN Model

    5.1.4 Simple Orthogonal Polynomial–Based Neural

    Network (SOPNN) Model Structure of the SOPNN Model Formulation of the SOPNN Model Gradient Computation of the SOPNN Model

    5.2 First-Order Linear ODEs

    5.3 Higher-Order ODEs

    5.4 System of ODEs


    6. Single-Layer Functional Link Artificial Neural Network

    with Regression-Based Weights

    6.1 ChNN Model with Regression-Based Weights

    6.1.1 Structure of the ChNN Model

    6.1.2 Formulation and Gradient Computation

    of the ChNN Model

    6.2 First-Order Linear ODEs

    6.3 Higher-Order ODEs


    7. Lane–Emden Equations

    7.1 Multilayer ANN-Based Solution of Lane–Emden Equations

    7.2 FLANN-Based Solution of Lane–Emden Equations

    7.2.1 Homogeneous Lane–Emden Equations

    7.2.2 Nonhomogeneous Lane–Emden Equation


    8. Emden–Fowler Equations

    8.1 Multilayer ANN-Based Solution of Emden–Fowler


    8.2 FLANN-Based Solution of Emden–Fowler Equations


    9. Duffing Oscillator Equations

    9.1 Governing Equation

    9.2 Unforced Duffing Oscillator Equations

    9.3 Forced Duffing Oscillator Equations


    10. Van der Pol–Duffing Oscillator Equation

    10.1 Model Equation

    10.2 Unforced Van der Pol–Duffing Oscillator Equation

    10.3 Forced Van der Pol–Duffing Oscillator Equation



    Dr. S. Chakraverty has over 25 years of experience as a researcher and teacher. Currently, he is working at the National Institute of Technology, Rourkela, Odisha as a full Professor and Head of the Department of Mathematics. Prior to this, he was with CSIRCentral Building Research Institute, Roorkee, India. After graduating from St. Columba’s College (Ranchi University), he obtained his M. Sc in Mathematics and M. Phil in Computer Applications from the University of Roorkee (now the Indian Institute of Technology Roorkee), earning First Position in the University honors. Dr. Chakraverty received his Ph. D. from IIT Roorkee in 1992. Afterwards, he did his post-doctoral research at Institute of Sound and Vibration Research (ISVR), University of Southampton, U.K. and at the Faculty of Engineering and Computer Science, Concordia University, Canada. He was also a visiting professor at Concordia and McGill Universities, Canada, during 1997-1999 and visiting professor of University of Johannesburg, South Africa during 2011-2014.

    Mrs. Susmita Mall received her M. Sc. degree in Mathematics from Ravenshaw University, Cuttack, Odisha, India in 2003. Currently she is a Senior Research Fellow in National Institute of Technology, Rourkela - 769 008, Odisha, India. She has been awarded Women Scientist Scheme-A (WOS-A) fellowship, under Department of Science and Technology (DST), Government of India to undertake her Ph. D. studies. Her current research interest includes Mathematical Modeling, Artificial Neural Network, Differential equations and Numerical analysis. To date, she has published seven research papers in international refereed journals and five in conferences.