Table Of ContentE A
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n li
e e
e d Applied Artificial Neural
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Applied Artificial Neural s A
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Network Methods for n ti Network Methods for
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Engineers and Scientists i
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Solving Algebraic Equations ie N
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The aim of this book is to handle different application problems of science ts ra
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and engineering using expert Artificial Neural Network (ANN). As such,
N Solving Algebraic Equations
the book starts with basics of ANN along with different mathematical
e
preliminaries with respect to algebraic equations. Then it addresses ANN tw
based methods for solving different algebraic equations viz. polynomial o
r
equations, diophantine equations, transcendental equations, system of k
M
linear and nonlinear equations, eigenvalue problems etc. which are the basic
e
equations to handle the application problems mentioned in the content of t
h
the book. Although there exist various methods to handle these problems, o
d
but sometimes those may be problem dependent and may fail to give a
s
converge solution with particular discretization. Accordingly, ANN based f
o
methods have been addressed here to solve these problems. Detail ANN r
architecture with step by step procedure and algorithm have been included.
Different example problems are solved with respect to various application
and mathematical problems. Convergence plots and/or convergence tables
of the solutions are depicted to show the efficacy of these methods. It is
worth mentioning that various application problems viz. Bakery problem,
Power electronics applications, Pole placement, Electrical Network
Analysis, Structural engineering problem etc. have been solved using the
ANN based methods. C
h
Ja
ek
swra Snehashish Chakraverty
v
ae
lr
t
y Sumit Kumar Jeswal
ISBN 978-981-123-020-2
World Scientific
www.worldscientific.com World Scientific
12097 hc
Applied Artificial Neural
Network Methods for
Engineers and Scientists
Solving Algebraic Equations
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Applied Artificial Neural
Network Methods for
Engineers and Scientists
Solving Algebraic Equations
Snehashish Chakraverty
Sumit Kumar Jeswal
National Institute of Technology Rourkela, India
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
APPLIED ARTIFICIAL NEURAL NETWORK METHODS FOR
ENGINEERS AND SCIENTISTS
Solving Algebraic Equations
Copyright © 2021 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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is not required from the publisher.
ISBN 978-981-123-020-2 (hardcover)
ISBN 978-981-123-021-9 (ebook for institutions)
ISBN 978-981-123-022-6 (ebook for individuals)
For any available supplementary material, please visit
https://www.worldscientific.com/worldscibooks/10.1142/12097#t=suppl
Desk Editor: Rhaimie Wahap
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Chennai - 600106, India
Printed in Singapore
Contents
Chapter 1: ANN Preliminaries ................................................................. 1
1.1 Introduction .................................................................................... 1
1.2 ANN Architectures ......................................................................... 3
1.3 Learning Rules................................................................................ 4
1.3.1 Backpropagation Algorithm or Delta Learning Rule .............. 5
1.4 Activation Function (or Transfer Function) ................................... 7
1.4.1 Sigmoid function ..................................................................... 8
1.4.2 Tanh function ........................................................................... 8
1.5 Conclusion ...................................................................................... 9
Chapter 2: Mathematical Preliminaries .................................................. 12
2.1 Introduction .................................................................................. 12
2.2 Polynomial Equations ................................................................... 12
2.3 Transcendental Equations ............................................................. 13
2.4 Diophantine Equations ................................................................. 13
2.5 Linear System of Equations (LSEs) ............................................. 14
2.6 Systems of Nonlinear Equations ................................................... 15
2.7 Eigenvalue Problems .................................................................... 15
2.8 Nonlinear Eigenvalue Problem (NEP) ......................................... 16
Chapter 3: Polynomial Equations with application in solving
Bakery problem ...................................................................................... 19
3.1 Introduction .................................................................................. 19
3.2 General model of a polynomial equation ..................................... 20
3.2.1 ANN procedure for solving polynomial equation ................. 20
3.3 Numerical Examples .................................................................... 25
3.4 Bakery Problem ............................................................................ 44
3.5 Conclusion .................................................................................... 45
v
Chapter 4: Transcendental equations in power electronics
applications ............................................................................................. 47
4.1 Introduction .................................................................................. 47
4.2 General model of a transcendental equation ................................. 48
4.3 Numerical Examples .................................................................... 49
4.4 Application Problem ..................................................................... 64
4.5 Conclusion .................................................................................... 67
Chapter 5: Diophantine Equations in Pole placement ............................ 69
5.1 Introduction .................................................................................. 69
5.2 General model of a Diophantine Equation ................................... 70
5.2.1 ANN Procedure for solving Diophantine equation ............... 70
5.3 Numerical Examples .................................................................... 75
5.4 Application Problem ..................................................................... 82
5.5 Conclusion .................................................................................... 84
Chapter 6: Systems of Linear Equations with application in static
structural problems ................................................................................. 87
6.1 Introduction .................................................................................. 87
6.2 ANN-based methodology ............................................................. 88
6.3 Numerical Examples .................................................................... 90
6.4 Static structural problem ............................................................... 98
6.5 Conclusion .................................................................................... 99
Chapter 7: Systems of Nonlinear Equations in Electrical Network
Analysis ................................................................................................ 103
7.1 Introduction ................................................................................ 103
7.2 Numerical Examples .................................................................. 104
7.3 Application Problem ................................................................... 108
7.4 Conclusion .................................................................................. 109
Chapter 8: Eigenvalue Problems with application in structural
dynamics ............................................................................................... 112
vi
8.1 Introduction ................................................................................ 112
8.2 ANN model for solving eigenvalue problems ............................ 113
8.3 Numerical Examples .................................................................. 115
8.4 Application Problem ................................................................... 121
8.4.1 Spring mass system ............................................................. 122
8.4.2 Multi-storey shear structure ................................................. 126
8.5 Conclusion .................................................................................. 131
Chapter 9: Nonlinear Eigenvalue Problems with application in
structural dynamics ............................................................................... 134
9.1 Introduction ................................................................................ 134
9.2 Nonlinear Eigenvalue Problem (NEP) ....................................... 135
9.3 Numerical Example .................................................................... 136
9.4 Application Problem ................................................................... 139
9.5 Conclusion .................................................................................. 150
Chapter 10: Definite Integrals in the fluid force on a vertical
surface .................................................................................................. 153
10.1 Introduction .............................................................................. 153
10.2 Preliminaries ............................................................................. 154
10.3 ChNN Methodology ................................................................. 155
10.4 Numerical Examples ................................................................ 158
10.5 Application Problem ................................................................. 160
10.6 Conclusion ................................................................................ 162
Chapter 11: Inverse Problems in structural dynamics .......................... 164
11.1 Introduction .............................................................................. 164
11.2 Numerical Examples ................................................................ 165
11.3 Conclusion ................................................................................ 171
vii
Preface
Algebraic equations play important and challenging roles in diversified
fields of science and engineering. These engineering and science
problems usually reduce to different algebraic equations. It is worth
mentioning that algebraic equations are of various types such as
polynomial, transcendental and Diophantine equations, linear and
nonlinear systems of equations, linear and nonlinear eigenvalue
problems, etc. There exist different analytical and numerical methods to
handle these algebraic equations. However, these equations may
sometimes be difficult to solve using traditional analytical/numerical
methods due to the presence of singularity or complexity of the
functions. Moreover, sometimes these methods may be problem-
dependent and may fail to give a converged solution with particular
discretization.
In recent decades, soft computing techniques such as Artificial Neural
Network (ANN)-based methods have been developed to handle algebraic
equations. These ANN-based methods may be seen as an alternative
approach and, in some cases, these methods may be advantageous over
the existing analytical/numerical methods.
The main aim of this book has been to introduce expert ANN methods to
handle different algebraic equations. As such, the book starts with the
basics of ANN along with different mathematical preliminaries with
respect to algebraic equations. Then it addresses ANN-based methods for
solving different algebraic equations viz. polynomial equations,
transcendental equations, Diophantine equations, linear and nonlinear
systems of equations, eigenvalue problems, definite integrals, etc. to
handle a variety of application problems. Different numerical examples
and application problems have been solved using the ANN procedure.
Convergence plots and/or convergence tables of the results have also
been depicted to show the efficacy of these ANN methods. It is worth
mentioning that various application problems viz. Bakery problem,
Power electronics applications, Pole placement, Electrical network
analysis, Structural engineering problems, etc. have also been
investigated applying the said method(s). As such, this book consists of
eleven chapters. Brief ideas related to each chapter are discussed as
follows.
viii
Chapter 1 includes preliminaries related to ANN. Different ANN
architectures, learning rules and activation functions have been presented
here. Mathematical preliminaries related to different algebraic equations
have been discussed in Chapter 2. Polynomial, transcendental and
Diophantine equations, linear and nonlinear systems of equations, linear
and nonlinear eigenvalue problems have been defined in Chapter 2. In
Chapter 3, a four-layer ANN architecture with a detailed procedure based
on the concept of backpropagation algorithm has been presented for
solving polynomial equations. Efficiency of the ANN procedure has been
shown by solving few numerical examples and an application problem
for a Bakery. ANN-based approach for solving transcendental equations
has been addressed in Chapter 4. Simple numerical examples with an
application problem of power electronics have been considered in this
context. A multi-layer ANN model along with a step-by-step description
for solving Diophantine equations has been incorporated in Chapter 5.
Standard numerical examples with an application problem of pole
placement have been included in this regard. Chapter 6 addresses a
single-layer ANN model for solving linear systems of equations.
Numerical examples and an application of static structural problems have
been investigated to show the efficiency of the ANN method. Further, an
ANN technique for handling nonlinear systems of equations has been
presented in Chapter 7, where numerical examples and an application
problem related to electrical network analysis have been examined. A
multi-layer ANN architecture for solving linear eigenvalue problems has
been constructed in Chapter 8. As such, numerical examples and
application of structural dynamics have been investigated here. Chapter 9
includes an ANN method for solving nonlinear eigenvalue problems and
incorporates a solution for a numerical example as well as an application
problem of structural dynamics. Chebyshev Neural Network (ChNN)
method for solving definite integrals has been addressed in Chapter 10,
where related numerical examples and an application problem of fluid
force on a vertical surface have been handled. Finally, Chapter 11
presents an ANN approach for solving inverse problems.
The present book addresses systematic understanding of ANN method(s)
in order to solve various algebraic equations. Application problems from
different subject areas are considered to demonstrate the applicability of
the ANN methods by solving the governing algebraic equations. The
authors do believe that the book will prove to be a benchmark for
graduate and postgraduate students, teachers, engineers, researchers and
ix
