C++ Solutions for Mathematical Problems

249 Pages·2005·1.42 MB·English

by  Arun Ghosh

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Copyright © 2005 New Age International (P) Ltd., Publishers
Published by New Age International (P) Ltd., Publishers
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ISBN : 978-81-224-2420-1
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Preface v
Preface
The objective of this book is to present a substantial introduction to the ideas, phenomena and methods
of the problems that are frequently observed in mathematics, mathematical physics and engineering
technology. The book can be appreciated at a considerable number of levels and is designed for every-
one from amateurs to research workmen.
Included throughout are applications with appropriate suggestions and discussions, whenever needed,
that form a significant and integral part of the text book.
In a word, the text directs at an all-embracing and practical treatment of differential equations with
some methods specifically developed for the purpose controlled by computer programs. The effects of
this treatment, the computerised solutions for each problem, represented in compact form, sometimes
with graphical figures, have been provided for further study.
The operation has been performed by the programming language C++ based on any MS-DOS
computer system (version 6.0) applying the classical methods, such as Euler, Simpson, Runge-Kutta,
Finite-Difference, etc.
Chapters 4 and 5 provide introductions to first order and second order initial-value problems dis-
cussing the possibilities for finding solutions, analytic and computerised. Non-linear types of differen-
tial equations have also been considered.
Emphasis has been placed on section 6.3 in Chapter 6 that deals with the problems concerning
Fourier series, because the subject matter of numerical evaluation for Fourier series is a predominant
topic.
Special attention is focussed on Chapter 7 in view of the fact that the differential equations with
boundary conditions are the kernels of the physical and technical problems. The Difference Method for
the solution of a two point second-order boundary value problem has been applied. Chapter 8 contains
numerically developed methods for the computer solution of elliptic, hyperbolic and parabolic partial
differential equations.
A variety of solved examples in each chapter has been given for the students who cannot get any
difficulty to understand the conceptual text. Chapter 11 entitled “A Short Review On C++” provides an
opportunity to recapitulate the fundamental points on C++.
I prepared the text on personal computer from Compaq (type Presario CDTV528) and on Laptop
from Toshiba (type Satellite 110CS) operated on MS-DOS 6.0 and Windows 95. MS-Workgroups (ver-
sion 3.0) have also been used in the preparation.
vi Preface
A software supplement consisting of a set of programs designed in C++ (Turbo) is provided for the
reader to work out the related mathematical models referred to the text. A Diskette (3.5 inch/1.44 MB)
in standard PC-compatible form containing this supplement will accompany the book.
I think, my anticipation to believe is not indifferent, that the students, instructors and other readers
who use this text can enjoy the development just as well the author has taken joy in the preparation.
I wish to express my very great thanks to my beloved Professor P. Sinharay of Calcutta University
who has motivated me with interest. I am grateful to Professor G. Bertram of Hannover University who
has guided me to complete this book also to the friends of the Computer center (RRZN) in Hannover for
their helpful suggestions for improving the edition. In this connection I should not forget to mention the
names of the twins Steven and Benjamin who have contributed much to prepare this volume.
Finally, I express my gratitude to Mr S. Gupta of New Age Int. (P) Ltd, publisher in New Delhi.
ARUN GHOSH
Contents vii
Contents
Preface v
1. Preliminary Mathematical Viewpoints 1
1.1 Environment Numerically Developed 1
1.2 Error Spread 2
1.2.1 Error Reduction 3
2. Computing Surface Areas 5
2.1 Evaluation of the Definite Integral 5
2.2 Areas of Curves 14
2.3 Surfaces in 2-Dimensional Space 19
2.4 Related Software for the Solution 23
3. Systems of Simultaneous Equations 25
3.1 Preliminary Concepts 25
3.2 Matrix Relationship 27
3.2.1 Matrix Product 28
3.2.2 Nonsingular Matrix: Inverse Matrix 30
3.2.3 Matrix Transposition 31
3.3 Systems of Linear Equations 34
3.3.1 Gauss Elimination 35
3.3.2 Cramer’s Rule 38
3.3.3 Matrix Inversion 41
3.3.4 LU-Decomposition 45
3.4 Related Software for the Solution 48
viii Contents
4. First Order Initial-Value Problems 50
4.1 Preliminary Concepts 50
4.2 Methods for Solution 55
4.3 Exact Equations 61
4.4 Linear Equations 63
4.5 Bernoulli’s Equation 66
4.6 Riccati Equation 69
4.7 System of Simultaneous Differential Equations 70
4.8 Related Software for the Solution 78
5. Second Order Initial-Value Problems 80
5.1 Preliminary Concepts 80
5.2 Linear Homogeneous Type 81
5.3 Linear Nonhomogeneous Type 88
5.3.1 Method of Undetermined Coefficients 88
5.3.2 Euler’s Equation 90
5.3.3 Method of Variation of Parameters 92
5.4 Nonlinear Equations 98
5.5 Related Software for the Solution 102
6. Computing Operational Series 103
6.1 Analytic Numerical Series 103
6.1.1 Infinite Series. Convergence and Divergence 103
6.1.2 Comparison Test. Condition for Convergence 105
6.1.3 D’Alembert Test (Test-Ratio Test) 105
6.2 Power Series Solution 108
6.2.1 Introduction 108
6.2.2 Series Expansion 109
6.2.3 First Order Equations 112
6.2.4 Second Order Equations 115
6.3 Fourier Series Solution 118
6.3.1 Fourier Coefficients 118
6.3.2 Dirichlet’s Principles 120
6.3.3 Even and Odd Functions 122
6.3.4 Any Arbitrary Interval [–1, 1] 124
6.4 Related Software for the Solution 126
7. Boundary-Value Problems for Ordinary Differential Equations 128
7.1 Introduction 128
7.2 Linear Boundary Value Problems 129
7.3 Nonlinear Equations 136
7.4 Related Software for the Solution 141
Contents ix
8. Partial Differential Equations 142
8.1 Introduction 142
8.2 Wave Equation 144
8.3 Heat Equation 145
8.4 Laplace Equation 146
8.5 Poisson Equation 150
8.6 Related Software for the Solution 152
9. Laplace Transforms 153
9.1 Introduction 153
9.2 Laplace Transforms of Functions 154
9.3 Evaluation of Derivatives and Integrals 157
9.4 Inverse Laplace Transforms 159
9.5 Convolutions 161
9.6 Application to Differential Equations 163
10. Problems to be Worked Out 167
11. A Short Review on C++ 200
11.1 Basic Structure of a C++ Program 200
11.2 The Keywords from C++ 201
11.3 Identifiers in C++ 201
11.4 The Features of C++ 202
11.5 Loop Statements 203
11.6 The For Loop 205
11.7 Pointers 206
Appendix I 221
Appendix II 229
References 237