Advanced Engineering Mathematics and Analysis

378 Pages·2021·5.16 MB·english

by  Rami A. Maher

MATHEMATICS RESEARCH DEVELOPMENTS

A E
DVANCED NGINEERING
M A
ATHEMATICS AND NALYSIS

V 1
OLUME

No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or
by any means. The publisher has taken reasonable care in the preparation of this digital document, but makes no
expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No
liability is assumed for incidental or consequential damages in connection with or arising out of information
contained herein. This digital document is sold with the clear understanding that the publisher is not engaged in
rendering legal, medical or any other professional services.
M R D
ATHEMATICS ESEARCH EVELOPMENTS

Additional books and e-books in this series can be found on Nova’s website
under the Series tab.
MATHEMATICS RESEARCH DEVELOPMENTS

A E
DVANCED NGINEERING
M A
ATHEMATICS AND NALYSIS

V 1
OLUME

RAMI A. MAHER
Copyright © 2022 by Nova Science Publishers, Inc.
https://doi.org/10.52305/THXL5007

All rights reserved. No part of this book may be reproduced, stored in a retrieval system
or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape,
mechanical photocopying, recording or otherwise without the written permission of the
Publisher.

We have partnered with Copyright Clearance Center to make it easy for you to obtain
permissions to reuse content from this publication. Simply navigate to this publication’s
page on Nova’s website and locate the “Get Permission” button below the title description.
This button is linked directly to the title’s permission page on copyright.com. Alternatively,
you can visit copyright.com and search by title, ISBN, or ISSN.

For further questions about using the service on copyright.com, please contact:
Copyright Clearance Center
Phone: [phone] Fax: [phone] E-mail: [e-mail].

NOTICE TO THE READER
The Publisher has taken reasonable care in the preparation of this book, but makes no
expressed or implied warranty of any kind and assumes no responsibility for any errors or
omissions. No liability is assumed for incidental or consequential damages in connection
with or arising out of information contained in this book. The Publisher shall not be liable
for any special, consequential, or exemplary damages resulting, in whole or in part, from
the readers’ use of, or reliance upon, this material. Any parts of this book based on
government reports are so indicated and copyright is claimed for those parts to the extent
applicable to compilations of such works.

Independent verification should be sought for any data, advice or recommendations
contained in this book. In addition, no responsibility is assumed by the Publisher for any
injury and/or damage to persons or property arising from any methods, products,
instructions, ideas or otherwise contained in this publication.

This publication is designed to provide accurate and authoritative information with regard
to the subject matter covered herein. It is sold with the clear understanding that the
Publisher is not engaged in rendering legal or any other professional services. If legal or
any other expert assistance is required, the services of a competent person should be
sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A
COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF
PUBLISHERS.

Additional color graphics may be available in the e-book version of this book.

Library of Congress Cataloging-in-Publication Data

ISBN: (cid:28)(cid:26)(cid:27)(cid:16)(cid:20)(cid:16)(cid:25)(cid:27)(cid:24)(cid:19)(cid:26)(cid:16)(cid:23)(cid:22)(cid:24)(cid:16)(cid:26)(cid:3)(cid:11)(cid:72)(cid:16)(cid:69)(cid:82)(cid:82)(cid:78)(cid:12)

Published by Nova Science Publishers, Inc. † New York
To the memory of my parents

Contents

Preface ..…………………………………...…………………………………………………….…..xi
Acknowledgments ………………..…………………………………………………………….xiii
Chapter 1 Linear Algebra and Matrices 1
1.1. Introduction ................................................................................. 1
1.2. Motivation ................................................................................... 2
1.3. Basics and Definitions of Matrices ............................................. 5
1.4. Computation by Elementary Operations .................................. 17
1.5. The Solution of Linear Algebraic Systems ............................... 23
1.6. Basics of Linear Algebra .......................................................... 37
1.7. Linear Transformations ............................................................ 43
1.8. Eigenvalue and Eigenvector Problems .................................... 49
1.9. Cayley-Hamilton Theorem and Minimum Polynomial
of a Matrix ........................................................................................ 57
1.10. Symmetric and Orthogonal Matrices ...................................... 65
1.11. Singular Values, Singular Vectors, and Norms
of Matrices ....................................................................................... 71
1.12. Matrix Factorizations .............................................................. 83
Exercises ......................................................................................... 89
Chapter 2 Differential Equations – Part I 101
2.1. Introduction ............................................................................. 101
2.2. Classification of Differential Equations ................................... 102
2.3. Ordinary Differential Equations in Engineering Problems...... 105
2.4. Solution Concepts .................................................................. 112
2.5. First-Order Ordinary Differential Equations ............................ 115
2.6. Second-Order Ordinary Differential Equations ...................... 129
2.7. High-Order Ordinary Differential Equations ........................... 140
2.8. The Solution of a First-Order System of
Differential Equations .................................................................... 143
Exercises ....................................................................................... 152
viii Contents
Chapter 3 Differential Equations – Part II 159
3.1. Introduction ............................................................................. 159
3.2. Second-Order Variable-Coefficients Ordinary Differential
Equations ....................................................................................... 159
3.3. Special Second-Order Ordinary Differential Equations ......... 177
3.4. Nonlinear Second-Order Differential Equations ..................... 193
3.5. Method of a Parameter ........................................................... 204
3.6. Solution Method Based on Jacobi Elliptic Functions ............. 209
Exercises ....................................................................................... 215
Chapter 4 Laplace Transforms 221
4.1. Introduction ............................................................................. 221
4.2. Definitions and Theorems ...................................................... 222
4.3. Laplace Inverse ...................................................................... 229
4.4. The Solution of Differential Equations Using Laplace
Transformation............................................................................... 233
4.5. Laplace Transform of Especial Functions .............................. 244
4.6. Convolution Theorem ............................................................. 252
4.7. Advanced Applications ........................................................... 261
Exercises ....................................................................................... 267
Chapter 5 Numerical Methods 277
5.1. Introduction ............................................................................. 277
5.2. Definitions and Concepts ....................................................... 278
5.3. The Solution of Nonlinear Algebraic Equations ..................... 280
5.4. Polynomial Approximation and Interpolation .......................... 296
5.5. Numerical Differentiation ........................................................ 311
5.6. Numerical Integration ............................................................. 316
5.7. Numerical Methods for Solving Ordinary Differential
Equations ....................................................................................... 326
5.8. Numerical Solution of Boundary Value Problem .................... 338
Exercises ....................................................................................... 347
Answers to Selected Exercises 357
Sections 1.1–1.5 ............................................................................ 357
Sections 1.6–1.7 ............................................................................ 358
Sections 1.8–1.9 ............................................................................ 358
Section 1.10–1.12 .......................................................................... 359
Section 2.1–2.4 .............................................................................. 360
Section 2.5 ..................................................................................... 360
Section 2.6 ..................................................................................... 361
Sections 2.7–2.8 ............................................................................ 361
Section 3.2 ..................................................................................... 361
Section 3.3 ..................................................................................... 362
Section 4.2 ..................................................................................... 364
Section 4.3–4.4 .............................................................................. 364
Section 5.3 ..................................................................................... 366