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
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MATHEMATICS RESEARCH DEVELOPMENTS

A E
DVANCED NGINEERING
M A
ATHEMATICS AND NALYSIS

V 1
OLUME

RAMI A. MAHER
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https://doi.org/10.52305/THXL5007

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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