[PDF] Advanced Engineering Mathematics by Merle C. Potter, Jack L. Lessing , Edward F. Aboufadel free download PDF

Description :-

The purpose of this book is to introduce students of the physical sciences to several mathemati-

cal methods often essential to the successful solution of real problems. The methods chosen are

those most frequently used in typical physics and engineering applications. The treatment is not

intended to be exhaustive; the subject of each chapter can be found as the title of a book that

treats the material in much greater depth. The reader is encouraged to consult such a book should

more study be desired in any of the areas introduced

Contents :- 

1 Ordinary Differential Equations 1

1.1 Introduction 1

1.2 Definitions 2

1.2.1 Maple Applications 5

1.3 Differential Equations of First Order 7

1.3.1 Separable Equations 7

1.3.2 Maple Applications 11

1.3.3 Exact Equations 12

1.3.4 Integrating Factors 16

1.3.5 Maple Applications 20

1.4 Physical Applications 20

1.4.1 Simple Electrical Circuits 21

1.4.2 Maple Applications 23

1.4.3 The Rate Equation 23

1.4.4 Maple Applications 25

1.4.5 Fluid Flow 26

1.5 Linear Differential Equations 28

1.5.1 Introduction and a Fundamental Theorem 28

1.5.2 Linear Differential Operators 31

1.5.3 Wronskians and General Solutions 33

1.5.4 Maple Applications 35

1.5.5 The General Solution of the Nonhomogeneous

Equation 36

1.6 Homogeneous, Second-Order, Linear Equations

with Constant Coefficients 38

1.6.1 Maple Applications 42

1.7 Spring–Mass System: Free Motion 44

1.7.1 Undamped Motion 45

1.7.2 Damped Motion 47

1.7.3 The Electrical Circuit Analog 52

1.8 Nonhomogeneous, Second-Order, Linear Equations

with Constant Coefficients 54

1.9 Spring–Mass System: Forced Motion 59

1.9.1 Resonance 61

1.9.2 Near Resonance 62

1.9.3 Forced Oscillations with Damping 64

1.10 Variation of Parameters 69

1.11 The Cauchy–Euler Equation 72

1.12 Miscellania 75

1.12.1 Change of Dependent Variables 75

1.12.2 The Normal Form 76

1.12.3 Change of Independent Variable 79

Table 1.1 Differential Equations 82

2 Series Method 85

2.1 Introduction 85

2.2 Properties of Power Series 85

2.2.1 Maple Applications 92

2.3 Solutions of Ordinary Differential Equations 94

2.3.1 Maple Applications 98

2.3.2 Legendre’s Equation 99

2.3.3 Legendre Polynomials and Functions 101

2.3.4 Maple Applications 103

2.3.5 Hermite Polynomials 104

2.3.6 Maple Applications 105

2.4 The Method of Frobenius: Solutions About

Regular Singular Points 106

2.5 The Gamma Function 111

2.5.1 Maple Applications 115

2.6 The Bessel–Clifford Equation 116

2.7 Laguerre Polynomials 117

2.8 Roots Differing by an Integer: The Wronskian Method 118

2.9 Roots Differing by an Integer: Series Method 122

2.9.1 s = 0 124

2.9.2 s = N, N a Positive Integer 127

2.10 Bessel’s Equation 130

2.10.1 Roots Not Differing by an Integer 131

2.10.2 Maple Applications 133

2.10.3 Equal Roots 134

2.10.4 Roots Differing by an Integer 136

2.10.5 Maple Applications 137

2.10.6 Basic Identities 138

2.11 Nonhomogeneous Equations 142

2.11.1 Maple Applications 146

3 Laplace Transforms 147

3.1 Introduction 147

3.2 The Laplace Transform 147

3.2.1 Maple Applications 158

3.3 Laplace Transforms of Derivatives and Integrals 162

3.4 Derivatives and Integrals of Laplace Transforms 167

3.5 Laplace Transforms of Periodic Functions 171

3.6 Inverse Laplace Transforms: Partial Fractions 175

3.6.1 Unrepeated Linear Factor (s − a) 175

3.6.2 Maple Applications 176

3.6.3 Repeated Linear Factor (s − a)

m 177

3.6.4 Unrepeated Quadratic Factor [(s − a)

2 + b2

] 178

3.6.5 Repeated Quadratic Factor [(s − a)

2 + b2

]

m 180

3.7 A Convolution Theorem 181

3.7.1 The Error Function 183

3.8 Solution of Differential Equations 184

3.8.1 Maple Applications 192

3.9 Special Techniques 195

3.9.1 Power Series 195

Table 3.1 Laplace Transforms 198

4 The Theory of Matrices 200

4.1 Introduction 200

4.1.1 Maple Applications 200

4.2 Notation and Terminology 200

4.2.1 Maple, Excel, and MATLAB Applications 202

4.3 The Solution of Simultaneous Equations by

Gaussian Elimination 207

4.3.1 Maple and MATLAB Applications 212

4.4 Rank and the Row Reduced Echelon Form 216

4.5 The Arithmetic of Matrices 219

4.5.1 Maple, Excel, and MATLAB Applications 222

4.6 Matrix Multiplication: Definition 225

4.6.1 Maple, Excel, and MATLAB Applications 229

4.7 The Inverse of a Matrix 233

4.8 The Computation of A−1 236

4.8.1 Maple, Excel, and MATLAB Applications 240

4.9 Determinants of n × n Matrices 243

4.9.1 Minors and Cofactors 249

4.9.2 Maple and Excel Applications 251

4.9.3 The Adjoint 252

4.10 Linear Independence 254

4.10.1 Maple Applications 258

4.11 Homogeneous Systems 259

4.12 Nonhomogeneous Equations 266

5 Matrix Applications 271

5.1 Introduction 271

5.1.1 Maple and Excel Applications 271

5.2 Norms and Inner Products 271

5.2.1 Maple and MATLAB Applications 276

5.3 Orthogonal Sets and Matrices 278

5.3.1 The Gram–Schmidt Process and

the Q–R Factorization Theorem 282

5.3.2 Projection Matrices 288

5.3.3 Maple and MATLAB Applications 291

5.4 Least Squares Fit of Data 294

5.4.1 Minimizing ||Ax − b|| 299

5.4.2 Maple and Excel Applications 300

5.5 Eigenvalues and Eigenvectors 303

5.5.1 Some Theoretical Considerations 309

5.5.2 Maple and MATLAB Applications 312

5.6 Symmetric and Simple Matrices 317

5.6.1 Complex Vector Algebra 320

5.6.2 Some Theoretical Considerations 321

5.6.3 Simple Matrices 322

5.7 Systems of Linear Differential Equations:

The Homogeneous Case 326

5.7.1 Maple and MATLAB Applications 332

5.7.2 Solutions with Complex Eigenvalues 337

5.8 Systems of Linear Equations: The Nonhomogeneous Case 340

5.8.1 Special Methods 345

5.8.2 Initial-Value Problems 349

5.8.3 Maple Applications 350

6 Vector Analysis 353

6.1 Introduction 353

6.2 Vector Algebra 353

6.2.1 Definitions 353

6.2.2 Addition and Subtraction 354

6.2.3 Components of a Vector 356

6.2.4 Multiplication 358

6.2.5 Maple Applications 366

6.3 Vector Differentiation 369

6.3.1 Ordinary Differentiation 369

6.3.2 Partial Differentiation 374

6.3.3 Maple Applications 376

6.4 The Gradient 378

6.4.1 Maple and MATLAB Applications 388

6.5 Cylindrical and Spherical Coordinates 392

6.6 Integral Theorems 402

6.6.1 The Divergence Theorem 402

6.6.2 Stokes’ Theorem 408

7 Fourier Series 413

7.1 Introduction 413

7.1.1 Maple Applications 414

7.2 A Fourier Theorem 416

7.3 The Computation of the Fourier Coefficients 419

7.3.1 Kronecker’s Method 419

7.3.2 Some Expansions 421

7.3.3 Maple Applications 426

7.3.4 Even and Odd Functions 427

7.3.5 Half-Range Expansions 432

7.3.6 Sums and Scale Changes 435

7.4 Forced Oscillations 439

7.4.1 Maple Applications 441

7.5 Miscellaneous Expansion Techniques 443

7.5.1 Integration 443

7.5.2 Differentiation 447

7.5.3 Fourier Series from Power Series 450

8 Partial Differential Equations 453

8.1 Introduction 453

8.1.1 Maple Applications 454

8.2 Wave Motion 455

8.2.1 Vibration of a Stretched, Flexible String 455

8.2.2 The Vibrating Membrane 457

8.2.3 Longitudinal Vibrations of an Elastic Bar 459

8.2.4 Transmission-Line Equations 461

8.3 Diffusion 463

8.4 Gravitational Potential 467

8.5 The D’Alembert Solution of the Wave Equation 470

8.5.1 Maple Applications 474

8.6 Separation of Variables 476

8.6.1 Maple Applications 488

8.7 Solution of the Diffusion Equation 490

8.7.1 A Long, Insulated Rod with Ends at Fixed Temperatures 491

8.7.2 A Long, Totally Insulated Rod 495

8.7.3 Two-Dimensional Heat Conduction in a Long,

Rectangular Bar 498

8.8 Electric Potential About a Spherical Surface 504

8.9 Heat Transfer in a Cylindrical Body 508

8.10 The Fourier Transform 512

8.10.1 From Fourier Series to the Fourier Transform 512

8.10.2 Properties of the Fourier Transform 517

8.10.3 Parseval’s Formula and Convolutions 519

8.11 Solution Methods Using the Fourier Transform 521

9 Numerical Methods 527

9.1 Introduction 527

9.1.1 Maple Applications 528

9.2 Finite-Difference Operators 529

9.2.1 Maple Applications 533

9.3 The Differential Operator Related to the Difference Operator 535

9.3.1 Maple Applications 540

9.4 Truncation Error 541

9.5 Numerical Integration 545

9.5.1 Maple and MATLAB Applications 550

9.6 Numerical Interpolation 552

9.6.1 Maple Applications 553

9.7 Roots of Equations 555

9.7.1 Maple and MATLAB Applications 558

9.8 Initial-Value Problems—Ordinary Differential Equations 560

9.8.1 Taylor’s Method 561

9.8.2 Euler’s Method 562

9.8.3 Adams’ Method 562

9.8.4 Runge–Kutta Methods 563

9.8.5 Direct Method 566

9.8.6 Maple Applications 569

9.9 Higher-Order Equations 571

9.9.1 Maple Applications 576

9.10 Boundary-Value Problems—Ordinary

Differential Equations 578

9.10.1 Iterative Method 578

9.10.2 Superposition 578

9.10.3 Simultaneous Equations 579

9.11 Numerical Stability 582

9.12 Numerical Solution of Partial Differential

Equations 582

9.12.1 The Diffusion Equation 583

9.12.2 The Wave Equation 585

9.12.3 Laplace’s Equation 587

9.12.4 Maple and Excel Applications 590

10 Complex Variables 597

10.1 Introduction 597

10.1.1 Maple Applications 597

10.2 Complex Numbers 597

10.2.1 Maple Applications 606

10.3 Elementary Functions 608

10.3.1 Maple Applications 613

10.4 Analytic Functions 615

10.4.1 Harmonic Functions 621

10.4.2 A Technical Note 623

10.4.3 Maple Applications 624

10.5 Complex Integration 626

10.5.1 Arcs and Contours 626

10.5.2 Line Integrals 627

10.5.3 Green’s Theorem 632

10.5.4 Maple Applications 635

10.6 Cauchy’s Integral Theorem 636

10.6.1 Indefinite Integrals 637

10.6.2 Equivalent Contours 639

10.7 Cauchy’s Integral Formulas 641

10.8 Taylor Series 646

10.8.1 Maple Applications 651

10.9 Laurent Series 653

10.9.1 Maple Applications 657

10.10 Residues 658

10.10.1 Maple Applications 667

11 Wavelets 670

11.1 Introduction 670

11.2 Wavelets as Functions 670

11.3 Multiresolution Analysis 676

11.4 Daubechies Wavelets and the Cascade Algorithm 680

11.4.1 Properties of Daubechies Wavelets 680

11.4.2 Dilation Equation for Daubechies Wavelets 681

11.4.3 Cascade Algorithm to Generate D4(t) 683

11.5 Wavelets Filters 686

11.5.1 High- and Low-Pass Filtering 686

11.5.2 How Filters Arise from Wavelets 690

11.6 Haar Wavelet Functions of Two Variables 693

For Further Study 699

Appendices 700

Appendix A

Table A U.S. Engineering Units, SI Units, and Their Conversion

Factors 700

Appendix B

Table B1 Gamma Function 701

Table B2 Error Function 702

Table B3 Bessel Functions 703

Appendix C

Overview of Maple 708

Answers to Selected Problems 714

Index 731

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