NUMERICAL METHODS FOR ENGINEERS AND SCIENTISTS AN INTRODUCTION WITH APPLICATIONS USING MATLABPDF电子书下载

Chapter 1 Introduction 1

1.1 Background 1

1.2 Representation of Numbers on a Computer 4

1.3 Errors in Numerical Solutions 10

1.3.1 Round-Off Errors 10

1.3.2 Truncation Errors 13

1.3.3 Total Error 14

1.4 Computers and Programming 15

1.5 Problems 18

Chapter 2 Mathematical Background 23

2.1 Background 23

2.2 Concepts from Pre-Calculus and Calculus 24

2.3 Vectors 28

2.3.1 Operations with Vectors 30

2.4 Matrices and Linear Algebra 32

2.4.1 Operations with Matrices 33

2.4.2 Special Matrices 35

2.4.3 Inverse of a Matrix 36

2.4.4 Properties of Matrices 37

2.4.5 Determinant of a Matrix 37

2.4.6 Cramers Rule and Solution of a System of Simultaneous Linear Equations 38

2.4.7 Norms 40

2.5 Ordinary Differential Equations （ODE） 41

2.6 Functions of Two or More Independent Variables 44

2.6.1 Definition of the Partial Derivative 44

2.6.2 Chain Rule 45

2.6.3 The Jacobian 46

2.7 Taylor Series Expansion of Functions 47

2.7.1 Taylor Series for a Function of One Variable 47

2.7.2 Taylor Series for a Function of Two Variables 49

2.8 Inner Product and Orthogonality 50

2.9 Problems 51

7.5 The Discrete Fourier Series and Discrete Fourier Transform 263

7.6 Complex Discrete Fourier Transform 268

7.7 Power （Energy） Spectrum 271

7.8 Aliasing and Nyquist Frequency 272

7.9 Alternative Forms of the Discrete Fourier Transform 278

7.10 Use of MATLAB Built-In Functions for Calculating Discrete Fourier Transform 278

7.11 Leakage and Windowing 284

7.12 Bandwidth and Filters 286

7.13 The Fast Fourier Transform （FFT） 289

7.14 Problems 298

Chapter 8 Numerical Differentiation 303

8.1 Background 303

8.2 Finite Difference Approximation of the Derivative 305

8.3 Finite Difference Formulas Using Taylor Series Expansion 310

8.3.1 Finite Difference Formulas of First Derivative 310

8.3.2 Finite Difference Formulas for the Second Derivative 315

8.4 Summary of Finite Difference Formulas for Numerical Differentiation 317

8.5 Differentiation Formulas Using Lagrange Polynomials 319

8.6 Differentiation Using Curve Fitting 320

8.7 Use of MATLAB Built-In Functions for Numerical Differentiation 320

8.8 Richardson’s Extrapolation 322

8.9 Error in Numerical Differentiation 325

8.10 Numerical Partial Differentiation 327

8.11 Problems 330

Chapter 9 Numerical Integration 341

9.1 Background 341

9.1.1 Overview ofApproaches in Numerical Integration 342

9.2 Rectangle and Midpoint Methods 344

9.3 Trapezoidal Method 346

9.3.1 Composite Trapezoidal Method 347

9.4 Simpson’s Methods 350

9.4.1 Simpsons 1/3 Method 350

9.4.2 Simpsons 3/8 Method 353

9.5 Gauss Quadrature 355

9.6 Evaluation of Multiple Integrals 360

9.7 Use of MATLAB Built-In Functions for Integration 362

9.8 Estimation of Error in Numerical Integration 364

9.9 Richardson’s Extrapolation 366

9.10 Romberg Integration 369

9.11 Improper Integrals 372

9.11.1 Integrals with Singularities 372

9.11.2 Integrals with Unbounded Limits 373

9.12 Problems 374

Chapter 10 Ordinary Differential Equations：Initial-Value Problems 385

10.1 Background 385

10.2 Euler’s Methods 390

10.2.1 Euler’s Explicit Method 390

10.2.2 Analysis of Truncation Error in Euler’s Explicit Method 394

10.2.3 Eulers Implicit Method 398

10.3 Modified Euler’s Method 401

10.4 Midpoint Method 404

10.5 Runge-Kutta Methods 405

10.5.1 Second-Order Runge-Kutta Methods 406

10.5.2 Third-Order Runge-Kutta Methods 410

10.5.3 Fourth-Order Runge-Kutta Methods 411

10.6 Multistep Methods 417

10.6.1 Adams-Bashforth Method 418

10.6.2 Adams-Moulton Method 419

10.7 Predictor-Corrector Methods 420

10.8 System of First-Order Ordinary Differential Equations 422

10.8.1 Solving a System ofFirst-Order ODEs Using Eulers Explicit Method 424

10.8.2 Solving a System of First-Order ODEs Using Second-Order Runge-Kutta Method （Modified Euler Version） 424

10.8.3 Solving a System ofFirst-Order ODEs Using the Classical Fourth-Order Runge-Kutta Method 431

10.9 Solving a Higher-Order Initial Value Problem 432

10.10 Use of MATLAB Built-In Functions for Solving Initial-Value Problems 437

10.10.1 Solving a Single First-Order ODE Using MATLAB 438

10.10.2 Solving a System of First-Order ODEs Using MATLAB 444

10.11 Local Truncation Error in Second-Order Range-Kutta Method 447

10.12 Step Size for Desired Accuracy 448

10.13 Stability 452

10.14 Stiff Ordinary Differential Equations 454

10.15 Problems 457

Chapter 11 Ordinary Differential Equations：Boundary-Value Problems 471

11.1 Background 471

11.2 The Shooting Method 474

11.3 Finite Difference Method 482

11.4 Use of MATLAB Built-In Functions for Solving Boundary Value Problems 492

11.5 Error and Stability in Numerical Solution of Boundary Value Problems 497

11.6 Problems 499

Appendix A Introductory MA TLAB 509

A.1 Background 509

A.2 Starting with MATLAB 509

A.3 Arrays 514

A.4 Mathematical Operations with Arrays 519

A.5 Script Files 524

A.6 Plotting 526

A.7 User-Defined Functions and Function Files 528

A.8 Anonymous Functions 530

A.9 Function functions 532

A.10 Subfunctions 535

A.11 Programming in MATLAB 537

A.11.1 Relational and Logical Operators 537

A.11.2 Conditional Statements，if-else Structures 538

A.11.3 Loops 541

A.12 Problems 542

Appendix B MATLAB Programs 547

Appendix C Derivation of the Real Discrete Fourier Transform（DFT） 551

C.1 Orthogonality of Sines and Cosines for Discrete Points 551

C.2 Determination of the Real DFT 553

Index 555