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CONTENTS3

PART ONE MODELING,COMPUTERS,AND ERROR ANALYSIS3

PT 1.1 Motivation3

PT 1.2 Mathematical Background5

PT 1.3 Orientation8

CHAPTER 1 Mathematical Modeling and Engineering Problem-Solving11

1.1 A Simple Mathematical Model11

1.2 Conservation Laws and Engineering18

Problems21

CHAPTER 2 Computers and Software24

2.1 Computing Environments25

2.2 The Software Development Process26

2.3 Algorithm Design30

2.4 Program Composition40

2.5 Quality Control43

2.6 Documentation and Maintenance47

2.7 Software Strategy48

Problems53

CHAPTER 3 Approximations and Round-Off Errors56

3.1 Significant Figures57

3.2 Accuracy and Precision59

3.3 Error Definitions60

3.4 Round-Off Errors63

Problems78

CHAPTER 4 Truncation Errors and the Taylor Series79

4.1 The Taylor Series79

4.2 Error Propagation95

4.3 Total Numerical Error99

4.4 Blunders,Formulation Errors,and Data Uncertainty101

Problems103

EpILOGUE:PART ONE105

PT 1.4 Trade-Offs105

PT 1.6 Advanced Methods and Additional References108

PT 1.5 Important Relationships and Formulas108

PART TWO ROOTS OF EQUATIONS111

PT 2.1 Motivation111

PT 2.2 Mathematical Background113

PT 2.3 Orientation114

CHAPTER 5 Bracketing Methods118

5.1 Graphical Methods118

5.2 The Bisection Method122

5.3 The False-Position Method132

5.4 Incremental Searches and Determining Initial Guesses137

Problems138

CHAPTER 6 Open Methods141

6.1 Simple Fixed-Point Iteration142

6.2 The Newton-Raphson Method147

6.3 The Secant Method153

6.4 Multiple Roots158

6.5 Systems of Nonlinear Equations161

Problems165

CHAPTER 7 Roots of Polynomials167

7.1 Polynomials in Engineering and Science167

7.2 Computing with Polynomials170

7.3 Conventional Methods173

7.4 Müeller's Method174

7.5 Bairstow's Method178

7.6 Other Methods183

7.7 Root Location with Libraries and Packages183

Problems193

CHAPTER 8 Engineering Applications:Roots of Equations194

8.1 Ideal and Nonideal Gas Laws(Chemicol/Petroleum Engineering)194

8.2 Open-Channel Flow(Civil/Environmental Engineering)197

8.3 Design of an Electric Circuit(Electrical Engineering)201

8.4 Vibration Analysis(Mechanical/Aerospace Engineering)203

Problems210

PT 2.4 Trade-Offs215

EPILOGUE:PART TWO215

PT 2.5 Important Relationships and Formulas216

PT 2.6 Advanced Methods and Additional References216

PART THREE LINEAR ALGEBRAIC EQUATIONS219

PT 3.1 Motivation219

PT 3.2 Mathematical Background221

PT 3.3 Orientation229

CHAPTER 9 Gauss Elimination233

9.1 Solving Small Numbers of Equations233

9.2 Naive Gauss Elimination240

9.3 Pitfalls of Elimination Methods246

9.4 Techniques for Improving Solutions252

9.5 Complex Systems259

9.6 Nonlinear Systems of Equations259

9.7 Gauss-Jordan261

Problems263

9.8 Summary263

CHAPTER 10 LU Decomposition and Matrix Inversion266

10.1 LU Decomposition266

10.2 The Matrix Inverse275

10.3 Error Analysis and System Condition279

Problems287

CHAPTER 11 Special Matrices and Gauss-Seidel288

11.1 Special Matrices288

11.2 Gauss-Seidel292

11.3 Linear Algebraic Equations with Libraries and Packages299

Problems307

CHAPTER 12 Engineering Applications:Linear Algebraic Equations309

12.1 Steady-State Analysis of a System of Reactors (Chemical/Petroleum Engineering)309

12.2 Analysis of a Statically Determinate Truss(Civil/Environmental Engineering)312

12.3 Currents and Voltages in Resistor Circuits(Electrical Engineering)316

12.4 Spring-Mass Systems (Mechanical/Aerospace Engineering)318

Problems321

PT 3.4 Trade-Offs327

EPlLOGUE:PART THREE327

PT 3.5 Important Relationships and Formulas328

PT 3.6 Advanced Methods and Additional References328

PART FOUR OPTIMIZATION331

PT 4.1 Motivation331

PT 4.2 Mathematical Background336

PT 4.3 Orientation337

CHAPTER 13 One-dimensional Unconstrained Optimization341

13.1 Golden-Section Search342

13.2 Quadratic Interpolation349

13.3 Newton's Method351

Problems353

CHAPTER 14 Multidimensional Unconstrained Optimization355

14.1 Direct Methods356

14.2 Gradient Methods360

Problems373

15.1 Linear Programming375

CHAPTER 15 Constrained Optimization375

15.2 Nonlinear Constrained Optimization386

15.3 Optimization with Packages387

Problems397

CHAPTER 16 Engineering Applications:Optimization399

16.1 Least-Cost Design of a Tank(Chemical/Petroleum Engineering)399

16.2 Least-Cost Treatment of Wastewater(Civil/Environmental Engineering)403

16.3 Maximum Power Transfer for a Circuit(Electrical Engineering)408

16.4 Mountain Bike Design (Mechanical/Aerospace Engineering)412

Problems414

EPILOGUE:PART FOUR420

PT 4.4 Trade-Offs420

PT 4.5 Additional References421

PART FIVE CURVE FITTING423

PT 5.1 Motivation423

PT 5.2 Mathematical Background425

PT 5.3 Orientation434

17.1 Linear Regression438

CHAPTER 17 Least-Squares Regression438

17.2 Polynomial Regression454

17.3 Multiple Linear Regression460

17.4 General Linear Least-Squares463

17.5 Nonlinear Regression468

Problems471

CHAPTER 18 Interpolation473

18.1 Newton's Divided-Difference Interpolating Polynomials474

18.2 Logrange Interpolating Polynomials485

18.3 Coefficients of an Interpolating Polynomial490

18.4 Inverse Interpolation490

18.5 Additional Comments491

18.6 Spline Interpolation494

Problems505

CHAPTER 19 Fourier Approximation507

19.1 Curve Fitting with Sinusoidal Functions508

19.2 Continuous Fourier Series514

19.3 Frequency and Time Domains517

19.4 Fourier Integral and Transform521

19.5 Discrete Fourier Transform (DFT)523

19.6 Fast Fourier Transform (FFT)525

19.7 The Power Spectrum532

19.8 Curve Fitting with Libraries and Packages533

Problems544

CHAPTER 20 Engineering Applications:Curve Fitting546

20.1 Linear Regression and Population Models(Chemical/Petroleum Engineering)546

20.2 Use of Splines to Estimate Heat Transfer (Civil/Environmental Engineering)550

20.3 Fourier Analysis(Electrical Engineering)552

20.4 Analysis of Experimental Data(Mechonical/Aerospace Engineering)553

Problems555

EPILOGUE:PART FIVE561

PT 5.4 Trade-Offs561

PT 5.5 Important Relationships and Formulas562

PT 5.6 Advanced Methods and Additional References564

PT 6.1 Motivation567

PART SIX NUMERICAL DIFFERENTIATION AND INTEGRATION567

PT 6.2 Mathematical Background576

PT 6.3 Orientation579

CHAPTER 21 Newton-Cotes Integration of Equations582

21.1 The Trapezoidal Rule584

21.2 Simpson's Rules595

21.3 Integration with Unequal Segments604

21.4 Open Integration Formulas607

Problems607

CHAPTER 22 Integration of Equations610

22.1 Newton-Cotes Algorithms for Equations610

22.2 Romberg Integration612

22.3 Gauss Quadrature617

22.4 Improper Integrals624

Problems628

23.1 High-Accuracy Differentiation Formulas629

CHAPTER 23 Numerical Differentiation629

23.2 Richardson Extrapolation632

23.3 Derivatives of Unequally Spaced Data634

23.4 Derivatives and Integrals for Data with Errors635

23.5 Numerical Integration/Differentiation Formulas with Libraries and Packages636

Problems641

CHAPTER 24 Engineering Applications:Numerical Integration and Differentiation643

24.1 Integration to Determine the Total Quantity of Heat (Chemical/Petroleum Engineering)643

24.2 Effective Force on the Mast of a Racing Sailboat (Civil/Environmental Engineering)645

24.3 Root-Mean-Square Current by Numerical Integration(Electrical Engineering)647

24.4 Numerical Integration to Compute Work (Mechanical/Aerospace Engineering)650

Problems654

EPILOGUE:PART SIX661

PT 6.4 Trade-Offs661

PT 6.5 Important Relationships and Formulas662

PT 6.6 Advanced Methods and Additional References662

PT 7.1 Motivation665

PART SEVEN ORDINARY DIFFERENTIAL EQUATIONS665

PT 7.2 Mathematical Background669

PT 7.3 Orientation671

CHAPTER 25 Runge-Kutta Methods675

25.1 Euler's Method676

25.2 Improvements of Euler's Method687

25.3 Runge-Kutta Method695

25.4 Systems of Equations705

25.5 Adaptive Runge-Kutta Method710

Problems718

CHAPTER 26 Stiffness and Multistep Methods719

26.1 Stiffness719

26.2 Multistep Methods723

Problems743

CHAPTER 27 Boundary-Value and Eigenvalue Problems745

27.1 General Methods of Boundary-Value Problems746

27.2 Eigenvalue Problems752

27.3 ODEs and Eigenvalues with Libraries and Packages768

Problems775

CHAPTER 28 Engineering Applications:Ordinary Differential Equations 　777

28.1 Using ODEs to Analyze the Transient Response of a Reactor(Chemical/Petroleum Engineering)777

28.2 Predator-Prey Models and Chaos(Civil/Environmental Engineering)784

28.3 Simulating Transient Current for an Electronic Circuit(Electrical Engineering)788

28.4 The Swinging Pendulum(Mechanicol/Aerospace Engineering)793

Problems796

EPILOGUE:PART SEVEN800

PT 7.4 Trade-Offs800

PT 7.5 Important Relationships and Formulas801

PT 7.6 Advanced Methods and Additional References801

PART EIGHT PARTIAL DIFFERENTIAL EQUATIONS805

PT 8.1 Motivation805

PT 8.2 Orientation808

29.1 The Laplace Equation812

CHAPTER 29 Finite Difference:Elliptic Equations812

29.2 Solution Techniques814

29.3 Boundary Conditions820

29.4 The Control Volume Approach826

29.5 Software to Solve Elliptic Equations829

Problems830

CHAPTER 30 Finite Difference:Parabolic Equations832

30.1 The Heat Conduction Equation832

30.2 Explicit Methods833

30.3 A Simple Implicit Method837

30.4 The Crank-Nicholson Method841

30.5 Parabolic Equations in Two Spatial Dimenstons844

Problems848

CHAPTER 31 Finite Element Method849

31.1 The General Approach850

31.2 Finite-Element Application in One Dimension854

31.3 Two-Dimensional Problems863

31.4 PDEs with Libraries and Packages867

Problems874

CHAPTER 32 Engineering Applications:Partial Differential Equations876

32.1 One-Dimensional Mass Balance of a Reactor (Chemical/Petroleum Engineering)876

32.2 Deflections of a Plate(Civil/Environmental Engineering)880

32.3 Two-Dimensional Electrostatic Field Problems(Electrical Engineering)882

32.4 Finite-Element Solution of a Series of Springs(Mechanical/Aerospace Engineering)885

Problems889

PT 8.4 Important Relationships and Formulas891

EPILOGUE:PART EIGHT891

PT 8.3 Trade-Offs891

PT 8.5 Advanced Methods and Additional References892

APPENDIX A:THE FOURIER SERIES893

APPENDIX B:GETTING STARTED WITH MATHCAD895

APPENDIX C:GETTING STARTED WITH MATLAB905

BIBLIOGRAPHY913

INDEX916