CONTENTS. ABOUTTHEAUTHORS xviii PART ONE MODELING, COMPUTERS, AND ERROR ANALYSIS 3
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1 -- PREFACE xvi ABOUTTHEAUTHORS xviii PART ONE MODELING, COMPUTERS, AND ERROR ANALYSIS 3 PT1. 1 Motivation 3 PT1.2 Mathematical Background 5 PT1.3 Orientation 8 CHAPTER 1 Mathematical Modeling and Engineering Problem Solving A Simple Mathematical Model Conservation Laws and Engineering 18 Problems 21 CHAPTER 2 Programming and Software Packages and Programming Structured Programming Modular Programming Excel MATLAB Other Languages and Libraries 45 Problems 46 CHAPTER 3 Approximations and Round-Off Errors Significant Figures Accuracy and Precision Error Definitions 54 vii
2 viii 3.4 Round-OffErrors 57 Problems 72 CHAPTER 4 Truncation Errors and the Taylor Series TheTaylorSeries Error Propagation TotalNumerical Error Blunders,FormulationErrors, and Data Uncertainty 95 Problems 97 EPILOGUE: PART ONE 99 PT1.4 Trade-Offs 99 PT1.5 Important Relationshipsand Formulas 102 PT1.6 Advanced Methods and Additional ReFerences 102 PART TWO ROOTS OF EQUATIONS 105 PT2.1 Motivation 105 PT2.2 Mathematical Background 107 PT2.3 Orientation 108 CHAPTER 5 Bracketing Methods Graphical Methods The BisectionMethod The False-PositionMethod IncrementalSearchesand Determining Initial Guesses 130 Problems 131 CHAPTER 6 Open Methods Simple Fixed-PointIteration The Newton-RaphsonMethod The SecantMethod Multiple Roots Systemsof Nonlinear Equations 153 Problems 157 CHAPTER 7 Roots of Polynomials Polynomialsin Engineeringand Science Computing with Polynomials Conventional Methods Muller's Method Bairstow'sMethod Other Methods 176
3 ix 7.7 Root location with libraries and Packages 176 Problems 185 CHAPTER 8 Engineering Applications: Roots of Equations Ideal and Nonideal Gas laws (Chemical/Bio Engineering) Open-Channel Flow (Civil/Environmental Engineering) Design of an Electric Circuit (Electrical Engineering) Vibration Analysis (Mechanical/Aerospace Engineering) 196 Problems 203 EPILOGUE: PART TWO 212 PT 2.4 Trade-Offs 212 PT 2.5 Important Relationships and Formulas 213 PT 2.6 Advanced Methods and Additional References 213 PART THREE LINEAR ALGEBRAIC EQUATIONS 217 PT 3.1 Motivation 217 PT 3.2 Mathematical Background 219 PT 3.3 Orientation 227 CHAPTER 9 Gauss Elimination Solving Small Numbers of Equations Na'ive Gauss Elimination Pitfallsof Elimination Methods Techniques for Improving Solutions Complex Systems Nonlinear Systems of Equations Gauss-Jordan Summary 261 Problems 261 CHAPTER 10 LU Decomposition and Matrix Inversion LUDecomposition The Matrix Inverse Error Analysis and System Condition 277 Problems 283 CHAPTER 11 Special Matrices and Gauss-Seidel Special Matrices Gauss-Seidel 289
4 x 11.3 Linear Algebraic Equations with Libraries and Packages 296 Problems 303 CHAPTER 12 Engineering Applications: Linear Algebraic Equations Steady-State Analysis of a System of Reactors (Chemical/Bio Engineering) Analysis of a Statically Determinate Truss (Civil/Environmental Engineering) Currents and Voltages in Resistor Circuits (Electrical Engineering) Spring-Mass Systems (Mechanical/Aerospace Engineering) 314 Problems 31 7 EPILOGUE: PART THREE 327 PT3.4 Trade-Offs 327 PT3.5 Important Relationshipsand Formulas 328 PT3.6 Advanced Methods and Additional References 328 PART FOUR OPTIMIZATION 331 PT4.1 Motivation 331 PT4.2 Mathematical Background 336 PT4.3 Orientation 337 CHAPTER 13 One-Dimensional Unconstrained Optimization Golden-SectionSearch Quadratic Interpolation Newton's Method 351 Problems 353 CHAPTER 14 Multidimensional Unconstrained Optimization Direct Methods Gradient Methods 360 Problems 373 CHAPTER 15 Constrained Optimization LinearProgramming Nonlinear Constrained Optimization Optimization with Packages 387 Problems 398 CHAPTER 16 Engineering Applications: Optimization Least-CostDesignof a Tank(Chemical/Bio Engineering) Least-Cost Treatmentof Wastewater (Civil/Environmental Engineering) 405
5 xi 16.3 Maximum Power Transfer for a Circuit (Electrical Engineering) Mountain Bike Design (Mechanical! Aerospace Engineering) 413 Problems 415 EPILOGUE: PART FOUR 422 PT4.4 Trade-Offs 422 PT4.5 Additional References 423 PARTFIVE CURVEFITTING 425 PT5.1 Motivation 425 PT5.2 Mathematical Background 427 PT5.3 Orientation 436 CHAPTER 17 Least-Squares Regression Linear Regression Polynomial Regression Multiple Linear Regression General Linear Least Squares Nonlinear Regression 468 Problems 471 CHAPTER 18 Interpolation Newton's Divided-DifferenceInterpolating Polynomials Lagrange Interpolating Polynomials Coefficientsof an Interpolating Polynomial InverseInterpolation Additional Comments Spline Interpolation 495 Problems 505 CHAPTER 19 Fourier Approximation Curve Fitting with Sinusoidal Functions Continuous Fourier Series Frequency and Time Domains Fourier Integral and Transform Discrete Fourier Transform (DFT) Fast Fourier Transform (FFT) The Power Spectrum Curve Fitting with Libraries and Packages 533 Problems 542
6 xii CHAPTER20 Engineering Applications: Curve Fitting linear Regressionand PopulationModels (Chemical/BioEngineering) Use of Splinesto EstimateHeat Transfer(Civil/EnvironmentalEngineering) FourierAnalysis(ElectricalEngineering) Analysisof ExperimentalData (Mechanical/Aerospace Engineering) 551 Problems 553 EPILOGUE:PARTFIVE 563 PT5.4 Trade-Offs 563 PT5.5 ImportantRelationshipsand Formulas 564 PT5.6 Advanced Methods and AdditionalReferences 566 PART SIX NUMERICAL DIFFERENTIATION AND INTEGRATION 569 PT6.1 Motivation 569 PT6.2 Mathematical Background 578 PT6.3 Orientation 581 CHAPTER21 Newton-Cotes Integration Formulas TheTrapezoidal Rule Simpson'sRules Integrationwith UnequalSegments Open IntegrationFormulas MultipleIntegrals 608 Problems 61 0 CHAPTER22 Integration of Equations Newton-CotesAlgorithmsfor Equations RombergIntegration Gauss Quadrature ImproperIntegrals 627 Problems 631 CHAPTER23 Numerical Differentiation High-AccuracyDifferentiationFormulas RichardsonExtrapolation Derivativesof UnequallySpaced Data Derivativesand Integralsfor Data with Errors Numericallntegration/Differentiationwith libraries and Packages 639 Problems 643
7 xiii CHAPTER 24 Engineering Applications: Numerical Integration and DiHerentiation Integration to Determine the Total Quantity of Heat (Chemical/Bio Engineering) Effective Force on the Mast of a Racing Sailboat (Civil/Environmental Engineering) Root-Mean-Square Current by Numerical Integration (Electrical Engineering) Numerical Integration to Compute Work (Mechanical/Aerospace Engineering) 653 Problems 657 EPILOGUE:PARTSIX 667 PT6.4 Trade-Offs 667 PT6.5 ImportantRelationshipsand Formulas 668 PT6.6 Advanced Methods and AdditionalReferences 668 PART SEVEN ORDINARY DIFFERENTIAL EQUATIONS 671 PT7.1 Motivation 671 PT7.2 Mathematical Background 675 PT7.3 Orientation 677 CHAPTER25 Runge-Kutta Methods Euler'sMethod Improvementsof Euler'sMethod Runge-KuttaMethods Systemsof Equations Adaptive Runge-KuttaMethods 716 Problems 724 CHAPTER26 StiHness and Multistep Methods Stiffness MultistepMethods 730 Problems 750 CHAPTER27 Boundary-Value and Eigenvalue Problems General Methods for Boundary-ValueProblems EigenvalueProblems ODEsand Eigenvalueswith Librariesand Packages Problems
8 xiv CHAPTER 28 Engineering Applications: Ordinary Differential Equations Using ODEs to Analyze the Transient Response of a Reactor (Chemical/Bio Engineering) Predator-Prey Models and Chaos (Civil/Environmental Engineering) Simulating Transient Current for an Electric Circuit (Electrical Engineering) The Swinging Pendulum (Mechanical/Aerospace Engineering) 797 Problems 801 EPILOGUE: PART SEVEN 808 PT7.4 Trade-Offs 808 PT7.5 Important Relationshipsand Formulas 809 PT7.6 Advanced Methods and Additional References 809 PART EIGHT PARTIAL DIFFERENTIAL EQUATIONS 813 PT8.1 Motivation 813 PT8.2 Orientation 816 CHAPTER 29 Finite Difference: Elliptic Equations The Laplace Equation SolutionTechniques Boundary Conditions The Control-VolumeApproach Software to Solve Elliptic Equations 837 Problems 838 CHAPTER 30 Finite Difference: Parabolic Equations The Heat Conduction Equation Explicit Methods A Simple Implicit Method The Crank-NicolsonMethod Parabolic Equationsin Two Spatial Dimensions 852 Problems 855 CHAPTER 31 Finite-Element Method TheGeneral Approach Finite-ElementApplication in One Dimension Two-DimensionalProblems Solving PDEswith Librariesand Packages 875 Problems 881
9 xv CHAPTER32 Engineering Applications: Partial DiHerential Equations One-DimensionalMass Balance of a Reactor(Chemical/Bio Engineering) Deflectionsof a Plate (Civil/EnvironmentalEngineering) Two-DimensionalElectrostaticFieldProblems(ElectricalEngineering) Finite-ElementSolutionof a Series of Springs (Mechanical/Aerospace Engineering) 893 Problems 797 EPILOGUE: PART EIGHT 899 PT 8.3 Trade-Offs 899 PT 8.4 Important Relationships and Formulas 899 PT 8.5 Advanced Methods and Additional References 900 APPENDIX A: THE FOURIER SERIES 901 APPENDIX B: GETTING STARTEDWITH MATLAB 903 BIBLIOGRAPHY 911 INDEX 915
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