Applied Mathematics for Science and Engineering

Preface viii 1 Problem Formulation and Model Development 1 Introduction 1 Algebraic Equations from Vapor–Liquid Equilibria (VLE) 3 Macroscopic Balances: Lumped-Parameter Models 4 Force Balances: Newton’s Second Law of Motion 6 Distributed Parameter Models: Microscopic Balances 6 Using the Equations of Change Directly 8 A Contrast: Deterministic Models and Stochastic Processes 10 Empiricisms and Data Interpretation 10 Conclusion 12 Problems 13 References 14 2 Algebraic Equations 15 Introduction 15 Elementary Methods 16 Newton–Raphson (Newton’s Method of Tangents) 16 Regula Falsi (False Position Method) 18 Dichotomous Search 19 Golden Section Search 20 Simultaneous Linear Algebraic Equations 20 Crout’s (or Cholesky’s) Method 21 Matrix Inversion 23 Iterative Methods of Solution 23 Simultaneous Nonlinear Algebraic Equations 24 Pattern Search for Solution of Nonlinear Algebraic Equations 26 Sequential Simplex and the Rosenbrock Method 26 An Example of a Pattern Search Application 28 Algebraic Equations with Constraints 28 Conclusion 29 Problems 30 References 32 3 Vectors and Tensors 34 Introduction 34 Manipulation of Vectors 35 Force Equilibrium 37 Equating Moments 37 Projectile Motion 38 Dot and Cross Products 39 Differentiation of Vectors 40 Gradient Divergence and Curl 40 Green’s Theorem 42 Stokes’ Theorem 43 Conclusion 44 Problems 44 References 46 4 Numerical Quadrature 47 Introduction 47 Trapezoid Rule 47 Simpson’s Rule 48 Newton–Cotes Formulae 49 Roundoff and Truncation Errors 50 Romberg Integration 51 Adaptive Integration Schemes 52 Simpson’s Rule 52 Gaussian Quadrature and the Gauss–Kronrod Procedure 53 Integrating Discrete Data 55 Multiple Integrals (Cubature) 57 Monte Carlo Methods 59 Conclusion 60 Problems 62 References 64 5 Analytic Solution of Ordinary Differential Equations 65 An Introductory Example 65 First-Order Ordinary Differential Equations 66 Nonlinear First-Order Ordinary Differential Equations 67 Solutions with Elliptic Integrals and Elliptic Functions 69 Higher-Order Linear ODEs with Constant Coefficients 71 Use of the Laplace Transform for Solution of ODEs 73 Higher-Order Equations with Variable Coefficients 75 Bessel’s Equation and Bessel Functions 76 Power Series Solutions of Ordinary Differential Equations 78 Regular Perturbation 80 Linearization 81 Conclusion 83 Problems 84 References 88 6 Numerical Solution of Ordinary Differential Equations 89 An Illustrative Example 89 The Euler Method 90 Modified Euler Method 91 Runge–Kutta Methods 91 Simultaneous Ordinary Differential Equations 94 Some Potential Difficulties Illustrated 94 Limitations of Fixed Step-Size Algorithms 95 Richardson Extrapolation 97 Multistep Methods 98 Split Boundary Conditions 98 Finite-Difference Methods 100 Stiff Differential Equations 100 Backward Differentiation Formula (BDF) Methods 101 Bulirsch–Stoer Method 102 Phase Space 103 Summary 105 Problems 106 References 109 7 Analytic Solution of Partial Differential Equations 111 Introduction 111 Classification of Partial Differential Equations and Boundary Conditions 111 Fourier Series 112 A Preview of the Utility of Fourier Series 114 The Product Method (Separation of Variables) 116 Parabolic Equations 116 Elliptic Equations 122 Application to Hyperbolic Equations 127 The Schrödinger Equation 128 Applications of the Laplace Transform 131 Approximate Solution Techniques 133 Galerkin MWR Applied to a PDE 134 The Rayleigh–Ritz Method 135 Collocation 137 Orthogonal Collocation for Partial Differential Equations 138 The Cauchy–Riemann Equations Conformal Mapping and Solutions for the Laplace Equation 139 Conclusion 142 Problems 143 References 146 8 Numerical Solution of Partial Differential Equations 147 Introduction 147 Finite-Difference Approximations for Derivatives 148 Boundaries with Specified Flux 149 Elliptic Partial Differential Equations 149 An Iterative Numerical Procedure: Gauss–Seidel 151 Improving the Rate of Convergence with Successive Over-Relaxation (SOR) 152 Parabolic Partial Differential Equations 154 An Elementary Explicit Numerical Procedure 154 The Crank–Nicolson Method 155 Alternating-Direction Implicit (ADI) Method 157 Three Spatial Dimensions 158 Hyperbolic Partial Differential Equations 158 The Method of Characteristics 160 The Leapfrog Method 161 Elementary Problems with Convective Transport 162 A Numerical Procedure for Two-Dimensional Viscous Flow Problems 165 MacCormack’s Method 170 Adaptive Grids 171 Conclusion 173 Problems 176 References 183 9 Integro-Differential Equations 184 Introduction 184 An Example of Three-Mode Control 185 Population Problems with Hereditary Infl uences 186 An Elementary Solution Strategy 187 VIM: The Variational Iteration Method 188 Integro-Differential Equations and the Spread of Infectious Disease 192 Examples Drawn from Population Balances 194 Particle Size in Coagulating Systems 198 Application of the Population Balance to a Continuous Crystallizer 199 Conclusion 201 Problems 201 References 204 10 Time-Series Data and the Fourier Transform 206 Introduction 206 A Nineteenth-Century Idea 207 The Autocorrelation Coeffi cient 208 A Fourier Transform Pair 209 The Fast Fourier Transform 210 Aliasing and Leakage 213 Smoothing Data by Filtering 216 Modulation (Beats) 218 Some Familiar Examples 219 Turbulent Flow in a Deflected Air Jet 219 Bubbles and the Gas–Liquid Interface 220 Shock and Vibration Events in Transportation 222 Conclusion and Some Final Thoughts 223 Problems 224 References 227 11 An Introduction to the Calculus of Variations and the Finite-Element Method 229 Some Preliminaries 229 Notation for the Calculus of Variations 230 Brachistochrone Problem 231 Other Examples 232 Minimum Surface Area 232 Systems of Particles 232 Vibrating String 233 Laplace’s Equation 234 Boundary-Value Problems 234 A Contemporary COV Analysis of an Old Structural Problem 236 Flexing of a Rod of Small Cross Section 236 The Optimal Column Shape 237 Systems with Surface Tension 238 The Connection between COV and the Finite-Element Method (FEM) 238 Conclusion 241 Problems 242 References 243 Index 245