Applied Mathematics for Science and Engineering

Larry A. Glasgow is Professor of Chemical Engineering at Kansas State University. He has taught many of the core courses in chemical engineering with particular emphasis upon transport phenomena, engineering mathematics, and process analysis. Dr. Glasgow's work in the classroom and his enthusiasm for teaching have been recognized many times with teaching awards. Glasgow is also the author of Transport Phenomena: An Introduction to Advanced Topics (Wiley, 2010).

• Preface viii1 Problem Formulation and Model Development 1Introduction 1Algebraic Equations from Vapor–Liquid Equilibria (VLE) 3Macroscopic Balances: Lumped-Parameter Models 4Force Balances: Newton’s Second Law of Motion 6Distributed Parameter Models: Microscopic Balances 6Using the Equations of Change Directly 8A Contrast: Deterministic Models and Stochastic Processes 10Empiricisms and Data Interpretation 10Conclusion 12Problems 13References 142 Algebraic Equations 15Introduction 15Elementary Methods 16Newton–Raphson (Newton’s Method of Tangents) 16Regula Falsi (False Position Method) 18Dichotomous Search 19Golden Section Search 20Simultaneous Linear Algebraic Equations 20Crout’s (or Cholesky’s) Method 21Matrix Inversion 23Iterative Methods of Solution 23Simultaneous Nonlinear Algebraic Equations 24Pattern Search for Solution of Nonlinear Algebraic Equations 26Sequential Simplex and the Rosenbrock Method 26An Example of a Pattern Search Application 28Algebraic Equations with Constraints 28Conclusion 29Problems 30References 323 Vectors and Tensors 34Introduction 34Manipulation of Vectors 35Force Equilibrium 37Equating Moments 37Projectile Motion 38Dot and Cross Products 39Differentiation of Vectors 40Gradient Divergence and Curl 40Green’s Theorem 42Stokes’ Theorem 43Conclusion 44Problems 44References 464 Numerical Quadrature 47Introduction 47Trapezoid Rule 47Simpson’s Rule 48Newton–Cotes Formulae 49Roundoff and Truncation Errors 50Romberg Integration 51Adaptive Integration Schemes 52Simpson’s Rule 52Gaussian Quadrature and the Gauss–Kronrod Procedure 53Integrating Discrete Data 55Multiple Integrals (Cubature) 57Monte Carlo Methods 59Conclusion 60Problems 62References 645 Analytic Solution of Ordinary Differential Equations 65An Introductory Example 65First-Order Ordinary Differential Equations 66Nonlinear First-Order Ordinary Differential Equations 67Solutions with Elliptic Integrals and Elliptic Functions 69Higher-Order Linear ODEs with Constant Coefficients 71Use of the Laplace Transform for Solution of ODEs 73Higher-Order Equations with Variable Coefficients 75Bessel’s Equation and Bessel Functions 76Power Series Solutions of Ordinary Differential Equations 78Regular Perturbation 80Linearization 81Conclusion 83Problems 84References 886 Numerical Solution of Ordinary Differential Equations 89An Illustrative Example 89The Euler Method 90Modified Euler Method 91Runge–Kutta Methods 91Simultaneous Ordinary Differential Equations 94Some Potential Difficulties Illustrated 94Limitations of Fixed Step-Size Algorithms 95Richardson Extrapolation 97Multistep Methods 98Split Boundary Conditions 98Finite-Difference Methods 100Stiff Differential Equations 100Backward Differentiation Formula (BDF) Methods 101Bulirsch–Stoer Method 102Phase Space 103Summary 105Problems 106References 1097 Analytic Solution of Partial Differential Equations 111Introduction 111Classification of Partial Differential Equations and Boundary Conditions 111Fourier Series 112A Preview of the Utility of Fourier Series 114The Product Method (Separation of Variables) 116Parabolic Equations 116Elliptic Equations 122Application to Hyperbolic Equations 127The Schrödinger Equation 128Applications of the Laplace Transform 131Approximate Solution Techniques 133Galerkin MWR Applied to a PDE 134The Rayleigh–Ritz Method 135Collocation 137Orthogonal Collocation for Partial Differential Equations 138The Cauchy–Riemann Equations Conformal Mapping and Solutions for the Laplace Equation 139Conclusion 142Problems 143References 1468 Numerical Solution of Partial Differential Equations 147Introduction 147Finite-Difference Approximations for Derivatives 148Boundaries with Specified Flux 149Elliptic Partial Differential Equations 149An Iterative Numerical Procedure: Gauss–Seidel 151Improving the Rate of Convergence with Successive Over-Relaxation (SOR) 152Parabolic Partial Differential Equations 154An Elementary Explicit Numerical Procedure 154The Crank–Nicolson Method 155Alternating-Direction Implicit (ADI) Method 157Three Spatial Dimensions 158Hyperbolic Partial Differential Equations 158The Method of Characteristics 160The Leapfrog Method 161Elementary Problems with Convective Transport 162A Numerical Procedure for Two-Dimensional Viscous Flow Problems 165MacCormack’s Method 170Adaptive Grids 171Conclusion 173Problems 176References 1839 Integro-Differential Equations 184Introduction 184An Example of Three-Mode Control 185Population Problems with Hereditary Infl uences 186An Elementary Solution Strategy 187VIM: The Variational Iteration Method 188Integro-Differential Equations and the Spread of Infectious Disease 192Examples Drawn from Population Balances 194Particle Size in Coagulating Systems 198Application of the Population Balance to a Continuous Crystallizer 199Conclusion 201Problems 201References 20410 Time-Series Data and the Fourier Transform 206Introduction 206A Nineteenth-Century Idea 207The Autocorrelation Coeffi cient 208A Fourier Transform Pair 209The Fast Fourier Transform 210Aliasing and Leakage 213Smoothing Data by Filtering 216Modulation (Beats) 218Some Familiar Examples 219Turbulent Flow in a Deflected Air Jet 219Bubbles and the Gas–Liquid Interface 220Shock and Vibration Events in Transportation 222Conclusion and Some Final Thoughts 223Problems 224References 22711 An Introduction to the Calculus of Variations and the Finite-Element Method 229Some Preliminaries 229Notation for the Calculus of Variations 230Brachistochrone Problem 231Other Examples 232Minimum Surface Area 232Systems of Particles 232Vibrating String 233Laplace’s Equation 234Boundary-Value Problems 234A Contemporary COV Analysis of an Old Structural Problem 236Flexing of a Rod of Small Cross Section 236The Optimal Column Shape 237Systems with Surface Tension 238The Connection between COV and the Finite-Element Method (FEM) 238Conclusion 241Problems 242References 243Index 245