Description
Book SynopsisBased on a decade s worth of the authors lecture notes detailing the topic of applied mathematics for scientists and engineers. The flow of the chapters is smooth and moves from one mathematical method to the next sustaining reader interest and easing the application of the techniques.
Table of ContentsPreface 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
