Applied Mathematics for Science and Engineering

Table of contents

1. Title page
2. Copyright page
3. Preface
4. 1: Problem Formulation and Model Development
   1. Introduction
   2. Algebraic Equations from Vapor–Liquid Equilibria (VLE)
   3. Macroscopic Balances: Lumped-Parameter Models
   4. Force Balances: Newton's Second Law of Motion
   5. Distributed Parameter Models: Microscopic Balances
   6. A Contrast: Deterministic Models and Stochastic Processes
   7. Empiricisms and Data Interpretation
   8. Conclusion
   9. Problems
   10. References
5. 2: Algebraic Equations
   1. Introduction
   2. Elementary Methods
   3. Simultaneous Linear Algebraic Equations
   4. Simultaneous Nonlinear Algebraic Equations
   5. Algebraic Equations with Constraints
   6. Conclusion
   7. Problems
   8. References
6. 3: Vectors and Tensors
   1. Introduction
   2. Manipulation of Vectors
   3. Green's Theorem
   4. Stokes' Theorem
   5. Conclusion
   6. Problems
   7. References
7. 4: Numerical Quadrature
   1. Introduction
   2. Trapezoid Rule
   3. Simpson's Rule
   4. Newton–Cotes Formulae
   5. Roundoff and Truncation Errors
   6. Romberg Integration
   7. Adaptive Integration Schemes
   8. Integrating Discrete Data
   9. Multiple Integrals (Cubature)
   10. Conclusion
   11. Problems
   12. References
8. 5: Analytic Solution of Ordinary Differential Equations
   1. An Introductory Example
   2. First-Order Ordinary Differential Equations
   3. Nonlinear First-Order Ordinary Differential Equations
   4. Higher-Order Linear ODEs with Constant Coefficients
   5. Higher-Order Equations with Variable Coefficients
   6. Bessel's Equation and Bessel Functions
   7. Power Series Solutions of Ordinary Differential Equations
   8. Regular Perturbation
   9. Linearization
   10. Conclusion
   11. Problems
   12. References
9. 6: Numerical Solution of Ordinary Differential Equations
   1. An Illustrative Example
   2. The Euler Method
   3. Runge–Kutta Methods
   4. Simultaneous Ordinary Differential Equations
   5. Limitations of Fixed Step-Size Algorithms
   6. Richardson Extrapolation
   7. Multistep Methods
   8. Split Boundary Conditions
   9. Finite-Difference Methods
   10. Stiff Differential Equations
   11. Bulirsch–Stoer Method
   12. Phase Space
   13. Summary
   14. Problems
   15. References
10. 7: Analytic Solution of Partial Differential Equations
    1. Introduction
    2. Classification of Partial Differential Equations and Boundary Conditions
    3. Fourier Series
    4. The Product Method (Separation of Variables)
    5. Applications of the Laplace Transform
    6. Approximate Solution Techniques
    7. The Cauchy–Riemann Equations, Conformal Mapping, and Solutions for the Laplace Equation
    8. Conclusion
    9. Problems
    10. References
11. 8: Numerical Solution of Partial Differential Equations
    1. Introduction
    2. Elliptic Partial Differential Equations
    3. Parabolic Partial Differential Equations
    4. Hyperbolic Partial Differential Equations
    5. Elementary Problems with Convective Transport
    6. A Numerical Procedure for Two-Dimensional Viscous Flow Problems
    7. MacCormack's Method
    8. Adaptive Grids
    9. Conclusion
    10. Problems
    11. References
12. 9: Integro-Differential Equations
    1. Introduction
    2. An Example of Three-Mode Control
    3. Population Problems with Hereditary Influences
    4. An Elementary Solution Strategy
    5. VIM: The Variational Iteration Method
    6. Integro-Differential Equations and the Spread of Infectious Disease
    7. Examples Drawn from Population Balances
    8. Conclusion
    9. Problems
    10. References
13. 10: Time-Series Data and the Fourier Transform
    1. Introduction
    2. A Nineteenth-Century Idea
    3. The Autocorrelation Coefficient
    4. A Fourier Transform Pair
    5. The Fast Fourier Transform
    6. Aliasing and Leakage
    7. Smoothing Data by Filtering
    8. Modulation (Beats)
    9. Some Familiar Examples
    10. Conclusion and Some Final Thoughts
    11. Problems
    12. References
14. 11: An Introduction to the Calculus of Variations and the Finite-Element Method
    1. Some Preliminaries
    2. Notation for the Calculus of Variations
    3. Brachistochrone Problem
    4. Other Examples
    5. A Contemporary COV Analysis of an Old Structural Problem
    6. Systems with Surface Tension
    7. The Connection between COV and the Finite-Element Method (FEM)
    8. Conclusion
    9. Problems
    10. Note
    11. References
15. Index
16. End User License Agreement