Applied Mathematics for Science and Engineering

This book was designed to prepare students in the applied sciences and engineering for both analytic and numerical solutions of problems arising in post–graduate studies and in industrial practice.  It includes examples and problems from biology, chemistry, and physics, as well as from most engineering disciplines and the presentation accommodates the learning styles of contemporary students.  The book covers topics not found in similar texts, including integro–differential equations, treatment of time–series data, and the calculus of variations.  It also includes some recently developed methods (both analytic and numerical) such as the variational iteration method (VIM) developed by J. H. He, which can be applied to ordinary differential equations, integro–differential equations, and differential–difference equations.  Although commercial software packages are mentioned (e.g., COMSOL™ and Mathcad™) and used for some examples, the presentation is not tied to the use of any particular software.

1. Problem Formulation and Model Development 1 Introduction Algebraic Equations from Vapor–Liquid Equilibria (VLE) Macroscopic Balances—Lumped–Parameter Models Force Balances—Newton’s Second Law of Motion Distributed Parameter models—Microscopic Balances A Contrast: Deterministic Models and Stochastic Processes Empiricisms and Data Interpretation Conclusion References Problems 2. Algebraic Equations 28 Introduction Elementary Methods Simultaneous Linear Algebraic Equations Simultaneous Nonlinear Algebraic Equations Algebraic Equations with Constraints Conclusion References Problems 3. Vectors and Tensors 64 Introduction Manipulation of Vectors Green’s Theorem Stokes’ Theorem Conclusion References Problems 4. Numerical Quadrature 90                 Introduction Trapezoid Rule Simpson’s Rule Newton–Cotes Formulae Roundoff and Truncation Errors Romberg Integration Adaptive Integration Schemes Integrating Discrete Data Multiple Integrals (Cubature) Conclusion References Problems 5. Analytic Solution of Ordinary Differential Equations 126 An Introductory Example First Order Ordinary Differential Equations Nonlinear First Order Ordinary Differential Equations Higher Order Linear ODE’s with Constant Coefficients Higher Order Equations with Variable Coefficients Bessel’s Equation and Bessel Functions Power Series Solutions of ODE’s Regular Perturbation Linearization Conclusion References Problems 6. Numerical Solution of Ordinary Differential Equations 176 An Illustrative Example The Euler Method Runge–Kutta Methods Simultaneous Ordinary Differential Equations Limitations of Fixed Step–Size Algorithms Richardson Extrapolation Multistep Methods Split Boundary Conditions Finite Difference Methods Stiff Differential Equations Bulirsch–Stoer Method Phase Space Summary References Problems 7. Analytic Solution of Partial Differential Equations 222 Introduction Classification of Partial Differential Equations and Boundary Conditions Fourier Series The Product Method (Separation of Variables) Applications of the Laplace Transform Approximate Solution Techniques The Cauchy–Riemann Equation, Conformal Mapping, and Solutions for the Laplace Equation Conclusion References Problems 8. Numerical Solution of Partial Differential Equations 300 Introduction Elliptic Partial Differential Equations Parabolic Partial Differential Equations Hyperbolic Partial Differential Equations Elementary Problems with Convective Transport A Numerical Procedure for Two–Dimensional Viscous Flow Problems MacCormack’s Method Adaptive Grids Conclusion References Problems 9. Integro–Differential Equations 370 Introduction An Example of Three–Mode Control Population Problems with Hereditary Influences An Elementary Solution Strategy VIM: The Variational Iteration Method Integro–Differential Equations and the Spread of Infectious Disease Examples Drawn from Population Balances Conclusion References Problems 10. Time Series Data and the Fourier Transform 414 Introduction A 19 th Century Idea The Autocorrelation Coefficient A Fourier Transform Pair The Fast Fourier Transform Smoothing Data by Filtering Modulation (Beats) Some Familiar Examples Conclusion and Some Final Thoughts References Problems 11. An Introduction to the Calculus of Variations and the Finite Element Method 461 Some Preliminaries Notation for the Calculus of Variations Brachistochrone Problem Other Examples A Contemporary COV Analysis of an Old Structural Problem Systems with Surface Tension The Connection Between COV and the Finite Element Method Conclusion References Problems

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).