The Art of Modeling in Science and Engineering with Mathematica

Chapter 1 A First Look at Modeling
1.1 The Physical Laws
1.1.1 Conservation Laws
1.1.2 Auxiliary Relations
1.1.3 The Balance Space and Its Geometry
1.2 The Rate of the Variables: Dependent and Independent Variables
1.3 The Role of Balance Space: Differential and Integral Balances
1.4 The Role of Time: Unsteady State and Steady State Balances
1.5 Information Derived from Model Solutions
1.6 Choosing a Model
1. 7 Kick-Starting the Modeling Process
1.8 Solution Analysis
Practice Problems
Chapter 2 Analytical Tools: The Solution of Ordinary Differential Equations
2.1 Definitions and Classifications
2.1.1 Order of an ODE
2.1.2 Linear and Nonlinear ODEs
2.1.3 ODEs with Variable Coefficients
2.1.4 Homogeneous and Nonhomogeneous ODEs
2.1.5 Autonomous ODEs
2.2 Boundary and Initial Conditions
2.2.1 Some Useful Hints on Boundary Conditions
2.3 Analytical Solutions of ODEs
2.3.1 Separation of Variables
2.3.2 The D-Operator Method. Solution of Linear
n-th-Order ODEs with Constant Coefficients
2.3.3 Nonhomogeneous Linear Second-Order ODEs
with Constant Coefficients
2.3.4 Series Solutions of Linear ODEs with Variable Coefficients
2.3.5 Other Methods
2.4 Nonlinear Analysis
2.4.1 Phase Plane Analysis: Critical Points
2. 5 Laplace Transformation
2.5.1 General Properties of the Laplace Transform
2.5.2 Application to Differential Equations
Practice Problems
Chapter 3 The Use of Mathematica in Modeling Physical Systems
3.1 Handling Algebraic . Expressions
3.2 A Ige brai c Eq uati on s
3.2.1 Analytical Solution to Algebraic Equations
3.2.2 Numerical Solution to Algebraic Equations
3.3 In te grati on
3.4 Ordinary Differential Equations
3.4.1 Analytical Solution to ODEs
3.4.2 Numerical Solution to Ordinary Differential Equation
3.5 Partial Differential Equations
Practice Problems
Chapter 4 Elementary Applications of the Conservation Laws
4.1 Application of Force Balances
4.2 Applications of Mass Balances
4.2.1 Compartmental Models
4.2.2 Distributed Systems1
4.3 Applications of Energy Balances
4.3.1 Compartmental Models
4.3.2 Distributed Mode1s
4.4 Simultaneous Applications of the Conservation Laws
Practice Problems
Chapter 5 Partial Differential Equations: Classification, Types, and Properties - Some Simple Transformations
5.1 Properties and Classes of PDEs
5.1.1 Order of a PDE
5.1.1.1 First-Order PDEs
5.1.1.2 Second-Order PD Es
5.1.1.3 Higher-Order PDEs
5.1.2 Homogeneous PDEs and BCs
5.1.3 PDEs with Variable Coefficients
5.1.4 Linear and Nonlinear PDEs: A New Category - Quasilinear PDEs
5.1.5 Another New Category: Elliptic, Parabolic, and Hyperbolic PDEs
5.1.6 Boundary and Initial Conditions
5.2 PDEs of Major Importance
5.2.1 First-Order Partial Differential Equations
5.2.2 Second-Order PDEs
5.3 Useful Simplifications and Transformations
5.3.1 Elimination of Independent Variables: Reduction to ODEs
5.3.2 Elimination of Dependent Variables: Reduction of Number of Equations
5.3.3 Elimination of Nonhomogeneous Terms
5.3.4 Change in Independent Variables: Reduction to Canonical Form
5.3.5 Simplification of Geometry
5.3.5.1 Reduction of a Radial Spherical Configuration into a Planar One
5.3.5.2 Reduction of a Radial Circular or Cylindrical Configuration into a Planar One
5.3.5.3 Reduction of a Radial Circular or Cylindrical Configuration to a Semi-Infinite One
5.3.5.4 Reduction of a Planar Configuration to a Semi-Infinite One
5.3.6 Nondimensionalization
5.4 PDEs PDQ: Locating Solutions in the Literature
Practice Problems
Chapter 6 Solution of Linear Systems by Superposition Methods
6.1 Superposition by Addition of Simple Flows: Solutions in Search of a Problem
6.2 Superposition by Multiplication: The Neumann Product Solutions
6.3 Solution of Source Problems: Superposition by Integration
6.4 More Superposition by Integration: Duhamel's Integral and the Superposition of Danckwerts.
Practice Problems
Chapter 7 Vector Calculus: Generalized Transport Equations
7.1 Vector Notation and Vector Calculus
7.1.1 Differential Operators and Vector Calculus
7.1.2 Integral Theorems of Vector Calculus
7.2 Superposition Revisited: Green's Functions and the Solution of PDEs by Green's Functions
7.3 Transport of Mass
7.4 Transport of Energy
7.4.1 Steady state Temperatures and Heat Flux in Multidimensional Geometries: The Shape Factor
7.5 Transport of Momentum
Practice Problems
Chapter 8 Analytical Solutions of Partial Differential Equations
8.1 Separation of Variables
8.1.1 Orthogonal Functions and Fourier Series
8.1.1.1 Orthogonal and Orthonormal Functions
The Sturm-Liouville Theorem
8.1.2 Historical Note
8.2 Laplace Transformation and Other Integral Transforms
8.2.1 General Properties
8.2.2 The Role of the Kemel
8.2.3 Pros and Cons of Integral Transforms
8.2.3.1 Advantages
8.2.3.2 Disadvantages  45
8.2.4 The Laplace Transformation of PDEs
Historical Note
8.3 The Method of Characteristics
8.3.1 General Properties
8.3.2 The Characteristics
Practice Problems
Selected References
Index