1st Edition
Numerical Methods and Analysis with Mathematical Modelling
What sets this book apart is the modeling aspects utilizing numerical analysis (methods) to obtain solutions. The authors cover the basic numerical analysis methods first with simple examples to illustrate the techniques and discuss possible errors. The modeling prospective reveals the practical relevance of the numerical methods in context to real world problems.
At the core of this text are the real-world modeling projects. Chapters are introduced and techniques are discussed with common examples. A modeling scenario is introduced that will be solved with these techniques later in the chapter. Often the modeling problems require more than one previously covered technique presented in the book.
Fundamental exercises to practice the techniques are included. Multiple modeling scenarios per numerical methods illustrate the applications of the techniques introduced. Each chapter has several modeling examples that are solved by the methods described within the chapter.
The use of technology is instrumental in numerical analysis and numerical methods. In our text, we illustrate MAPLE, EXCEL, R, and Python. Our goal is not to teach technology but illustrate its power and limitations to perform algorithms and reach conclusions.
This book fulfills a need in the education of all students who plan to use technology to solve problems whether using physical models or true creative mathematical modeling, like discrete dynamical systems.
Chapter 1 Review of Differential Calculus
1.1. Introduction
1.2 Limits
1.3 Continuity
1.3 Differentiation
1.3.1 Increasing and decreasing functions
Example 8
1.3.2 Higher Derivatives
1.4 Convex and Concave Functions
Example 13. The 2nd derivative theorem
Exercises
1.5 Accumulation and Integration
Exercises 1.5
1.6 Taylor Polynomials
Exercises 1.7
1.7 Errors
1.8. Algorithms Accuracy
References and Further Readings
Chapter 2 Mathematical Modeling and Introduction to Technology: Perfect Partners
2.1 OVERVIEW AND THE PROCESS OF MATHEMATICAL MODELING..
2.2 THE MODLEING PROCESS
2.3 Making ASSUMPTIONS
2.4 ILLUSTRATE EXAMPLES
2.5 Technology
Exercises Chapter 2
References and Additional Readings
Chapter 3 Modeling with Discrete Dynamical Systems and Modeling Systems of DDS
3.1 Introduction Modeling with Discrete Dynamical Systems
3.2 Equilibrium and Stability Values and Long-Term Behavior
3.3 Using Python for a drug problem
3.4 Introduction to Systems of Discrete Dynamical Systems
3.4.1 Iteration and Graphical Solution
3.5 Modeling of Predator - Prey model, SIR Model, and Military Models
3.6 Technology Examples for Discrete Dynamical Systems
3.6.1 Excel for Linear and Nonlinear DDS
3.6.2 Maple for Linear and Nonlinear DDS
3.6.3 R for Linear and Nonlinear DDS
Example 2. Population dynamics using R
Exercises Chapter 3
Projects
References and Suggested Future Readings
CHAPTER 4 Numerical Solutions to Equations in One Variable
4.1 Introduction and Scenario
4.2 Archimedes’ design of ships
4.3 Bisection Method
4.4 Fixed Point Algorithm
4.5 Newton's Method
4.6 Secant Method
4.6.1 Archimedes’ Example with secant method
Example 4.6.2 Buying a car using Secant method
4.7 Root Find as a DDS
4.7.1 Example of Newton’s Using EXCEL
4.7.1 Root finding with Python
Exercises
Projects
References and Further Readings
CHAPTER 5 Interpolation and Polynomial Approximation
5.1 Introduction
5.2 Methods
5.2.1 Lagrange Polynomials
5.3 Lagrange Polynomials
5.4 Divided Differences
5.5 Cubic Splines
5.6 Telemetry Modeling and Lagrange Polynomials
5.7 Method of Divided Differences with Telemetry Data
5.8 NATURAL CUBIC SPLINE INTERPOLATION to Telemetry Data
5.9 Comparisons for Methods
5.10 Estimating the Error
5.11 Radiation Dosage Model
Exercises
Projects
References and Further Readings
Chapter 6 Numerical Differentiation and Integration
6.1 Introduction and Scenario
6.2 Numerical Differentiation
6.3 Numerical Integration
6.3 Car traveling problem
6.4 Revisit a Telemetry Model
6.5 Volume of Water in a Tank
EXERCISES/Projects
CHAPTER 7 Modeling with Numerical Solutions to Differential Equations---IVP for ODEs
7.1 Introduction and Scenario
Bridge Bungee Jumping
Spread of a Contagious Disease
7.2 Numerical Methods
7.2.1 Euler’s Method
7.2.2 Improved Euler’s Method (Heun’s method)
7.2.3 Runge-Kutta Methods
7.3 Population Modeling
7.4 Spread of a contagious disease
7.5 Bungee Jumping
7.6 Revisit Bungee as a 2nd order ODE IVP
7.6 Harvesting a Species
EXERCISES
7.7 System of ODEs
Projects
CHAPTER 8 Iterative Techniques in Matrix Algebra
8.1 Gauss Seidel and Jacobi
8.1.1 Gauss-Seidel Iterative Method
8.1.2 Jacobi Method
8.2 A Bridge Too Far
8.2 The Leontief Input-Output Economic Model
8.3 Markov Chains with Eigenvalues and Eigenvectors
8.4 Cubic Splines with Matrices
Exercises
Projects
References and Further Readings
CHAPTER 9 Modeling with Single Variable Unconstrained Optimization and Numerical Methods
9.1 Introduction
9.2 Single Variable Optimization and Basic Theory
9..3 Models with Basic Applications of Max-Min Theory (calculus review)
9.3 Applied Single Variable Optimization Models
9.3.1 Oil Rig Location Problem
9.4 Single Variable Numerical Search Techniques
9.4.1 Unrestricted Search
9.4.2 Dichotomous Search
9.4.3 Golden Section Search
9.4.4 Fibonacci Search
9.5 INTERPLOATION WITH DERIVATIVES: NEWTON’S METHOD FOR NONLINEAR OPTIMZATION
Exercises 9.5
Projects
Reference and Further Readings
Chapter 10 Multivariable Numerical Search Methods
10.1 Introduction
10.1.1 Background theory
10.2 Gradient Search Methods
10.3 Modified Newton's Method
10.4 Applications
10.4.1 Manufacturing
10.4.2 TV Manufacturing
EXERCISES
Projects
References and FURTHER READING
CHAPTER 11 Boundary Value Problems in ODE
11.1 Introduction
11.2 Linear Shooting Method
11.3 Linear Finite Differences Method
11.4 Applications
11.4.1 Motorcycle suspension
11.4.2 Parachuting by skydiving Free Fall
11.4.3 Free Fall
11.4.4 Bungee Two
11.4.5 Heat transfer
11.6 Beam Deflection
Exercises
Projects
References and Further Readings
CHAPTER 12 Approximation Theory and Curve Fitting
12.1 Introduction
12.2 Model Fitting
12.3 Application of Planning and Production Control
12.3 Continuous Least Squares
12.4 Co-Sign Out a Cosine
Exercises
Projects
Exercises
References and Further readings
Chapter 13 Numerical Solutions to Partial Differential Equations
13.1 Introduction, Methods, and Applications
13.1.2 Methods
13.1.2 Application Scenario
13.2 Solving the Heat Equation with Homogeneous Boundary Conditions
13.3 Methods with Python
Exercises
Projects
References and Furthe Readings
Biography
Dr. William P. Fox is an Emeritus Professor in the Department of Defense Analysis at the Naval Postgraduate School. Currently, he is a Visiting Professor in the Department of Mathematics at the College of William and Mary. He received his Ph.D. in Industrial Engineering from Clemson University. He has taught at the United States Military Academy, Francis Marion University, and Naval Postgraduate School. He has many publications and scholarly activities including over twenty books, twenty-four chapters of books & technical reports, one hundred and fifty journal articles, and over one hundred and fifty conference presentations and mathematical modeling workshops.
Richard D. West is a Professor Emeritus of Francis Marion University and a retired Colonel of the United States Army. He received an MS in Applied Mathematics from the University of Colorado in Boulder, which launched his teaching interest in Numerical Analysis. and earned his PhD in college mathematics education from New York University. After a 30-yeaer career in the Army he taught at Francis Marion University in Florence, where he served as Professor of Mathematics.
