Numerical Methods and Analysis with Mathematical Modelling: (Textbooks in Mathematics)

What sets Numerical Methods and Analysis with Mathematical Modelling apart are the modelling aspects utilizing numerical analysis (methods) to obtain solutions. The authors cover first the basic numerical analysis methods with simple examples to illustrate the techniques and discuss possible errors. The modelling 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 modelling projects. Chapters are introduced and techniques are discussed with common examples. A modelling scenario is introduced that will be solved with these techniques later in the chapter. Often, the modelling problems require more than one previously covered technique presented in the book. Fundamental exercises to practice the techniques are included. Multiple modelling scenarios per numerical methods illustrate the applications of the techniques introduced. Each chapter has several modelling examples that are solved by the methods described within the chapter. The use of technology is instrumental in numerical analysis and numerical methods. In this text, Maple, Excel, R, and Python are illustrated. The goal is not to teach technology but to 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.

Table of Contents:
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

About the Author :
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.