Numerical Methods and Analysis with Mathematical Modelling

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.

• Chapter 1 Review of Differential Calculus1.1. Introduction1.2 Limits1.3 Continuity1.3 Differentiation1.3.1 Increasing and decreasing functionsExample 81.3.2 Higher Derivatives1.4 Convex and Concave FunctionsExample 13. The 2nd derivative theoremExercises1.5 Accumulation and IntegrationExercises 1.51.6 Taylor PolynomialsExercises 1.71.7 Errors1.8. Algorithms AccuracyReferences and Further ReadingsChapter 2 Mathematical Modeling and Introduction to Technology: Perfect Partners2.1 OVERVIEW AND THE PROCESS OF MATHEMATICAL MODELING..2.2 THE MODLEING PROCESS2.3 Making ASSUMPTIONS2.4 ILLUSTRATE EXAMPLES2.5 TechnologyExercises Chapter 2References and Additional ReadingsChapter 3 Modeling with Discrete Dynamical Systems and Modeling Systems of DDS 3.1 Introduction Modeling with Discrete Dynamical Systems3.2 Equilibrium and Stability Values and Long-Term Behavior3.3 Using Python for a drug problem3.4 Introduction to Systems of Discrete Dynamical Systems3.4.1 Iteration and Graphical Solution3.5 Modeling of Predator - Prey model, SIR Model, and Military Models 3.6 Technology Examples for Discrete Dynamical Systems3.6.1 Excel for Linear and Nonlinear DDS3.6.2 Maple for Linear and Nonlinear DDS3.6.3 R for Linear and Nonlinear DDSExample 2. Population dynamics using RExercises Chapter 3ProjectsReferences and Suggested Future ReadingsCHAPTER 4 Numerical Solutions to Equations in One Variable4.1 Introduction and Scenario4.2 Archimedes’ design of ships4.3 Bisection Method4.4 Fixed Point Algorithm4.5 Newton's Method4.6 Secant Method4.6.1 Archimedes’ Example with secant methodExample 4.6.2 Buying a car using Secant method4.7 Root Find as a DDS4.7.1 Example of Newton’s Using EXCEL4.7.1 Root finding with PythonExercisesProjectsReferences and Further ReadingsCHAPTER 5 Interpolation and Polynomial Approximation5.1 Introduction5.2 Methods5.2.1 Lagrange Polynomials5.3 Lagrange Polynomials5.4 Divided Differences5.5 Cubic Splines5.6 Telemetry Modeling and Lagrange Polynomials5.7 Method of Divided Differences with Telemetry Data5.8 NATURAL CUBIC SPLINE INTERPOLATION to Telemetry Data5.9 Comparisons for Methods5.10 Estimating the Error5.11 Radiation Dosage ModelExercisesProjectsReferences and Further ReadingsChapter 6 Numerical Differentiation and Integration6.1 Introduction and Scenario6.2 Numerical Differentiation6.3 Numerical Integration6.3 Car traveling problem6.4 Revisit a Telemetry Model6.5 Volume of Water in a TankEXERCISES/ProjectsCHAPTER 7 Modeling with Numerical Solutions to Differential Equations---IVP for ODEs7.1 Introduction and ScenarioBridge Bungee JumpingSpread of a Contagious Disease7.2 Numerical Methods7.2.1 Euler’s Method7.2.2 Improved Euler’s Method (Heun’s method)7.2.3 Runge-Kutta Methods7.3 Population Modeling7.4 Spread of a contagious disease7.5 Bungee Jumping7.6 Revisit Bungee as a 2nd order ODE IVP7.6 Harvesting a SpeciesEXERCISES7.7 System of ODEsProjectsCHAPTER 8 Iterative Techniques in Matrix Algebra8.1 Gauss Seidel and Jacobi8.1.1 Gauss-Seidel Iterative Method8.1.2 Jacobi Method8.2 A Bridge Too Far8.2 The Leontief Input-Output Economic Model8.3 Markov Chains with Eigenvalues and Eigenvectors8.4 Cubic Splines with MatricesExercisesProjectsReferences and Further ReadingsCHAPTER 9 Modeling with Single Variable Unconstrained Optimization and Numerical Methods9.1 Introduction9.2 Single Variable Optimization and Basic Theory9..3 Models with Basic Applications of Max-Min Theory (calculus review)9.3 Applied Single Variable Optimization Models9.3.1 Oil Rig Location Problem9.4 Single Variable Numerical Search Techniques9.4.1 Unrestricted Search9.4.2 Dichotomous Search 9.4.3 Golden Section Search9.4.4 Fibonacci Search9.5 INTERPLOATION WITH DERIVATIVES: NEWTON’S METHOD FOR NONLINEAR OPTIMZATIONExercises 9.5ProjectsReference and Further ReadingsChapter 10 Multivariable Numerical Search Methods10.1 Introduction10.1.1 Background theory10.2 Gradient Search Methods10.3 Modified Newton's Method10.4 Applications10.4.1 Manufacturing10.4.2 TV ManufacturingEXERCISESProjectsReferences and FURTHER READINGCHAPTER 11 Boundary Value Problems in ODE11.1 Introduction11.2 Linear Shooting Method11.3 Linear Finite Differences Method11.4 Applications11.4.1 Motorcycle suspension11.4.2 Parachuting by skydiving Free Fall11.4.3 Free Fall11.4.4 Bungee Two11.4.5 Heat transfer11.6 Beam DeflectionExercisesProjectsReferences and Further ReadingsCHAPTER 12 Approximation Theory and Curve Fitting12.1 Introduction12.2 Model Fitting12.3 Application of Planning and Production Control12.3 Continuous Least Squares12.4 Co-Sign Out a CosineExercisesProjectsExercisesReferences and Further readingsChapter 13 Numerical Solutions to Partial Differential Equations13.1 Introduction, Methods, and Applications13.1.2 Methods13.1.2 Application Scenario13.2 Solving the Heat Equation with Homogeneous Boundary Conditions13.3 Methods with PythonExercisesProjectsReferences and Furthe Readings