Numerical Methods and Analysis with Mathematical Modelling

Cover
Half Title
Series
Title
Copyright
Dedication
Contents
About the Authors
Preface
Acknowledgements
1 Review of Differential Calculus
	1.1 Introduction
	1.2 Limits
	1.3 Continuity
	1.4 Differentiation
	1.5 Convex and Concave Functions
	1.6 Accumulation and Integration
	1.7 Taylor Polynomials
	1.8 Errors
	1.9 Algorithm Accuracy
2 Mathematical Modelling and Introduction to Technology: Perfect Partners
	2.1 Overview and the Process of Mathematical Modelling
	2.2 The Modelling Process
	2.3 Making Assumptions
	2.4 Illustrated Examples
	2.5 Technology
3 Modelling with Discrete Dynamical Systems and Modelling Systems of Discrete Dynamical Systems
	3.1 Introduction Modelling 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 DDSs
	3.5 Modelling of Predator–Prey, SIR, and Military Models
	3.6 Technology Examples for DDSs
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
	4.7 Root Find as a DDS
5 Interpolation and Polynomial Approximation
	5.1 Introduction
	5.2 Methods
	5.3 Lagrange Polynomials
	5.4 Divided Differences
	5.5 Cubic Splines
	5.6 Telemetry Modelling 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
6 Numerical Differentiation and Integration
	6.1 Introduction and Scenario
	6.2 Numerical Differentiation
	6.3 Numerical Integration
	6.4 Car Traveling Problem
	6.5 Revisit a Telemetry Model
	6.6 Volume of Water in a Tank
7 Modelling with Numerical Solutions to Differential Equations—Initial Value Problems for Ordinary Differential Equations
	7.1 Introduction and Scenario
	7.2 Numerical Methods to Solve ODEs
	7.3 Population Modelling
	7.4 Spread of a Contagious Disease
	7.5 Bungee Jumping
	7.6 Revisit Bungee as a Second-Order ODE IVP
	7.7 Harvesting a Species
	7.8 System of ODEs
8 Iterative Techniques in Matrix Algebra
	8.1 The Gauss–Seidel and Jacobi Methods are Both Iterative Methods in Numerical Analysis
	8.2 A Bridge Too Far
	8.3 The Leontief Input–Output Economic Model
	8.4 Markov Chains with Eigenvalues and Eigenvectors (Optional)
	8.5 Cubic Splines with Matrices
9 Modelling 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)
	9.4 Applied Single-Variable Optimization Models
	9.5 Single-Variable Numerical Search Techniques
	9.6 Interpolation with Derivatives: Newton’s Method for Nonlinear Optimization
10 Multivariable Numerical Search Methods
	10.1 Introduction
	10.2 Gradient Search Methods
	10.3 Modified Newton’s Method
	10.4 Applications
11 Boundary Value Problems in Ordinary Differential Equations
	11.1 Introduction
	11.2 Linear Shooting Method
	11.3 Linear Finite Differences Method
	11.4 Applications
	11.5 Beam Deflection
12 Approximation Theory and Curve Fitting
	12.1 Introduction
	12.2 Model Fitting
	12.3 Application of Planning and Production Control
	12.4 Continuous Least Squares
	12.5 Co-Sign Out a Cosine
13 Numerical Solutions to Partial Differential Equations
	13.1 Introduction, Methods, and Applications
	13.2 Solving the Heat Equation with Homogeneous Boundary Conditions
	13.3 Methods with Python
Answers to Selected Exercises
Index