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

✍ Scribed by William P. Fox, Richard D. West

• Publisher: Chapman and Hall/CRC
• Year: 2024
• Tongue: English
• Leaves: 424
• Edition: 1
• Category: Library

✦ Synopsis

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

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