Numerical Methods and Analysis with Mathematical Modelling

Klappentext

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.

Zusammenfassung

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.

Inhalt

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