Mathematical Modelling with Differential Equations

Ronald E. Mickens

• Publisher: Chapman and Hall/CRC
• Year: 2022
• Print ISBN: 9781032015309
• eBook ISBN: 9781000585582
• Edition: 1
• Topic: Mathematics
• Subtopic: Differential Equations
• Index: Mathematics

Mathematical Modelling with Differential Equations aims to introduce various strategies for modelling systems using differential equations. Some of these methodologies are elementary and quite direct to comprehend and apply while others are complex in nature and require thoughtful, deep contemplation. Many topics discussed in the chapter do not appear in any of the standard textbooks and this provides users an opportunity to consider a more general set of interesting systems that can be modelled. For example, the book investigates the evolution of a "toy universe, " discusses why "alternate futures" exists in classical physics, constructs approximate solutions to the famous Thomas—Fermi equation using only algebra and elementary calculus, and examines the importance of "truly nonlinear" and oscillating systems. Features

• Introduces, defines, and illustrates the concept of "dynamic consistency" as the foundation of modelling.
• Can be used as the basis of an upper-level undergraduate course on general procedures for mathematical modelling using differential equations.
• Discusses the issue of dimensional analysis and continually demonstrates its value for both the construction and analysis of mathematical modelling.

CHAPTER 1 What Is the $\sqrt{N}$?

DOI: 10.1201/9781003178972-1

1.1INTRODUCTION

Suppose we have a real, positive number, N, and we wish to have the value of its square-root, denoted by $\sqrt{N}$, of course there are two values associated with this quantity, one positive, the second negative. However, our focus will only be on the positive square-root since the negative one differs only by a sign.

The purpose of this chapter is to construct three algorithms for carrying out this task. Generally, we would like to construct a procedure with the properties:

1. (i)Select the number N and a priori choose the number of significant places we wish to have for the value of an approximation to its square-root.
2. (ii)Construct a procedure which will allow $\sqrt{N}$ to be calculated to any finite number of significant places.
3. (iii)Use this procedure or algorithm to calculate $\sqrt{N}$ to the prior specified accuracy.

In the presentation to come, we will illustrate our algorithms by use of the specific number, $N=10$. However, it should be clear that the general methodology holds for any $N>0$. Also, while our demonstrations are done by hand, anyone with minimum knowledge of computer programming could easily convert these procedures into short programs for implementation on a digital computer.

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Table of contents

• Cover Page
• Half-Title Page
• Title Page
• Copyright Page
• Contents
• Preface
• Chapter 0 ◾ Preliminaries
• Chapter 1 ◾ What Is the N?
• Chapter 2 ◾ Damping/Dissipative Forces
• Chapter 3 ◾ The Thomas-Fermi Equation
• Chapter 4 ◾ Single Population Growth Models
• Chapter 5 ◾ 1+2+3+4+5+⋯=−(1/2)
• Chapter 6 ◾ A Truly Nonlinear Oscillator
• Chapter 7 ◾ Discretization of Differential Equations
• Chapter 8 ◾ SIR Models for Disease Spread
• Chapter 9 ◾ Dieting Model
• Chapter 10 ◾ Alternate Futures
• Chapter 11 ◾ Toy Model of the Universe
• Chapter 12 ◾ Diffusion and Heat Equations
• Appendix
• Bibliography
• Index