Mathematical Modelling With Differential Equations

✍ Scribed by Ronald E. Mickens

• Publisher: Chapman & Hall
• Year: 2022
• Tongue: English
• Leaves: 284
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Cover
Half Title
Title Page
Copyright Page
Contents
Preface
CHAPTER 0: Preliminaries
0.1. INTRODUCTION
0.2. MATHEMATICAL MODELLING
0.3. ELEMENTARY MODELLING EXAMPLES
0.3.1. Ball and Bat Prices
0.3.2. Planet-Rope Problem
0.3.3. The Traveler and the Mountain Climb
0.3.4. What is the Sum: 12 + 14 + 18 +


?
0.4. WHAT IS SCIENCE?
0.4.1. Definitions, Comments, and Statements on Science and Related Issues
0.5. SCALING OF VARIABLES
0.5.1. Mathematical and Physical Equations
0.5.2. Characteristic Scales and Dimensionless Variables
0.5.3. Examples of Scaling
0.5.3.1. Decay equation
0.5.3.2. Duffings equation
0.5.3.3. Fisher Equation
0.6. DOMINANT BALANCE AND APPROXIMATIONS
0.6.1. Dominant Balance
0.6.2. Approximations of Functions
0.6.3. Examples
0.6.3.1. A damped harmonic oscillator
0.6.3.2. Extension of a function
0.6.3.3. Modified decay equation
0.7. USE OF HANDBOOKS
0.8. THE USE OF WIKIPEDIA
0.9. DISCUSSION
0.10. NOTES AND REFERENCES
CHAPTER 1: What Is the p N?
1.1. INTRODUCTION
1.2. INTERATIVE GUESSING
1.3. SERIES EXPANSION METHOD
1.4. NEWTON METHOD ALGORITHM
1.5. DISCUSSION
PROBLEMS
NOTES AND REFERENCES
CHAPTER 2: Damping/Dissipative Forces
2.1. INTRODUCTION
2.2. PROPERTIES OF DDF FUNCTIONS
2.3. DIMENSIONAL ANALYSIS AND DDF FUNCTIONS
2.3.1. F(v) Quadratic in v
2.3.2. F(v) Linear in v
2.4. ONE TERM POWER-LAW DDF FUNCTIONS
2.5. TWO TERM POWER-LAW DDF FUNCTIONS
2.5.1. p = 1 and q = 2
2.5.2. p = 1 and q = 1 2
2.6. DISCUSSION
PROBLEMS
NOTES AND REFERENCES
CHAPTER 3: The Thomas-Fermi Equation
3.1. INTRODUCTION
3.2. EXACT RESULTS
3.3. DYNAMIC CONSISTENCY
3.4. TWO RATIONAL APPROXIMATIONS
3.5. DISCUSSION
PROBLEMS
NOTES AND REFERENCES
CHAPTER 4: Single Population Growth Models
4.1. INTRODUCTION
4.2. LOGISTIC EQUATION
4.2.1. Introduction
4.2.2. Solution of Logistic Equation
4.2.3. Harvesting
4.2.4. Hubbert Theory for Resources
4.3. GOMPERTZ MODEL
4.3.1. Gompertz Equation and Solution
4.3.2. Gompertz Model with Harvesting
4.3.3. von Bertalanffy Equation for Tumor Growth
4.4. NON-LOGISTIC MODELS
4.4.1. P0 =  􀀀 P
4.4.2. P0 = a1 p P 􀀀 b1P
4.4.3. P0 = a2P 􀀀 b2P3
4.5. ALLEE EFFECT
4.6. DISCUSSION
PROBLEMS
NOTES AND REFERENCES
CHAPTER 5: 1 + 2 + 3 + 4 + 5 +


= −(1/2)
5.1. INTRODUCTION
5.2. PRELIMINARIES
5.2.1. Functional Equations
5.2.2. Analytic Continuation
5.3. NUMERICAL VALUES OF DIVERGENT SERIES
5.4. ELEMENTARY FUNCTION DEFINED BY AN INTEGRAL
5.4.1. The Function I(z)
5.4.2. Functional Relationship for I(z)
5.5. GAMMA FUNCTION
5.5.1. Derivation and Properties of the Gamma Function
5.5.2. A Class of Important Integrals
5.6. RIEMANN ZETA FUNCTION
5.6.1. Definition of Riemann zeta Function
5.6.2. Applications
5.7. DISCUSSION
PROBLEMS
NOTES AND REFERENCES
CHAPTER 6: A Truly Nonlinear Oscillator
6.1. INTRODUCTION
6.2. GENERAL PROPERTIES OF EXACT SOLUTIONS
6.2.1. Preliminaries
6.2.2. Phase Space Analysis
6.2.3. Calculation of Exact Period
6.3. APPROXIMATE SOLUTIONS
6.3.1. Preliminaries
6.3.2. Harmonic Balance Approximation
6.3.3. Iteration Scheme
6.3.4. Comparison of the Harmonic Balance and Iteration Procedures
6.4. SUMMARY
PROBLEMS
NOTES AND REFERENCES
CHAPTER 7: Discretization of Differential Equations
7.1. INTRODUCTION
7.2. EXACT SCHEMES
7.2.1. Methodology
7.2.2. Examples of Exact Schemes
7.3. NSFD METHODOLOGY
7.3.1. Important Ideas and Concepts
7.3.2. Constructing NSFD Schemes
7.4. NSFD SCHEMES FOR ONE’S
7.4.1. Modified Newton Law of Cooling
7.4.2. Stellar Structure
7.4.3. SEIR Model
7.4.4. Time-Independent Schrödinger Equations
7.4.5. Modified Anderson-May Model
7.5. PARTIAL DIFFERENTIAL EQUATION APPLICATIONS
7.5.1. Linear Advection-Diffusion Equation
7.5.2. Fisher Equation
7.5.3. Combustion Model
7.6. DISCUSSION
PROBLEMS
NOTES AND REFERENCES
CHAPTER 8: SIR Models for Disease Spread
8.1. INTRODUCTION
8.2. SIR METHODOLOGY
8.3. STANDARD SIR MODEL
8.4. FLATTENING THE CURVE
8.5. SOLVEABLE SIR MODEL
8.5.1. Conditions for an Epidemic
8.5.2. First-Integral
8.5.3. Maximum Infectives Number
8.5.4. Final Susceptible Population Number
8.5.5. Exact, Explicit Solution
8.6. RÉSUMÉ
PROBLEMS
NOTES AND REFERENCES
CHAPTER 9: Dieting Model
9.1. INTRODUCTION
9.2. MATHEMATICAL MODEL
9.2.1. Assumptions
9.2.2. Simplification Condition
9.2.3. Time Scale
9.3. ANALYSIS OF MODEL
9.4. APPROXIMATE SOLUTIONS
9.5. DISCUSSION
PROBLEMS
NOTES AND REFERENCES
CHAPTER 10: Alternate Futures
10.1. INTRODUCTION
10.2. TWO SYSTEMS EXHIBITING ALTERNATIVE FUTURES
10.2.1. Coin Flipping
10.2.2. Going from A to B
10.3. RÉSUMÉ OF CONCEPTS AND DEFINITIONS
10.4. COUNTERFACTUAL HISTORIES
10.5. SUMMARY AND DISCUSSION
PROBLEMS
NOTES AND REFERENCES
CHAPTER 11: Toy Model of the Universe
11.1. INTRODUCTION
11.2. IN THE BEGINNING: LET THERE BE RULES
11.3. SOME “DULL” MODEL UNIVERSES
11.4. NONTRIVIAL TMOU
11.5. FIBONACCI EQUATION
11.6. DISCUSSION
PROBLEMS
NOTES AND REFERENCES
CHAPTER 12: Diffusion and Heat Equations
12.1. INTRODUCTION
12.2. HEAT EQUATION DERIVATION
12.3. DIFFUSION EQUATION AND RANDOM WALKS
12.4. DIFFUSION AND PROBABILITY
12.5. DERIVATION DIFFICULTIES
12.6. HEATED ROD PROBLEM
12.7. COMMENTS
PROBLEMS
NOTES AND REFERENCES
Appendix
A. ALGEBRAIC RELATIONS
A.1. Factors and Expansions
A.2. Quadratic Equations
A.3. Cubic Equations
A.4. Expansions of Selected Functions
B. TRIGONOMETRIC RELATIONS
B.1. Fundamental Properties
B.2. Sums of Angles
B.3. Other Trigonometric Relations
B.4. Derivatives and Integrals
B.5. Powers of Trigonometric Functions
C. HYPERBOLIC FUNCTIONS
C.1. Definitions and Basic Properties
C.2. Basic Properties
C.3. Derivatives and Integrals
C.4. Other Relations
C.5. Relations between Hyperbolic and Trigonometric Functions
D. RELATIONS FROM CALCULUS
D.1. Differentiation
D.2. Integration by Parts
D.3. Differentiation of a Definite Integral with Respect to a Parameter
D.4. Some Important Integrals
E. FOURIER SERIES
F. EVEN AND ODD FUNCTIONS
G. SOME NONSTANDARD BUT IMPORTANT FUNCTIONS
G.1. Absolute Value Function
G.2. Sign and Theta Functions
G.3. Ramp Function
G.4. Boxcar or Rectangular Function
G.5. Triangular Function
G.6. Delta Function
H. DIFFERENTIAL EQUATIONS
H.1. First-Order, Separable Ordinary Differential Equations
H.2. General Linear, First-Order, Ordinary Differential Equation
H.3. Bernoulli Equations
H.4. Linear, Second-Order, Homogeneous Differential Equations with Constant Coefcients
I. LINEARIZATION OF CERTAIN TYPES OF NONLINEAR DIFFERENTIAL
I.1. Riccati Equation
I.2. Quadratic Nonlinearities: Two Special Cases
I.3. Square-Root Nonlinearities
Bibliography
Index