Nonstandard Finite Difference Schemes: Methodology and Applications

Contents
Preface
0. A Second Edition . . .Why?
	0.1 Purpose
	0.2 Ambiguities with the Discretization Process
	0.3 The Nonstandard Finite Difference Methodology
	References
1. Introduction
	1.1 Numerical Integration
	1.2 Standard Finite Difference Modeling Rules
	1.3 Examples
		1.3.1 Decay Equation
		1.3.2 Logistic Equation
		1.3.3 Harmonic Oscillator
		1.3.4 Unidirectional Wave Equation
		1.3.5 Diffusion Equation
		1.3.6 Burgers\' Equation
	1.4 Critique
	References
2. Numerical Instabilities
	2.1 Introduction
	2.2 Decay Equation
	2.3 Harmonic Oscillator
	2.4 Logistic Differential Equation
	2.5 Unidirectional Wave Equation
	2.6 Burgers\' Equation
	2.7 Summary
	References
3. Nonstandard Finite Difference Schemes
	3.1 Introduction
	3.2 Exact Finite Difference Schemes
	3.3 Examples of Exact Schemes
	3.4 Nonstandard Modeling Rules
	3.5 Best Finite Difference Schemes
	References
4. First-Order ODE\'s
	4.1 Introduction
	4.2 A New Finite Difference Scheme
	4.3 Examples
		4.3.1 Decay Equation
		4.3.2 Logistic Equation
		4.3.3 ODE with Three Fixed-Points
	4.4 Nonstandard Schemes
		4.4.1 Logistic Equation
		4.4.2 ODE with Three Fixed-Points
	4.5 Discussion
	References
5. Second-Order, Nonlinear Oscillator Equations
	5.1 Introduction
	5.2 Mathematical Preliminaries
	5.3 Conservative Oscillators
	5.4 Limit-Cycle Oscillators
	5.5 General Oscillator Equations
	5.6 Response of a Linear System
	References
6. Two First-Order, Coupled Ordinary Differential Equations
	6.1 Introduction
	6.2 Background
	6.3 Exact Scheme for Linear Ordinary Differential Equations
	6.4 Nonlinear Equations
	6.5 Examples
		6.5.1 Harmonic Oscillator
		6.5.2 Damped Harmonic Oscillator
		6.5.3 Duffing Oscillator
		6.5.4 x + x + ϵx2 = 0
		6.5.5 van der Pol Oscillator
		6.5.6 Lewis Oscillator
		6.5.7 General Class of Nonlinear Oscillators
		6.5.8 Batch Fermentation Processes
	6.6 Summary
	References
7. Partial Differential Equations
	7.1 Introduction
	7.2 Wave Equations
		7.2.1 ut + ux = 0
		7.2.2 ut − ux = 0
		7.2.3 utt − uxx = 0
		7.2.4 ut + ux = u(1 − u)
		7.2.5 uk + ux = buxx
	7.3 Diffusion Equations
		7.3.1 ut = auxx + bu
		7.3.2 ut = uuxx
		7.3.3 ut = uuxx + -u(1 − u)
		7.3.4 ut = uxx + -u(1 − u)
	7.4 Burgers\' Type Equations
		7.4.1 ut + uux = 0
		7.4.2 ut + uux = -u(1 − u)
		7.4.3 ut + uux = uuxx
	7.5 Discussion
	References
8. Schrödinger Differential Equations
	8.1 Introduction
	8.2 Schrödinger Ordinary Differential Equations
		8.2.1 Numerov Method
		8.2.2 Mickens-Ramadhani Scheme
		8.2.3 Combined Numerov-Mickens Scheme
	8.3 Schrödinger Partial Differential Equations
		8.3.1 ut = iuxx
		8.3.2 ut = i[uxx + f(x)u]
		8.3.3 Nonlinear, Cubic Schrödinger Equation
	References
9. The NSFD Methodology
	9.1 Introduction
	9.2 The Modeling Process
	9.3 Intrinsic Time and Space Scales
	9.4 Dynamical Consistency
	9.5 Numerical Instabilities
	9.6 Denominator Functions
	9.7 Nonlocal Discretization of Functions
	9.8 Method of Sub-Equations
	9.9 Constructing NSFD Schemes
	9.10 Final Comments
	References
10. Some Exact Finite Difference Schemes
	10.1 Introduction
	10.2 General, Linear, Homogeneous, First-Order ODE
	10.3 Several Important Exact Schemes
		10.3.1 Decay Equation
		10.3.2 Harmonic Oscillator
		10.3.3 Logistic Equation
		10.3.4 Quadratic Decay Equation
		10.3.5 Nonlinear Equation
		10.3.6 Cubic Decay Equation
		10.3.7 Linear Velocity Force Equation
		10.3.8 Damped Harmonic Oscillator
		10.3.9 Unidirectional Wave Equations
		10.3.10 Full Wave Equation
		10.3.11 Nonlinear, Fisher-Type Unidirection Wave Equation
		10.3.12 Unidirectional, Spherical Wave Equation
		10.3.13 Steady-State Wave Equation with Spherical Symmetry
		10.3.14 Wave Equation having Spherical Symmetry
		10.3.15 Two-Dimensional, Linear Advection Equation
		10.3.16 Two-Dimensional, Nonlinear (Logistic) Advection Equation
	10.4 Two Coupled, Linear ODE\'s with Constant Coefficients
	10.5 Jacobi Cosine and Sine Functions
	10.6 Cauchy-Euler Equation
	10.7 Michaelis-Menten Equation
	10.8 Weierstrass Elliptic Function
	10.9 Modified Lotka-Volterra Equations
	10.10 Comments
	References
11. Applications and Related Topics
	11.1 Introduction
	11.2 Stellar Structures
		References
	11.3 The x − y − z Model
		References
	11.4 Mickens\' Modified Newton\'s Law of Cooling
		References
	11.5 NSFD Schemes for dx=dt = −λxα
		Reference
	11.6 Exact Scheme for Linear ODE\'s with Constant Coefficients
		References
	11.7 Discrete 1-Dim Hamiltonian Systems
		11.7.1 Discrete Hamiltonian Construction
		11.7.2 Discrete Equations of Motion for Eq. (11.7.32)
		11.7.3 Non-Polynomial Potential Energy
		11.7.4 Two Interesting Results
		References
	11.8 Cube Root Oscillators
		11.8.1 Cube Root Oscillator
		11.8.2 Inverse Cube-Root Oscillator
		References
	11.9 Alternative Methodologies for Constructing Discrete-Time Population Models
		11.9.1 Comments
		11.9.2 Modified Anderson-May Models
		11.9.3 Discrete Exponentialization
	11.10 Interacting Populations with Conservation Laws
		11.10.0 Comments
		11.10.1 Conservation Laws
		11.10.2 Chemostate Model
		11.10.3 SIR Model
		11.10.4 SEIR Model with Net Birthrate
		11.10.5 Criss-Cross Model
		11.10.6 Brauer-van den Driessche SIR Model
		11.10.7 Spatial Spread of Rabies
		11.10.8 Fisher Equation
		References
	11.11 Black-Scholes Equations
		References
	11.12 Time-Independent Schrödinger Equations
		References
	11.13 Linear, Damped Wave Equation
		References
	11.14 NSFD Constructions for Burgers and Burgers-Fisher PDE\'s
		11.14.1 Burgers Equations: ut + uux = uxx
		11.14.2 Burgers-Fisher Equations: ut +uux = uxx +u(1−u)
		References
	11.15 Cross-Diffusion
		References
	11.16 Delay Differential Equations
		References
	11.17 Fractional Differential Equations
		References
	11.18 Summary
Appendix A Difference Equations
	A.1 Linear Equations
	A.2 Riccati Equations
	A.3 Separation-of-Variables
	Reference
Appendix B Linear Stability Analysis
	B.1 Ordinary Differential Equations
	B.2 Ordinary Difference Equations
	References
Appendix C Discrete WKB Method
	References
Bibliography
Index