Nonstandard Finite Difference Schemes: Methodology and Applications

✍ Scribed by Ronald E. Mickens

• Publisher: World Scientific Publishing
• Year: 2020
• Tongue: English
• Leaves: 332
• Category: Library

✦ Synopsis

This second edition of Nonstandard Finite Difference Models of Differential Equations provides an update on the progress made in both the theory and application of the NSFD methodology during the past two and a half decades. In addition to discussing details related to the determination of the denominator functions and the nonlocal discrete representations of functions of dependent variables, we include many examples illustrating just how this should be done.Of real value to the reader is the inclusion of a chapter listing many exact difference schemes, and a chapter giving NSFD schemes from the research literature. The book emphasizes the critical roles played by the 'principle of dynamic consistency' and the use of sub-equations for the construction of valid NSFD discretizations of differential equations.

✦ Table of Contents

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