# Mathematics of Continuous and Discrete Dynamical Systems

Share this page *Edited by *
*Abba B. Gumel*

This volume contains the proceedings of the
AMS Special Session on Nonstandard Finite-Difference Discretizations
and Nonlinear Oscillations, in honor of Ronald Mickens's 70th
birthday, held January 9–10, 2013, in San Diego, CA.

Included are papers on design and analysis of discrete-time and
continuous-time dynamical systems arising in the natural and
engineering sciences, in particular, the design of robust nonstandard
finite-difference methods for solving continuous-time ordinary and
partial differential equation models, the analytical and numerical
study of models that undergo nonlinear oscillations, as well as the
design of deterministic and stochastic models for epidemiological and
ecological processes. Some of the specific topics covered in the book
include the analysis of deterministic and stochastic SIR-type models,
the assessment of cost-effectiveness of vaccination problems,
finite-difference methods for oscillatory dynamical systems (including
the Schrödinger equation and Brusselator system), the design of
exact and elementary stable finite-difference methods, the study of a
two-patch model with Allee effects and disease-modified fitness, the
study of the delay differential equation model with application to
circadian rhythm and the application of some special functions in the
solutions of some problems arising in the natural and engineering
sciences.

A notable feature of the book is the collection of some
relevant open problems, intended to help guide the direction of future
research in the area.

# Table of Contents

## Mathematics of Continuous and Discrete Dynamical Systems

- Preface vii8 free
- Biographical Summary of Ronald E. Mickens ix10 free
- Deterministic and Stochastic SIR Epidemic Models with Power Function Transmission and Recovery Rates 112 free
- Evaluating the Cost-Effectiveness of Vaccination Programs 1728
- A Simple Two-Patch Epidemiological Model with Allee Effects and Disease-Modified Fitness 4960
- Designing NSFD Methods for Models of Population Interactions 89100
- Nonstandard Discretizations of the SIS Epidemiological Model with and without Diffusion 113124
- Galerkin-Least Squares Approximations for Delay Differential Equations: Application to a Circadian Rhythm Model 133144
- Exact Finite Difference Schemes 147158
- Design and Analysis of NSFD Methods for the Diffusion-Free Brusselator 163174
- An Implicit Generalized Finite-Difference Time-Domain Scheme for Solving Nonlinear Schrödinger Equations 181192
- A Dynamically Consistent Mickens-Type Discretization of the Hodgkin-Huxley Partial Differential Equation with Non-Polynomial Reaction Law 195206
- Nonstandard Finite Difference Schemes for the Black–Scholes Equation 217228
- An Analytical Method for Truly Nonlinear Oscillators 229240
- 1. Introduction 229240
- 2. Exact Amplitude, Frequency and Velocity Method 232243
- 3. Truly nonlinear oscillator with additional small nonlinearity 236247
- 4. Van Der Pol oscillator of general type 238249
- 5. Example: Truly non-integer order nonlinear oscillator 239250
- 6. Conclusions 243254
- Acknowledgement 244255
- References 244255

- A Note on the Lambert 𝑊-Function: Applications in the Mathematical and Physical Sciences 247258
- Leah-Cosine and -Sine Functions: Definitions and Elementary Properties 265276
- On the Use of Special Functions for Studying Truly Nonlinear Conservative Oscillators 281292
- I Wish I Knew How to ... 299310