Nonlinear Physics with Maple for
Scientists and Engineers
R. Enns and G. McGuire,
Simon Fraser University & University College of the Fraser Valley
0-8176-3838-5 * 1997 * $54.50 * Hardcover * 400 pages * 165 Illustrations
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Nonlinear Physics with Maple for |
See Also -
Laboratory Manual for Nonlinear Physics for Scientists and Engineers, ISBN 0-8176-3841-5
2- Volume Set, ISBN 0-8176-3977-2One IBM Compatible Disk with 50 Maple Files are included w/the main text. This eliminates students' need to write their own files.
This text is not designed to teach Maple, but it is a text that uses Maple as a tool to learn nonlinear physics.
This textbook (together with its supplement Laboratory Manual for Nonlinear Physics for Scientists and Engineers) is designed and centered around Maple's ability to perform symbolic computations, plot, animate and permit students to investigate various nonlinear models. The primary concern is the problem of how to deal with physical phenomena described by nonlinear ordinary or partial differential equations, i.e., by equations which are nonlinear functions of the dependent variables.
The Level of the Text
The essential prerequisites for the last eight chapters of this text would normally be one semester of ordinary differential equations and an intermediate course in classical mechanics. The last four chapters assume that the student has some familiarity with partial derivatives and has encountered the wave, diffusion, and Schrodinger equations and knows something about their solutions.The Maple package permits one to perform a wealth of functions which include: algebraic manipulation of equations (including expanding, combining, collecting of terms,and factoring), analytical differentiation, analytical & numerical integration, production of 2 D plots, more complicated vector fields and 3-dim. Plots, Taylor expansions of functions to arbitrary order, animation of plots and solutions, and summation of series.
Maple has revolutionized the teaching of nonlinear physics. The use of Maple makes nonlinear physics more accessible to students in other fields, and makes it possible for students to take nonlinear physics earlier in their curriculum. Maple reduces the amount of time spent on graphing and numerically solving nonlinear equations, while making it easier to explore and extend new, novel solutions to the nonlinear models under study. The Maple software dissipates students' mathematical apprehensions, thus making nonlinear physics accessible to undergraduates of all levels. No prior knowledge of Maple or computer programming is assumed, the reader being gently introduced to Maple as an auxiliary tool as the concepts of nonlinear science are developed.
Contents:
1. Introduction
1.1 It's a Nonlinear World
1.2 Symbolic Computation
1.3 Nonlinear Activities
1.4 Structure of the Text2. Some Nonlinear Systems
2.1 Nonlinear Mechanics
2.1.1 The Simple Pendulum
2.1.2 The Eardrum
2.1.3 Nonlinear Damping
2.1.4 Nonlinear Lattice Dynamics
2.2 Competition Phenomena
2.2.1 Volterra-Lotka Competition Equations
2.2.2 Population Dynamics of Fox Rabies in Europe
2.2.3 Eigen and Schuster's Theory of the Selection and Evolution of Biological Molecules
2.2.4 Laser Beam Competition Equations
2.2.5 Rapoport's Model for the Arms Race
2.3 Nonlinear Electrical Phenomena
2.3.1 Nonlinear Inductance
2.3.2. An Electronic Oscillator (The Van der Pol Equation)
2.4 Chemical and Other Oscillators
2.4.1 Chemical Oscillators
2.4.2 The Beating Heart
2.5 Pattern Formation
2.5.1 Chemical Waves
2.5.2 Snowflakes and Other Fractal Structures
2.5.3 Rayleigh-Benard Convection
2.5.4 Cellular Automata and the Game of Life
2.6 Solitons
2.6.1 Shallow Water Waves (KdV and Other Equations)
2.6.2 Sine-Gordon Equation
2.6.3 Self-Induced Transparency
2.6.4 Optical Solitions
2.6.5 The Jovian Great Red Spot (GRS)
2.6.6 The Davydov Soliton
2.7 Chaos and Maps
2.7.1 Forced Oscillators
2.7.2 Lorenz and Rossler Systems
2.7.3 Poincare Sections and Maps
2.7.4 Examples of One and Two Dimensional Maps3. Topological Analysis
3.1 Introductory Remarks
3.2 Types of Simple Singular Points
3.3 Classifying Simple Singular Points
3.4 Examples of Phase Plane Analysis
3.4.1 The Simple Pendulum
3.4.2 The Laser Competition Equations
3.4.3 Example of a Higher Order Singularity
3.5 Isoclines
3.6 3-Dimensional Nonlinear Systems4. Analytic Methods
4.1 Introductory Remarks
4.2 Some Exact Methods
4.2.1 Separation of Variables
4.2.2 The Bernoulli Equation
4.2.3 The Riccati Equation
4.2.4 Equations of the Structure d2y/dx2 = F(y)
4.3 Some Approximate Methods
4.3.1 Maple Generated Taylor Series Solution
4.3.2 The Perturbation Approach: Poisson's Method
4.3.3 Lindstedt's Method
4.4 The Krylov-Bogoliubov (KB) Method
4.5 Ritz and Galerkin Methods5. The Numerical Approach
5.1 Finite-difference Approximations
5.2 Euler and Modified Euler Methods
5.2.1 Euler Method
5.2.2 The Modified Euler Method
5.3 Runge-Kutta (RK) Methods
5.3.1 The Basic Approach
5.3.2 Examples of Common RK Algorithms
5.4 Adaptive Step Size
5.4.1 A Simple Example
5.4.2 The Step Doubling Approach
5.4.3 The RKF 45 Algorithm
5.5 Stiff Equations
5.6 Implicit and Semi-Implicit Schemes6. Limit Cycles
6.1 Stability Aspects
6.2 Relaxation Oscillations
6.3 Bendixson's First Theorem: The Negative Criterion
6.3.1 Bendixson's Negative Criterion
6.3.2 Proof of Theorem
6.3.3 Applications
6.4 The Poincare-Bendixson Theorem
6.4.1. Poincare-Bendixson Theorem
6.4.2 Application of the Theorem
6.5 The Brusselator Model
6.5.1 Prigogine-LeFever(Brusselator)Model
6.5.2 Application of the Poincare-Bendixson Theorem
6.6 3-Dimensional Limit Cycles7. Forced Oscillators
7.1 Duffing's Equation
7.1.1 The Harmonic Solution
7.1.2 The Nonlinear Response Curves
7.2 The Jump Phenomena and Hysteresis
7.3 Subharmonic and Other Periodic Oscillations
7.4 Power Spectrum
7.5 Chaotic Oscillations
7.6 Entrainment and Quasiperiodicity
7.6.1 Entrainment
7.6.2 Quasiperiodicity
7.7 The Rossler and Lorenz Systems
7.7.1 The Rossler Attractor
7.7.2 The Lorenz Attractor8. Nonlinear Maps
8.1 Introductory Remarks
8.2 The Logistic Map
8.2.1 Introduction
8.2.2 Geometrical Representation
8.3 Fixed Points and Stability
8.4 The Period Doubling Cascade to Chaos
8.5 Period Doubling in the Real World
8.6 The Lyapunov Exponent
8.7 Stretching and Folding
8.8 The Circle Map
8.9 Chaos versus Noise
8.10 Two-Dimensional Maps
8.10.1 Introductory Remarks
8.10.2 Classification of Fixed Points
8.10.3 Examples
8.10.4 Nonconservative versus Conservative Maps9. Nonlinear PDE Phenomena
9.1 Introductory Remarks
9.2 Burger's Equation
9.3 Backlund Transformations
9.3.1 The Basic Idea
9.3.2 Examples
9.3.3 Nonlinear Superposition
9.4 Solitary Waves
9.4.1 The Basic Approach
9.4.2 Phase Plane Analysis
9.4.3 KdV Equation
9.4.4 The 3-Wave Problem10. Numerical Simulation
10.1 Finite Difference Approximations
10.2 Explicit Methods
10.2.1 Diffusion Equation
10.2.2 Fisher's Nonlinear Diffusion Equation
10.2.3 Klein-Gordon Equation
10.2.4 KdV Solitary Wave Collisions
10.3 Von Neumann Stability Analysis
10.3.1 Linear Diffusion Equation
10.3.2 Burger's Equation
10.4 Implicit Methods
10.5 Method of Characteristics
10.5.1 Colliding Laser Beams
10.5.2 General Equation
10.5.3 Sine-Gordon Equation
10.6 Higher Dimensions11 Inverse Scattering Method
11.1 Lax's Formulation
11.2 Application to KdV Equation
11.2.1 Direct Problem
11.2.2 Time Evolution of the Scattering Data
11.2.3 The Inverse Problem
11.3 Multi-soliton Solutions
11.4 General Input Shapes
11.5 The Zakharov-Shabat/AKNS Approach