Differential Equations With Maple

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ISBN: 9780817641818 | 0817641815
Cover: Hardcover
Copyright: 1/1/2001

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A key subject in mathematics, engineering, biology and physics, this book features an exciting interactive approach to differential equations by integrating the basics of ODEs and the symbolic computation package Maple. Covers all the essential topics of ODE theory in a user friendly presentation of rigorous mathematics balanced with excellent examples from many different fields of applications.

Preface

I Maple Use and Programming

(26)

Introduction to Maple

(24)

Calculator or What?

(3)

First Contact

(2)

Programming in Maple

(1)

Maple Variables

(7)

Variable Types

(1)

Maple Expressions

(3)

Aggregate Types

(2)

Maple Tables

(1)

Maple Syntax

(12)

Function Declarations

(2)

Assignments

(1)

Conditionals

(1)

Loops

(1)

Maple Evaluation

(3)

Details about Maple Procedures

(1)

Procedure Arguments

(2)

Programming Gotchas

(1)

Maple Program Development

(1)

II Differential Equations

(252)

Introduction to Differential Equations

(22)

Example Problems

(14)

Simple Rate Equations

(1)

Mechanical Problems

(1)

Electrical Models

(2)

Kinetics

(1)

Beam Models

(8)

Order, Dimension and Type

(3)

First and Higher Order

(1)

Partial Derivative Problems

(1)

Problems

(5)

First Order Equations

(30)

Separable Problems

(7)

Separable Examples

(4)

Level Sets and Exact Equations

(6)

Solving Exact Problems

(1)

Exact Equation Examples

(1)

Linear Equations

(6)

Linear Examples

(1)

Tricky Substitutions

(2)

Lagrange

(1)

Riccati

(1)

Mechanics

(1)

Do Solutions Exist?

(4)

The Issue

(2)

Existence Theorems

(2)

Problems

(4)

Introduction to Numerical Methods

(30)

The Game and Euler's Method

(3)

Solution Taylor Series

(3)

Runge-Kutta Methods

(3)

General Runge-Kutta Methods

(4)

Maple Numeric Routines

(2)

Calling all RK4's (and relatives)

(3)

Variable Step Size Methods

100

(8)

Running RK4 With RK3

102

(1)

Adjusting h

103

(5)

Serious Methods

108

(1)

Problems

109

(2)

Higher Order Differential Equations

111

(30)

Equations of Order N

111

(4)

Vector Equations

112

(1)

Equivalent Formulations

113

(2)

Linear Independence and Wronskians

115

(2)

Linear Independence

116

(1)

Wronskians

116

(1)

Fundamental Solutions

117

(7)

Existence of Fundamental Solutions

118

(5)

Constructing Fundamental Solutions

123

(1)

General and Particular Solutions

124

(1)

Constant Coefficient Problems

125

(11)

Homogeneous Case

125

(5)

Undetermined Coefficients

130

(6)

Problems

136

(5)

Laplace Transform Methods

141

(32)

Basic Definition

141

(4)

Examples

143

(2)

New Wine From Old

145

(8)

Algebra and Tables

148

(5)

Maple Facilities

153

(1)

Derivatives and Laplace

153

(3)

High Order Problems by Laplace

156

(7)

Fundamental Solutions

158

(2)

Undetermined Coefficients Redux

160

(3)

Convolutions

163

(6)

Inhomogeneous Problems

168

(1)

Problems

169

(4)

Systems of Equations

173

(24)

Linear Systems

173

(1)

Bases and Eigenvectors

174

(2)

Diagonalization

176

(5)

Using Maple

180

(1)

Jordan Forms

181

(7)

Using Maple

187

(1)

Matrix Exponentials

188

(5)

Laplace Transforms

190

(1)

Using Maple

191

(2)

Problems

193

(4)

Stability

197

(24)

Second Order Problems

197

(1)

Harmonic Oscillator Again

198

(1)

Nodes and Spirals

199

(4)

Higher Order Problems

203

(3)

The Easy Cases

204

(1)

Jordan Form

205

(1)

Linearization

206

(4)

Introduction to Lyapunov Theory

210

(6)

Linear Problems

212

(2)

Applications

214

(2)

Problems

216

(5)

Periodic Problems

221

(24)

Periodic Inputs

221

(1)

Phasors

222

(3)

Fourier Series

225

(2)

Time Domain Methods

227

(7)

Periodic Coefficients

234

(2)

Fundamental Matrices

236

(3)

Stability

239

(1)

Floquet Representation

240

(3)

Problems

243

(2)

Impedances and Differential Equations

245

(14)

Introduction to Impedances

245

(1)

AC Circuits

246

(4)

Transient Circuits

250

(1)

Circuit Impedance Examples

251

(3)

Loops and Nodes

254

(1)

Problems

255

(4)

Partial Differential Equations

259

(20)

Basic Problems

259

(7)

Heat Equation

259

(2)

Wave Equation

261

(1)

Laplace's Equation

262

(2)

Beam Vibrations

264

(2)

Boundary Conditions

266

(2)

Heat and Diffusion Problems

266

(1)

Wave Equations

267

(1)

Laplace's Equation

267

(1)

Separation of Variables

268

(8)

Cooling of a Slab

268

(3)

Standing Wave Solutions

271

(2)

Steady State Heat Conduction

273

(3)

Problems

276

(3)

III Maple Application Topics

279

(110)

Introduction to Maple Applications

281

(2)

Plotting With Maple

283

(14)

Maple Plotting Structures

283

(1)

Remember Tables

284

(3)

Plotting Numerical Results

287

(1)

Plotting Vector Variables

288

(2)

Further Plotting

290

(7)

Maple and Laplace Transforms

297

(10)

Walking Maple Through Problems

298

(3)

Coding a Beam Problem Solver

301

(6)

Maple Linear Algebra Applications

307

(16)

Canonical Forms

307

(5)

Eigenvalues and Eigenvectors

307

(3)

Jordan Form Calculations

310

(2)

Matrix Exponentials

312

(3)

Stability Examples

315

(2)

Periodic Solutions with Maple

317

(6)

Runge-Kutta Designs

323

(32)

Orientation and Overview

323

(1)

Notational Issues

324

(1)

True Solution Derivatives

325

(4)

Runge-Kutta Derivatives

329

(4)

Traps and Tricky Bits

333

(1)

A Sample Run

334

(4)

Order Condition Code

338

(17)

Maple Packages

355

(34)

Introduction

355

(1)

Constructing Packages

356

(1)

A Control Design Package

357

(19)

Root Locus

358

(2)

Nyquist Locus

360

(3)

Bode Plots

363

(1)

Classical vs. State Space

364

(1)

LQR Utility Code

365

(1)

Least Squares Optimal Control

366

(5)

Control Equation Generation

371

(3)

LTI Realization Calculations

374

(2)

Initialization

376

(1)

Package Creation and Installation

376

(1)

How About Help?

377

(3)

Maple Libraries

377

(1)

Creating and Installing Help Files

378

(2)

Control Package Demo

380

(9)

IV Appendices

389

(14)

A Review Problems

391

(10)

B Laplace Transform Table

401

(2)

References

403

(2)

Index

405

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Differential Equations With Maple

Summary

Table of Contents

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