Very Applied First Course in Partial Differential Equations, A

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ISBN: 9780130304179 | 0130304174
Cover: Hardcover
Copyright: 1/1/2002

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This extremely readable book illustrates how mathematics applies directly to different fields of study. Focuses on problems that require physical to mathematical translations, by showing readers how equations have actual meaning in the real world. Covers fourier integrals, and transform methods, classical PDE problems, the Sturm-Liouville Eigenvalue problem, and much more. For readers interested in partial differential equations.

Preface

xvii

Introduction

(4)

The One-Dimensional Heat Equation

(36)

Introduction

(1)

Derivation of Heat Conduction in a One-Dimensional Rod

(12)

Derivation of the Mathematical Model

(7)

Initial Temperature

(3)

Boundary Conditions for a One-Dimensional Rod

(6)

Boundary Conditions of the First Kind

(1)

Boundary Conditions of the Second Kind

(1)

Boundary Conditions of the Third Kind

(4)

The Maximum Principle and Uniqueness

(7)

Steady-State Temperature Distribution

(11)

The One-Dimensional Wave Equation

(32)

Introduction

(1)

Derivation of the One-Dimensional Wave Equation

(8)

Boundary Conditions

(5)

Boundary Conditions of the First Kind

(1)

Boundary Conditions of the Second Kind

(1)

Boundary Conditions of the Third Kind

(4)

Conservation of Energy for a Vibrating String

(3)

Method of Characteristics

(7)

D'Alembert's Solution to the One-Dimensional Wave Equation

(9)

The Essentials of Fourier Series

(58)

Introduction

(1)

Elements of Linear Algebra

(16)

Vector Space

(1)

Linear Dependence, Linear Independence, and Basis

(2)

Orthogonality and Inner Product

(5)

Eigenvalues and Eigenvectors

(2)

Significance

(5)

A New Space: The Function Space of Piecewise Smooth Functions

(29)

Inner Product, Orthogonality, and Basis in a Function Space

(8)

Definition of Trigonometric Fourier series

100

(3)

Fourier series Representation of Piecewise Smooth Functions

103

(16)

Even and Odd Functions and Fourier Series

119

(12)

Separation of Variables: The Homogeneous Problem

131

(60)

Introduction

131

(1)

Operators: Linear and Homogeneous Equations

132

(7)

Linear Operators

132

(4)

Linear Equations

136

(3)

Separation of Variables: Heat Equation

139

(14)

Spatial Problem Solution

141

(3)

Time Problem Solution

144

(1)

The Complete Solution

145

(8)

Separation of Variables: Wave Equation

153

(11)

Spatial Problem Solution

155

(1)

Time Problem Solution

156

(1)

The Complete Solution

156

(8)

The Multidimensional Spatial Problem

164

(16)

Spatial Problem for X (x)

167

(1)

Spatial Problem for Y (y)

168

(1)

Time Problem

168

(1)

The Complete Solution

168

(12)

Laplace's Equation

180

(11)

An Electrostatics Derivation of Laplace's Equation

180

(1)

Uniqueness of Solution

181

(1)

Laplace's Equation in Cartesian Coordinate System

182

(9)

The Calculus of Fourier Series

191

(32)

Introduction

191

(1)

Fourier Series Representation of a Function: Fourier Series as a Function

191

(4)

Differentiation of Fourier Series

195

(11)

Integration of Fourier Series

206

(7)

Fourier Series and the Gibbs Phenomenon

213

(10)

Separation of Variables: The Nonhomogeneous Problem

223

(56)

Introduction

223

(1)

Nonhomogeneous Pdes With Homogeneous BCS

224

(12)

Homogeneous PDE With Nonhomogeneous BCS

236

(22)

Homogeneous PDE---Nonhomogeneous Constant BCs

236

(6)

Homogeneous PDE---Nonhomogeneous Variable BCs

242

(16)

Nonhomogeneous PDE and BCs

258

(19)

Summary

277

(2)

The Sturm-Liouville Eigenvalue Problem

279

(38)

Introduction

279

(2)

Definition of the Sturm-Liouville Eignevalue Problem

281

(12)

Rayleigh Quotient

293

(5)

The General PDE Example

298

(4)

Problems Involving Homogeneous BCS of the Third Kind

302

(15)

Solution of Linear Homogeneous Variable-Coefficient ODE

317

(34)

Introduction

317

(1)

Some Facts About the General Second-Order ODE

318

(2)

Euler's Equation

320

(5)

Brief Review of Power Series

325

(2)

The Power Series Solution Method

327

(7)

Legender's Equation and Legendre Polynomials

334

(7)

Method of Frobenius and Bessel's Equation

341

(10)

Classical PDE Problems

351

(52)

Introduction

351

(1)

Laplace's Equation

351

(13)

Laplace's equation in the Polar Coordinate System

352

(4)

Laplace's equation in the Spherical Coordinate System

356

(8)

Transverse Vibrations of a Thin Beam

364

(9)

Derivation of the Beam Equation

364

(3)

Transverse Vibrations of a Simply Supported Thin Beam

367

(6)

Heat Conduction in a Circular Plate

373

(7)

Schrodinger's Equation

380

(9)

The Telegrapher's Equation

389

(9)

Development of the Telegrapher's Equation

389

(2)

Application of the Telegrapher's Equation to a Neuron

391

(7)

Interesting Problems in Diffusion

398

(5)

Fourier Integrals and Transform Methods

403

(59)

Introduction

403

(1)

The Fourier Integral

404

(17)

Development of the Fourier Integral

405

(9)

The Fourier Sine and Cosine Integrals

414

(7)

The Laplace Transform

421

(18)

Laplace transform Solution Method of ODEs

422

(2)

The Error Function

424

(1)

Laplace Transform Solution Method of PDEs

425

(14)

The Fourier Transform

439

(11)

Fourier Cosine and Sine Transforms

442

(2)

Fourier Transform Theorems

444

(6)

Fourier Transform Solution Method of PDES

450

(12)

Appendices

462

(39)

A Summary of the Spatial Problem

463

(2)

B Proofs of Related Theorems

465

(8)

B.1 Theorems From Chapter 2

465

(1)

B.1.1 Leibniz's Formula

465

(1)

B.1.2 Maximum-Minimum Theorem

466

(3)

B.2 Theorems From Chapter 4

469

(1)

B.2.1 Eigenvectors of Distinct Eigenvalues Are Linearly Independent

469

(1)

B.2.2 Eigenvectors of Distinct Eigenvalues of an n by n Matrix From a Basis for Rn

470

(1)

B.3 Theorem From Chapter 5

470

(3)

C Basics from Ordinary Differential Equations

473

(16)

C.1 Some Solution Methods for First-Order ODES

473

(1)

C.1.1 First-Order ODE Where k(t) Is a Constant

473

(2)

C.1.2 First-Order ODE Where k(t) Is a Function

475

(4)

C.2 Some Solution Methods of Second-Order ODES

479

(1)

C.2.1 Second-Order Linear Homogeneous ODEs

479

(6)

C.2.2 Second-Order Linear Nonhomogeneous ODEs

485

(4)

D Mathematical Notation

489

(2)

E Summary of Thermal Diffusivity of Common Materials

491

(2)

F Tables of Fourier and Laplace Transforms

493

(8)

F.1 Tables of Fourier, Fourier Cosine, and Fourier Sine Transforms

494

(4)

F.2 Table of Laplace Transforms

498

(3)

Bibliography

501

(3)

Index

504

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Very Applied First Course in Partial Differential Equations, A

Summary

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