Applied Partial Differential Equations

, by Ockendon, John; Howison, Sam; Lacey, Andrew; Movchan, Alexander

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ISBN: 9780198527718 | 0198527713
Cover: Paperback
Copyright: 8/7/2003

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Partial differential equations are a central concept in mathematics. They are used in mathematical models of a huge range of real-world phenomena, from electromagnetism to financial markets. This new edition of the well-known text by Ockendon et al., providing an enthusiastic and clear guideto the theory and applications of PDEs, provides timely updates on: transform methods (especially multidimensional Fourier transforms and the Radon transform); explicit representations of general solutions of the wave equation; bifurcations; the Wiener-Hopf method; free surface flows; Americanoptions; the Monge-Ampere equation; linear elasticity and complex characteristics; as well as numerous topical exercises.This book is ideal for students of mathematics, engineering and physics seeking a comprehensive text in the modern applications of PDEs

Introduction

(5)

1 First-order scalar quasilinear equations

(29)

1.1 Introduction

(2)

1.2 Cauchy data

(1)

1.3 Characteristics

(4)

1.3.1 Linear and semilinear equations

(2)

1.4 Domain of definition and blow-up

(2)

1.5 Quasilinear equations

(4)

1.6 Solutions with discontinuities

(3)

1.7 Weak solutions

(3)

1.8 More independent variables

(3)

1.9 Postscript

(1)

Exercises

(6)

2 First-order quasilinear systems

(41)

2.1 Motivation and models

(6)

2.2 Cauchy data and characteristics

(4)

2.3 The Cauchy-Kowalevski theorem

(3)

2.4 Hyperbolicity

(7)

2.4.1 Two-by-two systems

(1)

2.4.2 Systems of dimension n

(2)

2.4.3 Examples

(3)

2.5 Weak solutions and shock waves

(8)

2.5.1 Causality

(3)

2.5.2 Viscosity and entropy

(3)

2.5.3 Other discontinuities

(1)

2.6 Systems with more than two independent variables

(5)

Exercises

(8)

3 Introduction to second-order scalar equations

(17)

3.1 Preamble

(2)

3.2 The Cauchy problem for semilinear equations

(2)

3.3 Characteristics

(3)

3.4 Canonical forms for semilinear equations

(4)

3.4.1 Hyperbolic equations

(1)

3.4.2 Elliptic equations

(2)

3.4.3 Parabolic equations

(1)

3.5 Some general remarks

(2)

Exercises

(4)

4 Hyperbolic equations

(58)

4.1 Introduction

(1)

4.2 Linear equations: the solution to the Cauchy problem

(8)

4.2.1 An as hoc approach to Riemann functions

(2)

4.2.2 The rationale for Riemann functions

(4)

4.2.3 Implications of the Riemann function representation

100

(2)

4.3 Non-Cauchy data for the wave equation

102

(4)

4.3.1 Strongly discontinuous boundary data

105

(1)

4.4 Transforms and eigenfunction expansions

106

(7)

4.5 Applications to wave equations

113

(7)

4.5.1 The wave equation in one space dimension

113

(3)

4.5.2 Circular and spherical symmetry

116

(2)

4.5.3 The telegraph equation

118

(1)

4.5.4 Waves in periodic media

119

(1)

4.5.5 General remarks

119

(1)

4.6 Wave equations with more than two independent variables

120

(8)

4.6.1 The method of descent and Huygens' principle

120

(5)

4.6.2 Hyperbolicity and time-likeness

125

(3)

4.7 Higher-order systems

128

(7)

4.7.1 Linear elasticity

128

(3)

4.7.2 Maxwell's equations of electromagnetism

131

(4)

4.8 Nonlinearity

135

(6)

4.8.1 Simple waves

135

(2)

4.8.2 Holograph methods

137

(2)

4.8.3 Liouville's equation

139

(2)

4.8.4 Another method

141

(1)

Exercises

141

(10)

5 Elliptic equations

151

(90)

5.1 Models

151

(12)

5.1.1 Gravitation

151

(1)

5.1.2 Electromagnetism

152

(1)

5.1.3 Heat transfer

153

(2)

5.1.4 Mechanics

155

(5)

5.1.5 Acoustics

160

(1)

5.1.6 Aerofoil theory and fracture

161

(2)

5.2 Well-posed boundary data

163

(4)

5.2.1 The Laplace and Poisson equations

163

(3)

5.2.2 More general elliptic equations

166

(1)

5.3 The maximum principle

167

(1)

5.4 Variational principles

168

(1)

5.5 Green's functions

169

(5)

5.5.1 The classical formulation

169

(2)

5.5.2 Generalised function formulation

171

(3)

5.6 Explicit representations of Green's functions

174

(9)

5.6.1 Laplace's equation and Poisson's equation

174

(6)

5.6.2 Helmholtz' equation

180

(2)

5.6.3 The modified Helmholtz equation

182

(1)

5.7 Green's functions, eigenfunction expansions and transforms

183

(3)

5.7.1 Eigenvalues and eigenfunctions

183

(1)

5.7.2 Green's functions and transforms

184

(2)

5.8 Transform solutions of elliptic problems

186

(9)

5.8.1 Laplace's equation with cylindrical symmetry: Hankel transforms

187

(3)

5.8.2 Laplace's equation in a wedge geometry; the Mellin transform

190

(1)

5.8.3 Helmholtz' equation

191

(3)

5.8.4 Higher-order problems

194

(1)

5.9 Complex variable methods

195

(16)

5.9.1 Conformal maps

197

(2)

5.9.2 Riemann-Hilbert problems

199

(5)

5.9.3 Mixed boundary value problems and singular integral equations

204

(2)

5.9.4 The Wiener-Hopf method

206

(3)

5.9.5 Singularities and index

209

(2)

5.10 Localized boundary data

211

(1)

5.11 Nonlinear problems

212

(9)

5.11.1 Nonlinear models

212

(1)

5.11.2 Existence and uniqueness

213

(2)

5.11.3 Parameter dependence and singular behaviour

215

(6)

5.12 Liouville's equation again

221

(1)

5.13 Postscript V2 or -A?

222

(1)

Exercises

223

(18)

6 Parabolic equations

241

(64)

6.1 Linear models of diffusion

241

(4)

6.1.1 Heat and mass transfer

241

(2)

6.1.2 Probability and finance

243

(2)

6.1.3 Electromagnetism

245

(1)

6.1.4 General remarks

245

(1)

6.2 Initial and boundary conditions

245

(2)

6.3 Maximum principles and well-posedness

247

(2)

6.3.1 The strong maximum principle

248

(1)

6.4 Green's functions and transform methods for the heat equation

249

(13)

6.4.1 Green's functions: general remarks

249

(2)

6.4.2 The Green's function for the heat equation with no boundaries

251

(3)

6.4.3 Boundary value problems

254

(6)

6.4.4 Convection-diffusion problems

260

(2)

6.5 Similarity solutions and groups

262

(9)

6.5.1 Ordinary differential equations

264

(1)

6.5.2 Partial differential equations

265

(6)

6.5.3 General remarks

271

(1)

6.6 Nonlinear equations

271

(15)

6.6.1 Models

271

(4)

6.6.2 Theoretical remarks

275

(1)

6.6.3 Similarity solutions and travelling waves

275

(6)

6.6.4 Comparison methods and the maximum principle

281

(3)

6.6.5 Blow-up

284

(2)

6.7 Higher-order equations and systems

286

(5)

6.7.1 Higher-order scalar problems

287

(2)

6.7.2 Higher-order systems

289

(2)

Exercises

291

(14)

7 Free boundary problems

305

(54)

7.1 Introduction and models

305

(13)

7.1.1 Stefan and related problems

306

(4)

7.1.2 Other free boundary problems in diffusion

310

(4)

7.1.3 Some other problems from mechanics

314

(4)

7.2 Stability and well-posedness

318

(8)

7.2.1 Surface gravity waves

319

(2)

7.2.2 Vortex sheets

321

(1)

7.2.3 Hele-Shaw flow

322

(2)

7.2.4 Shock waves

324

(2)

7.3 Classical solutions

326

(3)

7.3.1 Comparison methods

326

(1)

7.3.2 Energy methods and conserved quantities

327

(1)

7.3.3 Green's functions and integral equations

328

(1)

7.4 Weak and variational methods

329

(9)

7.4.1 Variational methods

330

(5)

7.4.2 The enthalpy method

335

(3)

7.5 Explicit solutions

338

(7)

7.5.1 Similarity solutions

339

(2)

7.5.2 Complex variable methods

341

(4)

7.6 Regularisation

345

(2)

7.7 Postscript

347

(2)

Exercises

349

(10)

8 Non-quasilinear equations

359

(34)

8.1 Introduction

359

(1)

8.2 Scalar first-order equations

360

(18)

8.2.1 Two independent variables

360

(6)

8.2.2 More independent variables

366

(1)

8.2.3 The eikonal equation

366

(8)

8.2.4 Eigenvalue problems

374

(2)

8.2.5 Dispersion

376

(1)

8.2.6 Bicharacteristics

377

(1)

8.3 Hamilton-Jacobi equations and quantum mechanics

378

(2)

8.4 Higher-order equations

380

(3)

Exercises

383

(10)

9 Miscellaneous topics

393

(41)

9.1 Introduction

393

(2)

9.2 Linear systems revisited

395

(10)

9.2.1 Linear systems: Green's functions

396

(2)

9.2.2 Linear elasticity

398

(2)

9.2.3 Linear inviscid hydrodynamics

400

(3)

9.2.4 Wave propagation and radiation conditions

403

(2)

9.3 Complex characteristics and classification by type

405

(2)

9.4 Quasilinear systems with one real characteristic

407

(7)

9.4.1 Heat conduction with ohmic heating

407

(1)

9.4.2 Space charge

408

(1)

9.4.3 Fluid dynamics: the Navier-Stokes equations

409

(1)

9.4.4 Inviscid flow: the Euler equations

409

(3)

9.4.5 Viscous flow

412

(2)

9.5 Interaction between media

414

(1)

9.5.1 Fluid/solid acoustic interactions

414

(1)

9.5.2 Fluid/fluid gravity wave interaction

415

(1)

9.6 Gauges and invariance

415

(2)

9.7 Solitons

417

(9)

Exercises

426

(8)

Conclusion

434

(2)

References

436

(3)

Index

439

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Applied Partial Differential Equations

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Author Biography

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