Applied Partial Differential Equations,9780198527718

Applied Partial Differential Equations

by ; ; ;
Edition: Revised
ISBN13: 9780198527718
ISBN10: 0198527713
Format: Paperback
Pub. Date: 2003-08-07
Publisher(s): Oxford University Press
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Summary

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

Author Biography

John Ockendon and Sam Howison are university lecturers working in the Oxford Centre for Industrial and Applied Mathematics, University of Oxford Andrew Lacey is a professor at Heriot-Watt University Alexander Movchan is a professor at Liverpool University

Table of Contents

Introduction 1(5)
1 First-order scalar quasilinear equations 6(29)
1.1 Introduction
6(2)
1.2 Cauchy data
8(1)
1.3 Characteristics
9(4)
1.3.1 Linear and semilinear equations
11(2)
1.4 Domain of definition and blow-up
13(2)
1.5 Quasilinear equations
15(4)
1.6 Solutions with discontinuities
19(3)
1.7 Weak solutions
22(3)
1.8 More independent variables
25(3)
1.9 Postscript
28(1)
Exercises
29(6)
2 First-order quasilinear systems 35(41)
2.1 Motivation and models
35(6)
2.2 Cauchy data and characteristics
41(4)
2.3 The Cauchy-Kowalevski theorem
45(3)
2.4 Hyperbolicity
48(7)
2.4.1 Two-by-two systems
49(1)
2.4.2 Systems of dimension n
50(2)
2.4.3 Examples
52(3)
2.5 Weak solutions and shock waves
55(8)
2.5.1 Causality
56(3)
2.5.2 Viscosity and entropy
59(3)
2.5.3 Other discontinuities
62(1)
2.6 Systems with more than two independent variables
63(5)
Exercises
68(8)
3 Introduction to second-order scalar equations 76(17)
3.1 Preamble
76(2)
3.2 The Cauchy problem for semilinear equations
78(2)
3.3 Characteristics
80(3)
3.4 Canonical forms for semilinear equations
83(4)
3.4.1 Hyperbolic equations
83(1)
3.4.2 Elliptic equations
84(2)
3.4.3 Parabolic equations
86(1)
3.5 Some general remarks
87(2)
Exercises
89(4)
4 Hyperbolic equations 93(58)
4.1 Introduction
93(1)
4.2 Linear equations: the solution to the Cauchy problem
94(8)
4.2.1 An as hoc approach to Riemann functions
94(2)
4.2.2 The rationale for Riemann functions
96(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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