Applied Partial Differential Equations

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

• ISBN13: 9780198527701
• ISBN10: 0198527705
• Edition: Revised
• Format: Hardcover
• Copyright: 2003-08-07
• Publisher: Oxford University Press

Summary

This title includes the following features: New edition of the well-knowntext by Ockendon et al.; Contains numerous new and topical exercises; Focuses onrecent applications of PDEs; Timely updates on: transform methods (especiallymultidimensional Fourier transforms and the Radon transform), explicitrepresentations of general solutions of the wave equation, bifurcations, theWiener-Hopf method, free surface flows, American options, the Monge-Ampereequation, linear elasticity, and complex characteristics

Author Biography

Alexander Movchan is professor, Department of Mathematical Sciences, 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 ad 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 Localised 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	(4)
6.5.3 General remarks	269	(2)
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 Euter 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