Applied Partial Differential Equations

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

• ISBN13: 9780198532439
• ISBN10: 0198532431
• Format: Paperback
• Copyright: 1999-09-23
• Publisher: Oxford University Press

Summary

Partial differential equations (PDEs) are a central concept in mathematics. Their power lies in their universality: there is a huge and ever-growing range of real-world phenomena to which they can be applied, from fluid mechanics and electromagnetism to probability and finance. This is aclearly written guide to the theory and applications of PDEs.Its central aim is to set out, in an informal yet rigorous manner, a mathematical framework within which to assess any given PDE. Space is devoted as much to explicit methods of solution as to more general qualitative ideas, the most important of which is the concept of well-posedness. Thisattribute is vital in deciding the accuracy to which the problem can be solved numerically, and it becomes increasingly important as the power of computer software grows. Prerequisites here have been kept to a minimum: some familiarity with ordinary differential equations, functions of a singlecomplex variable, and the calculus of functions of several real variables are all that is needed.

Table of Contents

Introduction	1	(6)
First-order scalar quasilinear equations	6	(29)
Introduction	6	(2)
Cauchy data	8	(1)
Characteristics	9	(4)
Linear and semilinear equations	11	(2)
Domain of definition and blow-up	13	(2)
Quasilinear equations	15	(4)
Solutions with discontinuities	19	(3)
Weak solutions	22	(3)
More independent variables	25	(3)
Postscript	28	(7)
First-order quasilinear systems	35	(40)
Motivation and models	35	(6)
Cauchy data and characteristics	41	(4)
The Cauchy-Kowalevski theorem	45	(3)
Hyperbolicity	48	(7)
Two-by-two systems	49	(1)
Systems of dimension n	50	(2)
Examples	52	(3)
Weak solutions and shock waves	55	(8)
Causality	56	(4)
Viscosity and entropy	60	(1)
Other discontinuities	61	(2)
Systems with more than two independent variables	63	(12)
Introduction to second-order scalar equations	75	(17)
Preamble	75	(2)
The Cauchy problem for semilinear equations	77	(2)
Characteristics	79	(2)
Canonical forms for semilinear equations	81	(5)
Hyperbolic equations	81	(2)
Elliptic equations	83	(2)
Parabolic equations	85	(1)
Some general remarks	86	(6)
Hyperbolic equations	92	(52)
Introduction	92	(1)
The Cauchy problem	93	(8)
An ad hoc approach to Riemann functions	93	(2)
The rationale for Riemann functions	95	(4)
Implications of the Riemann function representation	99	(2)
Non-Cauchy data for the wave equation	101	(4)
Strongly discontinuous boundary data	104	(1)
Transforms and eigenfunction expansions	105	(5)
Applications to wave equations	110	(7)
One space dimension	110	(2)
Circular and spherical symmetry	112	(2)
The telegraph equation	114	(1)
Waves in periodic media	115	(1)
General remarks	116	(1)
More than two independent variables	117	(6)
The method of descent and Huygens' principle	117	(4)
Hyperbolicity and time-likeness	121	(2)
Higher-order systems	123	(6)
Linear elasticity	123	(3)
Maxwell's equations of electromagnetism	126	(3)
Nonlinearity	129	(15)
Simple waves	130	(2)
Hodograph methods	132	(2)
Liouville's equation	134	(1)
Another method	135	(9)
Elliptic equations	144	(82)
Models	144	(12)
Gravitation	144	(1)
Electromagnetism	145	(1)
Heat transfer	146	(2)
Mechanics	148	(5)
Acoustics	153	(1)
Aerofoil theory and fracture	154	(2)
Well-posed boundary data	156	(4)
The Laplace and Poisson equations	156	(3)
More general elliptic equations	159	(1)
The maximum principle	160	(1)
Variational principles	161	(1)
Green's functions	162	(5)
The classical formulation	162	(2)
Generalised function formulation	164	(3)
Explicit representations of Green's functions	167	(8)
Laplace's equation and Poisson's equation	167	(6)
Helmholtz' equation	173	(1)
The modified Helmholtz equation	174	(1)
Transforms	175	(3)
Eigenvalues and eigenfunctions	175	(1)
Green's functions and transforms	176	(2)
Transform solutions of elliptic problems	178	(9)
Laplace's equation with cylindrical symmetry: Hankel transforms	179	(2)
The Mellin transform	181	(2)
Helmholtz' equation	183	(3)
Higher-order problems	186	(1)
Complex variable methods	187	(14)
Conformal maps	188	(1)
Riemann-Hilbert problems	189	(6)
Mixed boundary value problems	195	(2)
The Wiener-Hopf method	197	(2)
Singularities and index	199	(2)
Localised boundary data	201	(1)
Nonlinear problems	202	(9)
Nonlinear models	203	(1)
Existence and uniqueness	204	(1)
Parameter dependence and singular behaviour	205	(6)
Liouville's equation again	211	(1)
Postscript: ∇2 or - Δ?	212	(14)
Parabolic equations	226	(64)
Linear models of diffusion	226	(4)
Heat and mass transfer	226	(2)
Probability and finance	228	(2)
Electromagnetism	230	(1)
General remarks	230	(1)
Initial and boundary conditions	230	(2)
Maximum principles and well-posedness	232	(2)
The strong maximum principle	233	(1)
The heat equation	234	(13)
Green's functions: general remarks	234	(2)
Explicit representations	236	(3)
Boundary value problems	239	(6)
Convection--diffusion problems	245	(2)
Similarity solutions and groups	247	(9)
Ordinary differential equations	250	(1)
Partial differential equations	250	(4)
General remarks	254	(2)
Nonlinear equations	256	(16)
Models	256	(4)
Theoretical remarks	260	(1)
Similarity solutions and travelling waves	260	(6)
Comparison methods and the maximum principle	266	(3)
Blow-up	269	(3)
Higher-order equations and systems	272	(18)
Higher-order scalar problems	272	(2)
Higher-order systems	274	(16)
Free boundary problems	290	(52)
Introduction and models	290	(13)
Stefan and related problems	291	(4)
Other free boundary problems in diffusion	295	(4)
Some other problems from mechanics	299	(4)
Stability and well-posedness	303	(8)
Surface gravity waves	305	(1)
Vortex sheets	306	(2)
Hele-Shaw flow	308	(1)
Shock waves	309	(2)
Classical solutions	311	(3)
Comparison methods	311	(1)
Energy methods and conserved quantities	312	(1)
Green's functions and integral equations	313	(1)
Weak and variational methods	314	(9)
Variational methods	315	(5)
The enthalpy method	320	(3)
Explicit solutions	323	(7)
Similarity solutions	323	(3)
Complex variable methods	326	(4)
Regularisation	330	(1)
Postscript	331	(11)
Non-quasilinear equations	342	(31)
Introduction	342	(1)
Scalar first-order equations	343	(16)
Two independent variables	343	(5)
More independent variables	348	(1)
The eikonal equation	349	(7)
Eigenvalue problems	356	(2)
Dispersion	358	(1)
Hamilton--Jacobi equations and quantum mechanics	359	(3)
Higher-order equations	362	(11)
Miscellaneous topics	373	(38)
Introduction	373	(2)
Linear systems revisited	375	(9)
Linear systems: Green's functions	376	(2)
Linear elasticity	378	(2)
Linear inviscid hydrodynamics	380	(3)
Wave propagation and radiation conditions	383	(1)
Quasilinear systems with one real characteristic	384	(7)
Heat conduction with ohmic heating	384	(1)
Space charge	385	(1)
Fluid dynamics: the Navier-Stokes equations	386	(1)
Inviscid flow; the Euler equations	386	(3)
Viscous flow	389	(2)
Interaction between media	391	(1)
Fluid/solid acoustic interactions	391	(1)
Fluid/fluid gravity wave interaction	392	(1)
Gauges and invariance	392	(2)
Solitons	394	(17)
Conclusion	411	(2)
References	413	(4)
Index	417