| |
| |
| Introduction | |
| |
| |
| |
| First-order scalar quasilinear equations | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| Cauchy data | |
| |
| |
| |
| Characteristics | |
| |
| |
| |
| Linear and semilinear equations | |
| |
| |
| |
| Domain of definition and blow-up | |
| |
| |
| |
| Quasilinear equations | |
| |
| |
| |
| Solutions with discontinuities | |
| |
| |
| |
| Weak solutions | |
| |
| |
| |
| More independent variables | |
| |
| |
| |
| Postscript | |
| |
| |
| Exercises | |
| |
| |
| |
| First-order quasilinear systems | |
| |
| |
| |
| Motivation and models | |
| |
| |
| |
| Cauchy data and characteristics | |
| |
| |
| |
| The Cauchy-Kowalevski theorem | |
| |
| |
| |
| Hyperbolicity | |
| |
| |
| |
| Two-by-two systems | |
| |
| |
| |
| Systems of dimension n | |
| |
| |
| |
| Examples | |
| |
| |
| |
| Weak solutions and shock waves | |
| |
| |
| |
| Causality | |
| |
| |
| |
| Viscosity and entropy | |
| |
| |
| |
| Other discontinuities | |
| |
| |
| |
| Systems with more than two independent variables | |
| |
| |
| Exercises | |
| |
| |
| |
| Introduction to second-order scalar equations | |
| |
| |
| |
| Preamble | |
| |
| |
| |
| The Cauchy problem for semilinear equations | |
| |
| |
| |
| Characteristics | |
| |
| |
| |
| Canonical forms for semilinear equations | |
| |
| |
| |
| Hyperbolic equations | |
| |
| |
| |
| Elliptic equations | |
| |
| |
| |
| Parabolic equations | |
| |
| |
| |
| Some general remarks | |
| |
| |
| Exercises | |
| |
| |
| |
| Hyperbolic equations | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| Linear equations: the solution to the Cauchy problem | |
| |
| |
| |
| An ad hoc approach to Riemann functions | |
| |
| |
| |
| The rationale for Riemann functions | |
| |
| |
| |
| Implications of the Riemann function representation | |
| |
| |
| |
| Non-Cauchy data for the wave equation | |
| |
| |
| |
| Strongly discontinuous boundary data | |
| |
| |
| |
| Transforms and eigenfunction expansions | |
| |
| |
| |
| Applications to wave equations | |
| |
| |
| |
| The wave equation in one space dimension | |
| |
| |
| |
| Circular and spherical symmetry | |
| |
| |
| |
| The telegraph equation | |
| |
| |
| |
| Waves in periodic media | |
| |
| |
| |
| General remarks | |
| |
| |
| |
| Wave equations with more than two independent variables | |
| |
| |
| |
| The method of descent and Huygens' principle | |
| |
| |
| |
| Hyperbolicity and time-likeness | |
| |
| |
| |
| Higher-order systems | |
| |
| |
| |
| Linear elasticity | |
| |
| |
| |
| Maxwell's equations of electromagnetism | |
| |
| |
| |
| Nonlinearity | |
| |
| |
| |
| Simple waves | |
| |
| |
| |
| Hodograph methods | |
| |
| |
| |
| Liouville's equation | |
| |
| |
| |
| Another method | |
| |
| |
| Exercises | |
| |
| |
| |
| Elliptic equations | |
| |
| |
| |
| Models | |
| |
| |
| |
| Gravitation | |
| |
| |
| |
| Electromagnetism | |
| |
| |
| |
| Heat transfer | |
| |
| |
| |
| Mechanics | |
| |
| |
| |
| Acoustics | |
| |
| |
| |
| Aerofoil theory and fracture | |
| |
| |
| |
| Well-posed boundary data | |
| |
| |
| |
| The Laplace and Poisson equations | |
| |
| |
| |
| More general elliptic equations | |
| |
| |
| |
| The maximum principle | |
| |
| |
| |
| Variational principles | |
| |
| |
| |
| Green's functions | |
| |
| |
| |
| The classical formulation | |
| |
| |
| |
| Generalised function formulation | |
| |
| |
| |
| Explicit representations of Green's functions | |
| |
| |
| |
| Laplace's equation and Poisson's equation | |
| |
| |
| |
| Helmholtz' equation | |
| |
| |
| |
| The modified Helmholtz equation | |
| |
| |
| |
| Green's functions, eigenfunction expansions and transforms | |
| |
| |
| |
| Eigenvalues and eigenfunctions | |
| |
| |
| |
| Green's functions and transforms | |
| |
| |
| |
| Transform solutions of elliptic problems | |
| |
| |
| |
| Laplace's equation with cylindrical symmetry: Hankel transforms | |
| |
| |
| |
| Laplace's equation in a wedge geometry; the Mellin transform | |
| |
| |
| |
| Helmholtz' equation | |
| |
| |
| |
| Higher-order problems | |
| |
| |
| |
| Complex variable methods | |
| |
| |
| |
| Conformal maps | |
| |
| |
| |
| Riemann--Hilbert problems | |
| |
| |
| |
| Mixed boundary value problems and singular integral equations | |
| |
| |
| |
| The Wiener--Hopf method | |
| |
| |
| |
| Singularities and index | |
| |
| |
| |
| Localised boundary data | |
| |
| |
| |
| Nonlinear problems | |
| |
| |
| |
| Nonlinear models | |
| |
| |
| |
| Existence and uniqueness | |
| |
| |
| |
| Parameter dependence and singular behaviour | |
| |
| |
| |
| Liouville's equation again | |
| |
| |
| |
| Postscript: [down triangle, open superscript 2] or -[Delta]? | |
| |
| |
| Exercises | |
| |
| |
| |
| Parabolic equations | |
| |
| |
| |
| Linear models of diffusion | |
| |
| |
| |
| Heat and mass transfer | |
| |
| |
| |
| Probability and finance | |
| |
| |
| |
| Electromagnetism | |
| |
| |
| |
| General remarks | |
| |
| |
| |
| Initial and boundary conditions | |
| |
| |
| |
| Maximum principles and well-posedness | |
| |
| |
| |
| The strong maximum principle | |
| |
| |
| |
| Green's functions and transform methods for the heat equation | |
| |
| |
| |
| Green's functions: general remarks | |
| |
| |
| |
| The Green's function for the heat equation with no boundaries | |
| |
| |
| |
| Boundary value problems | |
| |
| |
| |
| Convection--diffusion problems | |
| |
| |
| |
| Similarity solutions and groups | |
| |
| |
| |
| Ordinary differential equations | |
| |
| |
| |
| Partial differential equations | |
| |
| |
| |
| General remarks | |
| |
| |
| |
| Nonlinear equations | |
| |
| |
| |
| Models | |
| |
| |
| |
| Theoretical remarks | |
| |
| |
| |
| Similarity solutions and travelling waves | |
| |
| |
| |
| Comparison methods and the maximum principle | |
| |
| |
| |
| Blow-up | |
| |
| |
| |
| Higher-order equations and systems | |
| |
| |
| |
| Higher-order scalar problems | |
| |
| |
| |
| Higher-order systems | |
| |
| |
| Exercises | |
| |
| |
| |
| Free boundary problems | |
| |
| |
| |
| Introduction and models | |
| |
| |
| |
| Stefan and related problems | |
| |
| |
| |
| Other free boundary problems in diffusion | |
| |
| |
| |
| Some other problems from mechanics | |
| |
| |
| |
| Stability and well-posedness | |
| |
| |
| |
| Surface gravity waves | |
| |
| |
| |
| Vortex sheets | |
| |
| |
| |
| Hele-Shaw flow | |
| |
| |
| |
| Shock waves | |
| |
| |
| |
| Classical solutions | |
| |
| |
| |
| Comparison methods | |
| |
| |
| |
| Energy methods and conserved quantities | |
| |
| |
| |
| Green's functions and integral equations | |
| |
| |
| |
| Weak and variational methods | |
| |
| |
| |
| Variational methods | |
| |
| |
| |
| The enthalpy method | |
| |
| |
| |
| Explicit solutions | |
| |
| |
| |
| Similarity solutions | |
| |
| |
| |
| Complex variable methods | |
| |
| |
| |
| Regularisation | |
| |
| |
| |
| Postscript | |
| |
| |
| Exercises | |
| |
| |
| |
| Non-quasilinear equations | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| Scalar first-order equations | |
| |
| |
| |
| Two independent variables | |
| |
| |
| |
| More independent variables | |
| |
| |
| |
| The eikonal equation | |
| |
| |
| |
| Eigenvalue problems | |
| |
| |
| |
| Dispersion | |
| |
| |
| |
| Bicharacteristics | |
| |
| |
| |
| Hamilton--Jacobi equations and quantum mechanics | |
| |
| |
| |
| Higher-order equations | |
| |
| |
| Exercises | |
| |
| |
| |
| Miscellaneous topics | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| Linear systems revisited | |
| |
| |
| |
| Linear systems: Green's functions | |
| |
| |
| |
| Linear elasticity | |
| |
| |
| |
| Linear inviscid hydrodynamics | |
| |
| |
| |
| Wave propagation and radiation conditions | |
| |
| |
| |
| Complex characteristics and classification by type | |
| |
| |
| |
| Quasilinear systems with one real characteristic | |
| |
| |
| |
| Heat conduction with ohmic heating | |
| |
| |
| |
| Space charge | |
| |
| |
| |
| Fluid dynamics: the Navier--Stokes equations | |
| |
| |
| |
| Inviscid flow: the Euler equations | |
| |
| |
| |
| Viscous flow | |
| |
| |
| |
| Interaction between media | |
| |
| |
| |
| Fluid/solid acoustic interactions | |
| |
| |
| |
| Fluid/fluid gravity wave interaction | |
| |
| |
| |
| Gauges and invariance | |
| |
| |
| |
| Solitons | |
| |
| |
| Exercises | |
| |
| |
| Conclusion | |
| |
| |
| References | |
| |
| |
| Index | |