Research Catalog

Title

Applied partial differential equations / John Ockendon ... [et al.].

Publication

Oxford ; New York : Oxford University Press, c2003.

Items in the Library & Off-site

Details

Additional Authors

Ockendon, J. R.

Description

xi, 449 p. : ill.; 25 cm.

Series Statement

Oxford texts in applied and engineering mathematics

Uniform Title

Oxford texts in applied and engineering mathematics.

Subject

• Differential equations, Partial
• Equações diferenciais parciais
• Équations différentielles

Bibliography (note)

• Includes bibliographical references (p. 436-438) and index.

Contents

1 First-order scalar quasilinear equations 6 -- 1.2 Cauchy data 8 -- 1.3 Characteristics 9 -- 1.3.1 Linear and semilinear equations 11 -- 1.4 Domain of definition and blow-up 13 -- 1.5 Quasilinear equations 15 -- 1.6 Solutions with discontinuities 19 -- 1.7 Weak solutions 22 -- 1.8 More independent variables 25 -- 2 First-order quasilinear systems 35 -- 2.1 Motivation and models 35 -- 2.2 Cauchy data and characteristics 41 -- 2.3 The Cauchy-Kowalevski theorem 45 -- 2.4 Hyperbolicity 48 -- 2.4.1 Two-by-two systems 49 -- 2.4.2 Systems of dimension n 50 -- 2.5 Weak solutions and shock waves 55 -- 2.5.1 Causality 56 -- 2.5.2 Viscosity and entropy 59 -- 2.5.3 Other discontinuities 62 -- 2.6 Systems with more than two independent variables 63 -- 3 Introduction to second-order scalar equations 76 -- 3.1 Preamble 76 -- 3.2 The Cauchy problem for semilinear equations 78 -- 3.3 Characteristics 80 -- 3.4 Canonical forms for semilinear equations 83 -- 3.4.1 Hyperbolic equations 83 -- 3.4.2 Elliptic equations 84 -- 3.4.3 Parabolic equations 86 -- 4 Hyperbolic equations 93 -- 4.2 Linear equations: the solution to the Cauchy problem 94 -- 4.2.1 An ad hoc approach to Riemann functions 94 -- 4.2.2 The rationale for Riemann functions 96 -- 4.2.3 Implications of the Riemann function representation 100 -- 4.3 Non-Cauchy data for the wave equation 102 -- 4.3.1 Strongly discontinuous boundary data 105 -- 4.4 Transforms and eigenfunction expansions 106 -- 4.5 Applications to wave equations 113 -- 4.5.1 The wave equation in one space dimension 113 -- 4.5.2 Circular and spherical symmetry 116 -- 4.5.3 The telegraph equation 118 -- 4.5.4 Waves in periodic media 119 -- 4.6 Wave equations with more than two independent variables 120 -- 4.6.1 The method of descent and Huygens' principle 120 -- 4.6.2 Hyperbolicity and time-likeness 125 -- 4.7 Higher-order systems 128 -- 4.7.1 Linear elasticity 128 -- 4.7.2 Maxwell's equations of electromagnetism 131 -- 4.8 Nonlinearity 135 -- 4.8.1 Simple waves 135 -- 4.8.2 Hodograph methods 137 -- 4.8.3 Liouville's equation 139 -- 4.8.4 Another method 141 -- 5 Elliptic equations 151 -- 5.1 Models 151 -- 5.1.1 Gravitation 151 -- 5.1.2 Electromagnetism 152 -- 5.1.3 Heat transfer 153 -- 5.1.4 Mechanics 155 -- 5.1.5 Acoustics 160 -- 5.1.6 Aerofoil theory and fracture 161 -- 5.2 Well-posed boundary data 163 -- 5.2.1 The Laplace and Poisson equations 163 -- 5.2.2 More general elliptic equations 166 -- 5.3 The maximum principle 167 -- 5.4 Variational principles 168 -- 5.5 Green's functions 169 -- 5.5.1 The classical formulation 169 -- 5.5.2 Generalised function formulation 171 -- 5.6 Explicit representations of Green's functions 174 -- 5.6.1 Laplace's equation and Poisson's equation 174 -- 5.6.2 Helmholtz' equation 180 -- 5.6.3 The modified Helmholtz equation 182 -- 5.7 Green's functions, eigenfunction expansions and transforms 183 -- 5.7.1 Eigenvalues and eigenfunctions 183 -- 5.7.2 Green's functions and transforms 184 -- 5.8 Transform solutions of elliptic problems 186 -- 5.8.1 Laplace's equation with cylindrical symmetry: Hankel transforms 187 -- 5.8.2 Laplace's equation in a wedge geometry; the Mellin transform 190 -- 5.8.3 Helmholtz' equation 191 -- 5.8.4 Higher-order problems 194 -- 5.9 Complex variable methods 195 -- 5.9.1 Conformal maps 197 -- 5.9.2 Riemann -- Hilbert problems 199 -- 5.9.3 Mixed boundary value problems and singular integral equations 204 -- 5.9.4 The Wiener -- Hopf method 206 -- 5.9.5 Singularities and index 209 -- 5.10 Localised boundary data 211 -- 5.11 Nonlinear problems 212 -- 5.11.1 Nonlinear models 212 -- 5.11.2 Existence and uniqueness 213 -- 5.11.3 Parameter dependence and singular behaviour 215 -- 5.12 Liouville's equation again 221 -- 5.13 Postscript: [down triangle, open superscript 2] or -[Delta]? 222 -- 6 Parabolic equations 241 -- 6.1 Linear models of diffusion 241 -- 6.1.1 Heat and mass transfer 241 -- 6.1.2 Probability and finance 243 -- 6.1.3 Electromagnetism 245 -- 6.2 Initial and boundary conditions 245 -- 6.3 Maximum principles and well-posedness 247 -- 6.3.1 The strong maximum principle 248 -- 6.4 Green's functions and transform methods for the heat equation 249 -- 6.4.1 Green's functions: general remarks 249 -- 6.4.2 The Green's function for the heat equation with no boundaries 251 -- 6.4.3 Boundary value problems 254 -- 6.4.4 Convection -- diffusion problems 260 -- 6.5 Similarity solutions and groups 262 -- 6.5.1 Ordinary differential equations 264 -- 6.5.2 Partial differential equations 265 -- 6.6 Nonlinear equations 271 -- 6.6.1 Models 271 -- 6.6.2 Theoretical remarks 275 -- 6.6.3 Similarity solutions and travelling waves 275 -- 6.6.4 Comparison methods and the maximum principle 281 -- 6.6.5 Blow-up 284 -- 6.7 Higher-order equations and systems 286 -- 6.7.1 Higher-order scalar problems 287 -- 6.7.2 Higher-order systems 289 -- 7 Free boundary problems 305 -- 7.1 Introduction and models 305 -- 7.1.1 Stefan and related problems 306 -- 7.1.2 Other free boundary problems in diffusion 310 -- 7.1.3 Some other problems from mechanics 314 -- 7.2 Stability and well-posedness 318 -- 7.2.1 Surface gravity waves 319 -- 7.2.2 Vortex sheets 321 -- 7.2.3 Hele-Shaw flow 322 -- 7.2.4 Shock waves 324 -- 7.3 Classical solutions 326 -- 7.3.1 Comparison methods 326 -- 7.3.2 Energy methods and conserved quantities 327 -- 7.3.3 Green's functions and integral equations 328 -- 7.4 Weak and variational methods 329 -- 7.4.1 Variational methods 330 -- 7.4.2 The enthalpy method 335 -- 7.5 Explicit solutions 338 -- 7.5.1 Similarity solutions 339 -- 7.5.2 Complex variable methods 341 -- 7.6 Regularisation 345 -- 8 Non-quasilinear equations 359 -- 8.2 Scalar first-order equations 360 -- 8.2.1 Two independent variables 360 -- 8.2.2 More independent variables 366 -- 8.2.3 The eikonal equation 366 -- 8.2.4 Eigenvalue problems 374 -- 8.2.5 Dispersion 376 -- 8.2.6 Bicharacteristics 377 -- 8.3 Hamilton -- Jacobi equations and quantum mechanics 378 -- 8.4 Higher-order equations 380 -- 9 Miscellaneous topics 393 -- 9.2 Linear systems revisited 395 -- 9.2.1 Linear systems: Green's functions 396 -- 9.2.2 Linear elasticity 398 -- 9.2.3 Linear inviscid hydrodynamics 400 -- 9.2.4 Wave propagation and radiation conditions 403 -- 9.3 Complex characteristics and classification by type 405 -- 9.4 Quasilinear systems with one real characteristic 407 -- 9.4.1 Heat conduction with ohmic heating 407 -- 9.4.2 Space charge 408 -- 9.4.3 Fluid dynamics: the Navier -- Stokes equations 409 -- 9.4.4 Inviscid flow: the Euler equations 409 -- 9.4.5 Viscous flow 412 -- 9.5 Interaction between media 414 -- 9.5.1 Fluid/solid acoustic interactions 414 -- 9.5.2 Fluid/fluid gravity wave interaction 415 -- 9.6 Gauges and invariance 415 -- 9.7 Solitons 417.

ISBN

• 0198527705 (hbk. : acid-free paper)
• 9780198527701 (hbk. : acid-free paper)
• 0198527713 (pbk. : acid-free paper)
• 9780198527718 (pbk. : acid-free paper)

LCCN

• 2003276601
• 99948389915

OCLC

• ocm52486357\
• 52486357
• SCSB-4820157

Owning Institutions

Columbia University Libraries