Ockendon J., Howison S., Lacey A., Movchan A. Applied Partial Differential Equations
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448 INDEX
singular integral equation, 203. 204
slow viscous flow, 159
Airy stress function, 351
drop spreading under gravity, 273
drop spreading with surface tension, 282
free boundary problem, 3511
Hele-Shaw cell, 302
non-Newtonian, 213
soliton
KdV equation, 921
Lax formulation, 420
solution surface, 8
Sommerfeld problem, 192
radiation condition, 192
solution, 235
Sommerfeld radiation condition, 181
Sommerfeld representation, 226
sonic line, 82
space charge, 408
space-like, 65, 104
boundary, 104
heat equation, 24fi
specular reflection, 372
sphere in R'", surface area, 123
spinodal, 288
squeeze film, 309
stationary phase, 3611
Stefan condition, 30
Stefan problem
comparison method, 326
conserved quantities, 328
enthalpy method, 336
formulation, 3011
Green's function approach, 328
impurities, 302
in superconductivity, 302
integral equation formulation, 328
link with oxygen consumption problem,
333
mushy region, 3311
Neumann similarity solution, 3411
one-phase, 302
regularisation, 3411
supercooled, 323
blow-up, 328
travelling wave solution, 339
with volumetric heating, 336
Stefan--Boltzmann law, 155. 272
stoichiometric condition, 311. 312
stream function, 213
boundary layer equations, 224
in slow flow, 189
Navier-Stokes equations, 413
streamline, 410
stress intensification, 205
strong maximum principle, 248
subcharacteristics, 88
subsolution, 214
supersolution, 214
supersonic, 43
support, 276
surface gravity waves, 319
between two fluids, 415
'1hylor diffusion, 261
telegraph equation, 8fi
electromagnetism, 148
for regenerator model, 93
Fourier transform, 118
propagation of discontinuities, 119
Riemann function, 102, 18
similarity variable, 1D2
test function, 22, 106
distribution, 9Z
thermal diffusivity, 242
thermistor, 407, 429
thunder, 123
tides, model for, 92
time-like, 55 65, 104
boundary, 104. 196
heat equation, 246
direction
acoustics, 127
electromagnetic waves, 133
Euler equations, 412
tomography, 112
torsion bar, 156
Prandtl stress function, 1115
traffic model, L 2,9 32
transcritical bifurcation, 219
transform
complex Fourier, 109
finite Fourier, 180
Fourier, 102
Fourier sine, 186
Hankel, 117. 188
Hilbert, 129
Kontorovich--Lebedev, 193
Laplace, 108
Mellin, 190
Radon, 112
transonic flow, 78
hodograph transformation, 78
travelling wave
Burgers' equation, 277. ,280
Cahn-Allen equation, 277
Fisher equation, 276
semilinear parabolic equation, 276
Ilricomi equation, 78, 82, 90
characteristics, 82
Turing instability, 2811
unilateral minimisation problem, 331
unitary, 419
univalent, 198
upper solution
elliptic equation, 214
parabolic equation, 281
variational inequality, 331
INDEX 449
contact problem, 334
evolution, 333
obstacle problem, 331
variational principle
free boundary problem, 3311
minimal surface, 214
nonlinear elliptic equation, 2L4
p-Laplace equation, 214
patch of vorticity, 334
Poisson's equation, 168
unilateral minimisation, 331
variational inequality, 331
velocity potential, 77, 411
complex, 341
virtual dislocation, 205
viscosity, 6 37. 52
variable, 213
volcano growth, 273
vortex, 19. 1511
dynamics, 347
horseshoe, 403
starting, 402
vortex ring, 403
vortex sheet, 59, 3111
linear stability, 321
vorticity, 155. 212
conservation, 410, 412
matrix, 412
water-entry problem, 356
wave
electromagnetic, 131
anisotropic, 133
radiation condition, 4114
N-wave, 13Z
P-wave, 134
Rayleigh, 138
S-wave, 134
simple, 1,35
surface gravity, 319
wave equation, 83, 23
d'Alembert solution,
92, 100, 102,
114
canonical form, 84
characteristic variables, 84
Duhamel's method, 123
frequency domain, 134
general solution
one dimension, 84, 115
three dimensions, 125
two dimensions, 125
Goursat problem, 104
Hankel transform, LIZ
Huygens' principle, 122
in periodic media, 119
initial-boundary value problem, 142
interface conditions, 134
kinematic, s 16
method of descent, 123
more than two independent variables,
12(1
non-Cauchy data, 102
one-dimensional
transform methods, 113
two-point boundary value problem,
115
retarded potential solution, 121
Riemann function, 101
Riemann invariants, 894 103
strongly discontinuous data, 105
three-dimensional radially-symmetric,
117. 120
two-dimensional axially-symmetric, 1111
weak solution, 1115
weak solution, 22, 9Z
enthalpy method, 335
first-order quasilinear system, 55
first-order scalar linear equation, 22
free boundary problem, 322
Poisson's equation, 16$
porous medium equation, 338
scalar conservation law, 24
vector conservation law, M
wave equation, 145
well-posed, 1, 41
parabolic equation, 24Z
Poisson's equation, 168
Wiener--Hopf method, 206
WKB approximation, 359
dispersion, 376
year, coldest day of, 291
Young's modulus, 1.28
zone of silence, 1211
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