Strauss W.A. — Partial Differential Equations: An Introduction :: Электронная библиотека попечительского совета мехмата МГУ

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Strauss W.A. — Partial Differential Equations: An Introduction

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Название: Partial Differential Equations: An Introduction

Автор: Strauss W.A.

Аннотация:

Our understanding of the fundamental processes of the natural world is based to a large extent on partial differential equations (PDEs). The second edition of Partial Differential Equations provides an introduction to the basic properties of PDEs and the ideas and techniques that have proven useful in analyzing them. It provides the student a broad perspective on the subject, illustrates the incredibly rich variety of phenomena encompassed by it, and imparts a working knowledge of the most important techniques of analysis of the solutions of the equations.

In this book mathematical jargon is minimized. Our focus is on the three most classical PDEs, the wave, heat and Lapace equations. Advanced concepts are introduced frequently but with the least possible technicalities. The book is flexibly designed for juniors, seniors or beginning graduate students in science, engineering or mathematics.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 1st edition

Год издания: 1992

Количество страниц: 440

Добавлена в каталог: 13.09.2008

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Предметный указатель

Fourier series, pointwise convergence      125 132 140
Fourier series, triple      250 261
Fourier series, uniform convergence      124 135—136
Fourier sine series      84
Fourier sine series, and boundary conditions      111
Fourier sine series, coefficients in      101
Fourier sine series, double      158
Fourier transform      325—329
Fourier transform, convolution      328
Fourier transform, in spherical coordinates      332 357
Fourier transform, in three dimensions      329
Fourier transform, properties      327
Fourier’s law of heat      16 24
Fredholm alternative      301
frequency      85 256
Frequency variable      326
Full Fourier series      103—107
Full Fourier series, complex form      112—113
Function of two variables      3
fundamental frequency      85 256
Fundamental solution      48
future      231
Gamma function      268 338 395—396
Gas dynamics      22 25 343—345
Gauge, group      356
Gauge, invariance      356
Gauge, theory      355—356
Gauss — Seidel method      210
Gaussian and diffusion equation      48
Gaussian, transform of      326
Gauss’s theorem      393
Gelfand — Levitan method      371
General Fourier series      114 248
General solution      32
Geometric method      6 9 359
Ghost points      198
Gibb’s phenomenon      136—138
Grad, $\nabla$       169
Gradient      386
Gram — Schmidt orthogonalization      119 247
Graph of a function      392
Graphing method      92—95
Green      169
Green’s first identity      120 171
Green’s function      48 178—179
Green’s function, revisited      320—324
Green’s function, symmetry      179—181
Green’s second identity      115 176 246
Green’s Theorem      392
Green’s theorem, method using      73—74
Ground state      173 243 285
Guitar      11 265
Half-line      55—61
Half-space      181—183
Hammer blow      37 108—109
Hankel function      271 348
Harmonic function      16
Harmonic function, half-ball      187
Harmonic function, half-space      181—183
Harmonic function, infinitely differentiable      163
Harmonic function, maximum principle      162—163 172
Harmonic function, mean value property      162 171—172
Harmonic function, minimum energy      173—174 374
Harmonic function, Poisson’s formula      159—162
Harmonic function, quadrant      188
Harmonic function, rectangle      155—157
Harmonic function, representation formula      176—178
Harmonic function, rotationally invariant      151 153
Harmonic function, sphere      183—186
Harmonic oscillator, quantum mechanical      239—240
Heap of water      367
Heat conductivity      16
Heat equation      16
Heat exchange      18
Heat flow      15—16 19 22 65 298
Heat flow, convection      52
Heat flow, energy absorption      97
Heat flow, maximum principle      41
Heat flow, Robin condition      90
Heat flow, source function      49. 236 330—331
Heat flux      16 25
Heaviside function      318 326
Heisenberg uncertainty principle      327—328
Helium ion      17
Hermite polynomial      240
Hermite’s differential equation      239
Homogeneous linear equation      2 5
Hooke’slaw      21
Hopf maximum principle      172
Hot Spot      41 49
Hotrod      97
Huygens’s principle      39—40 222 227
Hydrogen atom      17—18 241—244 280—282 298
Hydrogen atom, continuous spectrum      351
Hyperbolic equation      28 30 53
Hyperbolic equation, Klein — Gordon equation      353 356
IC      see “Initial condition
Ill-conditioned matrix      26
Ill-posed problem      26
Impedance      23
Implicit function theorem      392
Implicit scheme      198
Incident wave      346
Incoming wave      346
Indicial equation      394
Infinite series      121—123 388—389
Infinitely differentiable solution      79 139 163
Inhomogeneous elliptic equation      301
Inhomogeneous linear equation      2 5 12 15
Inhomogeneous linear equation on finite interval      142
Inhomogeneous linear equation on half line      67 76
Inhomogeneous linear equation on whole line      65 69
Inhomogeneous medium      297 347 374
Inhomogeneous string      346—347
Initial condition      20 (see also “Boundary condition)
Inner product      114 116—117 246 255 298
Instability of matrix      26
Instability, diffusion equation      53
Insulation      22 88 90
Integral of derivatives      392—393
Intermediate Value Theorem      385
Invariance, diffusion equation      45—46
Invariance, KdV equation      374
Invariance, under dilations      40 46
Invariance, under Lorentz tranformations      221—222
Invariance, under rotations      150—153
Invariance, wave equation      40
Inverse scattering      369—372
Inversion formula, Fourier transform      326
Inversion formula, Laplace transform      335
Irrotational flow      23 147 345
Isotropic      150
Jacobi iteration      209—210
Jacobian      4 387
Jump conditions      24
Jump discontinuity      80 124 364 385
Kadomstev — Petviashvili equation      372
kinetic energy      38 218
Kinetic theory of gases      378
Kirchhoffs formula      222
Klein — Gordon equation      222 353—354
Klein — Gordon equation, solution of      356—357
Korteweg — deVries (KdV) equation      367
Kruskal, M.      368
Lagrange multiplier      284
Laguerre’s differential equation      242 281
Laplace equation      2 16 146
Laplace equation, annulus      165—166 299
Laplace equation, circle      159—162
Laplace equation, computation      208—210

Laplace equation, cube      157—158
Laplace equation, exterior of a circle      166 —167
Laplace equation, half-plane      187 332
Laplace equation, half-space      181—183
Laplace equation, maximum principle      148
Laplace equation, rectangle      155—156
Laplace equation, sphere      183—186
Laplace equation, wedge      164—165
Laplace transform      334
Laplace transform, techniques      334—337
Laplacian      14
Laplacian, four-dimensional      188
Laplacian, Green’s function      178 320
Laplacian, invariance      150—153
Laplacian, Neumann function      188
Laplacian, polar coordinates      151
Laplacian, spherical coordinates      153
Legendre function      274—278
Legendre function, associated      260 278
Legendre function, generating function      277
Legendre function, Legendre polynomial      118 275
Legendre function, normalizing constants      275
Legendre function, Rodrigues’ formula      276
Legendre function, zeros      276
Legendre’s differential equation      259—260 274
Legendre’s differential equation, associated      277
Light      22
Light cone      216—217
Light ray      230
LIMIT      384
Limit from the right, left      384
Linear elements      214
Linear operator      2
Linearity      2
Linearized Equation      22 345 380
Linearly dependent      5
Lorentz transformation      221
magnetic quantum number      279
Marching in time      192
Matrix equation      26
Maximin principle      307
Maximum of a function      385
Maximum principle, diffusion equation      41
Maximum principle, Hopf form      168 172
Maximum principle, Laplace equation      148 162
Maxwell’s equations      22 39 339—342
Mean value property      162 171—172
Mean value property, discrete      209
Mean-square convergence      122 124 249 295
Mesh size      189
Mesons      353
Metric,       127
Minimal surface equation      377
Minimax principle      291 294
Minimum principle, diffusion equation      41
Minimum principle, for first eigenvalue      284
Minimum principle, for nth eigenvalue      286
Minimum principle, Laplace equation      148
Minimum problem      284 374
Molecules      18
Momentum, conservation of      343 368
Momentum, density      40
Momentum, variable      327
Multiplicity      247—248 266
musical notes      85
N-soliton      372
Neumann condition      20 293—294 307
Neumann condition, Dirichlet’s principle      175
Neumann condition, discretization      197—198
Neumann condition, trial function      294
Neumann function for a domain      188
Neumann function for Bessel’s equation      271
Neumann problem, finite interval      87 112
Neumann problem, half-line      57
Neumann problem, uniqueness      174
Neumann problem, wedge      164
Newton’s Law of Cooling      18 22
Nodal set      264—268 313
nodes      243 264
Nonexistence      25
Nonlinear interaction      2 368
Nonlinear wave equation      2 205
Nonuniqueness      25
Norm,       126
Norm, uniform      70
Normal derivative      20
Normal mode      86
Normal vector      13 20 217
Normal velocity      23
Normalizing constants      131 270 275 370
Nth term test for divergence      388
Null vector      231
Observable      18
Odd extension      56 111 114
Odd function      55 110
ODE      1
ODE, arbitrary constants      3
ODE, Bessel      252 268
ODE, Euler      160 394
ODE, existence theorem      393
ODE, Hermite      239
ODE, Laguerre      242 281
ODE, method of power series      239 242 253 274 394
ODE, singular point      393
One-sided limit      124—125 384
Open interval      384
Open set      148 386
Open window, BC at      23
Operator      2 18
Operator, method      66 74 233
Orbital quantum number      279
Order      1
Orthogonal      114 116 117 247
Outgoing wave      348
Overdetermined      25 26
Overtone      85
Parabolic PDE      28 30
Parametrization of surface      391
Parseval’s equality      128 327
Partial sum      388
Particles, elementary      353
Past history      38 69 221 231
Pauli matrices      354
Perfect thermal conduction      22
Period      61 109
periodic extension      109
Periodic function      109 125
Periodic traveling wave      373
Phase plane      381
Phase shift      372
Piecewise       391
Piecewise continuous      124 385
Pitchfork      380
Plancherel’s theorem      327
Planck’s constant      17
Plane wave      221 347 351
plasma      368
Plucked string      35—36
Point mass      314
Point spectrum      351
Pointwise convergence      121 125 132 388
Poisson summation formula      330
Poisson’s equation      146 181 320
Poisson’s equation, finite element method      212
Poisson’s formula      159—162
Poisson’s formula in three dimensions      185—186
Polar coordinates      150 159
Population dynamics      15
Potential continuous spectrum      350—351
Potential electric      147 175

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