Weinberger H.F. — First course in partial defferential equations with complex variables and transform methods :: Электронная библиотека попечительского совета мехмата МГУ

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Weinberger H.F. — First course in partial defferential equations with complex variables and transform methods

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Название: First course in partial defferential equations with complex variables and transform methods

Автор: Weinberger H.F.

Аннотация:

This book is an attempt to present the materials usually covered in such courses in a framework where the general properties of partial differential equations such as characteristics, domains of dependence, and maximum principles can be clearly seen. It is intended for a one-year course in partial differential equations, including the elementary theory of complex variables. (The first seven chapters, or the first six and the last chapter form a one-semester course, and the first five chapters a one-quarter course.)

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Рубрика: Математика/Анализ/Дифференциальные уравнения/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1995

Количество страниц: 446

Добавлена в каталог: 18.04.2005

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Предметный указатель

Absolute value      204
Absolutely integrable function      305 306
Analytic at infinity      278
Analytic continuation      216
Analytic extension      215
Analytic function      103 213
Approximation, in the mean      70 72
Approximation, in the mean, pointwise      70
Argand diagram      201
Argument      204
Associated Legendre functions      192 193
Asymptotic behavior of the Laplace transform      349 350
Bessel function      149 179—181 340—344
Bessel function with imaginary argument      149 340—344
Bessel’s equation      176 179
Bessel’s inequality      73
Bilinear transformation      249
Boundary mesh point      380
Boundary value problem, two-point      120—126 168
Branch cut      227 293—296
Calculus of residues      282—292 303 304
Cauchy criterion      209 307
Cauchy integral representation      264 266
Cauchy principal value      284 287 290 291 313 332
Cauchy-Riemann equations      211 213 223 237
Cauchy’s Theorem      220 262
Chain rule for analytic functions      234
Change of scale      88—92 95
Characteristic (natural) frequency      154 186 187
Characteristic cone      153
Characteristic coordinates      46
Characteristic surface      154
Characteristic triangle      26
Characteristic values (eigenvalues)      65 124 160—168 171—178 180 186 187 391
Characteristics      18—21 38 42 43 45 59 115
Circle of convergence      210
Classification of partial differential operators      41—46 51 52 59
Closed curve      48 49
Complete set of functions      74 75 84 85 166 176—178
Completeness of eigenfunctions      166 176—178
Complex conjugate      203
complex derivative      214
complex numbers      201—206
Complex path integral      219
Conformal mapping      238—245 257—261
Conjugate harmonic function      212
Connected set      218
Conservation of energy      36
Continuity with respect to data      6 15 34 56 61 107 318 382
Contour integrals      220 261 262 263 271—277 282—296
convergence, in the mean      71 84 85 143 166 307 308 309
Convergence, in the mean, of Fourier series      79 80 81—88
Convergence, in the mean, pointwise      70
Convergence, in the mean, uniform      33 70 72 81 83 84 145 167 177 178 307
Convolution product      326 347
Convolution theorem      327 331 340 347
Cosine series      67 87 88
Cosine transform      321
Damped wave equation      112—115
Derivative of an analytic function      214
Derivatives of Bessel functions      179 180
Descent, method of      336
Difference quotient      375
Diffraction      369
Diffusion      59
Dimensional analysis      91
Dini’s test      79 80
Dirichlet’s principle      393 396
Discontinuities, propagation of      21 40 42 43 46 115 154
Divergence theorem      52 58
Domain      49 218
Domain of dependence      18 23 26 37 39 42 46 115 120 377 387 388
Domain of influence      20 23 26 40 42 46
Double Fourier series      141—145
Duhamel’s principle      371
d’Alembert solution      9 13 26 114
Eigenfunction      65 160—168 171—178 186
Eigenvalue      65 124 160—168 171—178 180 186 187 391
Elastic bar      6 91
Electrostatic potential      49 50
Elliptic coordinates      193
Elliptic differential operator      43 44 51 52
Energy      36 53 110 152—154
Entire function      217 266
Error bounds      86 97 98 102 145 266 386 389
Essential singularity      271
Existence      6
Exponential function      215 216 257
Extended plane      246
Finite difference approximation      375—383
Finite Fourier transform      129 130 186 197
Finite sine transform      127
Fixed boundary condition      23
Focusing      154
Fourier coefficients      72
Fourier cosine series      67 87 88
Fourier inversion theorems      314 315 317 321 332 338 359
Fourier series      65 66 67 77—88 141—145 298
Fourier sine series      65 66 67 87 88
Fourier transform      301 310—313 329—332 337—345
Free boundary condition      23
Frobenius method      179
Gauss’s theorem (divergence theorem)      52 58
Generalized circle      247
Green’s function      122—125 135—140 158 161 167 177 178 197 240 241
Green’s function, one-sided      119
Green’s Theorem      53 59
Harmonic conjugate      212
Harmonic function      52
Harmonic polynomials      104
Heat equation      58 60 66 92—95 108—110 126 318 322 327 329—332 355—361
Heat flow      50 58
Heat kernel      328
Heaviside function      347 367
Holder continuity      79 84
Holomorphic function      213
Homogeneous equation      30 31
Huyghen’s principle      336
Hyperbolic differential operator      42 44
Hyperbolic functions      227 258 259 260
Imaginary part      201
Implicit function theorem      230
Improperly posed problem      51 61
Infinite series evaluated by residues      289—292
Influence function      119 352
Initial value problems for ordinary differential equations      117—120
Integral equation      385
Integral of an analytic function      222
Interior mesh point      380
Inverse Fourier transform      314 315 317 319 321 332 338
Inverse function      229 230 239
Inverse Laplace transform      348
Inverse points      247 248
Inverse transformation      229 230 239
inversion      246 247
Isolated singularity      269
Iteration      383 384—391
Jacobian determinant      25 230
Jordan’s lemma      303
Kirchhoff’s formula      335

Laplace operator      49
Laplace operator in cylindrical coordinates      149
Laplace operator in polar coordinates      100
Laplace operator in spherical coordinates      188
Laplace transform      346 349 350
Laplace’s equation      43 49 63 95—107 110—111 149—151 194—196 236—245 253 380—383
Laplace’s solution of the heat equation      331
Laurent series      278—281
Least squares approximation      70 72
Legendre functions, associated      192 193
Legendre polynomials      191 192 193
Limit of a complex sequence      208
Linear fractional transformation      249
Linear operator      29
Linear partial differential equation      30
Linear problem      30
Linear transformation      246
Liouville’s theorem      266 267
Logarithm      224 230
L’Hopital’s Rule      271
Mapping      238
Mapping, conformal      238—245 257—261
mapping, one-to-one      238
Mathematical model      5
Maximum principle      55—57 59 61 94 97 101 108 111 265 320 381
Mean convergence      71 84 85 143 166 307 308 309
Mean value theorem      103 196
Mean-square deviation      71
Mellin inversion theorem      348
Membrane      48 54 182—184
Mesh point      376
Method of descent      336
Method of successive approximations      384—391
Minimum principle for eigenvalues      164—165 168
Moebius transformation      249
Monotonicity of eigenvalues      173 174 175
Morera’s theorem      223
Multiple Fourier series      141—145
Multiple Fourier transform      329—332
Multiple-valued analytic function      223 224 227 293—296
Multiply connected domain      220
Natural (characteristic) frequencies      154 186 187
nearest neighbors      380
Nonhomogeneous wave equation      25
Normal modes      154 186 187
Null function      308 309 310
One-sided Green’s function      119
One-to-one mapping      238
Open set      49 218
Operational formulas (rules)      324—325 331 340 346 351
Operator      29
Order of a pole      270
Order of a zero      270
Ordinary differential equations      117—126 167—168 351—354
Orthogonal functions      71 143 161
Oscillation theorem      174 175
Parabolic differential operator      42 44 59
Parseval’s equation      74 75 85 142 166 177 312 313 322 332
Partial differential operator      29
path integral      219
Phragmen — Lindelof theorem      107 111 112
Point at infinity      246
Pointwise convergence      70
Poisson’s equation      48 49 50 129 155—158 196—199
Poisson’s integral formula      102 103 139 196 199 252
Poisson’s solution of the wave equation      335
Polar representation of a complex number      204 216
Pole      270
Power function      227 260
Power series      179 190 209—219
Principal value      284 287 290 291 313 332
Principle of superposition      33
Propagation of discontinuities      21 40 42 43 46 115 154
Properly posed problem      6 51 61 125 377 382
Radius of convergence      210 213 217
Ratio Test      213
Rayleigh quotient      163
Rayleigh — Ritz method      393—395
Real part      201
Reciprocity law      122
Recursion formulas for Bessel functions      179 180
Removable singularity      269
Residue      273 274
Residue theorem      275 283 284 285 290 303 304
Resonance      187
Riemann — Leuesgue lemma      76 316
Riesz — Fischer theorem      308 309
Rodrigues formula      191
Rouche’s Theorem      278
Scaling      88—92 95
Schwarz’s inequality      82 83 84 306
Self-adjoint form of an ordinary differential equation      117
Separable partial differential operator      63 66—68
Separation of variables      63—69 92—103 379
Separation theorem      173
Separation theorem for eigenvalues      175
Series evaluated by residues      289—292
sgn      433
Shift formula (shift rule)      324 340
Simple closed contour      220
Simply connected domain      49 220
Sine series      65—67 87 88
Sine transform      321
Singular differential equation      176 177
Singularity      217 269
Singularity, essential      271
Singularity, isolated      269
Singularity, removable      269
Sound waves      7 336 362
Specific heat      58
Spherical harmonics      192 193 195
Square integrable function      306 310
Stability condition      378 379
Stokes rule      370
Stokes’ Theorem      53
String      1 5
Strong maximum principle      265
Successive approximations      384—391
Superposition      33
Taylor series      214 215 231—235
Tchebysheff polynomials      73
Telegrapher’s equation      115
Temperature      50 58
Term-by-term integration of Fourier series      75
thermal conductivity      58
Thomson’s principle      397
Three-dimensional wave equation      152—154 333—336
Triangle inequality      205 307
Trigonometric Fourier series      77—88
Trigonometric functions of a complex variable      227 260
Two-dimensional wave equation      362 —371 378—379
Two-point boundary value problem      120—126 168
Unbounded domains      253—256
Uniform convergence      33 70 72 81 83 84 145 167 177 178 307
Uniqueness      6 34 39 53 56 59—61 64 107 110 152—155 318 382
Vibrating membrane      48 54 182—184
Vibrating string      1 5
Wave equation, damped      112—115
Wave equation, damped, one-dimensional      9—27 66
Wave equation, damped, three dimensional      152—154 333—336
Wave equation, damped, two-dimensional      362—371 378—379

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