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Stakgold I. — Boundary value problems of mathematical physics
Stakgold I. — Boundary value problems of mathematical physics



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Название: Boundary value problems of mathematical physics

Автор: Stakgold I.

Аннотация:

For more than 30 years, this two-volume set has helped prepare graduate students to use partial differential equations and integral equations to handle significant problems arising in applied mathematics, engineering, and the physical sciences. Originally published in 1967, this graduate-level introduction is devoted to the mathematics needed for the modern approach to boundary value problems using Green's functions and using eigenvalue expansions. Now a part of SIAM's Classics series, these volumes contain a large number of concrete, interesting examples of boundary value problems for partial differential equations that cover a variety of applications that are still relevant today. For example, there is substantial treatment of the Helmholtz equation and scattering theory — subjects that play a central role in contemporary inverse problems in acoustics and electromagnetic theory.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1987

Количество страниц: 352

Добавлена в каталог: 24.11.2013

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Предметный указатель
Abel's formula for the Wronskian      60
Absolutely continuous functions      145
Action of distribution      28 29 31
Adjoint      see also "Self-adjoint operators" "Symmetry"
Adjoint boundary conditions      70
Adjoint for differential system      84
Adjoint for integral equation      81 212
Adjoint for linear equations      80
Adjoint Green's function      71
Adjoint of integral operator      212
Adjoint of matrix      80 152—154
Adjoint of transformation on Euclidean space      152—154
Adjoint of transformation on Hilbert space      170—180
Adjoint, formal      40 68
Adjoint, null space of      171
Admissible pair      170
Alternative Theorems      79—86
Alternative theorems for completely continuous operators      187
Alternative theorems for differential equations      82—86
Alternative theorems for equations in Euclidean space      153—154
Alternative theorems for integral equations      81 221
Alternative theorems for linear equations      79—81 153
Alternative theorems for operators with closed range      172
Banach spaces      106
Basis for Hilbert space      119
Basis for vector space      97
Basis of eigenvectors      160
Basis, dual      138
Basis, orthonormal      123—124 129
Basis, orthonormal, characterization of      127—128
Bessel equation      329—332
Bessel equation as singular problem      301 305—308 313—315 317 321
Bessel equation, eigenvalues of      282
Bessel equation, Green's function for      75 77
Bessel functions      329
Bessel functions, modified      331
Bessel's inequality      125
Bi-orthogonal set      256
Bilinear concomitant      69
Bilinear expansion for Green's function      261 273 290
Bilinear expansion for kernel      238—240
Bilinear series      see "Bilinear expansion"
boundary conditions      64—69 see
Boundary conditions, adjoint      70
Boundary conditions, unmixed      64 66
Boundary value problem regular      269 270
Boundary value problem regular for second order equation      64—91
Boundary value problem regular, singular      269 283—322
Cauchy principal value      49
Cauchy sequences      100
Closable operator      142
Closed operator      142 168
Closed set.      114
Compact operator      see "Completely continuous transformation"
Compact set      114 164
Compatibility conditions      see "Consistency conditions"
Complete orthonormal set      127 see
Completely continuous transformation      185
Completely continuous transformation, alternative theorem for      187
Completely continuous transformation, integral operator as      192
Completeness of metric space      101—105
Completeness of orthonormal set      127
Concentrated source      see "Dirac delta function"
Conjunct      69
Consistency conditions      80 83 85 154 201 221 265
Convergence in the mean      117—118 131
Convergence of distributions      42
Convergence of test functions      30
Convergence, Cauchy      100
Convergence, uniform      131
Convergence, weak      133
Courant minimax principle      225
Deficiency      181
Degenerate eigenvalue      213
Delta function      see "Dirac delta function"
Delta sequence      24—27
Dense set      114
Dependence of set of functions      59—60 118
Dependence of set of vectors      97 118
Dependence, criterion for      60
Dimension of vector space      97
Dipole      25—26 35
Dirac delta function      6—7 18—27 33
Dirac delta function, expansion of      273
Dirac delta function, sifting property of      22—24
Distance function      see "Metric"
Distributional solutions of differential equations      51—58
Distributions      28—58
Distributions as linear functionals      28 31
Distributions as solutions of differential equations      53
Distributions, action on test functions      28—29 31
Distributions, convergence of      42
Distributions, delta      33
Distributions, differentiation of      37 41
Distributions, dipole      35
Distributions, divergent integrals and      47—50
Distributions, Fourier series of      45—47
Distributions, fundamental solution and      54 56
Distributions, operations on      35—36
Distributions, order of      40
Distributions, pseudofunctions and      48—51
Distributions, regular      33
Distributions, singular      33
Distributions, values of      37
Domain of differential operator      76
Domain of function      93 95
Domain of transformation      95
Eigenfunctions of integral operator      213
Eigenfunctions of self-adjoint system      220 270
Eigenfunctions, completeness of      219 220
Eigenfunctions, normalization of      274 278
Eigenfunctions, orthogonality of      213 270
Eigenmanifold      156
Eigenvalues of integral operator      212—213
Eigenvalues of transformation on Euclidean space      154 156
Eigenvalues of transformation on Hilbert space      181
Eigenvalues, approximations for      228—236
Eigenvalues, degenerate      213
Eigenvalues, existence of      214
Eigenvalues, extended problem of      248—249
Eigenvalues, extremal principles for      162 187—190 214—217
Eigenvalues, multiplicity of      156—157 214
Eigenvalues, simple      213
Equal almost everywhere      33 105
Essentially regular transformation      166
Existence and uniqueness for inhomogeneous integral equation      221
Existence and uniqueness for initial value problems      58 59 61
Existence and uniqueness of Green's function      66
Expansion theorem      218
Extremal principles for eigenvalues      162 187—190 214—214 223—236
Extremal principles for inhomogeneous equation      245—250
Finite part of divergent integrals      48
Fourier coefficients      125
Fourier cosine transform      293
Fourier series of distributions      44—46 50 51
Fourier sine transform      287
Fourier sum      125
Fourier transform      47 294
Fourier — Bessel expansion      308
Fredholm integral equations of first kind      195 201
Fredholm integral equations of second kind      195
Functionals      134 245 see
Fundamental solution      54 62 see
Fundamental solution, causal      56
Fundamental solution, pole of      62
GALERKIN equations      244 245 248
Galerkin's method      244 245
Generalized function      34 see
Generalized solutions of differential equations      53
Gram — Schmidt procedure      122
Green's formula      70
Green's function      1—18 54—56 65—69 71—78 86—91 219—220 261—268 271—282 284—295 303—321
Green's function as integral operator      12 199 219
Green's function for Bessel's equation      75 77
Green's function for initial value problems      56 66
Green's function for regular boundary value problems      66 271
Green's function for singular problems      303—304
Green's function, adjoint      70
Green's function, existence and uniqueness of      66
Green's function, modified      86—91 316—319
Green's function, singularities of      274 304
Hankel functions      329
Hankel transforms      315—316
Heat conduction equation      327—328
Heaviside function      20 27 34 38
Heaviside sequence      27
Hermite equation      302 320—321
Hermite polynominals      321
Hermitian      see "Symmetric"
Hilbert spaces, definition of      110
Hilbert spaces, functionals on      135—139
Hilbert spaces, properties of      116—135
Hilbert spaces, separable      116
Hilbert spaces, transformations on      140—146 165—190
Hilbert — Schmidt operators      see "Integra] operators"
Impulse response      78 see
Impulsive source      see "Dirac delta function"
Indefinite operator      165 224
Independence of set of functions      59—60 118
Independence of set of vectors      97 118
Independence, criterion for      60
Influence function      see "Green's function"
Initial conditions      64 67
Initial value problem      59
Initial value problem, Green's function for      56 66
Inner product on complex linear space      109
Inner product on real linear space      107
Inner product spaces      107—114 see
Inner product spaces, complex      109
Inner product spaces, real      107
Integral equations      191—258
Integral equations, approximate methods for      241—250
Integral equations, differential equations and      199—206
Integral equations, eigenvalue problem for      195 212—220
Integral equations, Fredholm      195
Integral equations, Volterra      196
Integral operators of Hilbert — Schmidt type      193
Integral operators, adjoint of      212
Integral operators, complete continuity of      193
Integral operators, eigenvalues and eigenfunctions of      195—196
Integral operators, Green's functions and      199—206
Integral operators, indefinite      224
Integral operators, kernel of      192
Integral operators, negative and nonnegative      224
Integral operators, nonsymmetric      250—258
Integral operators, positive and nonpositive      224
Integral operators, real      225
Integral operators, spectrum of      212—220
Inverse operator      12 140 151 165—180
Kantorovich — Lebedev transform      317
Kernel of integral operator      144
Kernel, bilinear series for      238—240
Kernel, Hilbert — Schmidt      193
Kernel, iterated      206 239
Kernel, left iterate of      251
Kernel, nonsymmetric      250—258
Kernel, right iterate of      251
Kernel, separable      192 198
kernel, symmetric      212
Kohn — Kato method      235 236
Lagrange's identity      70
Least squares approximation      124 224
Lebesgue convergence theorem      105
Lebesgue integral      104—105
Legendre equation      302 319 320
Legendre polynomials      123 130 319
Limit circle case for singular problem      297 301 310—313
Limit point case for singular problem      297 301 310
Linear functionals on Euclidean space      135—139
Linear functionals on Hilbert space      135—139
Linear functionals on space of test functions      31
Linear functionals, boundedness of      135 137
Linear functionals, continuity of      135
Linear functionals, extension of      137
Linear functionals, norm of      135
Linear functionals, Riesz representation theorem for      136
Linear manifolds      120 121
Linear spaces      96—99 105—190 see
Linear transformations      see "Transformations"
Locally integrable      31
MacDonald functions      331
Mapping      see "Operator" "Transformations"
Matrix      79 146
Mellin cosine transform      316
Mellin sine transform      316 317
Mellin transform      308 309
Mercer's theorem      240
Metric      100 117
Metric spaces      99—105 114—116
Metric spaces, Cauchy sequences in      100
Metric spaces, closed sets in      114
Metric spaces, compact sets in      114
Metric spaces, completeness of      101
Metric spaces, convergence in      100
Metric spaces, dense sets in      114
Metric spaces, triangle inequality for      100
Metric, generated by norm      106
Minimax principle      225
Modified Green's function      86—91 317—319
Modified Green's function, symmetric      88
Multiplicity, algebraic      157
Multiplicity, geometric      156
Negative operator      165 224
Neumann functions      329
Neumann series      206—211
Neumann series for Volterra equation      209
Nonnegative operator      165 224
Nonpositive operator      165 224
Norm      105 117
Norm of functional      135
Norm of s type      296
Norm of transformation      140
Norm, generated by inner product      108
Norm, metric generated by      106
Normal form of differential equation      279
Normalization of eigenfunctions      274 278
Normed linear spaces      105—107
Nullity of transformation      153
One-to-one transformation      93
Operator      see also "Transformations"
Operator, bounded below      277
Operator, closable      142
Operator, closed      142 168
Operator, continuous spectrum of      181
Operator, differentiation      174
Operator, extremal principles for      162 214 226 245
Operator, point spectrum of      181
Operator, projection      149
Operator, regular value of      181
Operator, residual spectrum of      181
Operator, resolvent set of      181
Operator, shifts      169
Operator, spectrum of      180
Order of distribution      40
Orthogonal complement      121
Orthogonal sets      108 120
Orthogonalization by Gram — Schmidt procedure      122
Orthonormal basis      123—124 127—128 129
Parallelogram law      113
Parseval's identity      127
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