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Stakgold I. Boundary Value Problems of Mathematical Physics
Stakgold I.  Boundary Value Problems of Mathematical Physics







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: Boundary Value Problems of Mathematical Physics

: Stakgold I.

:

The present book is an outgrowth of a series of courses which I taught, first at Harvard and later at Northwestern, to classes consisting primarily of graduate students in engineering and in the physical sciences. Addressing myself to a similar audience, I have attempted here to present a modern, comprehensive treatment of boundary value problems. By definition, a boundary value problem consists of an ordinary or partial differential equation with associated boundary or initial conditions. When E. P. Wigner, a Nobel Laureate in Physics, spoke of "the unreasonable effectiveness of mathematics in the physical sciences," he must have had boundary value problemsinmind, for nearly every branch of the physical sciences has been enlightened by the mathematical theory of boundary value problems.


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Addition theorem for cylindrical waves      268
Addition theorem for Legendre functions      398
Addition theorem for spherical waves      290
Adjoint      9 198
Admissible functions      332
Approximation in subspace      33536
Aronszajn, N.      382
Asymptotic expansions for heat equation      206 222 236
Asymptotic expansions of eigenvalues of Laplacian      23134
Asymptotic expansions of integrals      399402
Asymptotic expansions,      23940
Base operators      382
Bazley, N.W.      382
Bessel equation      242 26870 295
Biharmonic equation      40 105 365
Boundary conditions, essential      353
Boundary conditions, natural      35255
Boundary value problem      88
Boundary value problem, well-posed      89
Capacity      17174 35052
Cauchy data      74
Cauchy problem      73
Causal function      (see Right-sided function)
Causal fundamental solution      (see Fundamental solution)
Characteristic      74
Characteristic of wave equation      46
Characteristic, surface      74
Classification of partial differential equations      7387
Comparison theorem      377
Composite medium      183 234293
Conductor      17183
Consistency condition      171
continuous dependence on data      89 91
Continuous dependence on data,      95 102103 226
Convolution      1820 49
Cylindrical wave      26768
Damped wave equation      6569 25759
Damped wave equation,      264
Delta function      5 20 27 34
Descent, method of      255
Diaz, J.      344
Difference kernel      311
Diffraction      (see Scattering)
Diffusion      230 24041 329
Dipole      6 111 201
Dipole layer      12 113
Dirac      5
Dirac of slow growth      31 37
Dirac, direct product of      17
Dirac, partial differential equations for      39
Dirac, product of functions and      7
Dirac, regular      5
Dirac, singular      5
Dirac, translation of      5
Dirac, values of      9
Dirichlet problem      90103 12225 135 142 172 296 348
Dissipative wave equation      (see Damped wave equation)
Distributions, action of      4
Distributions, convergence of      1016
Distributions, convolution of      18
Distributions, definition of      4 31 37
Distributions, differentiation of      7
Distributions, dipole      6 8
Divergence theorem      89
DAlemberts formula      85
Eigenfunction expansion for heat-conduction equation      21318
Eigenfunction expansion for Laplaces equation      15364
Eigenfunction expansion for wave equation      248 252
Eigenvalues of negative Laplacian      13642
Eigenvalues, asymptotic distribution of      231 23940
Eigenvalues, comparison theorem for      377
Eigenvalues, extremal principles for      36992
Eigenvalues, lower bounds to      38192
Elliptic equations      80
Energy flux      262 303
Energy inner product      342
Energy integral for heat conduction      225
Energy integral for wave equation      244 26162
Energy norm      343
Entire functions      315
Extremal principles for capacity      35052
Extremal principles for eigenvalues in Hilbert space      372
Extremal principles for eigenvalues in n space      36972
Extremal principles for functionals      337 40 35255 358-361
Extremal principles for torsional rigidity      34648
Extremal principles,      392
Extremal principles, complementary      34452
Fluid flow      18590
Fokker Planck equation      230
Fourier integral theorem      23
Fourier transforms and Wiener Hopf equations      31131
Fourier transforms of distributions      3039
Fourier transforms of functions      23
Fourier transforms of test function      31
Fox, D. W.      382
Free boundary      237
Friedrichs, K. O.      344
Functionals      3 332
Functionals, continuity of      3
Fundamental solution      48
Fundamental solution of damped wave equation      6569
Fundamental solution of heat-conduction equation      5860
Fundamental solution of Helmholtzs equation      5358 266-267
Fundamental solution of Laplaces equation      4953
Fundamental solution of wave equation      6165 249 25356
Fundamental solution, on Riemann surface      27072
Fundamental solution, pole of      48
Generalized functions      (see Distributions)
Generalized solution      42 (see also Fundamental solution)
Greens function; see also Fundamental for heat conduction      198 204 20918
Greens function; see also Fundamental for Helmholtzs equation      26590
Greens function; see also Fundamental for Laplaces equation      13071
Greens function; see also Fundamental for wave equation      24652
Greens Theorem      40 89
Hadamard, J.      255
Hankel functions      (see Bessel equation)
Hankel transform      27580
Harmonic functions, maximum principle for      101
Harmonic functions, mean value theorem for      99
Heat-conduction equation      81 194243
Heat-conduction equation,      28081
Heat-conduction equation, backward      229
Heat-conduction equation, causal fundamental solution      5860
Heat-conduction equation, causal Greens function for      197222
Heat-conduction equation, energy integral for      223
Heat-conduction equation, Greens theorem for      41 196
Heat-conduction equation, ill-posed problems for      229
Heat-conduction equation, in composite medium      23437
Heat-conduction equation, maximum principle for      22425
Heat-conduction equation, Stefan problem for      23738
Heat-conduction equation, uniqueness for      22526
Helmholtzs equation, fundamental solution of      5358
Helmholtzs equation, Greens function for      26585
Helmholtzs equation, half-plane problem for      28190 32127
Helmholtzs equation, in exterior domain      294311
Helmholtzs equation, in wedge      27273
Helmholtzs equation, mean value property      105
Hilbert Schmidt kernels      135 375
Huyghens principle      256
Hyperbolic equations      8085
Images      149 16669 204 209 211 251
Images,      252
Incident field      299
Initial data      72
Initial value problem      73
Integral equations for capacity      17274 351
Integral equations for scattering problems      301
Integral equations of potential theory      12230 146 171
Integral equations of Wiener Hopf type      31131
Integral equations with difference kernel      31131
Integral equations,      193
Integrodifferential equation      36667 389-390
Interior operator      74
Intermediate problems      382
Jones, D. S.      368
Kantorovich Lebedev transform      273
Kirchhoif s formula      263
Klein Gordon equation      70
Laplaces equation      40 4953 88192
Laplaces equation, eigenvalue problem for      13642 231
Laplaces equation, exterior Dirichlet problem for      123 129 142
Laplaces equation, fundamental solution of      4953
Laplaces equation, Greens function for      13071
Laplaces equation, Greens theorem for      40
Laplaces equation, interior Dirichlet problem for      122 128
Least squares      36163
Left-side function      28
Legendre functions      39398
Levine, H.      283 311 340 357
Limiting absorption      25961
Locally integrable      2
Macdonald function      266 279 321
Mapping function      164
Maximin theorem      371
Mehlers integral representation      284
Mellin transform      167 169
Minimax theorem      371
Monochromatic excitation      25961
Multiindex      2
Neumann problem      126 128 171 185
Neumann problem and fluid flow      18591
Neumann problem for      126 128
Neumann problem for, double      12 113
Neumann problem for, in two dimensions      128
Neumann problem for, Laplace transform      38 206 218 236
Neumann problem for, simple      7 39 112
Neumann problem for, surface      7 12 110121
Neumann problem, consistency condition for      171
Neumann problem, extremal principles for      36364
Null sequences      3 30 36
One-sided functions      28
Operators, base      382
Operators, bounded above      372
Operators, bounded below      372
Operators, completely continuous      13335 375
Operators, Hilbert Schmidt      135 375
Operators, indefinite      35557
Operators, integrodifferential      366 38990
Operators, interior      74
Operators, nonnegative      336
Operators, nonsymmetric      35557
Operators, positive      337 358
Operators, self-adjoint      9 376
Operators, semibounded      372
Operators, strongly positive      343 363
Operators, symmetric      337
Parabolic equations      80
Parseval formula      24
Partial differential equations      88311
Partial differential equations for distributions      3948
Partial differential equations of first order      7679
Partial differential equations of second order      7987
Partial differential equations, classification of      7387
Partial differential equations, elliptic      80
Partial differential equations, fundamental solutions of      4872
Partial differential equations, hyperbolic      80
Partial differential equations, parabolic      80
Plane wave      28586 302
Poisson equation      103 345
Poisson kernel      95
Poisson sum formula      212
Pole of fundamental solution      48
Potential theory      88193 267
Projection operator      33536
Propagation of discontinuities      46 77
Radiation condition      297
Rayleigh quotient      369
Reciprocity principle      303 342 368
Rellich, F.      297
Retarded potential      254
Riemann mapping theorem      164
Riemann surface      270
Right-side function      28
Ritz Rayleigh, equations      341 363 36668
Ritz Rayleigh, procedure      332 334 34043 362 377-378
Scattered amplitude      304
Scattered field      300
Scattering      299311 328
Scattering cross section      303
Scattering cross section, stationary principle for      309
Schwarz constants      380
Schwarz inequality      344
Schwinger Levine principle      311 340 357
Self-adjoint      9 376
Semigroups of operators      227
Slow growth, distribution of      31 37
Slow growth, functions of      29
Sokolnikoff, I.S.      346
Sommerfeld, A.      277 297
Spherical harmonics      109 126 127 144
Spherical harmonics,      290 295 39398
Spherical wave      267 290
Stationary principles for indefinite operators      356
Stationary principles for nonsymmetric operators      357 367368
Stationary principles for scattering cross section      30911
Steady heat conduction      88 183
Stefan Problem      237
Strict solution      42
Support      2
Symbolic functions      (see Distributions)
Tangential derivative      74
Telegraphy equation      65 258
Test functions      3
Test functions of rapid decay      30 36
Test functions, convergence of      3 30
Test functions, null sequences of      3 30 36
Theta function      212
Torsional rigidity      34648
Transversal      41
Uniqueness theorem for heat conduction      22526
Uniqueness theorem for Helmholtzs equation      29699
Uniqueness theorem for Laplaces equation      102
Uniqueness theorem for wave equation      243
Variation-iteiation      38081
Variational methods      (see Extremal principles and Stationary principles)
Wave equation      81 194 19697 24365
Wave equation, damped      6569 25759 264
Wave equation, dAlemberts solution of      85
Wave equation, fundamental solution of      6165
Wave equation, generalized solution of      44
Wave equation, Greens function of      24656
Wave equation, Greens theorem for      41 47 197
Wave equation, in composite medium      293
Wave equation, method of descent for      25556
Wave guide      291
Weber transform      242
Weinstein, A.      382
Well-posed problem      89
Wiener Hopf equation      31131
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