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| Авторизация |
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| Asmar N.H. — Partial Differential Equations with fourier series and boundary value problems |
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| Предметный указатель |
Overtone 121
Parametric form of Bessel’s equation 200 209 255
Parametric form of the modified Bessel equation 229 230
Parseval’s identity 56 65 332
Parseval’s identity for Bessel series 332
Parseval’s identity for complete orthogonal systems 331
Parseval’s identity, complex form of 65
Parseval’s identity, evaluating series with 57
Parseval’s theorem 590
Partial differential equation 2
Partial differential equation, change of variables in 4
Partial differential equation, classification of 107
Partial differential equation, elliptic 107
Partial differential equation, examples of 104
Partial differential equation, first order 2 3
Partial differential equation, general form of order two 105
Partial differential equation, homogeneous 105
Partial differential equation, hyperbolic 107
Partial differential equation, linear 104
Partial differential equation, linear homogeneous 3
Partial differential equation, nonhomogeneous 105
Partial differential equation, nonlinear 105 106
Partial differential equation, nonlinear of first order 108
Partial differential equation, order of 104
Partial differential equation, parabolic 107
Partial fractions 486 188
Partial sum of Fourier series 28 61
Partial sum of Fourier series, Dirichlet kernel representation 78
Particular solution A5
Period 18
Periodic boundary condition 336
Periodic driving force 69
Periodic electromotive force 69
periodic extension 50
Periodic function 18
Periodic function integral of 20
Periodic function, antiderivative of 24
Periodic function, describing a 18
Periodic function, operations on 23
Periodic function, sums of 23
Phase function 26 5
Plancherel’s theorem 590
Planck’s constant 575 591
Pointwise convergence 85
Poisson integral formula 429 558 642 613 662
Poisson integral formula for band-limited functions 558
Poisson integral formula for the first quadrant 680
Poisson integral formula on the disk 227 642
Poisson integral formula, justification of the 662
Poisson kernel 227 429 642
Poisson kernel, properties of the 433
Poisson kernel, semigroup property of the 432
Poisson problem 632 6 33
Poisson problem in a ball 295
Poisson problem in a box 177
Poisson problem in a disk 233 641
Poisson problem with Neumann condition 688
Poisson problem, decomposition into subproblems 171 234
Poisson problem, in a disk, solution by Green’s function 641
Poisson problem, solution 172 221 235
Poisson problem, solution by Green’s function 632
Poisson’s equation 170 464 633
Poisson’s equation in a ball 294
Poisson’s equation in a disk 233
Poisson’s equation in a rectangle 170
Poisson’s equation in spherical coordinates 291
Poisson’s equation in the upper half-plane 643
Poisson’s equation with a Dirac function 635
Poisson’s equation, discretization of 538
Poisson’s equation, three dimensional 177
Poisson’s equation, two dimensional 104
Polar coordinates 194
Polar form 646
Polygonal line 613
Position density 574
Power series A28
Power series integration of A31
Power series, differentiation of A31
Power series, examples of A29
Power series, identity principle for A33
Power series, method A40
Power series, radius of convergence A29
Power series, shifting the index in a A36
Power series, uniform convergence of 40
Principal branch of complex logarithm 647
Principle of conservation of energv 135
Probability density for the momentum 591
Probability density function 574
Product ,solution 115
Quantum harmonic oscillator 580
quantum mechanics 573 574
Radial equation for the hydrogen atom 583
Radial wave function 585
Radian 652
Radius of convergence A29
Ratio Test A30
Real part 645
Rectangular membrane 113
Rectangular membrane and the two dimensional wave equation 114
Rectangular membrane forced vibrations of 114
Rectangular membrane, modeling 113
Reduction of order formula A21 A27 A60
Reduction of order formula for nonhomogeneous equations A27
Region 613
Regular singular point A52 237
Removable singularitv 682
Resonance 76
Riemann - Lebesgne lemma 80
Riemann mapping theorem 675
Riemann zeta function 58
Riemann, Bernhard 103
RLC-circuit 69
Robin condition 148 180 181 221 225 341 316
Robin condition in polar coordinates 221
Robin condition on a rectangle 181
Rodrigues’ formula for associated Legendre functions 319
Rodrigues’ formula for generalized Laguerre polynomials 604
Rodrigues’ formula for Hermite polynomials 606
Rodrigues’ formula for Legendre polynomial 31 8
Sampling theorem 546 547
Sampling theorem and partial differential equations 555
Sampling theorem and the heat equation 557
Sampling theorem and the Poisson integral formula 558
Sampling theorem for band limited functions 549
Sampling theorem for time-limited functions 549
Sampling theorem, proof of 552
Sampling theorem, used to compute Fourier transforms 551 552
Sawtooth function, $ 2
Schr dinger’s equation 104 573
Schrdinger’s equation and spherical harmonics 583
Schrdinger’s equation for the hydrogen atom 582
Schrdinger’s equation one dimensional 577
Schrdinger’s equation, as an axiom in quantum mechanics 575
Schrdinger’s equation, connection with Sturm - Liouville theory 577
Schrdinger’s equation, eigenvalues in 576
Schrdinger’s equation, normalizable solution of 576
Schrdinger’s equation, solution by separation of variables 575
Schrdinger’s equation, time-dependent 575
Schrdinger’s equation, time-independent 576
Separated variables 115
Separation constant 116 124 136 156
Separation constant, a problem with positive and negative 153
Shannon, Claude 546
Signum function 396
Simple curve 612
Simply connected 613
Sine Fourier transform, table of A69
Sine integral function 393 396
Sine series 50 118
Singular point A51
| Singular point, regular A52
Siring see "Vibrating string"
Specific heat 143
Spectrum 577
Spherical Bessel differential equation 292
Spherical Bessel function 256 260 293
Spherical Bessel function and the Helmholtz equation 293
Spherical Bessel function of the first kind 256
Spherical Bessel function, orthogonality of 261
Spherical Bessel function, recurrence relations for 261
Spherical coordinates 196
Spherical harmonics 273 282 292 583
Spherical harmonics and the solution of Helmholtz equation 292
Spherical harmonics coefficient 284
Spherical harmonics, defined 282
Spherical harmonics, differential equation for the 288 292
Spherical harmonics, evaluation of 289
Spherical harmonics, list of 288
Spherical harmonics, orthogonality of 283
Spherical harmonies series 270 284
Spring-mass system 69
Spring-mass system with damping 69
Spring-mass system without damping 75 76
Spring-mass system, Fourier series ,solution 69
Spring-mass system, natural frequency 72
Spring-mass system, normal mock of vibration 70
Spring-mass system, steady-stale solution 69 71 74
Square integrable function 54
Square integrable with respect to a weight 331
Square summable 56
Square wave 34
Stability criterion for the heat equation 522
Stability criterion for the wave equation 527
Stationan point 265
Stationary phase 265
Stationary phase, method of 265
Stationary state 576 585
Steady-state problem with time independent heat source 170
Steady-state solution 69 71 74 139
Steady-state temperature 139
Steady-state temperature in a circular plate 216
Steady-state temperature in a cylinder 228
Steady-state temperature in a disk 216
Steady-state temperature in a square plate 165
Steady-state temperature in the upper half space 512
Steiner inversion 639 640
Step function 191
Sturm - Liouville form 333
Sturm - Liouville form of 601
Sturm - Liouville form of Legendre’s equation 334
Sturm - Liouville form of the parametric form of Bessel’s equation 334
Sturm - Liouville problem 333
Sturm - Liouville problem with periodic boundary conditions 336
Sturm - Liouville problem, fourth order 353 364
Sturm - Liouville problem, regular 333
Sturm - Liouville problem, singular 333
Sturm - Liouville theory 149 326 333
Sturm - Liouville theory and generalized Fourier series 149
Sturm - Liouville theory, fourth order 353
Summation by parts see "Abel’s summation by parts formula"
Sums of cosines 96
Sums of cosines, diverge 19
Sums of sines 96
Sums of sines, diverge 99
Superposition principle 11 106 201 210
Superposition principle for ordinary differential equations A5
Symmetric partial sum 61
Table of Fourier cosine transforms A68
Table of Fourier sine transforms A69
Table of Fourier transforms A66
Table of Hankel transforms 513
Table of Laplace transforms A70
Telegraph equation 15 104 107
Tempered distribution 446
Terminal point 612
Test function 117
thermal conductivity 143
Thermal diffusivity 144 424
Thermal energy density 1-13
Time-independent solution see "Steady-state solution"
Translate of argument function 654
Transverse vibrations of a beam 360 416 419
Triangular wave 25
Triangular wave with arbitrary period 40
Triangular wave, defined with the gieatest integer function 25
Trigonometric polynomial 59
Trigonometric series 97
Trigonometric system 21
Trigonometric system, arbitrary period 23
Trigonometric system, orthogonality of 21
Tukey, J. W. 564
Two dimensional wave equation 114
Uncertainty 592
Uniform convergence 85
Uniform convergence and differentiation term by term 92
Uniform convergence and integration term by term 91
Uniform convergence of continuous functions 89
uniform convergence of Fourier series 31 85
Uniform convergence of Fourier series, a necessary ami sufficient condition 90
Uniform convergence of Fourier series, a necessary condition 89
Uniform convergence of geometric series 40
Uniform convergence of power series 32 10
Uniform convergence, defined for scries 87
Uniform convergence, defined for sequences 86
Uniformly continuous 80
Uniqueness of eigenfunctions in Sturm - Liouville theory 336
Unit step function 491
Vandermonde determinant A20
Variation of parameters formula A23 176
Vibrating string 7
Vibrating string forced vibrations of 111 124
Vibrating string initial conditions 8
Vibrating string, boundary conditions 8
Vibrating string, conservation of energy 134
Vibrating string, d’Alembert’s solution 126
Vibrating string, energy of 134
Vibrating string, Fourier sint series solution 12
Vibrating string, geometric intierpretation of solution 127
Vibrating string, modeling 109
Vibrating string, normal mode 10 118 121
Vibrating string, snapshots of 10
Vibrating string, superposing solutions 11
Vibrating string, time period of motion 124
Vibrating string, wave equation for the 110
Wallis’s formulas 247 307
Wave equation 7 412 505 725
Wave equation in a ball 297
Wave equation in polar coordinates 199 207
Wave equation in polar coordinates, solution 202 213 215
Wave equation in sphcrical coordinates 291
Wave equation on a semi-infinite line 143 505
Wave equation on the infinite line 412
Wave equation one dimensional 7 104 114
Wave equation one dimensional with damping 124
Wave equation one dimensional, d’Alembert’s solution 126
Wave equation one dimensional, solution 119
Wave equation one dimensional, solution by the method of separation of variables 115
Wave equation with effect of gravitation 506
Wave equation with forcing term 505
Wave equation with time-dependent boundary conditions 443
Wave equation, characteristic interval 177
Wave equation, characteristic line 130
Wave equation, characteristic triangle 477
Wave equation, discretization of the 526
Wave equation, Duhamel’s principle for 475
Wave equation, nonhomogeneous 474
Wave equation, nonhomogeneous, solution using Duhamel’s principle 474 476
Wave equation, radially symmetiic 199
Wave equation, solution 14
Wave equation, two dimensional 104 155
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