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| Asmar N.H. — Partial Differential Equations with fourier series and boundary value problems |
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| Предметный указатель |
Discrete Fourier transform of convolutions 563
Discrete Fourier transform, approximation of the convolution by the 572
Discrete Fourier transform, approximation of the Fourier transform by the 571
Discrete Fourier transform, defined 560
Discrete Fourier transform, periodicity of 562
Discretization of Laplace’s equation 534 535
Discretization of Poisson’s equation 538
Discretization of the boundary condition 519 527
Discretization of the first derivative 518
Discretization of the heat equation 516 518
Discretization of the initial condition 519 527
Discretization of the second derivative 518
Discretization of the wave equation 526 527
Double Fourier series 157
Double Fourier sine series 158
Double Fourier sine series cofficient 158
Double Fourier sine series representation theorem 158
Duhamel’s principle 465 471
Duhamel’s principle, solution of nonhomogeneous heat equation 473
Duhamel’s principle, solution of nonhomogeneous wave equation 476
D’Alembert’s method 126 113 419 474
Eigenfunction 173 231 288 334 353
Eigenfunction expansion 232 338 354
Eigenfunction in Sturm - Liouville theory 334 336
Eigenfunction of the Laplacian 171
Eigenseries expansion 173
Eigenseries expansion for a rectangle 173
Eigenvalue 173 231 288 334 353
Eigenvalue in Sturm - Liouville theory 334 335
Eigenvalue of the Laplacmn 171
Eigenvalue problem 173
Eigenvalue problem one dimensional 174
Elastic bar 112
Electrostatic potential 657
Element of arc length 281
Element of surface area 281
Element of volume in spherical coordinates 293
Elliptic theta function 99
Energy in quantum systems 576
Entire function 650
Error function 122
Error function, basic properties of the 427
Euler load 368
Euler, Leonhard (1707-1783) 13
Euler’s constant 246
Euler’s equation A24 217 270 273 275 282
Euler’s equation, general solution of A25
Euler’s identity 62 67
Fast Fourier Transform 564
Fej r, Leopoldo 318
Fejr’s kernel 551
Feynman, Richard 193
FFT see "Fast Fourier transform"
Finite difference, approximation and nonhomogeneous boundary conditions 523
Finite difference, approximation of the heal equation 518
Finite difference, approximation, unstable 522
FLOOR see "Greatest integer function"
Forward difference approximation 518
Fourier - Bessel coefficient 201
Fourier - Bessel series 201
Fourier coefficient 27
Fourier coefficient, complex form of 61
Fourier coefficient, linearity 37
Fourier coefficient, square summable 56
Fourier cosine integral representation 434
Fourier cosine transform 435
Fourier cosine transform, operational properties 437
Fourier cosine transform, relationship to Fourier transform 437
Fourier integral, complex form 398
Fourier integral, relation to Fourier series 391
Fourier integral, representation theorem 390
Fourier series 17 26-28 281
Fourier series integration of 49
Fourier series of even functions 42 15
Fourier series of odd functions 42-45
Fourier series of sawtooth function 28
Fourier series of sawtooth function 2 -periodic 41
Fourier series of sawtooth function and uniform convergence 98
Fourier series of sawtooth function, fails termwise differentiation 49
Fourier series of sawtooth function, not uniformly convergent 90
Fourier series of square wave 34
Fourier series of triangular wave 31 40
Fourier series with decreasing coefficients 97
Fourier series, arbitrary period 38
Fourier series, changing variables in 33 34 41
Fourier series, completeness of 329
Fourier series, complex form of 61
Fourier series, continuity and uniform convergence 89
Fourier series, converge in the moan 55
Fourier series, differentiation of 18
Fourier series, diverge 31
Fourier series, Euler formulas 26-28
Fourier series, generalized 149
Fourier series, generating Bessel functions 263
Fourier series, Gibbs phenomenon 29
Fourier series, linear combinations of 33
Fourier series, linearity of 34 37
Fourier series, mean square convergence 55
Fourier series, operations on 33
Fourier series, partial sum of 28 61
Fourier series, particular solution of nonhomogeneous differential equations 70
Fourier series, pointwise convergence of 85
Fourier series, representation theorem 30 39 390
Fourier series, representation theorem, proof of 77
Fourier series, solution of biharmonic equation 375
Fourier series, solution of the heat equation 137
Fourier series, solution of the wave equation 119
Fourier series, summing series with 33
Fourier series, translating 33 34
Fourier series, uniform convergence of 31 85 97 101
Fourier series, uniform convergence of, neccesary and sufficient condition 90
Fourier series, used in the method of separation of variables 118
Fourier series, varying the period in 4
Fourier sine integral representation 131
Fourier sine series 12
Fourier sine transform 435
Fourier sine transform, operational properties 437
Fourier sine transform, relationship to Fourier transform 437
Fourier transform 398 517
Fourier transform and derivatives 401
Fourier transform and the heat equation 414 420
Fourier transform and the wave equation 412
Fourier transform inverse 398
Fourier transform of a piecewise polynomial function 453
Fourier transform of a shifted function 408
Fourier transform of convolutions 104
Fourier transform of dilations 408
Fourier transform of Dirac delta 151
Fourier transform of Dirac delta function 451
Fourier transform of generalised function 451
Fourier transform of Hermite functions 410
Fourier transform of partial derivatives 411 412
Fourier transform of the Gaussian function 406
Fourier transform of the Poisson kernel 429
Fourier transform, approximation by the discrete Fourier transform 507 569 571
Fourier transform, boundedness of 410
Fourier transform, continuity of 400
Fourier transform, defined 398
Fourier transform, derivative of 402
Fourier transform, linearity of 401
Fourier transform, operational properties 401
Fourier transform, reciprocity relation 407
Fourier transform, relationship to momentum of an electron 591
Fourier transform, shifting on the  -axis 108
Fourier transform, table of A66
Fourier, Joseph (1768-1830) 13 17
Fourier’s law of heat conduction 143 057
Fractional part 22
frequency 349
Fresnel integral 267 420
| Frobenius method A51 A54
Frobenius method, applied to Bessel’s equation 237
Frobenius method, proof A61
Frobenius method, summarized A55
Frobenius series A53
Function integrable 391
Function of exponential order 480
Function, average at a point 30
Function, band limited 547
Function, complex exponential 67
Function, continuous nowhere differentiable 94
Function, continuous on closed and bounded interval 80
Function, even 42
Function, harmonic 106 198
Function, odd 42
Function, periodic 18
Function, piecewise continuous 19-20 25
Function, piecewise smooth 19--20 25
Function, smooth 20
Function, square integrable 54
Function, triangular 53
Function, uniformly continuous 80
Fundamental mode 121
Fundamental period 18
Fundamental set of solutions A4
Fundamental solution of the heat equation 463
G 329
Gamma function 239 242 246
Gamma function incomplete 466
Gamma function, basic property of 242
Gamma function, defined 242
Gamma function, reciprocal of the 246
Gauss - Seidel iteration method 543
Gaussian function 405 428
Gaussian function, convolution of 428
Gauss’s kernel 121
Gauss’s kernel, properties of 428
Generalized double Fourier series 186
Generalized factorial function see "Gamma function"
Generalized Fourier coefficient 149 232 339
Generalized Fourier series 149 329 331
Generalized Fourier series, derivative of 448
Generalized Fourier series, equality between two 447
Generalized Fourier series, Fourier transform of 451
Generalized Fourier series, orthogonality of 149 152
Generalized function 446
Generalized Laguerre coefficient 603
Generalized Laguerre polynomial 581 602 609
Generalized Laguerre polynomial, generating function for 609
Generalized Laguerre polynomial, list of 589
Generalized Laguerre series 603
Generalized Laguerre’s differential equation 581 602
Geometric series A40
Geometric series, uniform convergence of 40
Ghost point 528 532
Gibbs phenomenon 29 258
Gibbs phenomenon for Bessel series 258
Gibbs phenomenon for Fourier integrals 393
Gibbs phenomenon for Fourier series 29
Gibbs phenomenon for the sawtooth function 37
GOO Laguerre seiies 601
Greatest integer function 22 24
Greatest integer function, constructing periodic functions with 22
Green’s first identity 617
Green’s formula for the integral of the Laplacian 618
Green’s function 629
Green’s function and Dirac delta function 635
Green’s function and eigenfunction expansions 633
Green’s function for a disk 634 638 677
Green’s function for a rectangle 633 634
Green’s function for a semi-infinite vertical strip 681
Green’s function for the first quadrant 680
Green’s function for the upper half-plane 638 643 679
Green’s function formula using conformal mapping 674 677 681
Green’s function, as a sum involving a logarithm 631
Green’s function, composed with conformal mapping 682
Green’s function, defined 630
Green’s function, harmonicity 631
Green’s function, method of images 638
Green’s function, representation formula 62 9
Green’s function, solution of Dirichlet problem 630
Green’s function, solution of Poisson problem 632
Green’s function, symmetry property 631 677
Green’s function, uniqueness 631
Green’s identities 617
Green’s second identity 617
Green’s Theorem 614
Green’s theorem for multiply connected regions 616
Green’s theorem, gridpoint 517
Ground energv 586
Ground state 586 588
Half-range expansion 50 118
Hanging chain 112 237 346
Hanging chain infinitely long 510
Hanging chain, fundamental mode of the 349
Hanging chain, modeling the vibrations 112
Hanging chain, normal mode of the 349
Hankel transform 509
Hankel transform inverse of 509
Hankel transform, defined 509
Hankel transform, operational properties 509
Hankel transform, table of 512
Harmonic conjugate 655
Harmonic conjugate, finding a 655
Harmonic conjugate, orthogonal trajectories 657
Harmonic function 106 198 622 655
Harmonic function independent of  660
Harmonic function, conjugate of 655
Harmonic function, dilating a 660
Harmonic function, examples of 652
Harmonic function, maximum-minimum principle 625
Harmonic function, mean value property 623
Harmonic function, real part of analytic function 655
Harmonic function, translating a 660
Harmonic measure of an interval 654
Harmonic measure of disjoint intervals 661
Heat equation 189 411 420 503
Heat equation in a ball 296
Heat equation in a polar coordinates, solution 216
Heat equation in polar coordinates 206
Heat equation in spherical coordinates 291 296 297
Heat equation on a semi-infinite line 440 503
Heat equation on the infinite line 114 420
Heat equation one dimensional 104 135
Heat equation one dimensional initial condition 135
Heat equation one dimensional, boundary condition 135
Heat equation one dimensional, solution by the method of separation of variables 136 437
Heat equation with constant heat source 504
Heat equation with convection 419
Heat equation with nonconstant coefficients 423
Heat equation with symmetry in a ball 299
Heat equation with time dependent heat source 299
Heat equation without a heat source 144
Heat equation, comparison principle for the 189
Heat equation, derivation of 142
Heat equation, discretization of 516
Heat equation, Duhamel’s principle 165
Heat equation, finite difference approximation 518
Heat equation, fundamental solution of 463
Heat equation, general form 144
Heat equation, maximum principle for the 188 192
Heat equation, nonhomogeneous 140 178 226 236 404 473
Heat equation, nonhomogeneous and Duhamel’s principle 465 473
Heat equation, nonhomogeneous in a ball 296
Heat equation, nonhomogeneous in polar coordinates 236
Heat equation, nonhomogeneous, solution using steady-state 142 226
Heat equation, nonhomogeneous, solution via the eigenfunction expansions method 178
Heat equation, Robin condition 341
Heat equation, shifting the initial data 428
Heat equation, solution with convolutions 421
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