Myint Tyn U., Debnath L. Linear Partial Differential Equations for Scientists and Engineers
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772 Index
consistent, 604
continuity equation, 85
continuity theorem, 333
convergence, 602
convolution, 448, 467
convolution property, 497
convolution theorem, 448, 467
Courant, 663
Courant parameter, 608
Courant–Friedrichs–Lewy convergence
criterion, 608
Crank–Nicolson formula, 626
Crank–Nicolson implicit formula, 625
cummulative truncation, 603
cylindrical wave equation, 155, 156
d’Alembert, 3
d’Alembert solution, 3, 123
d’Alembertian, 65
damped wave equation, 84
de Broglie wave, 542
Debnath, 482, 485, 515, 517, 519, 521,
588, 590
density, 580
differential operators, 16
differentiation, 445, 464
differentiation theorem, 209
diffusion equation, 514, 566, 569
diffusion equation in a finite medium,
482
diffusion in a finite medium, 483
diffusive, 97
dilatational, 73
Dirac delta function, 446, 470
Dirichlet, 5
Dirichlet boundary-value problem, 5
Dirichlet condition, 235
Dirichlet kernel, 203
Dirichlet problem, 330, 416
Dirichlet problem for a circle, 334
Dirichlet problem for a cylinder, 363
Dirichlet problem for a rectangle, 343,
348
Dirichlet problem for a sphere, 367
Dirichlet problem for circular annulus,
340
discontinuities, 44
discrete energy spectrum, 387
discrete values of energy, 392
discretization error, 603
dispersion relation, 541, 573
dispersion relation for water waves,
542
dispersive, 97, 542
dispersive wave, 540
domain of dependence, 123, 125
double Fourier series, 212, 213
Du Fort–Frankel explicit algorithm,
614
Duhamel formula, 481, 533
Duhamel principle, 165
eigenfunction, 237, 274
eigenvalue, 237
eigenvalue problem, 236
Einstein, 10
elliptic, 92
elliptic equation, 111
energy, 241
energy density, 547
energy equation, 163
energy flux, 547
energy integral, 163
envelope, 48
equation
Benjamin, Bona and Mahony
(BBM), 580
Bessel, 289, 375
biharmonic, 64, 404
biharmonic wave, 64
Boussinesq, 543, 675
Burgers, 81, 563, 569
Cauchy–Riemann, 5
characteristic, 33, 94, 120
continuity, 85
cylindrical wave, 155, 156
damped wave, 84
diffusion, 514, 566, 569
elliptic, 111
energy, 163
Euler, 38, 65, 88, 89, 100
Euler–Lagrange, 9, 629, 633, 634,
646
Euler–Poisson–Darboux, 598
Fokker–Planck, 62, 90, 264, 436
fractional axisymmetric wave-
diffusion, 518
fractional diffusion, 510, 514
Index 773
fractional diffusion-wave, 511
fractional partial differential
equation, 510
fractional Schr¨odinger, 520
fractional wave, 512
Hamilton–Jacobi, 9
heat, 64, 76, 84
Helmholtz, 64, 75, 85, 330, 372, 494
Hermite, 323
hyperbolic, 111
Klein–Gordon (KG), 64, 542
Korteweg–de Vries (KdV), 88, 542,
574, 593
Lagrange, 8
Laguerre, 323
Laplace, 3, 64, 330, 361
Lax, 592
Legendre, 302, 384
Legendre associated, 306
linear Schr¨odinger, 82, 582
linearized Korteweg–de Vries
(KdV), 83
Maxwell, 3, 84
Navier, 72, 73
nonhomogeneous wave, 139
nonlinear, 29
nonlinear diffusion, 80
nonlinear hyperbolic, 550
nonlinear Schr¨odinger (NLS), 83,
583, 585
parab olic , 111
partial differential, 12
Poisson, 64, 85, 330, 429
p ot ential, 3, 5
quasi-linear partial differential, 28,
89
Schr¨odinger, 64, 317, 330, 382, 404,
542, 581
spherical wave, 153
Sturm–Liouville, 274
Tchebycheff, 323
telegraph, 64, 84, 87, 147, 436
time dependent Schr¨odinger, 382
vorticity, 89
wave, 64, 67, 69, 73, 511, 535, 536,
542
wave and heat, 372
Whitham, 545, 548
equation for vibration of a beam, 542
error function, 452
Euler, 3
Euler equation, 38, 65, 88, 89, 100
Euler’s equation of motion, 85
Euler–Lagrange equation, 9, 629, 633,
634, 646
Euler–Lagrange variational principle,
634
Euler–Lagrange variational problem,
632
Euler–Poisson–Darboux equation, 598
even extension, 192
explicit finite difference methods, 608
Fermat, 7
Fermat principle in optics, 635
Fermi–Pasta–Ulam model, 88
fine structure corrections, 389
finite difference approximations, 602
finite element method, 663
finite Fourier cosine transform, 500
finite Fourier sine transform, 499
finite Hankel transform, 505
first canonical form, 95
Fitzgerald, 10
five-point formula, 621
flood wave, 552
flux, 580
Fokker–Planck equation, 62, 90, 264,
436
Forsyth, 6
forward first difference, 602
Fourier, 4
Fourier coefficients, 173
Fourier cosine transform, 443, 456, 500
Fourier integral representation, 218,
219
Fourier integral theorem, 4, 217
Fourier integrals, 2 14
Fourier method, 232
Fourier series, 171, 173
Fourier sine integral formula, 443
Fourier sine transform, 443, 457
Fourier transform, 440
Fourier–Legendre series, 284
fractional axisymmetric wave-diffusion
equation, 518
fractional derivative, 480, 482, 499
fractional diffusion equation, 510, 514
774 Index
fractional diffusion-wave equation, 511
fractional partial differential equation,
510
fractional Rayleigh problem, 517
fractional Schr¨odinger equation, 520
fractional Stokes and Rayleigh
problem, 515
fractional unsteady Couette flow, 517
fractional wave equation, 512
fundamental harmonic, 239
fundamental period, 169
fundamental solution, 451
Galerkin approximation method, 655
Gauss, 5
Gaussian probability density, 404
general Parseval relation, 227
general solution, 30, 107
generalized coordinates, 638
generalized solution, 130
Gibbs, 186
Gibbs phenomenon, 181, 185, 208, 225
global convergence, 174
Goursat, 6
Goursat problem, 149
gravitational potential, 76
Green, 5
Green’s function, 312, 407–409, 451
group velocities, 541
group velocity, 546
Hadamard, 3
Hadamard example, 121
Hamilton, 8
Hamilton equations of motion, 8, 643
Hamilton’s principle, 8, 636
Hamilton–Jacobi equation, 9
Hamiltonian, 382
Hamiltonian function H, 642
Hankel transform, 489
harmonic, 330
harmonic function, 5, 330
harmonic oscillator, 320
heat conduction in a rectangular
volume, 381
heat equation, 64, 76, 84
heat flow in a rectangular plate, 375
Heaviside unit step function, 453, 470
Helmholtz equation, 64, 75, 85, 330,
372, 494
Helmholtz operator, 415, 418
Hermite equation, 323
Hermite function, 265
Hermite polynomials, 320
Hilbert, 10
Hilbert space, 629
homogeneous, 13, 29
hydrogen atom, 382
hyperbolic, 92
hyperbolic equation, 111
ill-posed problem, 549
implicit finite difference method, 624
implicit Richardson formula, 626
impulse function, 454
infinite square well potential, 389
initial boundary-value problem, 15
initial condition, 15, 37
initial data, 37
initial-value problem, 3, 15, 36, 118
inner product, 629
integral surface, 33
integration, 465
integration theorem, 209
inverse Fourier cosine transform, 500
inverse Fourier sine transform, 500
inverse Fourier transform, 440
inverse Hankel transform, 489, 505
inverse Laplace transform, 461
inverse Mellin transformation, 496
irrotational, 73
Jacobi, 6, 9
Jacobian, 411
Kantorovich method, 659
Klein, 10
Klein–Gordon (KG) equation, 64, 542
Korteweg–de Vries (KdV) equation,
88, 542, 574, 593
Kowalewskaya, 6
Lagrange, 2, 3
Lagrange equation, 8
Lagrange equations of motion, 639
Lagrange identity, 279, 297
Lagrangian, 8
Laguerre equation, 323
Index 775
Laplace, 4
Laplace equation, 3, 64, 330, 361
Laplace operator, 65, 72, 372
Laplace transform, 461
law of the conservation of energy, 643
Lax equation, 592
Lax equivalence theorem, 604
Lax pair, 591
Lax–Wendroff explicit method, 605
Lax–Wendroff second-order finite
difference scheme, 606
leap frog algorithm, 607, 610
Legendre, 5
Legendre associated equation, 306
Legendre equation, 302, 384
Legendre function, 302, 370
Legendre function of the first kind,
304
Legendre function of the second kind,
305
Legendre polynomial, 304
Lerch, 465
Lie, 6
Liebmann iterative method, 619, 620
Lighthill, 446
linear, 13, 28, 630
linear integral superposition principle,
19
linear Schr¨odinger equation, 82, 582
linear superposition principle, 19
linearity, 444, 463
linearized Korteweg–de Vries (KdV)
equation, 83
Liouville, 6, 7
Liouville theorem, 352
loc al convergence, 174
loc al frequency, 546
loc al truncation error, 604
loc al wavenumber, 546
longitudinal wave velocity, 73
Lorentz, 10
lowest state energy, 319
Mach number, 86
magnetic quantum number, 385
Mainardi, 514
Maup ertius, 7
maximum principle, 332
Maxwell, 3
Maxwell equation, 3, 84
mean value theorem, 338, 618
mean-square convergence, 173, 174
Mellin transform, 496
metho d of characteristics, 35
metho d of images, 420
metho d of majorants, 6
metho d of separation of variables, 3,
51, 232
Michelson, 186
minimum principle, 332
Mittag-Leffler function, 510
mixed condition, 235
Monge, 2, 5
Monge axis, 33
Monge cone, 35
Navier, 6
Navier equations, 72, 73
Neumann, 5
Neumann boundary-value problem, 5
Neumann condition, 235
Neumann problem, 331, 341, 430
Newton, 2
Newton second law of motion, 637
no des, 239
No ether, 10
nonhomogeneous, 13, 29
nonhomogeneous wave equation, 139
nonlinear, 13
nonlinear diffusion equation, 80
nonlinear dispersion relation, 585
nonlinear equation, 29
nonlinear hyperbolic equation, 550
nonlinear initial-value problem, 539
nonlinear Schr¨odinger (NLS) equation,
83, 583, 585
norm, 170, 450, 629
normal modes, 239
normal stresses, 69
numerical characteristics, 609
o dd extension, 192
orbital quantum member, 385
order, 13
orthogonal, 170, 277
orthonormal system, 170
Ostrogradsky, 5, 8
overtones, 239
776 Index
parab olic, 92
parab olic equation, 111
Parseval formula, 195, 449, 497
Parseval relation, 176, 212, 491
Parseval’s equality, 286
partial differential equation, 12
particular solution, 30
p eriod, 169
p eriodic, 169
p eriodic extension, 190, 192
phase, 541
phase function, 459
Picard, 11
piecewise continuous, 168
piecewise smooth, 168
piston problem, 558, 559
Planck constant, 318, 382
Plateau, 7
Poincar´e, 8
Poincar´e–Cartan invariant, 9
p oint phase, 459
p ointwise approximations, 663
p ointwise convergence, 173
p ointwise convergence theorem, 202
Poisson, 4
Poisson equation, 64, 85, 330, 429
Poisson integral formula, 337, 428, 451
Poncelet, 5
p ositive definite, 631
p ot ential, 5
p otential energy, 382
p otential equation, 3, 5
p ower spectrum, 450
principal quantum number, 387
principle of conservation of energy, 58 1
principle of least action, 7, 636
principle of linear superposition, 19
principle of superposition, 451
progressive wave, 127
quantum number, 319
quasi-linear, 13
quasi-linear partial differential
equation, 28, 89
radial velocity, 154
range of influence, 124
Rayleigh, 10
Rayleigh layer, 478
Rayleigh problem, 477
Rayleigh–Ritz approximation method,
647
Rayleigh–Ritz method, 655
regular Sturm–Liouville (RLS) system,
274
relaxation factor, 620
residual, 656
residual displacement, 129
Richardson explicit finite difference
scheme, 614
Richardson iteration formula, 620
Riemann function, 148
Riemann invariants, 557, 558
Riemann method, 142
Riemann zeta function, 189, 533
Riemann–Lebesgue lemma, 201
Robin problem, 331, 422
Rodriguez formula, 305
rotational waves, 74
round-off error, 604
Rydberg, 388
sawtooth wave function, 180
scalar potential, 74
scaling, 445, 463, 492
Schr¨odinger equation, 64, 317, 330,
382, 404, 542, 581
second canonical form, 95
second shifting, 470
self-adjoint, 143, 296, 630
semilinear, 28
shear stresses, 69
shifting, 444, 463
sho ck, 540
sho ck condition, 562
sho ck thickness, 569
sho ck wave, 561
signum function, 460
similarity variable, 478
simple eigenvalue, 277
simple harmonic oscillator, 637
simple wave, 553
simple wave motions, 558
singular solution, 30
singular Sturm–Liouville ( SSL)
system, 297
skin friction, 479
smooth function, 31
Index 777
Snell law of refraction of light, 636
solitary wave, 576, 581, 585
soliton, 83, 573, 576
solution
complete, 29
d’Alembert, 3, 123
fundamental, 451
general, 30
generalized, 130
particular, 30
singular, 30
Stokes steady-state, 478
weak, 31
solution surface, 33
solutions, 12
Sommerfeld radiation condition, 494
source point, 429
spect ral function, 459
spherical harmonics, 385
spherical symmetric wave, 153
spherical wave equation, 153
stability, 602
stable, 604
standing wave, 239
stationary phase method, 458
stationary point, 459
Stokes, 6
Stokes expansion, 548
Stokes layer, 478
Stokes problem, 477
Stokes steady-state solution, 478
Stokes–Ekman problem, 528
Sturm, 7
Sturm–Liouville equation, 274
Sturm–Liouville operator, 279
Sturm–Liouville theory, 6, 273
Successive Over-Relaxation (SOR)
scheme, 620
superposition principle, 20
linear, 19
linear integral, 19
Tchebycheff equation, 323
telegraph equation, 64, 84, 87, 147,
436
thermal conductivity, 75, 481
time dependent Schr¨odinger equation,
382
time invariant function, 581
total kinetic energy, 240
total potential energy, 24 0
traffic flow model, 549
transform
finite Fourier cosine, 500
finite Fourier sine, 499
finite Hankel, 505
Fourier, 440
Fourier cosine, 443, 456, 500
Fourier sine, 443, 457
Hankel, 489
inverse Fourier, 440
inverse Fourier cosine, 500
inverse Fourier sine, 500
inverse Hankel, 489, 505
inverse Laplace, 461
Laplace, 461
Mellin, 496
transverse wave velocity, 72
triangular wave, 128
triangular wave function, 187, 188
truncation error, 60 2
undular bore, 579
uniform and absolute convergence
theorem, 208
uniform convergence, 173
uniqueness theorem, 333
variational method, 629
vector potential, 74
vibrating membrane, 67, 372
vibrating string problem, 136, 235
vibration of a circular membrane, 374
von Neumann stability, 610
von Neumann’s stability method, 605
vorticity equation, 89
wave
cnoidal, 578
de Broglie, 542
dispersive, 540
flo od, 552
progressive, 127
rotational, 74
sho ck, 561
simple, 553
solitary, 576, 581, 585
spherical symmetric, 153
standing, 239
778 Index
triangular, 128
wave and heat equations, 372
wave equation, 64, 67, 69, 73, 511, 535,
536, 542
waves in three dimensions, 379
waves of distortion, 72
weak solution, 31
Weierstrass, 6
Weierstrass approximation theorem,
223
well-posed problem, 16, 124, 166
Weyl, 10
Weyl fractional integral, 498
Whitham, 545, 588
Whitham equation, 545, 548
Wright function, 511
Zakharov–Shabat (ZS) scheme, 594
Zienkiewicz and Cheung, 664