500 Index
potential equation, 255–285
in disk, 275–279
limitations of product method,
280–282
Poisson equation, 268–269
polynomial solution for, 256
in rectangle, 259–269
soap films, 283
solutions to (harmonic functions),
255
in unbounded regions, 270–272
principle of superposition, 3, 10, 152
wave equation and standing waves,
220
probability density function, 203
product method (separation of
variables), 150, 166–167
limitations of, 280–282
potential in rectangle, 259–261, 266
R
radial heat flow, 39–40
radiation, 142
radical functions, integrals of, 438
rational functions, integrals of, 438
Rayleigh method, 239
Rayleigh quotient, 239
rectangle, potential equation in,
259–269
reduction or order, 8–9
regular singular points, 7, 38–40
regular Sturm–Liouville problems,
178–179
convergence theorem, 182
one-dimensional wave equation, 234
removable discontinuities, 74, 79
restoring term, forcing functions, 117
Revision Rule, 17
right-hand limits, 73–74
Robin conditions, 139
rod vibrations, 253
rods of heat-conducting material.See
heat conduction problems
S
sampling theorem, 121–124
sawtooth function, 81–82
second-order equations
general form, 205
heat.Seeheat conduction problems
homogeneous, 2–9
nonhomogeneous, 21–23
potential.Seepotential equation
wave.Seewave equation
sectionally continuous functions, 75–76
sectionally smooth functions, 76
semi-infinite intervals, 40–41
semi-infinite rods, 188–191
separation of variables (product
method), 150, 166–167
limitations of, 280–282
potential in rectangle, 259–261, 266
sine function, 66–68
Fourier sine integral representation,
109–110
hyperbolic, 4
integrals of, 439
singular boundary value problems,
38–41
singular eigenvalue problems, 189
singular points, 7, 38–40
soap films, 283
soliton (solitary) waves, 249
solutions, general
boundary value problems, 26
homogeneous differential equations,
158
nonhomogeneous linear equations,
15
one-dimensional wave equation, 228
second-order equations, 3, 205,
280–281
solutions, particular, 15–23
square-wave function, 75–77, 79–80
standing waves, 220
steady-state problems.See alsopotential
equation
temperature (heat conduction),
143–147
convection, 170–174
different end conditions
(example), 157–161
fixed end temperatures (example),
149–155