Powers D.L. Boundary Value Problems and Partial Differential Equations
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498 Index
higher-order equations, homogeneous,
9–11
homogeneous boundary conditions,
as requirement, 282
homogeneous linear equations, 1–11
first-order, 1–2
higher-order, 9–11
second-order, 2–9
hyperbolic equations, 281
hyperbolic functions, 4, 436, 438–439
I
infinite intervals, 40–41
infinite rods, 193–197
initial conditions. See also initial
value–boundary value problems
wave equation, 217, 220–221, 228
initial value–boundary value problems,
140
heat conduction problems, 138. See
also heat conduction problems
convection, 170–174
different end conditions
(example), 163–166
fixed end temperatures (example),
149–155
generalizations on, 184–187
infinite rods, 193–197
insulated ends (example), 157–161
semi-infinite rods, 184–187
wave equation, 215–247, 280
d’Alembert’s method, 227–231,
252
estimating eigenvalues for, 236–239
frequencies of vibration, 223–224,
234
one-dimensional, in general,
233–235
in unbounded regions, 239–244
vibrating string problem, 215–224
insulated bars, 157–161. See also heat
conduction problems
insulated surfaces, 139
integrals, table of, 438–440
integro-differential boundary value
problems, 54
irrotational vortex, 291
J
jump discontinuities, 74
K
kryptonite, 42
L
Lake Ontario, 105
Lake Placid, 105
Laplace’s equation. See potential
equation
Laplacian operator, 256–257
law of conservation of energy, 135–136
law of cooling (Newton), 30, 139
law of radiation (Stefan–Boltzmann),
142
left-hand limits, 73–74
Legendre polynomials, integrals of, 440
level curves, 261–262
L’Hospital’s rule, 98
linear density, 216
linear differential equations
homogeneous, 1–11
first-order, 1–2
higher-order, 9–11
second-order, 2–9
nonhomogeneous, 14–23
Fourier series applications with,
117–119
undetermined coefficients, 16–20
variation of parameters, 20–23
linear operations, 140
linear partial differential equations
general form, 205
heat. See heat conduction problems
potential. See potential equation
wave. See wave equation
linearly independent solution, 3
M
Massena, New York, 132
mass–spring–damper system, 5
forced vibrations system, 17–20
maximum principle, 255, 278
mean error, 90–94
Index 499
mean value property, 278
periodic functions, 61
membrane displacement, 255. See also
potential equation
method of characteristics, 246
mixed boundary conditions, 139
N
Neumann conditions, 139
Neumann’s problem, 256, 290. See also
potential equation
Newton’s law of cooling, 30, 139
nonhomogeneous linear equations,
14–23
Fourier series applications with,
117–119
undetermined coefficients, 16–20
variation of parameters, 20–23
nonremovable discontinuities, 74–75,
79
normal probability density function,
203
normalized eigenfunctions, 183
normalizing constants, 183
nuclear fuel rods. See heat conduction
problems
numerical determination of Fourier
coefficients, 100–104
O
odd functions, 67
continuity behavior of, 83
extensions of functions, 69–71
ODEs (ordinary differential equations),
1–51
boundary value problems, 26–34
Green’s functions, 43–49
homogeneous, 1–11
first-order, 1–2
higher-order, 9–11
second-order, 2–9
nonhomogeneous, 14–23
Fourier series applications with,
117–119
undetermined coefficients, 16–20
variation of parameters, 20–23
singular boundary value problems,
38–41
one-dimensional wave equation,
233–235
ordinary differential equations, 1–51
boundary value problems, 26–34
Green’s functions, 43–49
homogeneous, 1–11
first-order, 1–2
higher-order, 9–11
second-order, 2–9
nonhomogeneous, 14–23
Fourier series applications with,
117–119
undetermined coefficients, 16–20
variation of parameters, 20–23
singular boundary value problems,
38–41
ordinary limits, 73–74
organ pipes, 225
orthogonality, 60–61, 73
of eigenfunctions, 175–177
oxygen removal from aluminum nitride,
156
P
parabolic equations, 281
Parseval’s equality, 92–93
partial differential equation
classifications, 280–282
particular solutions, nonhomogeneous
linear equations, 15–23
penetration of heat into earth, 192
periodic functions, 59–63
arbitrary periods, 64–65
extensions of, 65–71
endpoints of, 76–77
uniform convergence, 82–83
piecewise continuous functions, 75–76
piecewise smooth functions, 76
plate, flow past, 289
Poiseuille flow, 36
Poisson equation, 268–269
polar coordinates, potential equation in,
256–257, 275–279
polynomial solution for potential
equation, 256
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. See heat conduction problems
homogeneous, 2–9
nonhomogeneous, 21–23
potential. See potential equation
wave. See wave 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 also potential
equation
temperature (heat conduction),
143–147
convection, 170–174
different end conditions
(example), 157–161
fixed end temperatures (example),
149–155
Index 501
generalizations on, 184–187
insulated ends (example), 157–161
semi-infinite rods, 184–187,
193–197
wave equation, 218, 232
Stefan–Boltzmann law of radiation, 142
Stokes derivative, 250
stream function, 284
stresses due to thermal effects, 214
string, vibrating, 215–224. See also wave
equation
frequencies of vibration, 223–224,
234
one-dimensional wave equation,
233–235
Sturm–Liouville problems, 178–179
expansions in series of
eigenfunctions, 181–182
generalizations on heat conduction
problems, 184–187
one-dimensional wave equation, 234
sulphur dioxide, diffusion of, 55–56,
192–193
superposition, principle of, 3, 10, 152
wave equation and standing waves,
220
surfaces, insulated, 157–161. See also
heat conduction problems
suspension bridge (hanging cable
system), 26–29
symmetry of sine and cosine functions,
66–68
T
table of integrals, 438–440
Taylor series, 114
temperature (heat conduction)
steady-state, 143–147
three-dimensional steady-state
solution. See potential equation
two-dimensional steady-state
equation, 255
term, forcing functions, 117
thermal conductivity, 137
thermal diffusivity, 137
thermal stresses, 214
transient temperature distribution, 146
fixed end temperatures, 149–155
transverse displacement. See wave
equation
trapezoidal function, 125
traveling wave solution (d’Alembert’s
method), 227–231, 252
triangle function, 123
trigonometric functions, 435
trigonometric series, history of, 124
trivial solutions, 150
truncated Fourier series, 92
U
unbounded conditions
potential equation, 270–272
wave equation, 239–244
undetermined coefficients,
nonhomogeneous linear
equations, 16–20
uniform convergence, 79–83
V
variation of parameters, 20–23
velocity potential function, 258, 284
vibrating string problem, 215–224. See
also wave equation
frequencies of vibration, 223–224,
234
one-dimensional wave equation,
233–235
W
water hammers, 250
wave equation, 215–247, 280
d’Alembert’s method, 227–231, 252
estimating eigenvalues for, 236–239
frequencies of vibration, 223–224,
234
one-dimensional, in general, 233–235
in unbounded regions, 239–244
vibrating string problem, 215–224,
234
whirling speeds, 55
windows, 111
Wronskian, 3
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