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 alsoinitial
value–boundary value problems
wave equation, 217, 220–221, 228
initial value–boundary value problems,
140
heat conduction problems, 138.See
alsoheat 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 alsoheat
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, 42L
Lake Ontario, 105
Lake Placid, 105
Laplace’s equation.Seepotential
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.Seeheat conduction problems
potential.Seepotential equation
wave.Seewave 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