Powers D.L. Boundary Value Problems and Partial Differential Equations
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488 Answers to Odd-Numbered Exercises
9. 25(u
i+1
−2u
i
+u
i−1
) − 25u
i
=−25, i = 1, 2, 3, 4, 5; u
0
= 2, u
5
+(u
6
−
u
4
)/(2/5) =1. When the equation for i = 5 and the boundary condition
are combined, they become 2u
4
− 3.4u
5
=−1.4. Solution: u
1
= 1.382,
u
2
=1.146, u
3
=1.057, u
4
=1.023, u
5
=1.014.
11. 9(u
i+1
− 2u
i
+ u
i−1
) + (3/2)(u
i+1
− u
i−1
) − u
i
=−(1/3)i, i = 0, 1, 2;
u
3
= 1, (u
1
− u
−1
)/(2/3) = 0. When u
−1
is eliminated and coefficients
are collected, the equations to solve are
−19u
0
+18u
1
= 0,
7
1
2
u
0
−19u
1
+10
1
2
u
2
=−
1
3
,
7
1
2
u
1
−19u
2
=−11
1
6
.
Solution: u
0
=0.795, u
1
=0.839, u
2
=0.919.
Section 7.2
1. Line m of the solution should be exactly the same as line m +1ofTable4.
3. r = 2/5, t = 1/40.
i
m 01 2 3 4
0 00000
1 1 0. 0. 0. 0.
2 1 0.40. 0. 0.
3 1 0.48 0.16 0. 0.
4 1 0.56 0.224 0.064 0.
5 1 0.6016 0.2944 0.1024 0.0512
5. t = 1/32. Remember that u
4
(m) = u
0
(m) = mt.Allnumbersinthe
table should be multiplied by t.
i
m 01 2 3 4
0 00 0 0 0
1 1 0001
2 2 1/20 1/2 2
3 3 11/21 3
4 4 7/41 7/4 4
5 5 5/27/45/2 5
7. t = 1/32. All numbers in this table should be multiplied by t.
Chapter 7 489
i
m 01 2 3 4
0 00 0 0 0
1 0 1110
2 0 3/22 3/2 0
3 0 25/22 0
4 0 9/43 9/4 0
.
.
.
∞ 0 3430
9. t =
1
32
. Remember u
−1
= u
1
. All entries in this table should be multi-
plied by x =
1
4
.
i
m 01234
0 01234
11 1 2 3 4
21 3/22 3 4
33/23/29/43 4
43/215/89/425/8 4
515/815/85/225/8 4
Section 7.3
1.
i
m 01234
0 00 0 0 0
1 0 1/41/41/4 0
2 0 1/41/21/4 0
3 0 1/41/41/4 0
4 0 0000
5 0 −1/4 −1/4 −1/4 0
3. In this table, α = 1/
√
2.
i
m 01 2 3 4
0 00 0 0 0
1 0 α/41/4 α/4 0
2 0 1/4 α/21/4 0
3 0 α/41/4 α/4 0
4 0 0000
5 0 −α/4 −1/4 −α/4 0
490 Answers to Odd-Numbered Exercises
5.
i
t
m
m 01 2 3 4
0001/21 1/20
0.177 1 0 1/23/41/2 0
0.354 2 0 3/81/43/8 0
0.530 3 0 0 −1/800
0.707 4 0 −7/16 −3/8 −7/16 0
0.884 5 0 −5/8 −11/16 −5/8 0
7.
i
m 01 2 3 4
0 00000
1 0 0001
2 0 0011
3 0 0111
4 0 1110
5 0 110−1
6 0 00−1 −1
7 0 −1 −2 −1 −1
8 0 −2 −2 −2 0
9. Run: u
i
(m + 1) = (2 −2ρ
2
−16 t
2
) u
i
(m) + ρ
2
u
i−1
(m) + ρ
2
u
i+1
(m) −
u
i
(m − 1).Start:u
i
(1) =
1
2
((2 − 2ρ
2
− 16 t
2
) u
i
(0) + ρ
2
u
i−1
(0) +
ρ
2
u
i+1
(0)). Longest stable time step: t =1/
√
24(ρ
2
=2/3).
i
m 01 2 34
0 00.50 1.00 0.50 0
1 0 0.33 0.33 0.33 0
2 0 −0.28 −0.56 −0.28 0
3 0 −0.70 −0.70 −0.70 0
4 0 −0.19 −0.38 −0.19 0
5 0 0.45 0.45 0.45 0
6 0 0.49 0.98 0.49 0
7 0 0.21 0.21 0.21 0
8 0 −0.35 −0.71 −0.35 0
Section 7.4
1. At (1/4, 1/4),11/256; at (1/2, 1/4),14/256; at (1/2, 1/2),18/256.
3. In both this exercise and Exercise 4, the exact solution is u(x, y) =xy,and
the numerical solutions are exact.
Chapter 7 491
5. Coordinates and values of the corresponding u
i
are: (1/7, 1/7),5α;
(2/7, 1/7),10α; (3/7, 1/7),14α; (1/7, 2/7),21α; (2/7, 2/7),32α.Here
α = 19/1159.
7. u
1
= 0.670, u
2
= 0.721, u
3
= 0.961, u
4
= 1.212, u
5
= 0.954, u
6
= 0.651.
The remaining values are found by symmetry.
9. u
1
=0.386, u
2
=0.542, u
3
=0.784, u
4
=0.595. The remaining values are
found by symmetry.
Section 7.5
1. Use Eq. (8) with r =1/4.
i
m 1234 5 6
0 0000 0 0
10 0 0 1/41/41/4
21/16 1/16 1/16 5/16 3/85/16
33/32 1/83/32 23/64 27/64 23/64
3. Note that u
1
=u
2
=u
4
=u
5
; replacement equations become
u
1
(m +1) = u
3
(m)/4, u
3
(m +1) = u
1
(m).
i
m 13
0 11
11/41
21/41/4
31/16 1/4
41/16 1/16
5. Use Eq. (8) with r =1/4. Note that u
4
=u
2
, u
7
=u
3
, u
8
=u
6
.
i
m 12356 9
0 00000 0
10 0 1/40 1/41/2
20 1/16 5/16 1/87/16 5/8
31/32 7/64 3/81/417/64 23/32
7. Use the same numbering as for Exercise 5. Note that u
1
= u
3
= u
7
= u
9
and u
2
=u
4
=u
6
=u
8
. The running equations become
u
1
(m +1) =
1
2
u
2
(m) −u
1
(m −1),
492 Answers to Odd-Numbered Exercises
u
2
(m +1) =
1
2
u
1
(m) +
1
4
u
5
(m) −u
2
(m −1),
u
5
(m +1) = u
2
(m) −u
5
(m −1).
m
012 3 4 5 6 7
u
1
0 00 1/80 −5/16 0 15/32
u
2
0 01/40 −3/80 5/16 0
u
5
0 10 −3/40 3/80 −1/16
9.
i
m 125
0 111
11/23/41
2 −1/40 1/2
3 −1/2 −3/4 −3/2
4 −1/2 −5/4 −2
5 −3/4 −3/4 −1
11. See Fig. 12 below for numbering of points.
i
m 123 456
0 100 000
10 1/41/4000
2 −3/40 0 1/41/80
30 −1/2 −7/16 0 0 3/16
Figure 12 Solution of Exercise 11, Section 7.5.
Chapter 7 493
Chapter 7 Miscellaneous Exercises
1.
u
i+1
−2u
i
+u
i−1
(x)
2
−
√
24x
i
u
i
=0, i = 0, 1, 2,
u
1
−u
−1
2x
=1, u
3
=1;
−18u
0
+18u
1
= 6,
9u
0
−20.83u
1
+9u
2
= 0,
9u
1
−22u
2
=−9,
u
0
=−0.248, u
1
=0.08, u
2
=0.44.
3.
u
i+1
−2u
i
+u
i−1
(x)
2
+
1
1 +x
i
u
i+1
−u
i−1
2 x
=−(1 +x
i
),
i =1, 2, 3;u
0
=1, u
4
=0;
−32u
1
+17.60u
2
=−15.65,
14.67u
1
−32u
2
+17.33u
3
=−1.5,
14.86u
2
−32u
3
=−1.75,
u
1
=0.822, u
2
=0.606, u
3
=0.335.
5. u
i
(m + 1) = (u
i−1
(m) + u
i+1
(m))/2. Note that u
3
(m) = u
1
(m) and
u
4
(m) =u
0
(m).
i
m 01 2
0 00 0
10.030 0
2 0.06 0.015 0
3 0.09 0.03 0.15
4 0.12 0.053 0.03
5 0.14 0.075 0.053
6 0.17 0.1 0.075
7 0.20 0.122 0.10
8 0.22 0.15 0.122
7. First problem: u
i
(m+1) = (u
i+1
(m)+u
i
(m)+u
i−1
(m))/3; second prob-
lem: u
i
(m +1) = (u
i+1
(m) +u
i−1
(m))/3.
494 Answers to Odd-Numbered Exercises
First Problem Second Problem
ii
m 0123401234
0 0111001110
1 0 2/31 2/3 001/32/31/3 0
2 0 5/97/95/9 002/92/92/9 0
3 0 14/27 17/27 14/27 002/27 4/27 2/27 0
4 0 31/81 45/81 31/81 004/81 4/81 4/81 0
9. u
i
(m +1) = (u
i+1
(m) +u
i−1
(m))/2.
i
m 01 2 3 4 5
0 00 0 0 0 0
1 1/2 00000
2 1 1/40 0 0 0
3 3/2 1/21/80 0 0
4 2 13/16 1/41/16 0 0
5 5/2 9/87/16 1/81/32 0
6 3 87/32 5/815/64 1/16 0
11.
i
m 01234
0 00000
1 0 0001
2 0 0011
3 0 0111
4 0 1111
5 0 1111
6 0 0111
7 0 0011
8 0 0001
13. Let u
ij
∼
=
u(x
i
, y
j
).Thenu
11
=u
22
=u
33
=0.5, u
12
= 0.698, u
13
= 0.792,
u
21
=0.302, u
23
=0.624, u
21
=0.209, u
32
=0.376.
15. NumberasinChapter7,Fig.4.Thenu
1
= u
3
= u
7
= u
9
and u
2
= u
4
=
u
6
=u
8
.
m
01 2 3 4 5
u
1
1 1/23/81/43/16 1/8
u
2
1 3/41/23/81/43/16
u
5
1 13/41/23/81/4
Index
A
acoustic vibrations, 235
aluminum nitride, oxygen removal
from, 156
antenna vibrations, 226
approximation by Fourier series, 91–94
arbitrary periods, 64–65
B
bad discontinuities, 74–75, 79
band-limited functions, 121–123
bars, insulated, 157–161. See also heat
conduction problems
beam vibrations, 225
Bessel functions, integrals of, 440
Bessel inequality, 92–93
boundary conditions. See also initial
value–boundary value problems
of the first kind. See entries at
Dirichlet
limitations of product method,
281–282
mixed, 139
potential equation. See potential
equation
of the second kind. See entries at
Neumann
of the third kind, 139
wave equation, 217–220, 229
boundary value problems, 26–34. See
also initial value–boundary value
problems
Fourier series applications with,
119–120
Green’s functions, 23, 43–49
potential equation, 256–257
singular, 38–41
boundedness, 40–41
Boussinesq equation, 211
Brownian motion, 204
buckling of a column, 32–34
Burger’s equation, 209
C
cable, hanging, 26–29
calculus, 436–438
cantilevered beam, 226
car antenna vibrations, 226
catenaries, 35
Cauchy–Euler equation, 6–7, 38, 277
Cesaro summability, 131
chain rule, 232
495
496 Index
characteristic equation, 3, 10
characteristics, method of, 246
classifications of partial differential
equations, 280–282
coefficients of Fourier series. See Fourier
series
column buckling, 32–34
complementary error function, 200, 202
complex coefficients, potential equation
analysis, 284
complex Fourier coefficients, 113–115
sampling theorem, 121–124
conditions. See boundary conditions;
initial condition
conduction of heat. See heat conduction
problems
conservation of energy, law of, 135–136
continuity behavior, 73–76
convection, 170–174. See also heat
conduction problems
convergence
expansions in series of
eigenfunctions, 182
Fourier series, 73–77, 124
in the mean, 93–94
uniform, 79–83
cooling, Newton’s law of, 30, 139
cooling fins, 40–41
cosine function, 66–68
Fourier cosine integral
representation, 109–110
hyperbolic, 4
integrals of, 439
critical radius, 42
cutoff frequency, 121
D
d’Alembert’s method (traveling wave),
227–231, 252
damping term, 117
delta functions, 113
diffusion equations. See heat
conduction problems
diffusion of sulphur dioxide, 55–56,
192–193
diffusivity, 141
dimensions of heat flux, 135
Dinac’s delta function, 113
Dirichlet conditions, 139
Dirichlet’s problem, 256. See also
potential equation
in a disk, 275–279
in a rectangle, 259–269
soap films, 283
discontinuity, 74
disk, potential equation in, 275–279
E
eigenfunctions, 158–159
expansions in series of, 181–182
orthogonality of, 175–177
eigenvalue problems, 34, 158–159
estimating eigenvalues for wave
equations, 236–239
one-dimensional wave equation, 234
singular, 189
Sturm–Liouville problems, 178–179
expansions in series of
eigenfunctions, 181–182
generalizations on heat conduction
problems, 184–187
one-dimensional wave equation,
234
elliptic equations, 281
endpoints of periodic extensions, 76–77
energy conservation, law of, 135–136
enzyme electrodes, 212–214
equilibrium problems. See steady-state
problems
error function, heat conduction
problems, 199–202
even functions, 67
continuity behavior of, 83
extensions of functions, 69–71
exponential functions, integrals of,
438–439
exponential growth, 2
extensions of periodic functions, 65–71
endpoints of, 76–77
uniform convergence, 82–83
F
fast Fourier transform (FFT), 124
Fick’s law, 55, 141
Index 497
finite Fourier series, 91–94
first-order equations
homogeneous, 1–2
nonhomogeneous, 20–21
Fisher’s equation, 252
Fitzhugh–Nagumo equations, 239–244
fixed end temperatures (heat equation),
149–155
flat enzyme electrodes, 212–214
flow (fluid), 284–285, 289
fluid flows, 284–285, 289
Fokker–Planck equation, 205
forced vibrations of strings, 232
forced vibrations system, 17–20
forcing function, 117, 226
Fourier integrals, 106–111, 124, 190,
194
applications of, 117–123
coefficient functions, 108
complex coefficients. See complex
Fourier coefficients
Fourier transforms, 115
Fourier’s single integral, 112–113
history of, 124
representational theorem, 108
wave equation in unbounded regions,
239–244
Fourier series, 62–63, 124
applications of, 117–123
arbitrary periods, 64–65
complex coefficients. See complex
Fourier coefficients
convergence, 73–77
proof of, 95–99
uniform convergence, 79–83
cosine integral representation, 109
history of, 124
means of, 90–94
numerical determination of
coefficients, 100–104
operations on, 85–89
periodic extensions, 65–71
endpoints of, 76–77
uniform convergence, 82–83
potential in rectangle, 260–261
sine integral representation, 109
Fourier transforms, 115
Fourier’s law, 30
Fourier’s method (separation of
variables), 150, 166–167
freezing lake, temperature of, 204
frequencies of vibration, 223–224, 234
functions. See specific function by name
G
Gaussian probability density function,
203
general solutions
boundary value problems, 26
homogeneous differential equations,
158
nonhomogeneous linear equations,
15
one-dimensional wave equation, 228
second-order homogeneous
equations, 3
second-order linear partial
differential equations, 205,
280–281
generalized rectangles, 281
generation rate functions, 141
Gibbs’ phenomenon, 82
Green’s functions, 23, 43–49
groundwater flow, 52, 211–212
H
half-range extensions, 70–71
hanging cable system, 26–29
harmonic functions, 255. See also
potential equation
heat conduction problems, 29–31,
135–206, 280
convection, 170–174
cooling fins, 40–41
derivation of, 135–141
different end conditions (example),
157–161
error function, 199–202
fixed end temperatures (example),
149–155
generalizations on, 184–187
insulated ends (example), 157–161
radial heat flow, 39–40
steady-state temperatures, 143–147