Chapter 28 Metodos Numericos para Ingenieros
July 29, 2022 | Author: Anonymous | Category: N/A
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CHAPTER 28 28.1 The solution with the 2 nd-order RK (Heun without corrector) can be laid out as t
c 0 10
k 1 10 25
c
2 1.25
k 2 30 37.5
1 0.625
1.5 0.9375
20 34.375 0.78125 42.187 875 5 0.390625 30 40.2 40.234 3438 38 0.48 0.4882 8281 81 45 45.1 .11 1719 0.24 0.2441 4141 41 40 43.8 43.896 9648 48 0.30 0.3051 5176 76 46 46.9 .948 4824 24 0.15 0.1525 2588 88 50 46 46.1 .185 853 3 0.19 0.1907 0735 35 48.0 48.092 9265 65 0.09 0.0953 5367 67
0.585938 0. 0.36 36621 6211 1 0. 0.22 2288 8882 82 0. 0.14 1430 3051 51
For the 4th-order RK, the solution is t
c
0 10 20 30 40 50
10 25.7 25 .729 2917 17 35 35.2 .273 7317 17 41 41.0 .064 6419 19 44.5780 44.57 801 1 46 46.7 .710 1009 09
k 1
2 1.21 1.2135 3542 42 0.73 0.7363 6342 42 0.44 0.4467 6791 91 0.27 0.271 11 0.16 0.1644 4495 95
c mid mid 20 31 31.7 .796 9688 88 38 38.9 .954 5487 87 43 43.2 .298 9814 14 45.93 45.9335 351 1 47 47.5 .532 3257 57
k 2
1.5 0. 0.91 9101 0156 56 0.55 0.5522 2256 56 0.33 0.3350 5093 93 0.20 0.2033 3325 25 0.12 0.1233 3371 71
c mid mid 17.5 30.2 30.279 7995 95 38 38.0 .0344 3445 5 42 42.7 .7396 3965 5 45.59 45.5946 463 3 47 47.3 .3269 2695 5
k 3 1.625 0.98 0.9860 6003 03 0. 0.59 5982 8278 78 0. 0.36 3630 3017 17 0. 0.22 2202 0268 68 0. 0.13 1336 3652 52
c end end 26.25 35 35.5 .589 8919 19 41 41.2 .255 5594 94 44 44.6 .694 9436 36 46 46.7 .780 8069 69 48 48.0 .046 4662 62
k 4 1.1875 0. 0.72 7205 054 4 0. 0.43 43720 7203 3 0. 0.26 26528 5282 2 0. 0.16 1609 0965 65 0. 0.09 09766 7669 9
A plot of both solutions alon with the anal!tical result is displa!ed below"
28.2 The #ass-balance e$uations can be written as
dc1 dt
dc 2 dt dc dt
= −%'1&c1 + %'%&c + 1
= %'2c1 − %'2c 2 = %'%c 2 − %'2c + 4
PROPRIETARY MATERIAL' * The The +craw +craw-Hill -Hill o# o#panies, panies, ..nc' nc' All rihts reser/ed' reser/ed' 0o part of this +anual #a! be displa!ed, reproduced or distributed in an! for# or b! an! #eans, without the prior written per#ission of the publisher, or used be!ond the li#ited distribution to teachers and educators per#itted b! +craw-Hill for their indi/idual course preparation' .f !ou are a student usin this +anual, !ou are usin it wit without hout per#ission'
1.572917 0. 0.95 9544 44 0. 0.57 5791 9102 02 0. 0.35 3513 1382 82 0.2 0.213 1320 208 8 0. 0.12 1293 9369 69
2
dc 4 dt
dc dt
c 4 + %'%c = %'%c − %'12
= %'%4c1 + %'%2 c 2 − %'%&c
3elected solution results (uler5s #ethod) are displa!ed below, alon with a plot of the results' t 0 1 2 3 4 5
c 1 0.00 0.0000 00 1.00 1.0000 00 2.08 2.0800 00 3.16 3.1672 72 4.21 4.2160 60 5.19 5.1999 99
c 2 c 3 0.00 0.0000 00 0.00 0.0000 00 0.00 0.0000 00 4.00 4.0000 00 0.20 0.2000 00 7.00 7.0000 00 0.57 0.5760 60 9.26 9.2600 00 1.09 1.0942 42 1 10 0.973 .9738 8 1.71 1.7186 86 1 12 2.285 .2851 1
c 4 0.0 0.0000 000 0.0 0.0000 000 0.3 0.3500 500 0.9 0.9203 203 1.6 1.6201 201 2.3 2.3874 874
c 5 0.00 0.0000 00 0.00 0.0000 00 0.04 0.0400 00 0.12 0.1248 48 0.25 0.2555 55 0.43 0.4307 07
dc 1 /d /dt 1. 1.00 0000 00 1. 1.08 0800 00 1. 1.08 0872 72 1. 1.04 0488 88 0. 0.98 9839 39 0. 0.90 9051 51
dc2 /d /dt dc3 /d /dt dc4 /d /dt 0.0 .000 000 0 4. 4.00 0000 00 0. 0.00 000 00 0.2 .200 000 0 3. 3.00 0000 00 0. 0.35 350 00 0.3 .376 760 0 2. 2.26 2600 00 0. 0.57 570 03 0.5 .518 182 2 1. 1.71 7138 38 0. 0.69 699 99 0.6 .624 244 4 1.3 .31 113 0. 0.76 767 73 0.6 .696 963 3 1. 1.01 0147 47 0. 0.79 792 27
dc5 /dt /dt 0. 0.0 0000 000 0. 0.0 0400 400 0. 0.0 0848 848 0. 0.1 1307 307 0.1 .175 752 2 0.2 .216 165 5
16 16.8 .851 515 5 16 16.8 .862 621 1 16 16.8 .872 720 0 16 16.8 .881 813 3 16 16.8 .890 901 1
12 12.9 .9430 430 12 12.9 .9609 609 12 12.9 .9777 777 12 12.9 .9935 935 13 13.0 .0084 084
0. 0.00 0002 02 0. 0.00 0002 02 0. 0.00 0001 01 0. 0.00 0001 01 0. 0.00 0001 01
0. 0.00 0004 04 0. 0.00 0004 04 0. 0.00 0003 03 0. 0.00 0003 03 0. 0.00 0003 03
0. 0.01 0179 79 0. 0.01 0168 68 0. 0.01 0158 58 0. 0.01 0149 49 0. 0.01 0140 40
• • •
76 77 78 79 80
13 13.2 .241 416 6 13 13.2 .241 418 8 13 13.2 .241 419 9 13 13.2 .242 421 1 13 13.2 .242 422 2
13. 13.23 2395 95 13. 13.23 2399 99 13. 13.24 2403 03 13. 13.24 2406 06 13. 13.24 2409 09
18 18.6 .6473 473 18 18.6 .6475 475 18 18.6 .6476 476 18 18.6 .6477 477 18 18.6 .6478 478
0. 0.00 0001 01 0. 0.00 0001 01 0. 0.00 0001 01 0. 0.00 0001 01 0. 0.00 0001 01
0. 0.01 0106 06 0. 0.00 0099 99 0. 0.00 0093 93 0. 0.00 0088 88 0. 0.00 0082 82
Finall!, +AT6A7 can be used to deter#ine the eien/alues and eien/ectors" >> a=[.16 -.06 0 0 0;-.2 .2 0 0 0;0 -.05 .25 0 0;0 0 -.0875 .125 -.0375;-.04 -.02 0 0 .06] a = 0.1600 -0.0600 0 0 0 -0.2000 0.2000 0 0 0 0 -0.0500 0.2500 0 0 0 0 -0.0875 0.1250 -0.0375 -0.0400 -0.0200 0 0 0.0600 >> [v,d]=eig(a)
PROPRIETARY MATERIAL' * The The +craw +craw-Hill -Hill o# o#panies, panies, ..nc' nc' All rihts reser/ed' reser/ed' 0o part of this +anual #a! be displa!ed, reproduced or distributed in an! for# or b! an! #eans, without the prior written per#ission of the publisher, or used be!ond the li#ited distribution to teachers and educators per#itted b! +craw-Hill for their indi/idual course preparation' .f !ou are a student usin this +anual, !ou are usin it wit without hout per#ission'
v = 0 0 0 1.0000 0 d = 0.1250 0 0 0 0
0 0 0.8192 -0.5735 0
0 0 0 0.4997 0.8662
-0.1039 -0.1582 -0.0436 0.4956 0.8466
0.2604 -0.5701 0.6892 -0.3635 0.0043
0 0.2500 0 0 0
0 0 0.0600 0 0
0 0 0 0.0686 0
0 0 0 0 0.2914
28.3 3ubstitutin the para#eters into the differential e$uation i/es dc dt
2 c − %'1c = 14' − %'%
The #id-point #ethod can be applied with the result"
The results are approachin a stead!-state /alue of 8'&2&' Challenge question:
The stead! state for# (i'e', dc9 dc9dt : : %) of the e$uation is % : 14' ; %'%c %'%c ; %'1c %'1c2, which can be sol/ed for 8'&2& and ;1%'141' Thus, there is a neati/e root' .f we put in the initial y initial y /alue /alue as −1%'141 (or hiher precision), the solution will sta! at the neati/e root for awhile until roundoff errors #a! lead to the solution oin unstable' .f we use an initial /alue less than the neati/e root, the solution will also o unstable' Howe/er, if we pic< a /alue that is slihtl! hiher (as per #achine precision), it will ra/itate towards the positi/e root' For e=a#ple if we use −1%'14
PROPRIETARY MATERIAL' * The The +craw +craw-Hill -Hill o# o#panies, panies, ..nc' nc' All rihts reser/ed' reser/ed' 0o part of this +anual #a! be displa!ed, reproduced or distributed in an! for# or b! an! #eans, without the prior written per#ission of the publisher, or used be!ond the li#ited distribution to teachers and educators per#itted b! +craw-Hill for their indi/idual course preparation' .f !ou are a student usin this +anual, !ou are usin it wit without hout per#ission'
4
This "$an=[0,50]; >> %0=[1,1,0]; >> [,%]=!de45(&dcd"iff,"$an,%0);
PROPRIETARY MATERIAL' * The The +craw +craw-Hill -Hill o# o#panies, panies, ..nc' nc' All rihts reser/ed' reser/ed' 0o part of this +anual #a! be displa!ed, reproduced or distributed in an! for# or b! an! #eans, without the prior written per#ission of the publisher, or used be!ond the li#ited distribution to teachers and educators per#itted b! +craw-Hill for their indi/idual course preparation' .f !ou are a student usin this +anual, !ou are usin it wit without hout per#ission'
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.f this is done, the solution will ta> [,%]=!de23"(&dcd"iff,"$an,%0);
.n either case, a plot of the results can be de/eloped as >> $'!(,%)
28.13 !a" The first few steps of uler5s #ethod are shown in the followin table t 0 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5
x
y
2 2.12 2.24 2.2497 9744 44 2.38 2.3895 9598 98 2.53 2.5398 9874 74 2.70 2.7008 0807 07
1 0.98 0.96 0.9639 3928 28 0.95 0.9518 1871 71 0.94 0.9439 3959 59 0.94 0.9403 0369 69
A plot of the entire si#ulation is shown below"
PROPRIETARY MATERIAL' * The The +craw +craw-Hill -Hill o# o#panies, panies, ..nc' nc' All rihts reser/ed' reser/ed' 0o part of this +anual #a! be displa!ed, reproduced or distributed in an! for# or b! an! #eans, without the prior written per#ission of the publisher, or used be!ond the li#ited distribution to teachers and educators per#itted b! +craw-Hill for their indi/idual course preparation' .f !ou are a student usin this +anual, !ou are usin it wit without hout per#ission'
12
0otice that because the the uler #ethod is lower lower order, the pea> >> >>
"$an=[0,30]; %0=[2,1]; [,%]=!de45(&$ed$e%,"$an,%0); $'!(,%)
PROPRIETARY MATERIAL' * The The +craw +craw-Hill -Hill o# o#panies, panies, ..nc' nc' All rihts reser/ed' reser/ed' 0o part of this +anual #a! be displa!ed, reproduced or distributed in an! for# or b! an! #eans, without the prior written per#ission of the publisher, or used be!ond the li#ited distribution to teachers and educators per#itted b! +craw-Hill for their indi/idual course preparation' .f !ou are a student usin this +anual, !ou are usin it wit without hout per#ission'
14
The state-space plot can be enerated as >> $'!(%(*,1),%(*,2))
28.14 Dsin the step si>e of %'1, !a" and !#" both i/e unstable results' !(" The 4th-order RK #ethod !ields a stable solution' The first few /alues are shown in the followin table' A ti#e series plot of the results is also shown below' 0otice how after about t : &, this solution di/eres fro# the double precision /ersion in Fi' 2'8' t
0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5
x
y
z
5 5 5 9.78 9.7814 1469 69 17 17.0 .079 7946 46 10.4 10.439 3947 47 17. 17.7029 70297 7 20.87 0.8741 41 35 35..8968 89687 7 10 10.8 .810 1088 88 -2.5 -2.529 2924 24 39. 39.30 3074 745 5 0.54 0.5495 9577 77 -5.5 -5.544 4419 19 28. 28.07 0746 462 2 -3. -3.16 1646 461 1 -5.8 -5.841 4129 29 22 22.3 .368 6889 89
PROPRIETARY MATERIAL' * The The +craw +craw-Hill -Hill o# o#panies, panies, ..nc' nc' All rihts reser/ed' reser/ed' 0o part of this +anual #a! be displa!ed, reproduced or distributed in an! for# or b! an! #eans, without the prior written per#ission of the publisher, or used be!ond the li#ited distribution to teachers and educators per#itted b! +craw-Hill for their indi/idual course preparation' .f !ou are a student usin this +anual, !ou are usin it wit without hout per#ission'
1
!)" .n order to sol/e the proble# in +AT6A7, +AT6A7, a function is first de/eloped to hold the @s, funci!n %$='!en+(,%) %$=[-10#%(1)10#%(2);28#%(1)-%(2)-%(1)#%(3);2.666667#%(3)%(1)#%(2)];
Then, an @ sol/er li> "$an=[0,20]; >> %0=[5,5,5]; >> [,%]=!de45(&'!en+,"$an,%0); >> $'!(,%)
28.1 The second-order e$uation can be e=pressed as a pair of first-order e$uations, dy dz dw dz
=w =
f 2 EI
( L − z ) 2
?e used uler5s #ethod with h : 1 to obtain the solution"
PROPRIETARY MATERIAL' * The The +craw +craw-Hill -Hill o# o#panies, panies, ..nc' nc' All rihts reser/ed' reser/ed' 0o part of this +anual #a! be displa!ed, reproduced or distributed in an! for# or b! an! #eans, without the prior written per#ission of the publisher, or used be!ond the li#ited distribution to teachers and educators per#itted b! +craw-Hill for their indi/idual course preparation' .f !ou are a student usin this +anual, !ou are usin it wit without hout per#ission'
1&
z 0 1 2 3 4 5
y w dy /d /dz 0 0 0 0 0. 0.00432 0 0..00432 0.00 0.0043 432 2 0.00 0.0083 8357 57 0. 0.00 008 8357 357 0.012677 0.01212 0.0121 212 2 0.02 0.0247 4797 97 0.0 0.015 1561 619 9 0.01 0.0156 5619 19 0.040416 0.018864 0.018864
dw /dz /dz 0.00432 0.004037 0. 0.00 0037 3763 63 0. 0.003499 0.003 0.00324 245 5 0.003
0.8112 0.04524 0.04524 0.85 0.8564 644 4 0.04 0.0453 5317 17 0. 0.04 045 5317 317 0.901757 0.04536 0.0453 536 6 0.94 0.9471 711 17 0.04 0.0453 5379 79 0. 0.04 045 5379 379 0.992496 0.045384 0.045384
7. 7.68E-05 4.32 4.32EE-05 05 1.92 1.92E E-05 4.8E.8E-0 06 0
• • •
26 27 28 29 30
30
z
20
10
y
0 %1
0
1
28.1$ The second-order e$uation can be e=pressed as a pair of first-order e$uations, −2 z 9 %
dy dz
=w
dw dz
=
2%% ze ( L − z ) 2 ( + z )2 EI
?e used uler5s #ethod with h : 1 to obtain the solution" z 0 1 2 3 4 5
f &z ' y 0 0 31.18357 0 50.0099 0 61 61.4 .404 0481 81 0.00 0.0020 2098 98 68 68.0 .082 8252 52 0.00 0.0073 7333 33 71 71.6 .653 5313 13 0.01 0.0161 6148 48
w 0 0 0.002098 0.00 0.0052 5235 35 0.00 0.0088 8816 16 0.01 0.0124 2498 98
dy /d /dz 0 0 0.002098 0.00 0.0052 5235 35 0.00 0.0088 8816 16 0.01 0.0124 2498 98
dw / /dz dz 0 0.002098 0.003137 0. 0.00 0035 3581 81 0. 0.00 0036 3682 82 0. 0.00 0035 3583 83
• •
PROPRIETARY MATERIAL' * The The +craw +craw-Hill -Hill o# o#panies, panies, ..nc' nc' All rihts reser/ed' reser/ed' 0o part of this +anual #a! be displa!ed, reproduced or distributed in an! for# or b! an! #eans, without the prior written per#ission of the publisher, or used be!ond the li#ited distribution to teachers and educators per#itted b! +craw-Hill for their indi/idual course preparation' .f !ou are a student usin this +anual, !ou are usin it wit without hout per#ission'
1
•
26 27 28 29 30
29 29.6 .639 3907 07 27 27.8 .894 9419 19 26 26.2 .241 4164 64 24. 24.6781 67818 8 23.20033
30
0.70 0.7009 0979 79 0.74 0.7416 1693 93 0.78 0.7824 2444 44 0. 0.82 8232 3216 16 0.863995
0.04 0.0407 0713 13 0.04 0.0407 0751 51 0.040 0.04077 771 1 0.04 0.0407 078 8 0.040782
0.04 0.0407 0713 13 3. 3.79 79EE-05 05 0.04 0.0407 0751 51 2. 2.01 01EE-05 05 0.04 0.0407 0771 71 8. 8.4E 4E-0 -06 6 0.04 0.0407 078 8 1. 1.97 97EE-0 06 0.040782 0
z
20
10
y %1
0
1
28.1% This proble# was sol/ed usin the =cel spreadsheet in a fashion si#ilar to the last e=a#ple in 3ec' 2'1' ?e set up uler5s #ethod to sol/e the @s usin uesses for the diffusion coefficients' coefficients' Then we for#ed a colu#n containin the s$uared residuals between our predictions and the #easured /alues' AdCustin AdCustin the diffusion coefficients with the 3ol/er tool #ini#i>ed the su# of the s$uares' At first, we assu#ed that the diffusion coefficients coefficients were >ero' For this case the 3ol/er did not con/ere on a credible answer' ? ?ee then #ade & uesses of 1 ×1% for both' This #anitude was based on the fact that the /olu#es were of this order of #anitude' The resultin si#ulation did not fit the data /er! well, but was #uch better than when we had uessed uessed >ero' ?hen we we used 3ol/er, it co con/ered n/ered on E on E 1122 : & & 1'24×1% and and E E 11 : 2'2&4×1% which corresponded to a su# of the s$uares of residuals of '4' 3o#e of the uler calculations are displa!ed below alon with a plot of the fit' t 0 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5
c 1 0 2. 2.2639108 4.25 4.2553 5374 743 3 6.00 6.0022 2240 409 9 7.52 7.5296 9616 162 2 8.86 8.8601 0129 294 4
c 2 0 0 0.03 0.0351 5165 65 0.10 0.1007 0717 17 0.19 0.1923 2385 85 0.30 0.3063 6353 53
c 3
100 95.47218 91 91.2 .251 5187 87 87.3 87.312 1283 83 83 83.6 .631 3123 23 80 80.1 .185 8548 48
dc 1 /d /dt 22.63911 19.91464 17 17.4 .468 6867 67 15.2 15.273 7375 75 13 13.3 .305 0513 13 11 11.5 .540 4045 45
dc2 /d / dt 0 0.351651 0. 0.65 6555 5521 21 0. 0.91 9166 6677 77 1. 1.13 1396 9684 84 1. 1.32 3286 8649 49
dc3 /dt / dt -45.2782 -42.203 -39. -39.39 3905 05 -36. -36.81 816 6 -34. -34.45 4575 75 -32. -32.29 2948 48
PROPRIETARY MATERIAL' * The The +craw +craw-Hill -Hill o# o#panies, panies, ..nc' nc' All rihts reser/ed' reser/ed' 0o part of this +anual #a! be displa!ed, reproduced or distributed in an! for# or b! an! #eans, without the prior written per#ission of the publisher, or used be!ond the li#ited distribution to teachers and educators per#itted b! +craw-Hill for their indi/idual course preparation' .f !ou are a student usin this +anual, !ou are usin it wit without hout per#ission'
1
.t should be noted that we #ade up the E#easure#ents for this proble# usin the 4 th-order RK #ethod with /alues for diffusi/e #i=in of E 1122 : 1×1%& and and E E 11 : 2×1%&' ?e then used a rando# nu#ber enerator to add so#e error to this Edata' 28.18 The Heun #ethod without corrector can be used to co#pute t 0
p 6000
k 1
450
pend 6225
k 2 4 46 66.875 458.4375
0.5 0.1 5 1.5 1.5 2
6229 6267.1 29.2 .219 19 6467 64 .194 94 6714 6714.2 .261 61 6970 6970.7 .767 67
467. 465.03 7.19 1914 14 485. 48 0396 96 503. 503.56 5696 96 522. 522.80 8075 75
64 6462 62.8 .814 14 50 48 484. 4.71 711 11 494. 475. 474.13 5.95 9512 12 67 6709 09.7 .714 14 503. 3.22 2286 86 49 1341 41 69 6966 66.0 .046 46 52 522. 2.45 4535 35 51 513. 3.01 011 15 72 7232 32.1 .171 71 54 542. 2.41 4128 28 532. 532.61 6102 02
2313 23137. 7.43 43 2402 24021. 1.36 36 24 2493 939. 9.05 05 25 2589 891. 1.8 8 2688 26880. 0.94 94
1735 1735.3 .308 08 1801 1801.6 .602 02 18 1870 70.4 .429 29 1941 1941.8 .885 85 2016 2016.0 .071 71
24 2400 005. 5.09 09 24 2492 922. 2.16 16 2587 25874. 4.26 26 26 2686 862. 2.74 74 27 2788 888. 8.98 98
• • •
18 18.5 18.5 19 19.5 19.5 20
18 1800 00.3 .382 82 18 1869 69.1 .162 62 19 1940 40.5 .57 7 20 2014 14.7 .705 05 20 2091 91.6 .673 73
1767 1767.8 .845 45 1835 1835.3 .382 82 19 1905 05.4 .499 99 1978 1978.2 .295 95 2053 2053.8 .872 72
The results can be plotted' .n addition, linear reression can be used to fit a straiht line to ln p p /ersus /ersus t to to i/e ln p ln p : : '&88 G %'%t %'%t ' Thus, as would be e=pected fro# a first-order #odel, the slope is e$ual to the rowth rate of the population'
PROPRIETARY MATERIAL' * The The +craw +craw-Hill -Hill o# o#panies, panies, ..nc' nc' All rihts reser/ed' reser/ed' 0o part of this +anual #a! be displa!ed, reproduced or distributed in an! for# or b! an! #eans, without the prior written per#ission of the publisher, or used be!ond the li#ited distribution to teachers and educators per#itted b! +craw-Hill for their indi/idual course preparation' .f !ou are a student usin this +anual, !ou are usin it wit without hout per#ission'
18
p
ln p 10.4
20000
10
15000
(.6 10000 (.2 5000
8.8
0
8.4 0
10
20
0
10
20
28.1& The Heun #ethod can be used to co#pute t 0 0.5 0.5
p k 1 pend k 2 6000 840 6420 871.836 855.918 64 6427 27.9 .959 59 87 872. 2.40 4052 52 6864 6864..162 162 901. 901.66 6652 52 88 887. 7.03 0352 52
1.5 1.15
6871 68 71. ..085 477 47 7 902. 90 2.12 1234 34 7793 73 732 22.53 2..417 538 8 951. 928. 92 8.31 312 2 915. 91 5.21 2177 77 73 7329 29.0 85 92 928. 8.66 6622 22 77 93. 41 7 95 1.30 3099 99 93 939. 9.98 986 6
• • •
18 18 18.5 .5 19 19 19.5 .5 20
1879 18798. 8.01 01 18 1890 905. 5.98 98 1900 19004. 4.77 77 19 1909 095. 5.06 06 19 1917 177. 7.51 51
225.9 25.95 5 20 206. 6.83 8344 44 189. 189.14 1412 12 17 172. 2.79 7988 88 15 157. 7.73 7327 27
18 189 910.9 10.99 9 20 205. 5.94 9432 32 21 215. 5.94 9466 66 19 1900 009. 9.4 4 188. 188.30 3068 68 19 197. 7.57 5706 06 19 190 099.3 99.34 4 172. 172.02 02 180. 180.58 5806 06 1918 19181. 1.46 46 15 157. 7.00 008 8 16 164. 4.90 9034 34 1925 19256. 6.38 38 143. 143.19 1946 46 15 150. 0.46 4637 37
The results can be plotted'
The cur/e is s-shaped' This shape occurs because initiall! the population is increasin e=ponentiall! since p since p is is #uch less than p than p#a=' Howe/er, as p as p approaches approaches p p#a=, the rowth rate decreases and the population le/els off'
PROPRIETARY MATERIAL' * The The +craw +craw-Hill -Hill o# o#panies, panies, ..nc' nc' All rihts reser/ed' reser/ed' 0o part of this +anual #a! be displa!ed, reproduced or distributed in an! for# or b! an! #eans, without the prior written per#ission of the publisher, or used be!ond the li#ited distribution to teachers and educators per#itted b! +craw-Hill for their indi/idual course preparation' .f !ou are a student usin this +anual, !ou are usin it wit without hout per#ission'
2%
28.2' !a" 0onlinear reression (e'', usin the =cel sol/er option) can be used to #ini#i>e the su# of the s$uares of the residuals between the data and the si#ulation' The resultin esti#ates are" a : %'22, b : %'%121, c : %'2244, and d : : %'%%%28' %'%%%28' The fit is"
1200 #oose
800 400
wol/es
0 1(60
1((0
2020
!#" The results in state space are,
300 s 200 e v * o ) 100
0 0
500
1000
1500
+oose !("
1200 800 400 0 1(60
1((0
2020
PROPRIETARY MATERIAL' * The The +craw +craw-Hill -Hill o# o#panies, panies, ..nc' nc' All rihts reser/ed' reser/ed' 0o part of this +anual #a! be displa!ed, reproduced or distributed in an! for# or b! an! #eans, without the prior written per#ission of the publisher, or used be!ond the li#ited distribution to teachers and educators per#itted b! +craw-Hill for their indi/idual course preparation' .f !ou are a student usin this +anual, !ou are usin it wit without hout per#ission'
21
300 s e 200 v * o ) 100
0 0
500
1000
15 1500
+oose !)" 1200 800 400 0 1(60
1((0
2020
300
s e v * o )
200
100
0 0
500
1000 1500 +oose
28.21 Main Program: anging "aic ca'e - /=/() aa!'ic "!'ui!n /=/() ni" (',f,") / = /!(1"in($i2#') nde$enden aia'e , "="a , f=end iniia' c!ndii!n" [%(1)=ca'e %-c!!dinae, %(2)=ca'e "'!$e]; e"=0.5e-7; "$an=[0,200]; ic=[0 0]; g'!a' /! /!=0.0025; [,%]=!de45(&"'$,"$an,ic);
PROPRIETARY MATERIAL' * The The +craw +craw-Hill -Hill o# o#panies, panies, ..nc' nc' All rihts reser/ed' reser/ed' 0o part of this +anual #a! be displa!ed, reproduced or distributed in an! for# or b! an! #eans, without the prior written per#ission of the publisher, or used be!ond the li#ited distribution to teachers and educators per#itted b! +craw-Hill for their indi/idual course preparation' .f !ou are a student usin this +anual, !ou are usin it wit without hout per#ission'
22 %f(1)=%('eng()); /!(1)=/!; ea(1)=1; /!=0.002; [,%]=!de45(&"'$,"$an,ic); %f(2)=%('eng()); /!(2)=/!; ea(2)=a"( (%f(2)-%f(1))%f(2) ); f! :=3*10
/!(:)=/!(:-1)(/!(:-1)-/!(:-2))(%f(:-1)-%f(:-2))#(50-%f(:-1)); /!=/!(:); [,%]=!de45(&"'$,"$an,ic); %f(:)=%('eng()); ea(:)=a"( (%f(:)-%f(:-1))%f(:) ); if (ea(:)=e") $'!(,%(*,1)); gid; 'ae'( [v,d] = eig() v = -0.5879 0.7307 -0.3471 d = 0.6986 0 0
-0.6344 -0.3506 0.6890
0.2913 0.5725 0.7664
0 0.3395 0
0 0 0.0569
Therefore, the eien/alues are %'&8&, %'8 and %'%&8' The correspondin eien/ectors are (nor#ali>in so that the a#plitude for the third floor is one),
1'&84 − 2'1%1& 1
− %'82% − %'% 1
%'%%8 %'4&888 1
A raph can be #ade showin the three #odes
PROPRIETARY MATERIAL' * The The +craw +craw-Hill -Hill o# o#panies, panies, ..nc' nc' All rihts reser/ed' reser/ed' 0o part of this +anual #a! be displa!ed, reproduced or distributed in an! for# or b! an! #eans, without the prior written per#ission of the publisher, or used be!ond the li#ited distribution to teachers and educators per#itted b! +craw-Hill for their indi/idual course preparation' .f !ou are a student usin this +anual, !ou are usin it wit without hout per#ission'
2
%2
%1
0
1
2
+ode 1
%1
0
1
+ode 2
0
1
+ode 3
28.2$ The second-order e$uation can be e=pressed as a pair of first-order e$uations,
d" dt di dt
=i = −%'%2i − 4" + sin 1'%t
The para#eters can be substituted and the s!ste# sol/ed with the 4 th-order RK #ethod in double-precision double-precisio n with h : %'1' A table showin the first few steps and a raph of the entire solution are shown below' t % %'1 %'1 %'2 %'2 %' %' %'4 %'4
i %'%% %'%%%% %% %'%% %'%%8 8 %'% %'%&4 &4 %'% %'%8 8 %'1 %'14 4
q %' %'%% %%%% %% %' %'%% %%% % %' %'%% %%2 2 %' %'%% %%1 1 %' %'%1 %1
k11 %'%% %'%%%% %% %'1 %'14 4 %' %'4 4 %'48 %'48 %'&% %'&%21 21
k12 %'%% %'%%%% %% %'%% %'%%8 8 %'% %'%&4 &4 %'% %'%8 8 %'1 %'14 4
imid %' %'%% %%%% %% %' %'%1 %1 %' %'% %42 42 %' %'1% 1%42 42 %' %'1& 1&4 4
qm qmid %'%% %'%%%% %% %'%% %'%%% % %'%% %'%%4 4 %'%1 %'%121 21 %'%2 %'%2
k21 %' %'%8 %84 4 %' %'2 24 4 %' %'4 424 24 %' %' % % %' %'& &88 88
k22 %'%% %'%%%% %% %'%1 %'%1 %'% %'%42 42 %'1% %'1%42 42 %'1& %'1&4 4
imid %' %'%% %%4 4 %' %'%2 %2% % %' %'% %% % %' %'1% 1%2 2 %' %'1& 1&& &
qm qmid %'%% %'%%%% %% %'%% %'%%12 12 %'%% %'%%2 2 %'%1 %'%1 %'%2 %'%2% %
k31 %'%8 %'%8 %'2 %' 214 14 %'42 %' 42 %' %' % % %'& %' &
k32 %'%% %'%%4 4 %'%2 %'%2% % %'% %'%% % %'1% %'1%2 2 %'1& %'1&& &
iend %'%% %'%%8 8 %'% %'%&4 &4 %'% %'%8 8 %'1 %'14& 4& %'18 %'181 1
qe q end %'%% %'%%% % %'%% %'%%2& 2& %'%% %'%% %'%1 %'%1 %'% %'%4 4
k41 %'18 %'1 8 %' %' 42 42 %'48 %' 482 2 %'&% %' &%1 1 %'& %' &
k42 %'%% %'%%8 8 %'% %'%&4 &4 %'% %'%8 8 %'1 %'14& 4& %'18 %'181 1
phi1 %'%828 %'%8 28 %'2 %' 21 1 %'42 %' 428% 8% %' %' & & %'& %' &4& 4&
28.2% The second-order e$uation can be e=pressed as a pair of first-order e$uations,
PROPRIETARY MATERIAL' * The The +craw +craw-Hill -Hill o# o#panies, panies, ..nc' nc' All rihts reser/ed' reser/ed' 0o part of this +anual #a! be displa!ed, reproduced or distributed in an! for# or b! an! #eans, without the prior written per#ission of the publisher, or used be!ond the li#ited distribution to teachers and educators per#itted b! +craw-Hill for their indi/idual course preparation' .f !ou are a student usin this +anual, !ou are usin it wit without hout per#ission'
phi2 %'%% %'%%1 1 %'%2 %'%214 14 %'% %'%& & %'1% %'1%&1 &1 %'1& %'1&8 8
2
d" dt di dt
=i =−
$ L
i−
" #L
The para#eters can be substituted and the s!ste# sol/ed with the 4 th-order RK #ethod with h : %'%%' A table showin the first few steps and a raph of the entire solution are shown below' t %'%%% %'%% %'%1% %'%1 %'%2%
i -'22 -4'%1% -4'% -4'& -4'4
q %'1%% %'%2 %'%&% %'% %'%1
i
k11 -1&'1 -12'2& -'488 -2&'8& 21'84
k12 -'22 -4'%1% -4'% -4'& -4'4
iim mid -'&88 -4'1 -4'&8& -4'2 -4'2%
qmid k21 %'%82 -14&'&% %'%2 -1%%'1 %'%48 -1'% %'%2 -1'82 %'%%1 44'82
k22 -'&88 -4'1 -4'&8& -4'2 -4'2%
imid -'&4 -4'2&% -4'& -4'& -4'&&1
qmid k31 %'%81 -14'%2 %'%1 -88'1 %'%48 -%'4 %'%2 -2'21 %'%%1 44'%
k32 -'&4 -4'2&% -4'& -4'& -4'&&1
iend -4'%% -4'%& -4'&1 -4'4 -4'
qend k41 %'%2 -12'4 %'%&% -'% %'% -2&'&1 %'%1 21'8 -%'%1% &&'%%
k42 phi1 -4'%% -14'& -4'%& -88'&% -4'&1 -%'8& -4'4 -2'21 -4' 44'2&
4
t 0 %4 0
q
0.1
0.2
0.3
0.4
0.5
0.1
t 0
%0.1 0
0.1
0.2
0.3
0.4
0.5
28.28 The e$uation can be sol/ed anal!ticall! as
PROPRIETARY MATERIAL' * The The +craw +craw-Hill -Hill o# o#panies, panies, ..nc' nc' All rihts reser/ed' reser/ed' 0o part of this +anual #a! be displa!ed, reproduced or distributed in an! for# or b! an! #eans, without the prior written per#ission of the publisher, or used be!ond the li#ited distribution to teachers and educators per#itted b! +craw-Hill for their indi/idual course preparation' .f !ou are a student usin this +anual, !ou are usin it wit without hout per#ission'
phi2 -'&&4 -4'28 -4'& -4' -4'&1
28
di dt di
=− =−
i
$
i
L $
dt
L
ln i = −( $ 9 L)t + # # = ln %' ln(i 9 %') = −( $ 9 L)t i
= %'e − 2t
The nu#erical solution can be obtained b! e=pressin the e$uation as
di dt
= −1'i
and usin uler5s #ethod with h : %'% to sol/e for the current' 3o#e selected /alues are shown, alon with a plot of both the anal!tical and nu#erical result' A better #atch would be obtained b! usin a s#aller step or a hiher-order #ethod' t 0 0.05 0.05 0.1 0.1 0.15 0.15 0.2 0.2 0.25 0.25
analytical 0.5 0.50000 00000 0 0.46 0.4638 3872 72 0.43 0.4303 0354 54 0.39 0.3992 9258 58 0.37 0.3704 0409 09 0.34 0.3436 3645 45
Euler 0.50 0.500 000 000 0 0.46 0.4625 2500 00 0.42 0.4278 7813 13 0.39 0.3957 5727 27 0.36 0.3660 6047 47 0.33 0.3385 8594 94
di /dt / dt -0.75 0.7500 0000 00 -0.6 -0.693 9375 750 0 -0.6 -0.641 4171 719 9 -0.5 -0.593 9359 590 0 -0.5 -0.549 4907 071 1 -0.5 -0.507 0789 890 0
28.2& The nu#erical solution can be obtained b! e=pressin the e$uation as di dt
= −1'(i − i )
PROPRIETARY MATERIAL' * The The +craw +craw-Hill -Hill o# o#panies, panies, ..nc' nc' All rihts reser/ed' reser/ed' 0o part of this +anual #a! be displa!ed, reproduced or distributed in an! for# or b! an! #eans, without the prior written per#ission of the publisher, or used be!ond the li#ited distribution to teachers and educators per#itted b! +craw-Hill for their indi/idual course preparation' .f !ou are a student usin this +anual, !ou are usin it wit without hout per#ission'
%
and usin uler5s #ethod with h : %'% to sol/e for the current' 3o#e selected /alues are shown, alon with a plot of the nu#erical result' 0ote that the table and plot also show the anal!tical solution for the linear case co#puted in Jrob' 2'18' t 0 0.05 0.05 0.1 0.15 1 0.15 0. 0.2 0.2 0.25 0.25
analytical Euler di /dt / dt 0.5 0.50000 00000 0 0.50 0.500 000 000 0 -0.56 0.5625 2500 00 0.46 0.4638 3872 72 0.47 0.4718 1875 75 -0.5 -0.550 5020 207 7 0.43 0. 0354 54 0.39 0.4303 3992 9258 58 0.37 0.3704 0409 09 0.34 0.3436 3645 45
0.44 0. 4365 65 0.41 0.4443 4176 7618 18 0.39 0.3917 1759 59 0.36 0.3668 6887 87
-0.5 -0 3493 931 1 -0.5 -0.534 .517 1717 175 5 -0.4 -0.497 9745 451 1 -0.4 -0.476 7625 253 3
28.3' Dsin an approach si#ilar to 3ec' 2', the s!ste# can be e=pressed in #atri= for# as
1 − λ − 1 i1 = K%L − 1 2 − λ i 2 A pac a=[1 -1;-1 2]; >> [v,d]=eig(a) v = 0.8507
-0.5257
0.5257
0.8507
d = 0.3820 0
0 2.6180
Thus, we can see that the eien/alues are λ : : %'2 and 2'&1 or natural fre$uencies of ω : : %'&19 L# and 1'&19 L# ' The eien/ectors tell us that these correspond to oscillations that coincide (%'% %'2) and which run counter to each other ( −%'2 %'%)'
PROPRIETARY MATERIAL' * The The +craw +craw-Hill -Hill o# o#panies, panies, ..nc' nc' All rihts reser/ed' reser/ed' 0o part of this +anual #a! be displa!ed, reproduced or distributed in an! for# or b! an! #eans, without the prior written per#ission of the publisher, or used be!ond the li#ited distribution to teachers and educators per#itted b! +craw-Hill for their indi/idual course preparation' .f !ou are a student usin this +anual, !ou are usin it wit without hout per#ission'
1
ω =
%'&1
ω =
L#
1'&1 L#
28.31 The differential e$uations to be sol/ed are
linear " d θ =% dt d% 8'1
dt
=−
1
nonlinear " d θ =% dt 8'1 d%
θ
dt
=−
1
sin θ
A few steps for the 4 th-order RK solution of the nonlinear s!ste# are contained in the followin table and a plot of both solutions is shown below' t theta
v
k11 k 12 12 theta
v
k21 k 22 22 theta
v
k31 k 3322 theta
v
k41 k 42 42 1 2 % %' %'%%% %'%%% -&'8 %' -%'% -%'% -&'8 %' -%'% -%'% -&'8& %' -%'%&8 -%'%&8 -&'84 -%'% -&'8& %'%1
%' -%'%&8 -%'%&8 -&'84
%' -%'1%4 -%'1%4 -&'82
%' -%'1%4 -%'1%4 -&'81
%'4 -%'18 -%'18 -&'82 -%'1%4 -&'81
%'%2
%'4 -%'18 -%'18 -&'82
%' -%'1 -%'1 -&'822
%' -%'1 -%'1 -&'821
%'2 -%'2% -%'2% -&'81 -%'1 -&'821
%'%
%'2 -%'2% -%'2% -&'81
%'1 -%'242 -%'242 -&'8%
%'1 -%'242 -%'242 -&'8%
%'% -%'2 -%'2 -&'8 -%'242 -&'8%
%'%4
%'% -%'2 -%'2 -&'8
%' -%'11 -%'11 -&'
%' -%'11 -%'11 -&'
%' -%'4& -%'4& -&'& -%'11 -&'
4 "o"*i"%v
*i"%v
2
0 0
2
%2 *i"%t!et
4
6
"o"*i"%t!et
%4
28.32 The differential e$uations to be sol/ed are
dx dt d% dt
=% =−
c m
%−
k m
x
PROPRIETARY MATERIAL' * The The +craw +craw-Hill -Hill o# o#panies, panies, ..nc' nc' All rihts reser/ed' reser/ed' 0o part of this +anual #a! be displa!ed, reproduced or distributed in an! for# or b! an! #eans, without the prior written per#ission of the publisher, or used be!ond the li#ited distribution to teachers and educators per#itted b! +craw-Hill for their indi/idual course preparation' .f !ou are a student usin this +anual, !ou are usin it wit without hout per#ission'
2
A few steps for the 2 nd-order RK solution (Heun without iteration) are shown in the followin table and plots of displace#ent and /elocit! are shown below' t
x
v
k11
k 1122
x
v
k 21 21
k 22 22
0 0.4000 0.0000 0.0000 0.00 0.001 1 0.39 0.3998 98 -0 -0..4149 4149 -0 -0..4149 4149
-416.67 0.4000 -4 -412 12.9 .99 9 0. 0.39 3994 94
-0.4167 -0 -0..82 8279 79
-0.4167 -0 -0.8 .82 279
-413.19 -4 -409 09..12
0.00 0.002 2 0.00 0.003 3 0.00 0.004 4 0.00 0.005 5
-4 -408 08.9 .92 2 -4 -404 04.4 .46 6 -3 -399 99.6 .61 1 -3 -394 94.3 .39 9
-1 -1..23 2349 49 -1 -1..63 6372 72 -2 -2..03 0345 45 -2 -2..42 4264 64
-1 -1.2 .23 349 -1 -1.6 .63 372 -2 -2.0 .03 345 -2 -2.4 .42 264
-4 -404 04..65 -3 -399 99..80 -3 -394 94..58 -3 -388 88..99
0.39 0.3992 92 0.39 0.3981 81 0.39 0.3967 67 0.39 0.3949 49
-0 -0..8260 8260 -1 -1..2328 2328 -1 -1..6349 6349 -2 -2..0320 0320
-0 -0..8260 8260 -1 -1..2328 2328 -1 -1..6349 6349 -2 -2..0320 0320
0. 0.39 3983 83 0. 0.39 3969 69 0. 0.39 3951 51 0. 0.39 3928 28
-0.2083 -0 -0.6 .621 214 4
-414.93 -41 -411.05
-1 -1.0 .030 304 4 -1 -1.4 .435 350 0 -1 -1.8 .834 347 7 -2 -2.2 .229 292 2
-406 -406..79 -402 -402..13 -397 -397..09 -391 -391..69
1
2
isplace#ent"
Melocit!"
28.33 The differential e$uation to be sol/ed is dT dt
− T ) = %'2(2%
A few steps for the 2 nd-order RK solution (Heun without iteration) are shown in the followin table and a plot of te#perature /ersus ti#e is shown below' t 0 0.1 0.1
T 90 88. 88.2718 27188 8
k 1111
T eend nd
-17.5 88.25 -1 -17. 7.06 068 8 86 86..5650 56508 8
k 2211
-17.0625 -1 -16. 6.64 6413 13
-17.2813 -1 -16. 6.85 8546 46
PROPRIETARY MATERIAL' * The The +craw +craw-Hill -Hill o# o#panies, panies, ..nc' nc' All rihts reser/ed' reser/ed' 0o part of this +anual #a! be displa!ed, reproduced or distributed in an! for# or b! an! #eans, without the prior written per#ission of the publisher, or used be!ond the li#ited distribution to teachers and educators per#itted b! +craw-Hill for their indi/idual course preparation' .f !ou are a student usin this +anual, !ou are usin it wit without hout per#ission'
0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5
86 86.5 .586 8641 41 84 84.9 .942 4256 56 83 83.3 .339 3929 29 81 81.7 .775 7560 60
-16. -16.64 6466 66 -16. -16.23 2356 56 -15. -15.83 8348 48 -15. -15.44 4439 39
84.9 84.921 2175 75 83.3 83.319 1900 00 81.7 81.755 5581 81 80.2 80.231 3121 21
-16. -16.23 2304 04 -15. -15.82 8297 97 -15. -15.43 4390 90 -15. -15.05 0578 78
-16. -16.43 4385 85 -16. -16.03 0327 27 -15. -15.63 6369 69 -15. -15.25 2509 09
The te#perature will drop to 4% o in appro=i#atel! #inutes' 28.34 The differential e$uation to be sol/ed is
dQ
= %'(12)
1%%( 2% − 2')(2% − t )
dt
1%% − 2't
A few steps for the 2 nd-order RK solution (Heun without iteration) are shown in the followin table and a plot of heat flow /ersus ti#e is shown below' t 0 0.1 0.2 0.3 0.4 0.5
Q 0 209.7368 418.946 627.625 835.7709 1043.381
k 1111
t eend nd
2100 2 20 094.737 2089.447 2084.131 2 20 078.788 2073.418
0.1 0.2 0.3 0.4 0.5 0.6
k 2211 2094.737 2089.447 2084.131 2078.788 2073.418 2068.02
2097.368 2092.092 2086.789 2081.459 2076.103 2070.719
28.3 The differential e$uations to be sol/ed are
PROPRIETARY MATERIAL' * The The +craw +craw-Hill -Hill o# o#panies, panies, ..nc' nc' All rihts reser/ed' reser/ed' 0o part of this +anual #a! be displa!ed, reproduced or distributed in an! for# or b! an! #eans, without the prior written per#ission of the publisher, or used be!ond the li#ited distribution to teachers and educators per#itted b! +craw-Hill for their indi/idual course preparation' .f !ou are a student usin this +anual, !ou are usin it wit without hout per#ission'
4
nonlinear" d% dt
= 8' −
%'22 2 % &'1
linear" d% dt
= 8' − 12' % &'1
A few steps for the solution (uler) are shown in the followin table, which also includes the anal!tical solution fro# =a#ple 1'1' A plot of the result is also shown below' below' 0ote, the nonlinear solution is the bolder line' linear t 0 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5
v 0 0.98 1.94 1.9420 2012 12 2.88 2.8863 6365 65 3.81 3.8133 3385 85 4.72 4.7233 3389 89
dvdt 9.8 9.620117 9.44 9.4435 3537 37 9.27 9.2701 0197 97 9.10 9.1000 0039 39 8.93 8.9330 3005 05
nonlinear analytical v dvdt v 0 9.8 0 0.98 9.796827 0.971061 1.95 1.9596 968 83 9.78 9.7873 731 12 1. 1.92 9244 446 6 2.93 2.9384 8414 14 9.77 9.7714 1473 73 2. 2.86 8605 0518 18 3.91 3.9155 5561 61 9.74 9.7493 9345 45 3. 3.77 7795 9552 52 4.89 4.8904 0496 96 9.72 9.7209 0979 79 4. 4.68 6818 1871 71
28.3$ The differential e$uations to be sol/ed are
t N N 1 s"
d% dt d% dt
t ≥ 1 s"
= 8' − 12' % &'1
= 8' −
&'1
%
The first few steps for the solution (Heun) are shown in the followin table, alon with the steps when the parachute opens' A plot of the result is also shown below' t 0 0.1 0.2
v
k 1111
t eend nd
0 9.8 0.971006 9 9..621768 1.924352 9 9..446778
0.1 0.2 0.3
v eend nd
k 21 21
0.98 9.620117 9.710059 1.933183 9 9..445 451 157 9.533463 2.86903 9 9..273379 9. 9.360079
PROPRIETARY MATERIAL' * The The +craw +craw-Hill -Hill o# o#panies, panies, ..nc' nc' All rihts reser/ed' reser/ed' 0o part of this +anual #a! be displa!ed, reproduced or distributed in an! for# or b! an! #eans, without the prior written per#ission of the publisher, or used be!ond the li#ited distribution to teachers and educators per#itted b! +craw-Hill for their indi/idual course preparation' .f !ou are a student usin this +anual, !ou are usin it wit without hout per#ission'
0.3 0.4 0.5
2.86036 9 9..274971 3.779345 9 9..106288 4.681616 8 8..940673
0.4 0.5 0.6
3.787857 9 9..104725 9. 9.189848 4.689974 8 8..939 391 138 9.022713 5.575683 8 8..776 765 563 8.858618
12.8
48.2953 0. 0.935223
12.9
48.38883 0 0..918 180 057
12.9 13 13.1 13.2 13.3
48.38797 0. 0.918215 46.96618 -28.1316 44.26662 -25.9513 41.77629 -23.94 39.47896 -22.0846
13.0 13.1 13.2 13.3 13.4
48.47979 44.15302 41.67149 39.38228 37.2705
• • •
-29.354 -25.8596 -23.8554 -22.0065 -20.301
0. 0.9 92664 -14.2179 -26.9956 -24.9033 -22.9733 -21.1928
28.3% This proble# can be sol/ed with +AT6A7" a?$ed "$ing ?a"" "%"e? ?a""* ?=1 :g da?$ing, n!n'inea* a a"(dd) (dd), a=2 @(?")A2 "$ing, n!n'inea* A3, =5 @?A3 BCD 5 ve"i!n nde$enden aia'e , "$an=["a "!$] iniia' c!ndii!n" [(1)=ve'!ci%, (2)=di"$'ace?en]; 0=0; f=8; "$an=[0 8]; ic=[1 0.5]; a) 'inea "!'ui!n [,]=!de45(
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