PolymathPlus Report
📉   Differential Equations 2022-04-01 12:46 

# Example 25- Boundary Value ODE System
# Diffusion with Reaction
# Verified Final Values: CA = 0.140461, y = 2.76438
# Ref.: Comput. Appl. Eng. Educ. 6: 175-176, 1998

d(CA)/d(z)=y
d(y)/d(z)=k*CA/DAB
d(CA1)/d(z)=y1
d(y1)/d(z)=k*CA1/DAB
k=0.001
DAB=1.2E-9
err=y-0
err1=y1-0
y0=-130
L=.001
delta=0.0001
CAanal=0.2*cosh(L*(k/DAB)^.5*(1-z/L))/(cosh(L*(k/DAB)^.5))
derr=(err1-err)/(delta*y0)
ynew=y0-err/derr
z(0)=0
CA(0)=0.2
y(0)=-130
CA1(0)=0.2
y1(0)=-130.013
z(f)=0.001


Calculated values

  Variable Initial value Minimal value Maximal value Final value
1 CA 0.2 0.14044176 0.2 0.14046056
2 CA1 0.2 0.1404276 0.2 0.14044568
3 CAanal 0.2 0.13827265 0.2 0.13827265
4 DAB 1.2E-09 1.2E-09 1.2E-09 1.2E-09
5 delta 0.0001 0.0001 0.0001 0.0001
6 derr 1 1 1.4464177 1.4464177
7 err -130 -130 2.7643829 2.7643829
8 err1 -130.013 -130.013 2.7455795 2.7455795
9 k 0.001 0.001 0.001 0.001
10 L 0.001 0.001 0.001 0.001
11 y -130 -130 2.7643829 2.7643829
12 y0 -130 -130 -130 -130
13 y1 -130.013 -130.013 2.7455795 2.7455795
14 ynew -5.2268E-11 -131.91119 -5.2268E-11 -131.91119
15 z 0 0 0.001 0.001

Integration chart

 -140 -120 -100 -80 -60 -40 -20 0 20 0 0.0002 0.0004 0.0006 0.0008 0.001 z CA y CA1 y1

Formatted equations

  d C A ( ) d z ( ) = y
  d y ( ) d z ( ) = k C A D A B
  d C A 1 ( ) d z ( ) = y 1
  d y 1 d z ( ) = k C A 1 D A B
  k = 0 . 0 0 1
  D A B = 1 . 2 E - 9
  e r r = y 0
  e r r 1 = y 1 0
  y 0 = 1 3 0
  L = . 0 0 1
  δ = 0 . 0 0 0 1
  C A a n a l = 0 . 2 c o s h L k D A B . 5 1 z L c o s h L k D A B . 5
  d e r r = e r r 1 e r r δ y 0
  y n e w = y 0 e r r d e r r

Differential equations

1 d(CA)/d(z) = y
2 d(y)/d(z) = k*CA/DAB
3 d(CA1)/d(z) = y1
4 d(y1)/d(z) = k*CA1/DAB

Explicit equations

1 k = 0.001
2 DAB = 1.2E-9
3 err = y-0
4 err1 = y1-0
5 y0 = -130
6 L = .001
7 delta = 0.0001
8 CAanal = 0.2*cosh(L*(k/DAB)^.5*(1-z/L))/(cosh(L*(k/DAB)^.5))
9 derr = (err1-err)/(delta*y0)
10 ynew = y0-err/derr

General

Total number of equations 14
Number of differential equations 4
Number of explicit equations 10
Reporting digits 10
Elapsed time 0.00 sec
Solution method RKF_45
Step size guess. h 1E-06
Truncation error tolerance. eps 1E-06
Calculated Intermediate data points 50

Calculated data points

    z CA y CA1 y1 k DAB err err1 y0 L delta CAanal derr ynew
1 0 0.2 -130 0.2 -130.013 0.001 1.2E-09 -130 -130.013 -130 0.001 0.0001 0.2 1 -5.2268E-11
2 0.000049 0.19382799 -121.96069 0.19382736 -121.9737 0.001 1.2E-09 -123.25617 -123.26918 -130 0.001 0.0001 0.19473048 1.0007005 -6.8301088
3 0.000065 0.19189723 -119.38923 0.19189638 -119.40225 0.001 1.2E-09 -120.67177 -120.68479 -130 0.001 0.0001 0.19274848 1.001354 -9.4914001
4 0.000081 0.1900074 -116.84324 0.19000635 -116.85628 0.001 1.2E-09 -118.11312 -118.12615 -130 0.001 0.0001 0.1908076 1.0022212 -12.148653
5 0.000113 0.18634896 -111.82552 0.18634749 -111.83859 0.001 1.2E-09 -113.07087 -113.08393 -130 0.001 0.0001 0.18704756 1.0045973 -17.446567
6 0.000129 0.18457957 -109.3527 0.18457789 -109.36579 0.001 1.2E-09 -110.58619 -110.59927 -130 0.001 0.0001 0.18522759 1.0061066 -20.085013
7 0.000145 0.18284955 -106.90322 0.18284766 -106.91633 0.001 1.2E-09 -108.12511 -108.13821 -130 0.001 0.0001 0.18344714 1.0078306 -22.714994
8 0.000161 0.18115854 -104.47654 0.18115644 -104.48968 0.001 1.2E-09 -105.68709 -105.70022 -130 0.001 0.0001 0.18170583 1.0097696 -25.335435
9 0.000193 0.17789212 -99.68954 0.1778896 -99.702743 0.001 1.2E-09 -100.87819 -100.89137 -130 0.001 0.0001 0.17833913 1.0142943 -30.543474
10 0.000209 0.176316 -97.328195 0.17631327 -97.341432 0.001 1.2E-09 -98.506272 -98.519492 -130 0.001 0.0001 0.17671303 1.016881 -33.129007
11 0.000225 0.1747775 -94.987613 0.17477456 -95.000888 0.001 1.2E-09 -96.155371 -96.168627 -130 0.001 0.0001 0.17512463 1.0196846 -35.700872
12 0.000241 0.17327629 -92.667296 0.17327313 -92.680611 0.001 1.2E-09 -93.824983 -93.838278 -130 0.001 0.0001 0.17357358 1.0227058 -38.258089
13 0.000273 0.17038445 -88.085478 0.17038087 -88.098884 0.001 1.2E-09 -89.223764 -89.237146 -130 0.001 0.0001 0.17058227 1.0294033 -43.324777
14 0.000289 0.16899321 -85.823001 0.16898941 -85.836456 0.001 1.2E-09 -86.951951 -86.965381 -130 0.001 0.0001 0.16914135 1.0330812 -45.832411
15 0.000305 0.16763802 -83.578832 0.167634 -83.592339 0.001 1.2E-09 -84.698688 -84.712169 -130 0.001 0.0001 0.16773652 1.0369794 -48.321725
16 0.000321 0.16631859 -81.352494 0.16631436 -81.366057 0.001 1.2E-09 -82.463494 -82.477028 -130 0.001 0.0001 0.16636748 1.0410988 -50.791868
17 0.000353 0.16378591 -76.951414 0.16378124 -76.965095 0.001 1.2E-09 -78.045411 -78.059061 -130 0.001 0.0001 0.16373558 1.050005 -55.671389
18 0.000369 0.16257211 -74.775733 0.16256723 -74.789477 0.001 1.2E-09 -75.861579 -75.875292 -130 0.001 0.0001 0.16247216 1.0547936 -58.079216
19 0.000385 0.161393 -72.616004 0.16138789 -72.629815 0.001 1.2E-09 -73.693932 -73.707709 -130 0.001 0.0001 0.16124341 1.0598072 -60.464773
20 0.000401 0.16024832 -70.471766 0.16024299 -70.485647 0.001 1.2E-09 -71.542006 -71.555851 -130 0.001 0.0001 0.16004906 1.0650469 -62.827365
21 0.000433 0.15806128 -66.22794 0.1580555 -66.241969 0.001 1.2E-09 -67.283486 -67.297476 -130 0.001 0.0001 0.15776253 1.0762091 -67.48103
22 0.000449 0.15701845 -64.127446 0.15701245 -64.141553 0.001 1.2E-09 -65.175983 -65.190051 -130 0.001 0.0001 0.15666988 1.082134 -69.770877
23 0.000465 0.15600913 -62.040632 0.1560029 -62.054821 0.001 1.2E-09 -63.082385 -63.096533 -130 0.001 0.0001 0.15561065 1.0882898 -72.035305
24 0.000481 0.15503308 -59.967054 0.15502663 -59.981328 0.001 1.2E-09 -61.002244 -61.016475 -130 0.001 0.0001 0.15458461 1.0946777 -74.273786
25 0.000513 0.15318002 -55.857839 0.1531731 -55.87229 0.001 1.2E-09 -56.880565 -56.894971 -130 0.001 0.0001 0.15263127 1.1081556 -78.670958
26 0.000529 0.1523026 -53.821324 0.15229545 -53.83587 0.001 1.2E-09 -54.838146 -54.852644 -130 0.001 0.0001 0.15170354 1.1152484 -80.828761
27 0.000545 0.15145767 -51.796292 0.15145029 -51.810934 0.001 1.2E-09 -52.807427 -52.82202 -130 0.001 0.0001 0.15080817 1.1225792 -82.958839
28 0.000561 0.15064506 -49.782309 0.15063744 -49.797051 0.001 1.2E-09 -50.787973 -50.802665 -130 0.001 0.0001 0.14994498 1.1301494 -85.060833
29 0.000593 0.14911608 -45.785778 0.14910799 -45.80073 0.001 1.2E-09 -46.781142 -46.79604 -130 0.001 0.0001 0.14831439 1.1460149 -89.179288
30 0.000609 0.14839939 -43.802377 0.14839106 -43.817438 0.001 1.2E-09 -44.79291 -44.807916 -130 0.001 0.0001 0.14754664 1.1543134 -91.195194
31 0.000625 0.14771435 -41.82832 0.14770578 -41.843494 0.001 1.2E-09 -42.814233 -42.829351 -130 0.001 0.0001 0.14681037 1.1628583 -93.1819
32 0.000641 0.14706083 -39.863188 0.14705202 -39.878477 0.001 1.2E-09 -40.844691 -40.859923 -130 0.001 0.0001 0.14610542 1.1716512 -95.139207
33 0.000673 0.14584778 -35.958017 0.14583847 -35.973549 0.001 1.2E-09 -36.931329 -36.946799 -130 0.001 0.0001 0.14478888 1.1899889 -98.96498
34 0.000689 0.14528799 -34.017146 0.14527843 -34.032804 0.001 1.2E-09 -34.986675 -35.002269 -130 0.001 0.0001 0.14417701 1.1995376 -100.8332
35 0.000705 0.14475919 -32.083533 0.14474938 -32.099319 0.001 1.2E-09 -33.049484 -33.065206 -130 0.001 0.0001 0.1435959 1.2093421 -102.67152
36 0.000721 0.14426128 -30.156764 0.14425121 -30.172683 0.001 1.2E-09 -31.119344 -31.135197 -130 0.001 0.0001 0.14304543 1.2194047 -104.47989
37 0.000753 0.14335768 -26.322117 0.1433471 -26.338311 0.001 1.2E-09 -27.278571 -27.294695 -130 0.001 0.0001 0.14203592 1.2403124 -108.00669
38 0.000769 0.1429518 -24.413421 0.14294097 -24.429758 0.001 1.2E-09 -25.367118 -25.383383 -130 0.001 0.0001 0.14157667 1.251162 -109.72515
39 0.000785 0.14257642 -22.509934 0.14256532 -22.526417 0.001 1.2E-09 -23.461077 -23.477487 -130 0.001 0.0001 0.14114762 1.2622786 -111.41371
40 0.000801 0.14223146 -20.611248 0.14222009 -20.627881 0.001 1.2E-09 -21.560041 -21.576599 -130 0.001 0.0001 0.14074869 1.2736645 -113.07243
41 0.000833 0.1416325 -16.826665 0.14162059 -16.843608 0.001 1.2E-09 -17.771364 -17.788228 -130 0.001 0.0001 0.14004082 1.2972538 -116.30078
42 0.000849 0.14137837 -14.939959 0.14136619 -14.957063 0.001 1.2E-09 -15.882913 -15.899936 -130 0.001 0.0001 0.13973173 1.3094624 -117.87066
43 0.000865 0.1411544 -13.056441 0.14114195 -13.073709 0.001 1.2E-09 -13.997852 -14.015037 -130 0.001 0.0001 0.13945246 1.3219503 -119.41121
44 0.000881 0.14096055 -11.175708 0.14094782 -11.193144 0.001 1.2E-09 -12.115776 -12.133128 -130 0.001 0.0001 0.13920294 1.3347202 -120.92261
45 0.000913 0.14066302 -7.4209939 0.14064973 -7.4387766 0.001 1.2E-09 -8.3589786 -8.3766732 -130 0.001 0.0001 0.13879293 1.361117 -123.85874
46 0.000929 0.14055928 -5.5462119 0.14054571 -5.5641737 0.001 1.2E-09 -6.483455 -6.5013268 -130 0.001 0.0001 0.13863237 1.3747496 -125.2839
47 0.000945 0.14048554 -3.6726131 0.14047167 -3.6907578 0.001 1.2E-09 -4.6093145 -4.6273673 -130 0.001 0.0001 0.13850138 1.3886754 -126.68078
48 0.000961 0.14044176 -1.7997977 0.1404276 -1.8181293 0.001 1.2E-09 -2.7361574 -2.7543951 -130 0.001 0.0001 0.13839993 1.4028976 -128.04964
49 0.000993 0.14044408 1.9450805 0.14042933 1.9263635 0.001 1.2E-09 1.0088052 0.99018604 -130 0.001 0.0001 0.13828561 1.4322427 -130.70435
50 0.001 0.14046056 2.7643829 0.14044568 2.7455795 0.001 1.2E-09 2.7643829 2.7455795 -130 0.001 0.0001 0.13827265 1.4464177 -131.91119