187 lines
20 KiB
Plaintext
187 lines
20 KiB
Plaintext
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Analytical Results
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Analytical Displacement
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x α = 0.25 α = 0.50 α = 1.00 α = 2.00 α = 4.00 α = 10.00 α = 20.00
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0.000 0.0000 0.0000 0.0000 0.0000 -0.0000 -0.0000 0.0000
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0.125 12.6291 13.0279 14.8163 27.2083 -63.3488 -174.4389 65.5540
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0.159 16.0783 16.5810 18.8341 34.4180 -78.5516 -183.7768 -4.5451
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0.250 25.2459 26.0050 29.4014 52.7248 -111.1877 -110.0090 -105.0363
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0.375 37.8380 38.8806 43.5276 74.9632 -131.8039 105.0623 102.7444
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0.500 50.3932 51.6043 56.9747 92.5408 -120.1499 176.2660 -59.5897
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0.625 62.8992 64.1264 69.5327 104.3646 -79.0790 6.0989 -7.2646
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0.750 75.3437 76.3982 81.0056 109.6995 -18.6469 -172.4198 71.2297
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0.875 87.7147 88.3716 91.2145 108.2139 46.3507 -114.8345 -106.8658
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1.000 100.0000 100.0000 100.0000 100.0000 100.0000 100.0000 100.0000
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Analytical Strain
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x α = 0.25 α = 0.50 α = 1.00 α = 2.00 α = 4.00 α = 10.00 α = 20.00
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0.000 101.0493 104.2915 118.8395 219.9500 -528.5395 -1838.1640 2190.7119
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0.125 101.0000 104.0879 117.9123 213.1123 -463.8370 -579.6142 -1755.0748
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0.159 100.9693 103.9614 117.3376 208.9010 -425.0038 38.1449 -2188.8251
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0.250 100.8520 103.4778 115.1451 193.0243 -285.5711 1472.6333 621.4221
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0.375 100.6056 102.4636 110.5811 160.9350 -37.3874 1508.3226 759.3781
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0.500 100.2609 101.0493 104.2915 118.8395 219.9500 -521.4176 -1838.1640
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0.625 99.8183 99.2404 96.3745 69.3552 423.4360 -1837.1519 2185.8885
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0.750 99.2783 97.0440 86.9535 15.5586 523.2501 -637.1725 -1664.2573
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0.875 98.6413 94.4687 76.1758 -39.2052 494.9543 1435.3224 480.7297
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1.000 97.9079 91.5244 64.2093 -91.5315 345.4765 1542.3510 893.9902
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No artists with labels found to put in legend. Note that artists whose label start with an underscore are ignored when legend() is called with no argument.
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Convergence Results
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2nd Order FDM and P=1 FEM Convergence of U(x=1/(2pi))
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Convergence Table for α = 0.25
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dx Analytical Value Method Value Extrap Value % Error β % Extrap Error β Extrap
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---------------------------------------------------------------------------------------------------------------------------------------
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0.5 16.078254 16.078452 16.078452 0.000012 nan nan nan
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0.25 16.078254 16.078310 16.078310 0.000003 1.814778 nan nan
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0.125 16.078254 16.078268 16.078268 0.000001 2.068323 0.000001 1.725316
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0.0625 16.078254 16.078258 16.078258 0.000000 1.998164 0.000000 2.091062
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0.03125 16.078254 16.078255 16.078255 0.000000 2.004412 0.000000 1.993886
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0.015625 16.078254 16.078255 16.078255 0.000000 2.000002 0.000000 2.233528
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0.0078125 16.078254 16.078254 16.078254 0.000000 1.999997 0.000000 0.752269
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0.00390625 16.078254 16.078254 16.078254 0.000000 1.999987 0.000000 -0.082906
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0.001953125 16.078254 16.078254 16.078254 0.000000 2.000812 0.000000 -0.003978
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0.0009765625 16.078254 16.078254 16.078254 0.000000 1.992182 0.000000 0.005319
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0.00048828125 16.078254 16.078254 16.078254 0.000000 2.201742 0.000003 -0.000020
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Convergence Table for α = 0.5
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dx Analytical Value Method Value Extrap Value % Error β % Extrap Error β Extrap
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---------------------------------------------------------------------------------------------------------------------------------------
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0.5 16.580992 16.584311 16.584311 0.000200 nan nan nan
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0.25 16.580992 16.581930 16.581930 0.000057 1.823229 nan nan
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0.125 16.580992 16.581215 16.581215 0.000013 2.069082 0.000018 1.736830
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0.0625 16.580992 16.581048 16.581048 0.000003 1.998625 0.000003 2.091852
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0.03125 16.580992 16.581006 16.581006 0.000001 2.004462 0.000001 1.996719
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0.015625 16.580992 16.580995 16.580995 0.000000 2.000028 0.000000 2.006221
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0.0078125 16.580992 16.580993 16.580993 0.000000 2.000007 0.000000 1.994125
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0.00390625 16.580992 16.580992 16.580992 0.000000 2.000035 0.000000 1.891296
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0.001953125 16.580992 16.580992 16.580992 0.000000 2.000010 0.000000 1.843723
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0.0009765625 16.580992 16.580992 16.580992 0.000000 1.999286 0.000000 0.668336
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0.00048828125 16.580992 16.580992 16.580992 0.000000 2.003076 0.000000 0.007129
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Convergence Table for α = 1.0
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dx Analytical Value Method Value Extrap Value % Error β % Extrap Error β Extrap
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0.5 18.834148 18.899840 18.899840 0.003488 nan nan nan
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0.25 18.834148 18.852241 18.852241 0.000961 1.860259 nan nan
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0.125 18.834148 18.838449 18.838449 0.000228 2.072629 0.000299 1.787081
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0.0625 18.834148 18.835222 18.835222 0.000057 2.000605 0.000052 2.095846
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0.03125 18.834148 18.834415 18.834415 0.000014 2.004686 0.000014 1.999248
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0.015625 18.834148 18.834214 18.834214 0.000004 2.000145 0.000004 2.006198
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0.0078125 18.834148 18.834164 18.834164 0.000001 2.000036 0.000001 2.000161
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0.00390625 18.834148 18.834152 18.834152 0.000000 2.000039 0.000000 1.999982
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0.001953125 18.834148 18.834149 18.834149 0.000000 2.000002 0.000000 1.998332
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0.0009765625 18.834148 18.834148 18.834148 0.000000 2.000046 0.000000 2.109196
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0.00048828125 18.834148 18.834148 18.834148 0.000000 2.000014 0.000000 2.133319
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Convergence Table for α = 2.0
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dx Analytical Value Method Value Extrap Value % Error β % Extrap Error β Extrap
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---------------------------------------------------------------------------------------------------------------------------------------
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0.5 34.417979 37.552156 37.552156 0.091062 nan nan nan
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0.25 34.417979 35.150695 35.150695 0.021289 2.096760 nan nan
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0.125 34.417979 34.588873 34.588873 0.004965 2.100152 0.004985 2.095727
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0.0625 34.417979 34.460351 34.460351 0.001231 2.011902 0.001107 2.128103
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0.03125 34.417979 34.428526 34.428526 0.000306 2.006294 0.000304 2.013755
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0.015625 34.417979 34.420614 34.420614 0.000077 2.000814 0.000076 2.008115
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0.0078125 34.417979 34.418638 34.418638 0.000019 2.000204 0.000019 2.001018
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0.00390625 34.417979 34.418144 34.418144 0.000005 2.000073 0.000005 2.000246
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0.001953125 34.417979 34.418020 34.418020 0.000001 2.000013 0.000001 2.000097
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0.0009765625 34.417979 34.417989 34.417989 0.000000 2.000004 0.000000 1.999770
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0.00048828125 34.417979 34.417981 34.417981 0.000000 2.000059 0.000000 2.001842
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Convergence Table for α = 4.0
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dx Analytical Value Method Value Extrap Value % Error β % Extrap Error β Extrap
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---------------------------------------------------------------------------------------------------------------------------------------
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0.5 -78.551640 -44.521329 -44.521329 0.433222 nan nan nan
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0.25 -78.551640 -72.147403 -72.147403 0.081529 2.409722 nan nan
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0.125 -78.551640 -76.519925 -76.519925 0.025865 1.656329 0.010631 2.659493
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0.0625 -78.551640 -78.029415 -78.029415 0.006648 1.959954 0.010096 1.534405
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0.03125 -78.551640 -78.419329 -78.419329 0.001684 1.980741 0.001729 1.952832
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0.015625 -78.551640 -78.518501 -78.518501 0.000422 1.997361 0.000431 1.975144
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0.0078125 -78.551640 -78.543351 -78.543351 0.000106 1.999341 0.000106 1.996700
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0.00390625 -78.551640 -78.549567 -78.549567 0.000026 1.999770 0.000026 1.999198
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0.001953125 -78.551640 -78.551121 -78.551121 0.000007 1.999957 0.000007 1.999708
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0.0009765625 -78.551640 -78.551510 -78.551510 0.000002 1.999982 0.000002 1.999958
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0.00048828125 -78.551640 -78.551607 -78.551607 0.000000 1.999999 0.000000 1.999993
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4th Order FDM and P=2 FEM Convergence of U(x=1/(2pi))
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Convergence Table for α = 0.25
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dx Analytical Value Method Value Extrap Value % Error β % Extrap Error β Extrap
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0.5 16.078254 16.078287 16.078287 0.000002 nan nan nan
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0.25 16.078254 16.078253 16.078253 0.000000 4.811274 nan nan
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0.125 16.078254 16.078254 16.078254 0.000000 5.424900 0.000000 4.730211
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0.0625 16.078254 16.078254 16.078254 0.000000 4.272605 0.000000 0.909708
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0.03125 16.078254 16.078254 16.078254 0.000000 7.515896 0.000000 -0.004096
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0.015625 16.078254 16.078254 16.078254 0.000000 2.391135 0.000003 0.000000
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0.0078125 16.078254 16.078254 16.078254 0.000000 -1.007072 nan nan
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0.00390625 16.078254 16.078254 16.078254 0.000000 -2.314502 nan nan
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0.001953125 16.078254 16.078254 16.078254 0.000000 -1.799075 nan nan
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0.0009765625 16.078254 16.078254 16.078254 0.000000 -2.029648 0.000025 0.000001
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0.00048828125 16.078254 16.078254 16.078254 0.000000 0.207580 0.000011 -0.000000
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Convergence Table for α = 0.5
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dx Analytical Value Method Value Extrap Value % Error β % Extrap Error β Extrap
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0.5 16.580992 16.581525 16.581525 0.000032 nan nan nan
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0.25 16.580992 16.580973 16.580973 0.000001 4.786253 nan nan
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0.125 16.580992 16.580992 16.580992 0.000000 5.521549 0.000000 4.806285
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0.0625 16.580992 16.580992 16.580992 0.000000 4.118326 0.000000 5.636740
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0.03125 16.580992 16.580992 16.580992 0.000000 9.365088 0.000000 -0.509052
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0.015625 16.580992 16.580992 16.580992 0.000000 3.536667 0.000000 -0.000026
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0.0078125 16.580992 16.580992 16.580992 0.000000 -0.410172 nan nan
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0.00390625 16.580992 16.580992 16.580992 0.000000 -1.704184 nan nan
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0.001953125 16.580992 16.580992 16.580992 0.000000 -2.067930 nan nan
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0.0009765625 16.580992 16.580992 16.580992 0.000000 -1.875636 0.000006 0.000002
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0.00048828125 16.580992 16.580992 16.580992 0.000000 -2.081360 0.000004 -0.000016
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Convergence Table for α = 1.0
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dx Analytical Value Method Value Extrap Value % Error β % Extrap Error β Extrap
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0.5 18.834148 18.843758 18.843758 0.000510 nan nan nan
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0.25 18.834148 18.833772 18.833772 0.000020 4.677470 nan nan
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0.125 18.834148 18.834153 18.834153 0.000000 5.985224 0.000001 4.710179
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0.0625 18.834148 18.834147 18.834147 0.000000 3.448116 0.000000 5.882540
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0.03125 18.834148 18.834148 18.834148 0.000000 6.498357 0.000000 4.031577
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0.015625 18.834148 18.834148 18.834148 0.000000 4.002814 0.000000 -0.047321
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0.0078125 18.834148 18.834148 18.834148 0.000000 4.262898 0.000001 0.000029
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0.00390625 18.834148 18.834148 18.834148 0.000000 0.326859 0.000019 -0.000000
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0.001953125 18.834148 18.834148 18.834148 0.000000 -2.024489 nan nan
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0.0009765625 18.834148 18.834148 18.834148 0.000000 -2.030078 0.000013 0.000001
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0.00048828125 18.834148 18.834148 18.834148 0.000000 -1.997818 0.000003 0.000017
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Convergence Table for α = 2.0
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dx Analytical Value Method Value Extrap Value % Error β % Extrap Error β Extrap
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0.5 34.417979 34.663528 34.663528 0.007134 nan nan nan
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0.25 34.417979 34.402614 34.402614 0.000446 3.998268 nan nan
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0.125 34.417979 34.417862 34.417862 0.000003 7.037851 0.000024 4.096854
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0.0625 34.417979 34.417945 34.417945 0.000001 1.808584 0.000000 7.511752
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0.03125 34.417979 34.417978 34.417978 0.000000 4.687210 0.000001 1.381023
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0.015625 34.417979 34.417979 34.417979 0.000000 4.000567 0.000000 4.592474
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0.0078125 34.417979 34.417979 34.417979 0.000000 4.000935 0.000000 -5.512228
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0.00390625 34.417979 34.417979 34.417979 0.000000 4.056899 0.000000 -0.001227
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0.001953125 34.417979 34.417979 34.417979 0.000000 1.119402 0.000001 0.000016
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0.0009765625 34.417979 34.417979 34.417979 0.000000 -2.128619 nan nan
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0.00048828125 34.417979 34.417979 34.417979 0.000000 -2.005417 0.000000 0.000241
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Convergence Table for α = 4.0
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dx Analytical Value Method Value Extrap Value % Error β % Extrap Error β Extrap
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---------------------------------------------------------------------------------------------------------------------------------------
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0.5 -78.551640 -133.563988 -133.563988 0.700334 nan nan nan
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0.25 -78.551640 -78.603880 -78.603880 0.000665 10.040387 nan nan
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0.125 -78.551640 -78.578508 -78.578508 0.000342 0.959249 0.000000 11.080952
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0.0625 -78.551640 -78.552404 -78.552404 0.000010 5.135275 0.011709 -0.041042
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0.03125 -78.551640 -78.551711 -78.551711 0.000001 3.412791 0.000000 5.235885
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0.015625 -78.551640 -78.551644 -78.551644 0.000000 4.002608 0.000000 3.363585
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0.0078125 -78.551640 -78.551640 -78.551640 0.000000 4.001591 0.000000 3.942435
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0.00390625 -78.551640 -78.551640 -78.551640 0.000000 4.071443 0.000000 -1.652138
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0.001953125 -78.551640 -78.551640 -78.551640 0.000000 4.137881 0.000000 -0.003593
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0.0009765625 -78.551640 -78.551640 -78.551640 0.000000 1.177470 0.000000 0.000789
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0.00048828125 -78.551640 -78.551640 -78.551640 0.000000 -2.229793 0.000015 -0.000002
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