Next: 4. Predictor - Corrector Up: Main Previous: 3.3 Algorithm (Runge-Kutta) method
Example
Use the Runge-Kutta method of order 4 to find the approximate solution of the IVP.
with the step size (1) 0.1, (2) 0.05 and (3) 0.025. Calculate the error and tabulate the results.
Solution: It is an exercise. See the tables 7,8 and 9.
Table 7 |
|||||||||
Initial x |
Initial y |
Stepsize h |
Appx. Y |
Exa. Y |
Error |
k1 |
k2 |
k3 |
k4 |
0.000000000 |
1.000000000 |
0.100000000 |
1.000000000 |
1.000000000 |
0.000000000 |
0.100000000 |
0.110250000 |
0.111328877 |
0.123505187 |
0.100000000 |
1.000000000 |
0.100000000 |
1.111110490 |
1.111111111 |
0.000000621 |
0.100000000 |
0.146678862 |
0.140292162 |
1.566008596 |
0.200000000 |
1.111110490 |
0.100000000 |
1.234449921 |
1.250000000 |
0.015550079 |
0.123456652 |
0.184391026 |
0.175998811 |
2.210405014 |
0.300000000 |
1.234449921 |
0.100000000 |
1.388373760 |
1.428571429 |
0.040197668 |
0.152386661 |
0.237394267 |
0.227126268 |
3.221717202 |
0.400000000 |
1.388373760 |
0.100000000 |
1.583264224 |
1.666666667 |
0.083402442 |
0.192758170 |
0.315425554 |
0.303100092 |
4.940348000 |
0.500000000 |
1.583264224 |
0.100000000 |
1.837356086 |
2.000000000 |
0.162643914 |
0.250672560 |
0.006283673 |
0.338743260 |
7.497402655 |
Table 8 |
|||||||||
Initial x |
Initial y |
Stepsize h |
Appx. Y |
Exa. Y |
Error |
k1 |
k2 |
k3 |
k4 |
0.000000000 |
1.000000000 |
0.050000000 |
1.000000000 |
1.000000000 |
0.000000000 |
0.050000000 |
0.052531250 |
0.052661057 |
0.055404765 |
0.050000000 |
1.000000000 |
0.050000000 |
1.052631563 |
1.052631579 |
0.000000016 |
0.050000000 |
0.060789818 |
0.058647317 |
1.234940749 |
0.100000000 |
1.052631563 |
0.050000000 |
1.108026472 |
1.111111111 |
0.003084639 |
0.055401660 |
0.067678251 |
0.065192852 |
1.448887960 |
0.150000000 |
1.108026472 |
0.050000000 |
1.169566428 |
1.176470588 |
0.006904160 |
0.061386133 |
0.075762210 |
0.072896477 |
1.710476056 |
0.200000000 |
1.169566428 |
0.050000000 |
1.238140600 |
1.250000000 |
0.011859400 |
0.068394281 |
0.085351670 |
0.082024537 |
2.038362460 |
0.250000000 |
1.238140600 |
0.050000000 |
1.315013575 |
1.333333333 |
0.018319758 |
0.076649607 |
0.096836321 |
0.092947305 |
2.454432772 |
0.300000000 |
1.315013575 |
0.050000000 |
1.401756539 |
1.428571429 |
0.026814889 |
0.086463035 |
0.110739875 |
0.106160878 |
2.990097511 |
0.350000000 |
1.401756539 |
0.050000000 |
1.500357994 |
1.538461538 |
0.038103544 |
0.098246070 |
0.127776748 |
0.122343335 |
3.691048693 |
0.400000000 |
1.500357994 |
0.050000000 |
1.613369403 |
1.666666667 |
0.053297264 |
0.112553706 |
0.148940529 |
0.142440137 |
4.625404360 |
0.450000000 |
1.613369403 |
0.050000000 |
1.744116520 |
1.818181818 |
0.074065298 |
0.130148042 |
0.175643382 |
0.167799881 |
5.897549758 |
0.500000000 |
1.744116520 |
0.050000000 |
1.897012232 |
2.000000000 |
0.102987768 |
0.152097122 |
0.001156677 |
0.180042499 |
7.524391368 |
Table 9 |
|||||||||
Initial x |
Initial y |
Stepsize h |
Appx. Y |
Exa. Y |
Error |
k1 |
k2 |
k3 |
k4 |
0.000000000 |
1.000000000 |
0.025000000 |
1.000000000 |
1.000000000 |
0.000000000 |
0.025000000 |
0.025628906 |
0.025644828 |
0.026298683 |
0.025000000 |
1.000000000 |
0.025000000 |
1.025641025 |
1.025641026 |
0.000000000 |
0.025000000 |
0.026943420 |
0.026993882 |
0.027701006 |
0.050000000 |
1.025641025 |
0.025000000 |
1.052403627 |
1.052631579 |
0.000227952 |
0.026298488 |
0.028385073 |
0.028440684 |
0.029205611 |
0.075000000 |
1.052403627 |
0.025000000 |
1.080596229 |
1.081081081 |
0.000484852 |
0.027688835 |
0.029945008 |
0.030006771 |
0.030835976 |
0.100000000 |
1.080596229 |
0.025000000 |
1.110334291 |
1.111111111 |
0.000776821 |
0.029192205 |
0.031636710 |
0.031705495 |
0.032606372 |
0.125000000 |
1.110334291 |
0.025000000 |
1.141748121 |
1.142857143 |
0.001109021 |
0.030821056 |
0.033475403 |
0.033552235 |
0.034533273 |
0.150000000 |
1.141748121 |
0.025000000 |
1.174983056 |
1.176470588 |
0.001487532 |
0.032589719 |
0.035478577 |
0.035564665 |
0.036635645 |
0.175000000 |
1.174983056 |
0.025000000 |
1.210201697 |
1.212121212 |
0.001919515 |
0.034514630 |
0.037666391 |
0.037763169 |
0.038935408 |
0.200000000 |
1.210201697 |
0.025000000 |
1.247586556 |
1.250000000 |
0.002413444 |
0.036614704 |
0.040062185 |
0.040171363 |
0.041458011 |
0.225000000 |
1.247586556 |
0.025000000 |
1.287343191 |
1.290322581 |
0.002979389 |
0.038911805 |
0.042693097 |
0.042816721 |
0.044233135 |
0.250000000 |
1.287343191 |
0.025000000 |
1.329703954 |
1.333333333 |
0.003629379 |
0.041431312 |
0.045590828 |
0.045731364 |
0.047295558 |
0.275000000 |
1.329703954 |
0.025000000 |
1.374932496 |
1.379310345 |
0.004377849 |
0.044202815 |
0.048792593 |
0.048953027 |
0.050686250 |
0.300000000 |
1.374932496 |
0.025000000 |
1.423329214 |
1.428571429 |
0.005242215 |
0.047260984 |
0.052342310 |
0.052526283 |
0.054453736 |
0.325000000 |
1.423329214 |
0.025000000 |
1.475237865 |
1.481481481 |
0.006243617 |
0.050646651 |
0.056292097 |
0.056504080 |
0.058655835 |
0.350000000 |
1.475237865 |
0.025000000 |
1.531053671 |
1.538461538 |
0.007407867 |
0.054408169 |
0.060704181 |
0.060949699 |
0.063361868 |
0.375000000 |
1.531053671 |
0.025000000 |
1.591233304 |
1.600000000 |
0.008766696 |
0.058603134 |
0.065653332 |
0.065939270 |
0.068655523 |
0.400000000 |
1.591233304 |
0.025000000 |
1.656307281 |
1.666666667 |
0.010359386 |
0.063300586 |
0.071230019 |
0.071565026 |
0.074638568 |
0.425000000 |
1.656307281 |
0.025000000 |
1.726895488 |
1.739130435 |
0.012234947 |
0.068583845 |
0.077544527 |
0.077939565 |
0.081435739 |
0.450000000 |
1.726895488 |
0.025000000 |
1.803726783 |
1.818181818 |
0.014455035 |
0.074554201 |
0.084732382 |
0.085201482 |
0.089201250 |
0.475000000 |
1.803726783 |
0.025000000 |
1.887663979 |
1.904761905 |
0.017097926 |
0.081335758 |
0.092961594 |
0.093522900 |
0.098127536 |
0.500000000 |
1.887663979 |
0.025000000 |
1.979736026 |
2.000000000 |
0.020263974 |
0.089081882 |
0.000049597 |
0.097986323 |
0.107923254 |
Next: 4. Predictor-Corrector Up: Main Previous: 3.3 Algorithm (Runge-Kutta) method