Table 1 Comparison the approximate frequencies of Eq. (7) between the present method and the usual HBM method with the exact frequency \( \dot{\varphi }_{Ex} \), obtained by direct numerical integration
From: The rapidly convergent solutions of strongly nonlinear oscillators
a | \( \dot{\varphi }_{Ex} \) | \( \dot{\varphi }_{3(q,Usual)} \) Er (%) | \( \dot{\varphi }_{3(q,Present)} \) Er (%) |
|---|---|---|---|
0.5 | 1.0192663 | 1.0192661 | 1.0192663 |
0.00002 | 0.00000 | ||
0.7 | 1.0714295 | 1.0714202 | 1.0714295 |
0.00087 | 0.00000 | ||
1 | 1.26471 | 1.26446 | 1.26469 |
0.020 | 0.002 | ||
2 | 3.16666 | 3.16223 | 3.16639 |
0.140 | 0.008 | ||
3 | 6.80379 | 6.79391 | 6.80382 |
0.145 | 0.000 | ||
4 | 11.9959 | 11.9785 | 11.9963 |
0.145 | 0.003 | ||
5 | 18.7007 | 18.6736 | 18.7014 |
0.145 | 0.003 | ||
10 | 74.6909 | 74.5829 | 74.6941 |
0.145 | 0.004 | ||
50 | 1867.09 | 1864.39 | 1867.17 |
0.145 | 0.004 | ||
100 | 7468.34 | 7457.55 | 7468.66 |
0.145 | 0.004 |