Table 2 Comparison the approximate frequencies of Eq. (16) between the present method and truncation HBM Hosen et al. (2012), 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 | \( \dot{\varphi }_{Ex} \) | \( \dot{\varphi }_{3(c,Usual(trunc))} \) (Hosen et al. 2012) Er (%) | \( \dot{\varphi }_{3(c,Usual)} \) Er (%) | \( \dot{\varphi }_{3(c,Present)} \) Er (%) |
|---|---|---|---|---|
0.5 | 1.0891582 | 1.0891582 | 1.0891582 | 1.0891582 |
0.00000 | 0.00000 | 0.00000 | ||
0.7 | 1.1676370 | 1.1676370 | 1.1676374 | 1.1676370 |
0.00000 | 0.00004 | 0.00000 | ||
1 | 1.31778 | 1.31778 | 1.31778 | 1.31778 |
0.000 | 0.000 | 0.000 | ||
2 | 1.97602 | 1.97601 | 1.97607 | 1.97602 |
0.000 | 0.003 | 0.000 | ||
3 | 2.73849 | 2.73847 | 2.73862 | 2.73849 |
0.000 | 0.005 | 0.000 | ||
4 | 3.53924 | 3.53921 | 3.53946 | 3.53926 |
0.000 | 0.006 | 0.000 | ||
5 | 4.35746 | 4.35741 | 4.35777 | 4.35748 |
0.001 | 0.007 | 0.000 | ||
10 | 8.53359 | 8.53347 | 8.5343 | 8.53363 |
0.002 | 0.008 | 0.000 | ||
50 | 42.3730 | 42.3724 | 42.3767 | 42.3732 |
0.002 | 0.009 | 0.000 | ||
100 | 84.7275 | 84.7262 | 84.7349 | 84.7279 |
0.002 | 0.009 | 0.000 |