Theory and Modern Applications
Table 3 The comparison of numerical result of \(Q(x,t)\) obtained by NIM [35], HPM [44], q-HATM, and the exact solution Equation (32), also the absolute (ABS) errors when \(\alpha =1\), \(\hbar =-0.99\), \(k=\eta =0.1\), \(p=1.5\), \(r=-1.5\), and \(n=1\) for Example 4.1
From: A reliable technique to study nonlinear time-fractional coupled Korteweg–de Vries equations
t | x | HPM [44] | NIM [35] | q-HATM (\(Q^{(3)}\)) | Exact | ABS error (NIM) [35] | ABS error (q-HATM) |
|---|---|---|---|---|---|---|---|
0.2 | 0.0 | 0.49335133 | 0.49335133 | 0.49335133 | 0.49335132 | 1.07975 × 10−08 | 5.43049 × 10−09 |
0.25 | 0.49339376 | 0.49339372 | 0.49339370 | 0.49339371 | 9.74133 × 10−09 | 8.67541 × 10−09 | |
0.50 | 0.49346087 | 0.49346079 | 0.49346080 | 0.49346079 | 8.52453 × 10−09 | 7.26523 × 10−09 | |
0.75 | 0.49355233 | 0.49355223 | 0.49355223 | 0.49355222 | 7.09823 × 10−09 | 8.02863 × 10−09 | |
1 | 0.49366771 | 0.49366757 | 0.49366757 | 0.49366756 | 5.42089 × 10−09 | 8.68105 × 10−09 | |
0.4 | 0.0 | 0.49340533 | 0.49340533 | 0.49340531 | 0.49340516 | 1.72448 × 10−07 | 1.50992 × 10−07 |
0.25 | 0.49347759 | 0.49347730 | 0.49347730 | 0.49347714 | 1.62910 × 10−07 | 1.57735 × 10−07 | |
0.50 | 0.49357413 | 0.49357355 | 0.49357356 | 0.49357339 | 1.51173 × 10−07 | 1.62699 × 10−07 | |
0.75 | 0.49369446 | 0.49369359 | 0.49369362 | 0.49369345 | 1.36904 × 10−07 | 1.65861 × 10−07 | |
1 | 0.49383799 | 0.49383684 | 0.49383689 | 0.49383672 | 1.19824 × 10−07 | 1.67226 × 10−07 | |
0.6 | 0.0 | 0.49349533 | 0.49349533 | 0.49349529 | 0.49349446 | 8.70802 × 10−07 | 8.22526 × 10−07 |
0.25 | 0.49359735 | 0.49359635 | 0.49359636 | 0.49359552 | 8.33414 × 10−07 | 8.40136 × 10−07 | |
0.50 | 0.49372303 | 0.49372105 | 0.49372112 | 0.49372027 | 7.85653 × 10−07 | 8.48677 × 10−07 | |
0.75 | 0.49387178 | 0.49386883 | 0.49386895 | 0.49386810 | 7.26548 × 10−07 | 8.48209 × 10−07 | |
1 | 0.49404286 | 0.49403896 | 0.49403915 | 0.49403831 | 6.55344 × 10−07 | 8.38955 × 10−07 |