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Table 2 Comparative study in terms of absolute error between ADM [28], VIM [29], CFRDTM [26] and q-HATM for the approximate solution \(v(x,t)\) at \(\omega = 0.005\), \(\ell =0.1\), \(c=10\), \(\hslash =-1\), \(n=1\), \(\hslash =-1\) and \(\alpha =1\) for Example 6.1

From: A reliable technique for fractional modified Boussinesq and approximate long wave equations

(x,t) \(\vert {v}_{\mathrm{Exact}} - {v}_{\mathrm{ADM}} \vert \) \(\vert {v}_{\mathrm{Exact}} - {v}_{\mathrm{VIM}} \vert \) \(\vert {v}_{\mathrm{Exact}} - {v}_{\mathrm{CRFDTM}} \vert \) \(\vert {v}_{\mathrm{Exact}} - {v}_{{q}\text{-}\mathrm{HATM}}^{ ( {3} )} \vert \)
(0.1,0.1) 5.88676 × 10−5 1.65942 × 10−5 3.46945 × 10−18 3.46945 × 10−18
(0.1,0.3) 5.56914 × 10−5 4.98691 × 10−5 5.55112 × 10−17 5.55112 × 10−17
(0.1,0.5) 5.27169 × 10−5 8.32598 × 10−5 5.55112 × 10−16 5.55112 × 10−16
(0.2,0.1) 1.18213 × 10−4 1.06813 × 10−5 6.93889 × 10−18 6.93889 × 10−18
(0.2,0.3) 1.11833 × 10−4 4.83269 × 10−5 5.55112 × 10−17 5.55112 × 10−17
(0.2,0.5) 1.05858 × 10−4 8.06837 × 10−5 5.55112 × 10−16 5.55112 × 10−16
(0.3,0.1) 1.78041 × 10−4 1.55880 × 10−5 6.93889 × 10−18 6.93889 × 10−18
(0.3,0.3) 1.68429 × 10−4 4.68440 × 10−5 5.55112 × 10−17 5.55112 × 10−17
(0.3,0.5) 1.59428 × 10−4 7.82068 × 10−5 5.55112 × 10−16 5.55112 × 10−16
(0.4,0.1) 2.38356 × 10−4 1.51135 × 10−5 5.20417 × 10−18 5.20417 × 10−18
(0.4,0.3) 2.25483 × 10−4 4.54174 × 10−5 5.55112 × 10−17 5.55112 × 10−17
(0.4,0.5) 2.13430 × 10−4 7.58243 × 10−5 5.55112 × 10−16 5.55112 × 10−16
(0.5,0.1) 2.99162 × 10−4 1.46569 × 10−5 1.73472 × 10−18 1.73472 × 10−18
(0.5,0.3) 2.83001 × 10−4 4.40448 × 10−5 5.55112 × 10−17 5.55112 × 10−17
(0.5,0.5) 2.67868 × 10−4 7.35317 × 10−5 5.55112 × 10−16 5.55112 × 10−16