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

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

(x,t) \(\vert {u}_{\mathrm{Exact}} - {u}_{\mathrm{ADM}} \vert \) \(\vert {u}_{\mathrm{Exact}} - {u}_{\mathrm{VIM}} \vert \) \(\vert {u}_{\mathrm{Exact}} - {u}_{\mathrm{CRFDTM}} \vert \) \(\vert {u}_{\mathrm{Exact}} - {u}_{{q}\text{-}\mathrm{HATM}}^{ ( {3} )} \vert \)
(0.1,0.1) 8.16297 × 10−7 6.35269 × 10−5 5.55112 × 10−17 5.55112 × 10−17
(0.1,0.3) 7.64245 × 10−7 1.90854 × 10−4 5.55112 × 10−17 5.55112 × 10−17
(0.1,0.5) 7.16083 × 10−7 3.18549 × 10−4 5.55112 × 10−16 5.55112 × 10−16
(0.2,0.1) 3.26243 × 10−6 6.18930 × 10−5 5.55112 × 10−16 5.55112 × 10−16
(0.2,0.3) 3.05458 × 10−6 1.85945 × 10−4 1.11022 × 10−16 1.11022 × 10−16
(0.2,0.5) 2.86226 × 10−6 3.10352 × 10−4 7.77156 × 10−16 7.77156 × 10−16
(0.3,0.1) 7.33445 × 10−6 6.03095 × 10−5 0 0
(0.3,0.3) 6.86758 × 10−6 1.81187 × 10−4 1.66533 × 10−16 1.66533 × 10−16
(0.3,0.5) 6.43557 × 10−6 3.02408 × 10−4 6.666134 × 10−16 6.666134 × 10−16
(0.4,0.1) 1.30286 × 10−5 5.87746 × 10−5 5.55112 × 10−17 5.55112 × 10−17
(0.4,0.3) 1.22000 × 10−5 1.76574 × 10−4 5.55112 × 10−17 5.55112 × 10−17
(0.4,0.5) 1.14333 × 10−5 2.94707 × 10−4 5.55112 × 10−16 5.55112 × 10−16
(0.5,0.1) 2.03415 × 10−5 5.72867 × 10−5 0 0
(0.5,0.3) 1.90489 × 10−5 1.72102 × 10−4 1.11022 × 10−16 1.11022 × 10−16
(0.5,0.5) 1.78528 × 10−5 2.87241 × 10−4 6.10623 × 10−16 6.10623 × 10−16