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Table 3 Comparative study between ADM [28], VIM [29], LADM [30], CRFDTM [26] and q-HATM for the approximate solution \(u(x,t)\) at \(\omega =0.005\), \(\ell= 0.1\), \(c=10\), \(\alpha =1 \) and \(\hslash =-1\) for Example 6.2

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

(x,t) \(\vert {u}_{\mathrm{Exact}} - {u}_{\mathrm{ADM}} \vert \) \(\vert {u}_{\mathrm{Exact}} - {u}_{\mathrm{VIM}} \vert \) \(\vert {u}_{\mathrm{Exact}} - {u}_{\mathrm{LADM}} \vert \) \(\vert {u}_{\mathrm{Exact}} - {u}_{\mathrm{CRFDTM}} \vert \) \(\vert {u}_{\mathrm{Exact}} - {u}_{{q}\text{-}\mathrm{HATM}}^{ ( {3} )} \vert \)
(0.1,0.1) 8.02989 × 10−6 1.23033 × 10−4 7.10000 × 10−9 2.77556 × 10−17 2.77556 × 10−17
(0.1,0.3) 7.38281 × 10−6 3.69597 × 10−4 6.50000 × 10−9 2.77556 × 10−17 2.77556 × 10−17
(0.1,0.5) 6.79923 × 10−6 4.92780 × 10−4 5.90000 × 10−9 3.33067 × 10−16 3.33067 × 10−16
(0.2,0.1) 3.23228 × 10−5 1.69274 × 10−5 2.82000 × 10−8 2.77556 × 10−17 2.77556 × 10−17
(0.2,0.3) 297172 × 10−5 1.89210 × 10−4 2.59000 × 10−8 4.16334 × 10−17 4.16334 × 10−17
(0.2,0.5) 2.73673 × 10−5 1.55176 × 10−4 2.41000 × 10−8 3.60822 × 10−17 3.60822 × 10−17
(0.3,0.1) 7.32051 × 10−5 1.12345 × 10−5 6.33670 × 10−8 1.38778 × 10−17 1.38778 × 10−17
(0.3,0.3) 6.73006 × 10−5 6.55176 × 10−5 5.85000 × 10−8 2.77556 × 10−17 2.77556 × 10−17
(0.3,0.5) 6.19760 × 10−5 2.12346 × 10−5 5.40000 × 10−8 3.19189 × 10−16 3.19189 × 10−16
(0.4,0.1) 1.31032 × 10−4 7.36513 × 10−5 1.12400 × 10−7 1.38778 × 10−17 1.38778 × 10−17
(0.4,0.3) 1.20455 × 10−4 9.5016 × 10−5 1.03900 × 10−7 2.77556 × 10−17 2.77556 × 10−17
(0.4,0.5) 1.10919 × 10−4 8.23160 × 10−4 9.61000 × 10−8 3.19189 × 10−16 3.19189 × 10−16
(0.5,0.1) 2.06186 × 10−4 5.55176 × 10−5 1.75500 × 10−7 0 0
(0.5,0.3) 1.89528 × 10−4 3.21715 × 10−6 1.62200 × 10−7 5.55112 × 10−17 5.55112 × 10−17
(0.5,0.5) 1.74510 × 10−4 2.00176 × 10−5 1.5010 × 10−7 3.19189 × 10−16 3.19189 × 10−16