# Table 4 Comparative study between ADM , VIM , LADM , CRFDTM  and q-HATM for the approximate solution $$v(x,t)$$ at $$\omega =0.005$$, $$\ell= 0.1$$, $$c=10$$, $$\alpha =1$$ and $$\hslash =-1$$ for Example 6.2

(x,t) $$\vert {v}_{\mathrm{Exact}} - {v}_{\mathrm{ADM}} \vert$$ $$\vert {v}_{\mathrm{Exact}} - {v}_{\mathrm{VIM}} \vert$$ $$\vert {v}_{\mathrm{Exact}} - {v}_{\mathrm{LADM}} \vert$$ $$\vert {v}_{\mathrm{Exact}} - {v}_{\mathrm{CRFDTM}} \vert$$ $$\vert {v}_{\mathrm{Exact}} - {v}_{{q}\text{-}\mathrm{HATM}}^{ ( {3} )} \vert$$
(0.1,0.1) 4.81902 × 10−4 1.23033 × 10−4 9.5512 × 10−10 1.73472 × 10−18 1.73472 × 10−18
(0.1,0.3) 4.50818 × 10−4 1.7600 × 10−4 8.0600 × 10−10 2.60209 × 10−17 2.60209 × 10−17
(0.1,0.5) 4.22221 × 10−4 2.69597 × 10−4 6.7700 × 10−10 1.80411 × 10−16 1.80411 × 10−16
(0.2,0.1) 9.76644 × 10−4 2.69597 × 10−4 3.8210 × 10−9 3.46945 × 10−18 3.46945 × 10−18
(0.2,0.3) 9.13502 × 10−4 2.69597 × 10−4 3.224 × 10−9 2.34188 × 10−17 2.34188 × 10−17
(0.2,0.5) 8.55426 × 10−4 2.69597 × 10−4 2.7060 × 10−9 1.73472 × 10−16 1.73472 × 10−16
(0.3,0.1) 1.48482 × 10−3 2.69597 × 10−4 8.597 × 10−9 3.46945 × 10−18 3.46945 × 10−18
(0.3,0.3) 1.38858 × 10−3 2.69597 × 10−4 7.252 × 10−9 1.99493 × 10−17 1.99493 × 10−17
(0.3,0.5) 1.30009 × 10−3 2.69597 × 10−4 6.0910 × 10−9 1.61329 × 10−16 1.61329 × 10−16
(0.4,0.1) 2.00705 × 10−3 2.69597 × 10−4 1.5284 × 10−8 2.60209 × 10−18 2.60209 × 10−18
(0.4,0.3) 1.87661 × 10−3 2.69597 × 10−4 1.2893 × 10−8 1.73472 × 10−17 1.73472 × 10−17
(0.4,0.5) 1.75670 × 10−3 2.69597 × 10−4 1.0827 × 10−8 1.52656 × 10−16 1.52656 × 10−16
(0.5,0.1) 2.54396 × 10−3 2.69597 × 10−4 2.3880 × 10−8 8.67362 × 10−19 8.67362 × 10−19
(0.5,0.3) 2.37815 × 10−3 2.69597 × 10−4 2.0144 × 10−8 2.08167 × 10−17 2.08167 × 10−17
(0.5,0.5) 2.22578 × 10−3 2.69597 × 10−4 1.6916 × 10−8 1.43982 × 10−16 1.43982 × 10−16 