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Theory and Modern Applications

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