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Theory and Modern Applications

Table 8 Some numerical results of \(\mathcal{I}_{q}^{\alpha } [m_{2}] (t)\), \(\mathcal{I}_{q}^{\alpha -1} [m_{2}] (1)\), \(\mathcal{I}_{q}^{\alpha -2} [m_{2}] (b)\), and \(\mathcal{I}_{q}^{\alpha -\zeta } [m_{2}] (t)\) in Example 2 for \(t \in \overline{J}\) and \(q=\frac{1}{8}, \frac{1}{2}, \frac{6}{7}\)

From: Solutions of two fractional q-integro-differential equations under sum and integral boundary value conditions on a time scale

n

\(\sup \mathcal{I}_{q}^{\alpha } [m_{2}] (t)\)

\(\sup \mathcal{I}_{q}^{\alpha -1} [m_{2}] (1)\)

\(\sup \mathcal{I}_{q}^{\alpha -2} [m_{2}] (b)\)

\(\sup \mathcal{I}_{q}^{\alpha -\zeta } [m_{2}] (t)\)

\(q = \frac{1}{8} \)

1

0.0343

0.0391

0.0357

0.0349

2

0.0343

0.0391

0.0357

0.0349

3

0.0343

0.0391

0.0357

0.0349

\(q = \frac{1}{2} \)

1

0.0167

0.0299

0.0373

0.018

2

0.0179

0.0311

0.0368

0.0193

3

0.0183

0.0314

0.0368

0.0197

4

0.0184

0.0315

0.0367

0.0198

5

0.0184

0.0315

0.0367

0.0198

6

0.0184

0.0315

0.0367

0.0198

\(q = \frac{6}{7} \)

1

0.0029

0.0144

0.0434

0.0036

2

0.0045

0.018

0.0407

0.0054

3

0.0058

0.0206

0.0394

0.0069

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13

0.0107

0.0267

0.0375

0.0122

14

0.0108

0.0268

0.0374

0.0122

15

0.0108

0.0268

0.0374

0.0123

16

0.0109

0.0269

0.0374

0.0123

17

0.0109

0.0269

0.0374

0.0124

18

0.0109

0.0269

0.0374

0.0124