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Theory and Modern Applications

Table 9 Some numerical results of \(\mathcal{I}_{q}^{\alpha } [m_{3}] (t)\), \(\mathcal{I}_{q}^{\alpha -1} [m_{3}] (1)\), \(\mathcal{I}_{q}^{\alpha -2} [m_{3}] (b)\), and \(\mathcal{I}_{q}^{\alpha -\zeta } [m_{3}] (t)\) 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

\(\sup \mathcal{I}_{q}^{\alpha } [m_{3}] (t)\)

\(\sup \mathcal{I}_{q}^{\alpha -1} [m_{3}] (1)\)

\(\sup \mathcal{I}_{q}^{\alpha -2} [m_{3}] (b)\)

\(\sup \mathcal{I}_{q}^{\alpha -\zeta } [m_{3}] (t)\)

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

1

0.0231

0.0262

0.0288

0.0234

2

0.0231

0.0262

0.0288

0.0234

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

1

0.0123

0.0213

0.0297

0.0133

2

0.0136

0.0226

0.0293

0.0146

3

0.0139

0.0229

0.0292

0.0149

4

0.014

0.023

0.0292

0.015

5

0.014

0.023

0.0292

0.015

6

0.014

0.023

0.0292

0.015

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

1

0.0021

0.0101

0.0346

0.0026

2

0.0034

0.0131

0.0322

0.0041

3

0.0046

0.0154

0.031

0.0054

4

0.0057

0.017

0.0303

0.0066

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13

0.0093

0.0213

0.029

0.0105

14

0.0094

0.0213

0.029

0.0106

15

0.0094

0.0214

0.029

0.0106

16

0.0095

0.0214

0.029

0.0107

17

0.0095

0.0215

0.029

0.0107

18

0.0095

0.0215

0.029

0.0107

19

0.0095

0.0215

0.029

0.0108

20

0.0096

0.0215

0.029

0.0108

21

0.0096

0.0215

0.029

0.0108