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Theory and Modern Applications

Table 4 Some numerical results of \(\mathcal{I}_{q}^{\alpha }[m_{1}] (t)\) in Example 1 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

\(\mathcal{I}_{q}^{\alpha }[m_{1}] (t)\)

\(\mathcal{I}_{q}^{\alpha +1} [m_{1}] (1)\)

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

t = 0

t = 1

sup

s = 1

s = a

s = b

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

1

0

0.0516

0.0516

0.0454

0.0556

0.0404

0.0561

2

0

0.0527

0.0527

0.0465

0.0564

0.0408

0.0569

3

0

0.0529

0.0529

0.0466

0.0565

0.0409

0.0570

4

0

0.0529

0.0529

0.0466

0.0566

0.0409

0.057

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

1

0

0.0347

0.0347

0.0196

0.0515

0.0362

0.0514

2

0

0.0446

0.0446

0.0265

0.0586

0.0399

0.0576

3

0

0.0499

0.0499

0.0305

0.062

0.0416

0.0605

4

0

0.0526

0.0526

0.0325

0.0636

0.0424

0.062

5

0

0.0539

0.0539

0.0336

0.0644

0.0428

0.0627

6

0

0.0546

0.0546

0.0341

0.0649

0.043

0.063

7

0

0.055

0.055

0.0344

0.0651

0.0431

0.0632

8

0

0.0552

0.0552

0.0345

0.0652

0.0432

0.0633

9

0

0.0552

0.0552

0.0346

0.0652

0.0432

0.0633

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

1

0

0.0067

0.0067

0.0013

0.0293

0.0216

0.0298

2

0

0.011

0.011

0.0025

0.0363

0.0261

0.0366

3

0

0.0154

0.0154

0.004

0.0418

0.0294

0.0418

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37

0

0.0551

0.0551

0.0257

0.0705

0.0446

0.0673

38

0

0.0551

0.0551

0.0257

0.0705

0.0447

0.0673

39

0

0.0552

0.0552

0.0257

0.0705

0.0447

0.0673

40

0

0.0552

0.0552

0.0257

0.0705

0.0447

0.0674

41

0

0.0552

0.0552

0.0258

0.0705

0.0447

0.0674

42

0

0.0552

0.0552

0.0258

0.0705

0.0447

0.0674

43

0

0.0553

0.0553

0.0258

0.0706

0.0447

0.0674