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Table 7 Some numerical results of \(\tau (\alpha , b)<1\) in equation (29) in Example 1 for \(q=\frac{1}{7}, \frac{1}{2}, \frac{8}{9}\). Note that \((a)=\varGamma _{q}(\beta _{2}+1)\), \((b)= \varGamma _{q}(2-\beta _{1})\), and \((c) = M(\alpha , b)\)

From: On the existence of solutions for a multi-singular pointwise defined fractional q-integro-differential equation

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\(q =\frac{1}{7}\)

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

\(q =\frac{8}{9}\)

(a)

(b)

(c)

Ď„

(a)

(b)

(c)

Ď„

(a)

(b)

(c)

Ď„

1

0.9661

0.9656

4.8544

0.1742

0.9965

0.9772

2.8302

0.1005

1.6936

1.389

0.5474

0.016

2

0.9644

0.9642

4.8685

0.1749

0.9565

0.9493

3.0915

0.1117

1.4923

1.2733

0.7566

0.0231

3

0.9641

0.964

4.8705

0.175

0.9382

0.9364

3.2248

0.1175

1.3628

1.1967

0.9602

0.0302

4

0.9641

0.964

4.8708

0.175

0.9294

0.9302

3.292

0.1205

1.2721

1.1419

1.1541

0.0373

5

0.9641

0.964

4.8709

0.175

0.9251

0.9272

3.3258

0.122

1.205

1.1007

1.336

0.044

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10

0.9641

0.964

4.8709

0.175

0.921

0.9243

3.3586

0.1234

1.031

0.9905

2.0511

0.072

11

0.9641

0.964

4.8709

0.175

0.9209

0.9242

3.3591

0.1235

1.0124

0.9783

2.158

0.0763

12

0.9641

0.964

4.8709

0.175

0.9209

0.9242

3.3594

0.1235

0.9967

0.9681

2.2544

0.0802

13

0.9641

0.964

4.8709

0.175

0.9209

0.9242

3.3595

0.1235

0.9833

0.9593

2.3413

0.0838

14

0.9641

0.964

4.8709

0.175

0.9209

0.9242

3.3596

0.1235

0.9719

0.9517

2.4194

0.087

15

0.9641

0.964

4.8709

0.175

0.9209

0.9242

3.3596

0.1235

0.962

0.9452

2.4896

0.09

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62

0.9641

0.964

4.8709

0.175

0.9209

0.9242

3.3597

0.1235

0.8929

0.8989

3.072

0.1146

63

0.9641

0.964

4.8709

0.175

0.9209

0.9242

3.3597

0.1235

0.8929

0.8989

3.0723

0.1147

64

0.9641

0.964

4.8709

0.175

0.9209

0.9242

3.3597

0.1235

0.8928

0.8989

3.0725

0.1147