Table 1 Numerical analysis among the real part of the accurate and our method solutions.

From: On the analysis and deeper properties of the fractional complex physical models pertaining to nonsingular kernels

\(\aleph\)

\({\xi }\)

Re(Exact Solution)

\(Re(NDM_{CF}\) Solution)

\(Re(NDM_{ABC}\) Solution)

Re(Error by \(NDM_{CF})\)

Re(Error by \(NDM_{ABC})\)

0.001

0.2

1.0234574450

1.0236514600

1.0236514600

1.9401418410E−04

1.9401418410E−04

0.4

0.9618394838

0.9620218174

0.9620218174

1.8233362610E−04

1.8233362610E−04

0.6

0.8618760180

0.8620394017

0.8620394017

1.6338370930E−04

1.6338370930E−04

0.8

0.7275522751

0.7276901954

0.7276901954

1.3792029620E−04

1.3792029620E−04

1

0.5642233190

0.5643302774

0.5643302774

1.0695840740E−04

1.0695840740E−04

0.002

0.2

1.0234562830

1.0238450500

1.0238450500

3.8876672930E−04

3.8876672930E−04

0.4

0.9618383924

0.9622037527

0.9622037527

3.6536031900E−04

3.6536031900E−04

0.6

0.8618750400

0.8622024286

0.8622024286

3.2738856400E−04

3.2738856400E−04

0.8

0.7275514495

0.7278278144

0.7278278144

2.7636494640E−04

2.7636494640E−04

1

0.5642226787

0.5644370022

0.5644370022

2.1432351590E−04

2.1432351590E−04

0.003

0.2

1.0234543470

1.0240386040

1.0240386040

5.8425663510E−04

5.8425663510E−04

0.4

0.9618365724

0.9623856536

0.9623856536

5.4908117860E−04

5.4908117860E−04

0.6

0.8618734091

0.8623654246

0.8623654246

4.9201546410E−04

4.9201546410E−04

0.8

0.7275500728

0.7279654075

0.7279654075

4.1533465070E−04

4.1533465070E−04

1

0.5642216111

0.5645437068

0.5645437068

3.2209572580E−04

3.2209572580E−04

0.004

0.2

1.0234516360

1.0242321210

1.0242321210

7.8048490190E−04

7.8048490190E−04

0.4

0.9618340247

0.9625675200

0.9625675200

7.3349530500E−04

7.3349530500E−04

0.6

0.8618711262

0.8625283897

0.8625283897

6.5726350950E−04

6.5726350950E−04

0.8

0.7275481457

0.7281029744

0.7281029744

5.5482870910E−04

5.5482870910E−04

1

0.5642201166

0.5646503912

0.5646503912

4.3027463670E−04

4.3027463670E−04

0.005

0.2

1.0234481510

1.0244256020

1.0244256020

9.7745052980E−04

9.7745052980E−04

0.4

0.9618307494

0.9627493520

0.9627493520

9.1860259820E−04

9.1860259820E−04

0.6

0.8618681913

0.8626913240

0.8626913240

8.2313270020E−04

8.2313270020E−04

0.8

0.7275456682

0.7282405153

0.7282405153

6.9484712160E−04

6.9484712160E−04

1

0.5642181953

0.5647570554

0.5647570554

5.3886014880E−04

5.3886014880E−04

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