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Theory and Modern Applications

Table 10 Pointwise absolute errors, computed rates of convergence, CPU time in second executed by the modified integral discretization scheme [21] and Newton’s method. We set \(s_{0}=0.2\), \(\mathit{Tol}=10^{-6}\)

From: An efficient numerical approach for solving two-point fractional order nonlinear boundary value problems with Robin boundary conditions

N

\(\alpha _{2} = 1.1\)

\(\alpha _{2} = 1.3\)

\(\alpha _{2} = 1.5\)

\(\alpha _{2} = 1.7\)

\(\alpha _{2} = 1.9\)

Max. error

Rate

Time

Max. error

Rate

Time

Max. error

Rate

CPU time

Max. error

Rate

Time

Max. error

Rate

Time

Modified integral discretization scheme [21]

64

9.56E−03

0.028

5.14E−03

0.023

3.91E−03

0.102

3.76E−03

0.024

4.07E−03

0.023

128

3.89E−03

1.298

0.048

1.60E−03

1.686

0.048

1.05E−03

1.895

0.038

9.44E−04

1.995

0.049

1.02E−03

1.996

0.049

256

1.65E−03

1.236

0.162

5.20E−04

1.618

0.163

2.87E−04

1.872

0.100

2.37E−04

1.997

0.163

2.55E−04

1.998

0.163

512

7.22E−04

1.193

0.566

1.78E−04

1.548

0.578

8.01E−05

1.840

0.328

5.92E−05

1.998

0.583

6.39E−05

1.999

0.586

1024

3.22E−04

1.165

1.220

6.36E−05

1.484

1.221

2.30E−05

1.803

0.738

1.51E−05

1.971

1.239

1.60E−05

1.999

1.228

2048

1.45E−04

1.148

3.304

2.36E−05

1.432

3.338

6.77E−06

1.762

2.735

3.90E−06

1.955

3.363

4.00E−06

2.000

3.364

Newton’s method

64

4.42E−03

0.026

1.31E−03

0.012

1.14E−03

0.152

1.27E−03

0.012

1.42E−03

0.012

128

1.89E−03

1.225

0.012

3.83E−04

1.775

0.012

2.95E−04

1.955

0.013

3.17E−04

1.999

0.011

3.55E−04

2.000

0.011

256

8.10E−04

1.224

0.036

1.27E−04

1.588

0.023

7.62E−05

1.955

0.031

7.92E−05

2.002

0.023

8.87E−05

2.000

0.022

512

3.48E−04

1.219

0.047

6.31E−05

1.012

0.045

1.97E−05

1.949

0.049

1.98E−05

2.004

0.045

2.22E−05

2.000

0.045

1024

1.50E−04

1.213

0.109

2.88E−05

1.130

0.103

5.15E−06

1.938

0.113

4.92E−06

2.006

0.150

5.54E−06

2.000

0.103

2048

6.91E−05

1.120

0.268

1.26E−05

1.191

0.263

1.36E−06

1.922

0.287

1.22E−06

2.008

0.307

1.38E−06

2.001

0.281