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Advances in Continuous and Discrete Models

Theory and Modern Applications

Table 11 Pointwise absolute errors and computed rates of convergence obtained by the modified integral discretization scheme [21] and proposed methods: Newton’s method, Halley’s method. We set \(s_{0}=0.2\), \(\mathit{Tol}=10^{-5}\)

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

Time

Max. error

Rate

Time

Max. error

Rate

Time

Modified integral discretization scheme [21]

10

1.93E−03

–

0.02

1.67E−03

–

0.02

1.43E−03

–

0.02

1.29E−03

–

0.02

1.21E−03

–

0.02

20

5.48E−04

1.816

0.02

4.68E−04

1.831

0.02

3.97E−04

1.847

0.02

3.54E−04

1.862

0.02

3.32E−04

1.870

0.02

40

1.46E−04

1.907

0.09

1.24E−04

1.915

0.09

1.04E−04

1.925

0.09

9.28E−05

1.932

0.09

8.68E−05

1.937

0.1

80

3.78E−05

1.948

0.34

3.21E−05

1.954

0.34

2.68E−05

1.962

0.35

2.38E−05

1.966

0.34

2.22E−05

1.969

0.34

160

9.67E−06

1.968

1.35

8.16E−06

1.974

1.45

6.80E−06

1.980

1.34

6.01E−06

1.983

1.36

5.60E−06

1.984

1.35

320

2.45E−06

1.979

5.43

2.06E−06

1.984

5.44

1.71E−06

1.990

5.33

1.51E−06

1.991

5.32

1.41E−06

1.992

5.39

Newton’s method

10

7.27E−03

–

0.01

1.97E−03

–

0.02

9.45E−04

–

0.01

6.80E−04

–

0.02

5.80E−04

–

0.19

20

1.78E−03

2.033

0.02

3.92E−04

2.329

0.02

1.97E−04

2.263

0.02

1.56E−04

2.124

0.02

1.40E−04

2.047

0.02

40

3.92E−04

2.180

0.07

7.80E−05

2.328

0.07

4.38E−05

2.170

0.06

3.75E−05

2.055

0.07

3.47E−05

2.016

0.08

80

8.38E−05

2.226

0.24

1.62E−05

2.272

0.24

1.03E−05

2.093

0.23

9.25E−06

2.021

0.24

8.64E−06

2.005

0.24

160

1.78E−05

2.236

0.93

3.50E−06

2.205

0.93

2.49E−06

2.046

0.89

2.30E−06

2.007

0.94

2.16E−06

2.001

0.93

320

3.79E−06

2.232

3.58

7.93E−07

2.144

3.57

6.12E−07

2.021

3.55

5.74E−07

2.002

3.62

5.39E−07

2.000

3.56

Halley’s method

10

1.40E−03

–

0.02

2.79E−04

–

0.02

1.08E−04

–

0.02

6.95E−05

–

0.02

5.51E−05

–

0.02

20

1.55E−04

3.169

0.03

2.57E−05

3.441

0.03

1.06E−05

3.339

0.03

7.70E−06

3.174

0.03

6.56E−06

3.070

0.03

40

1.65E−05

3.237

0.11

2.44E−06

3.397

0.11

1.15E−06

3.211

0.11

9.19E−07

3.066

0.11

8.12E−07

3.014

0.11

80

1.73E−06

3.253

0.37

2.43E−07

3.325

0.40

1.33E−07

3.111

0.37

1.13E−07

3.023

0.37

1.02E−07

3.000

0.37

160

1.82E−07

3.252

1.40

2.56E−08

3.248

1.42

1.60E−08

3.055

1.38

1.41E−08

3.007

1.41

1.27E−08

2.998

1.40

320

1.92E−08

3.244

5.47

2.82E−09

3.180

5.44

1.96E−09

3.026

5.71

1.76E−09

3.002

5.44

1.59E−09

2.999

5.45

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