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Advances in Continuous and Discrete Models

Theory and Modern Applications

Table 3 Pointwise absolute errors in the maximum norm, computed rates of convergence obtained by the proposed schemes, number of iterations k, and CPU time in second executed with various fractional orders versus the number of sub-intervals in Example 4.1. We set \(s_{0}=0.2\), \(\mathit{Tol}=10^{-5}\) for shooting with Newton’s method and \(s_{0}=0.2\), \(\mathit{Tol}=10^{-10}\) for Halley’s method

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

N

 

Newton’s method \((\alpha _{1}, \alpha _{2})\)

Halley’s method

\((\alpha _{1}, \alpha _{2})\)

 

(0.9,1.1)

(0.7,1.3)

(0.5,1.5)

(0.3,1.7)

(0.1,1.9)

(0.9,1.1)

(0.7,1.3)

(0.5,1.5)

(0.3,1.7)

(0.1,1.9)

64

Max. error

1.17E − 03

9.45E − 05

6.06E − 04

7.72E − 04

7.94E − 04

5.05E − 05

4.17E − 06

9.12E − 06

1.03E − 05

7.17E − 06

Rate

–

–

–

–

–

–

–

–

–

–

k

8

5

4

4

4

13

7

6

5

5

CPU time (sec)

0.044

0.030

0.024

0.024

0.024

0.186

0.214

0.086

0.092

0.076

128

Max. error

2.55E − 04

6.60E − 05

1.72E − 04

2.05E − 04

2.09E − 04

5.61E − 06

2.42E − 07

1.38E − 06

1.46E − 06

1.05E − 06

Rate

2.196

5.176

1.817

1.911

1.925

3.168

4.105

2.728

2.826

2.777

k

8

4

4

4

4

11

7

6

5

5

CPU time (sec)

0.098

0.051

0.051

0.050

0.048

0.346

0.487

0.190

0.158

0.166

256

Max. error

5.46E − 05

2.44E − 05

4.59E − 05

5.29E − 05

5.38E − 05

6.24E − 07

7.59E − 08

1.89E − 07

1.95E − 07

1.43E − 07

Rate

2.223

1.435

1.907

1.955

1.959

3.170

1.673

2.866

2.905

2.871

k

7

4

4

4

4

12

6

5

5

5

CPU time (sec)

0.255

0.149

0.153

0.159

0.148

0.988

1.456

0.405

0.406

0.412

512

Max. error

1.16E − 05

7.58E − 06

1.19E − 05

1.35E − 05

1.37E − 05

6.85E − 08

1.39E − 08

2.48E − 08

2.53E − 08

1.90E − 08

Rate

2.236

1.687

1.949

1.976

1.976

3.186

2.449

2.929

2.942

2.914

k

8

4

4

4

4

12

6

5

5

5

CPU time (sec)

0.954

0.470

0.485

0.478

0.473

2.950

3.712

1.208

1.241

1.220

1024

Max. error

2.41E − 06

2.15E − 06

3.03E − 06

3.39E − 06

3.45E − 06

7.45E − 09

2.12E − 09

3.19E − 09

3.25E − 09

2.48E − 09

Rate

2.261

1.818

1.970

1.988

1.987

3.202

2.712

2.960

2.962

2.936

k

8

4

4

4

4

11

6

5

5

5

CPU time (sec)

2.371

1.176

1.215

1.243

1.192

6.751

3.088

3.088

2.992

3.121

2048

Max. error

4.95E − 07

5.31E − 07

7.67E − 07

8.49E − 07

8.63E − 07

8.03E − 10

3.02E − 10

4.01E − 10

4.18E − 10

3.21E − 10

Rate

2.288

2.018

1.984

1.998

1.999

3.214

2.811

2.988

2.961

2.949

k

8

4

4

4

4

11

6

5

5

5

CPU time (sec)

7.238

3.638

3.889

3.689

3.584

20.233

11.132

8.983

8.978

9.281

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