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Theory and Modern Applications

Table 9 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 4.3. We set \(s_{0}=0.2\) for both shooting techniques but \(\mathit{Tol} = 10^{-15} (10^{-16})\) for Newton’s method (Halley’s method), respectively

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

2.26E − 05

6.92E − 07

6.70E − 06

8.99E − 06

9.95E − 06

7.65E − 07

5.55E − 08

1.33E − 07

1.84E − 07

1.95E − 07

Rate

k

20

10

9

8

6

22

10

9

8

7

CPU time (sec)

3.969

1.227

1.427

1.230

0.941

5.149

2.296

2.055

1.896

1.680

128

Max. error

5.04E − 06

6.51E − 07

1.88E − 06

2.35E − 06

2.57E − 06

8.54E − 08

2.93E − 09

1.95E − 08

2.43E − 08

2.55E − 08

Rate

2.166

8.900

1.833

1.937

1.954

3.164

4.244

2.767

2.920

2.933

k

20

10

9

8

7

22

11

9

8

7

CPU time (sec)

3.969

4.636

5.262

4.543

4.066

18.971

9.424

7.705

6.906

6.112

256

Max. error

1.10E − 06

2.50E − 07

4.99E − 07

6.00E − 07

6.54E − 07

9.41E − 09

8.67E − 10

2.64E − 09

3.11E − 09

3.27E − 09

Rate

2.192

1.381

1.914

1.968

1.974

3.181

1.754

2.888

2.962

2.964

k

20

10

9

8

7

22

11

9

8

7

CPU time (sec)

15.179

17.853

20.053

17.621

15.541

73.376

33.136

29.642

26.633

23.345

512

Max. error

2.38E − 07

7.81E − 08

1.29E − 07

1.52E − 07

1.65E − 07

1.03E − 09

1.72E − 10

3.43E − 10

3.94E − 10

4.14E − 10

Rate

2.211

1.679

1.955

1.983

1.984

3.194

2.331

2.944

2.981

2.979

k

20

10

9

8

7

22

10

9

8

7

CPU time (sec)

60.494

69.994

80.021

69.867

62.629

289.130

131.007

116.497

106.946

92.665

1024

Max. error

5.08E − 08

2.23E − 08

3.27E − 08

3.82E − 08

4.17E − 08

1.12E − 10

2.71E − 11

4.37E − 11

4.96E − 11

5.23E − 11

Rate

2.228

1.810

1.975

1.990

1.989

3.205

2.669

2.971

2.990

2.986

k

20

10

9

8

7

22

10

9

8

7

CPU time (sec)

239.810

275.430

313.605

276.502

244.386

1132.549

515.751

461.774

414.636

363.701

2048

Max. error

1.07E − 08

6.05E − 09

8.26E − 09

9.59E − 09

1.05E − 08

1.20E − 11

3.90E − 12

5.50E − 12

6.25E − 12

6.58E − 12

Rate

2.246

1.880

1.986

1.994

1.991

3.215

2.798

2.990

2.990

2.991

k

20

10

9

8

7

22

11

9

8

7

CPU time (sec)

961.302

1099.115

1235.384

1096.546

963.071

4517.411

2247.280

1837.934

1634.163

1436.284