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Theory and Modern Applications

Table 7 Errors of shooting techniques \(\vert \tilde{F}(s_{m}) \vert \) in (41) versus the maximum number of iterations m with various initial approximations \(s_{0}\) in Example 4.3 as we set \(\alpha _{1} = 0.4\), \(\alpha _{2} = 1.7\), \(h = 0.01\)

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

m

Newton’s method

Halley’s method

\(s_{0}=0.2\)

\(s_{0}=0.4\)

\(s_{0}=0.6\)

\(s_{0}=0.8\)

\(s_{0}=1.0\)

\(s_{0}=0.2\)

\(s_{0}=0.4\)

\(s_{0}=0.6\)

\(s_{0}=0.8\)

\(s_{0}=1.0\)

1

0.487617

0.891569

1.202993

1.407814

1.485109

0.487722

0.891834

1.203474

1.408578

1.486244

2

0.101684

3.44E−01

0.653431

9.74E−01

1.258505

0.106265

3.56E−01

0.668984

9.90E−01

1.270771

3

0.004491

5.07E−02

0.182979

4.14E−01

0.72559

0.005185

5.68E−02

0.199315

4.40E−01

0.757881

4

1.26E−05

1.14E−03

0.014409

0.073158

0.226095

1.69E−05

1.51E−03

1.80E−02

8.68E−02

0.255888

5

1.13E−08

1.57E−06

0.000101

0.002343

0.021949

1.54E−08

2.39E−06

1.63E−04

3.48E−03

2.96E−02

6

1.01E−11

1.41E−09

9.49E−08

4.43E−06

0.000225

1.39E−11

2.16E−09

1.59E−07

8.62E−06

4.22E−04

7

9.06E−15

1.26E−12

8.48E−11

3.97E−09

2.22E−07

1.25E−14

1.95E−12

1.44E−10

7.82E−09

4.62E−07

8

6.94E−18

1.12E−15

7.58E−14

3.55E−12

1.99E−10

6.94E−18

1.76E−15

1.30E−13

7.06E−12

4.17E−10

9

0

1.39E−17

6.94E−17

3.16E−15

1.78E−13

6.94E−18

6.94E−18

1.25E−16

6.37E−15

3.77E−13

10

0

6.94E−18

6.94E−18

6.94E−18

1.67E−16

6.94E−18

0

6.94E−18

6.94E−18

3.33E−16