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Theory and Modern Applications

Table 8 The results obtained for the absolute errors of \(\mathfrak{W}(\tau )\) and \(\mathfrak{J}\) with various values of α where \(N=8\) for Example 2

From: A computational method based on the generalized Lucas polynomials for fractional optimal control problems

τ

α = 0.5

α = 0.6

α = 0.7

α = 0.8

α = 0.9

α = 0.95

α = 1

0.1

9.131 × 10−6

4.345 × 10−6

1.058 × 10−6

9.630 × 10−7

1.591 × 10−6

1.157 × 10−6

6.879 × 10−20

0.2

2.088 × 10−5

1.105 × 10−5

5.484 × 10−6

3.003 × 10−6

1.845 × 10−6

1.144 × 10−6

8.012 × 10−20

0.3

1.636 × 10−5

9.146 × 10−6

4.195 × 10−6

1.027 × 10−6

5.267 × 10−7

5.820 × 10−7

9.709 × 10−20

0.4

1.353 × 10−5

7.694 × 10−6

3.480 × 10−6

7.930 × 10−7

4.785 × 10−7

5.038 × 10−7

1.156 × 10−19

0.5

1.374 × 10−5

7.210 × 10−6

3.602 × 10−6

1.932 × 10−6

1.118 × 10−6

6.768 × 10−7

1.064 × 10−19

0.6

8.078 × 10−6

4.724 × 10−6

2.528 × 10−6

1.404 × 10−6

8.307 × 10−7

5.069 × 10−7

6.959 × 10−20

0.7

1.468 × 10−5

7.651 × 10−6

3.406 × 10−6

1.061 × 10−6

1.579 × 10−8

1.510 × 10−7

4.455 × 10−20

0.8

2.270 × 10−6

1.870 × 10−6

9.835 × 10−7

2.235 × 10−7

1.541 × 10−7

1.544 × 10−7

5.468 × 10−20

0.9

9.783 × 10−6

5.434 × 10−6

2.728 × 10−6

1.234 × 10−6

4.675 × 10−7

2.197 × 10−7

5.632 × 10−20

1.0

3.57 × 10−18

1.160 × 10−18

8.20 × 10−19

1.09 × 10−19

6.5 × 10−20

4.89 × 10−20

5.769 × 10−20

\(\lvert J-J^{*} \rvert \)

6.119 × 10−9

2.198 × 10−9

6.730 × 10−10

1.643 × 10−10

2.661 × 10−11

6.432 × 10−12

1.233 × 10−38