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

Advances in Continuous and Discrete Models

Theory and Modern Applications

Table 3 Approximate solutions of \(\mathfrak{W}(\tau )\) for various values of α where \(N=8 \) in Example 1

τ	α = 0.5	α = 0.6	α = 0.7	α = 0.8	α = 0.9	α = 0.95	α = 1	Exact
0.1	−0.29024517	−0.30334744	−0.31328561	−0.32047676	−0.32530759	−0.32692973	−0.32806014	−0.32806014
0.2	−0.25869970	−0.26522810	−0.27043720	−0.27421650	−0.27639207	−0.27684564	−0.27687383	−0.27687383
0.3	−0.23231436	−0.23389564	−0.23494522	−0.23499330	−0.23378057	−0.23267242	−0.23123424	−0.23123424
0.4	−0.20601442	−0.20579033	−0.20392189	−0.20055293	−0.19590475	−0.19317829	−0.19022704	−0.19022704
0.5	−0.18767675	−0.18323648	−0.17711352	−0.16978015	−0.16162988	−0.15736423	−0.15303073	−0.15303073
0.6	−0.17370140	−0.16299753	−0.15204853	−0.14090284	−0.12975078	−0.12426686	−0.11890014	−0.11890014
0.7	−0.15198700	−0.13868339	−0.12529307	−0.11195858	−0.09909087	−0.09297956	−0.08715152	−0.08715152
0.8	−0.12438243	−0.11082380	−0.09631388	−0.08197536	−0.06872931	−0.06270956	−0.05714883	−0.05714883
0.9	−0.10332343	−0.08222159	−0.06406149	−0.04913442	−0.03733057	−0.03249359	−0.02829103	−0.02829103
1.0	2.13 × 10⁻¹⁶	1.76 × 10⁻¹⁶	−7.7 × 10⁻¹⁷	4.4 × 10⁻¹⁸	−1.9 × 10⁻¹⁸	−2.5 × 10⁻¹⁹	4.485 × 10⁻²¹	5 × 10⁻²⁰