Fractional-order Boubaker functions and their applications in solving delay fractional optimal control problems

In this paper, a new set of functions called fractional-order Boubaker functions is defined for solving the delay fractional optimal control problems with a quadratic performance index. To solve the problem, first we obtain the operational matrix of the Caputo fractional derivative of these functions and the operational matrix of multiplication to solve the nonlinear problems for the first time. Also, a general formulation for the delay operational matrix of these functions has been achieved. Then we utilized these matrices to solve delay fractional optimal control problems directly. In fact, the delay fractional optimal control problem converts to an optimization problem, which can then be easily solved with the aid of the Gauss-Legendre integration formula and Newton's iterative method. Convergence of the algorithm is proved. The applicability of the method is shown by some examples; moreover, a comparison with the existing results shows the preference of this method.