Discrete Chebyshev Polynomials for Solving Fractional Variational Problems

In ‎the current study, a‎ general formulation of the discrete Chebyshev polynomials is given. ‎The operational matrix of fractional integration for these discrete polynomials is also derived. ‎Then,‎ a numerical scheme based on the discrete Chebyshev polynomials and their operational matrix has been developed to solve fractional variational problems‎. In this method, the need for using Lagrange multiplier during the solution procedure is eliminated.‎ The performance of the proposed scheme is validated through some illustrative examples. ‎Moreover, ‎the obtained numerical results ‎‎‎‎were compared to the previously acquired results by the classical Chebyshev polynomials. Finally, a comparison for the required CPU time is presented, which indicates more efficiency and less complexity of the proposed method.

A discrete polynomials approach for optimal control of fractional Volterra integro-differential equations

This study develops an efficient numerical method for optimal control problems governed by fractional Volterra integro-differential equations. A new type of polynomials orthogonal with respect to a discrete norm, namely discrete Hahn polynomials, is introduced and its properties investigated. Fractional operational matrices for the orthogonal polynomials are also derived. A direct numerical algorithm supported by the discrete Hahn polynomials and operational matrices is used to approximate solution of optimal control problems governed by fractional Volterra integro-differential equations. Several examples are analyzed and the results compared with those of other methods. The required CPU time assesses the computational cost and complexity of the proposed method.