Fractional-order Horadam functions: A new set of functions to solve fractional optimal control problems

Abstract

The primary aim of this paper is to present an approximation method based on a new set of functions named fractional-order Horadam functions (FHFs). Horadam polynomials have four free parameters a, b, p, q, and this feature distinguishes them from other known classical polynomials. By selecting different values of these parameters, we can construct various types of known orthogonal and non-orthogonal polynomials such as the first and second kinds of Chebyshev polynomials, Pell polynomials, Lucas polynomials, and more. First, we define the fractional order of these polynomials which are called FHFs and we present a new relation for FHFs. These functions, in addition to the aforementioned property, have one more free parameter α than the classical Horadam polynomials to their structure. Next, based on this impressive feature, we design a new efficient scheme to solve a class of fractional order optimal control problems (FOCPs). To do this, we construct a Riemann–Liouville integral operational matrix (RLIOM) for FHFs. Next, employing this matrix and Galerkin method, the problem reduces to a system of algebraic equations. Also, we find the upper bound of the error vector for the RLIOM and we indicate a convergence of approximations of FHFs. Moreover, some numerical examples show the efficiency and accuracy of the present technique.