Modified wavelets technique for multitype 2D fractional optimal control problems

Abstract

This manuscript is devoted to presenting a new approximate method to solve two-dimensional (2D) fractional optimal control problems, which include Caputo and Atangana–Riemann–Liouville fractal (ARL-F) fractional derivatives. Due to our purpose, we design a numerical algorithm based on the generalized Lucas wavelets (GLWs) and the least squares and collocation methods. GLWs are good choices in this scheme because they have two more parameters (α and β), compared with some existing classical wavelets. On the other hand, we have various types of wavelets (orthogonal and non-orthogonal) by selecting different parameters α and β. According to the impressive feature of the GLWs, we derive Caputo and ARL-F fractional derivative pseudo-operational matrices for the generalized Lucas polynomials and then, for GLWs. The combination of Caputo and ARL-F pseudo-operational matrices with the least squares and collocation methods makes that the considered problems lead to systems of algebraic equations. A brief discussion of the error estimation is investigated. We provide some numerical examples to demonstrate the validity, effectiveness, and applicability of the suggested technique. The proposed algorithm is easy to implement and presents accurate results.