The numerical treatment of fractal‐fractional 2D optimal control problems by Müntz–Legendre polynomials

Abstract

AbstractIn this work, we introduce a method based on the Müntz–Legendre polynomials (M‐LPs) for solving fractal‐fractional 2D optimal control problems that the fractal‐fractional derivative is described in Atangana‐Riemann‐Liouville's sense. First, we obtain operational matrices of fractal‐fractional‐order derivative, integer‐order integration, and derivative of the M‐LPs. Second, the under study problem is converted into an equivalent variational problem. Then, by applying the M‐LPs, their operational matrices and Gauss–Legendre integration, the mentioned problem is converted to a system of algebraic equations. Finally, this system is solved by Newton's iterative method. Also, we introduce an error bound for the described method. Two examples are included to test the applicability and validity of the present scheme.