|
چکیده
|
A novel numerical algorithm utilizing hybrid moving least squares and neural network (MLS-NN) is introduced in this paper to solve nonlinear fractal–fractional optimal control problems (FF-OCPs). The fractal–fractional derivative is defined in the Atangana–Riemann–Liouville sense, and the FF-OCPs are transformed into an equivalent variational problem. The proposed approach is independent of domain geometry and does not require meshing, making it an effectivemeshlessmethod. It uses the Legendre–Gauss quadrature rule to efficiently approximate the integral expressions of the fractional–fractal operators, which feature nonlocal and weakly singular kernels, and the integral term in the cost functional containing strongly nonlinear terms. Moreover, the Genocchi polynomials and their properties play a crucial role in our scheme, serving as the basis functions in theMLS approximation. TheMLS–NN approximations, along with the Gauss–Legendre integration rule, convert the problem into a system of algebraic equations, which is solved via Newton’s iterative method. Alongside the numerical approach, we investigate the convergence of the proposed method in Hilbert space. Four examples are provided to demonstrate the accuracy of the proposed scheme.
|