Chebyshev polynomials for the numerical solution of fractal–fractional model of nonlinear Ginzburg–Landau equation

Abstract

This paper introduces a new version for the nonlinear Ginzburg–Landau equation derived from fractal–fractional derivatives and proposes a computational scheme for their numerical solutions. The fractal–fractional derivative is defined in the Atangana–Riemann–Liouville sense with Mittage–Leffler kernel. The proposed approach is based on the shifted Chebyshev polynomials (S-CPs) and the collocation scheme. Through the way, a new operational matrix (OM) of fractal–fractional derivative is derived for the S-CPs and used in the presented method. More precisely, the unknown solution is separated into their real and imaginary parts, and then, these parts are expanded in terms of the S-CPs with undetermined coefficients. These expansions are substituted into the main equation and the generated operational matrix is utilized to extract a system of nonlinear algebraic equations. Thereafter, the yielded system is solved to obtain the approximate solution of the problem. The accuracy of the proposed approach is examined through some numerical examples. Numerical results confirm the suggested approach is very accurate to provide satisfactory results.