Discrete Legendre polynomials method to solve the coupled nonlinear Caputo–Hadamard fractional Ginzburg–Landau equations

This paper employs the Caputo–Hadamard derivative to create the coupled nonlinear fractional Ginzburg–Landau equations. An orthonormal version of the discrete Legendre polynomials is utilized to generate a numerical strategy for this system. For this purpose, a fractional derivative matrix corresponding to these polynomials is obtained. The idea of the strategy is that by approximating the system’s solution by means of these polynomials, a system of algebraic equations is achieved, from the solution of which the solution of the main system is determined. To show the accuracy of the presented scheme, two examples are studied. • A coupled Caputo-Hadamard fractional Ginzburg-Landau equations system is introduced. • The orthonormal discrete Legendre polynomials are defined on an arbitrary domain. • The Caputo-Hadamard fractional derivative matrix of these polynomials is obtained. • A collocation method via the mentioned polynomials is developed for the expressed system. • The validity of the developed method are checked by solving some numerical examples.