Numerical solution of coupled fractional Ginzburg–Landau equations under Caputo–Hadamard derivative

This paper introduces a high-performance spectral collocation method for solving coupled fractional Ginzburg–Landau equations involving the Caputo–Hadamard (CH) derivative. The numerical scheme employees two families of shifted Chebyshev polynomials (CPs) for spatial and temporal approximations. A key contribution is the novel construction of an operational matrix for the CH derivative, which facilitates a systematic transformation of the coupled nonlinear fractional system into an easily solvable algebraic one. The solution is approximated by expanding the unknown functions with CPs and employing collocation points derived from their roots. The accuracy and efficiency of the proposed method are rigorously validated through two numerical examples. A comparative analysis with a recent discrete Legendre polynomials technique demonstrates the superior precision and convergence rate of our approach. The results confirm that this established method is a powerful and reliable tool for tackling complex fractional dynamical systems.