Shifted Chebyshev polynomials method for Caputo-Hadamard fractional Ginzburg–Landau equation

Abstract

This paper introduces a fractional version of the Ginzberg–Landau equation utilizing the Caputo-Hadamard derivative. To address this problem, a numerical method based on the shifted Chebyshev polynomials is developed. To employ this approach, a formula for calculating the Hadamard fractional integral of these polynomials is derived. Using this formula, an operational matrix associated with the Hadamard fractional integral of the shifted Chebyshev polynomials is constructed. By expressing the solution of the problem in terms of its real and imaginary parts, the fractional differential equation is transformed into a system of fractional differential equations with real solutions, corresponding to the real and imaginary components of the original problem. Next, the fractional terms in the resulting system are expanded using the expressed polynomials. The presented fractional integral operational matrix is then employed to obtain finite expansions for the solution of the aforementioned system. Utilizing the ordinary second-order derivative operational matrix of the applied shifted polynomials and the collocation method, the fractional system is solved by addressing a system of nonlinear algebraic equations. This approach directly yields the solution to the Ginzberg–Landau equation. The convergence of the established method is rigorously examined both theoretically and numerically, supported by three illustrative numerical examples.