Piecewise fractional Chebyshev cardinal functions: Application for time fractional Ginzburg-Landau equation with a non-smooth solution

In this paper, two important subjects are investigated. First, a fruitful family of functions with the interpolation property called the orthonormal piecewise fractional Chebyshev cardinal functions are generated, and a formulation for their fractional derivative matrix is provided. Second, these functions together with their expressed matrix are used to construct a numerical method for fractional Ginzburg-Landau equation with a non-smooth solution. In fact, the proposed method approximates the problem solution by the expressed fractional functions (in the temporal direction) and classical Chebyshev cardinal polynomials (in the space direction). This method is able to get highly accurate solutions for the problem under investigation if its solution is in the piecewise form or includes terms with fractional powers, or includes both of them. The ability and validity of the method are examined by solving some numerical examples.

Automated summary

In this paper, the authors investigate two important subjects. Firstly, they generate a fruitful family of functions called the orthonormal piecewise fractional Chebyshev cardinal functions, which have the interpolation property, and provide a formulation for their fractional derivative matrix. Secondly, they use these functions and their expressed matrix to construct a numerical method for the fractional Ginzburg-Landau equation with a non-smooth solution. The proposed method approximates the problem solution using the expressed fractional functions in the temporal direction and classical Chebyshev cardinal polynomials in the space direction, and it is able to obtain highly accurate solutions for problems with piecewise form, fractional powers, or both. The ability and validity of the method are demonstrated through the solution of numerical examples.