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In this research, the Hadamard fractional derivative is used to define the time fractional coupled nonlinear Schrödinger–Hirota equations. The logarithmic Chebyshev cardinal functions, as a new category of cardinal functions, are introduced to build a numerical method to solve this system. To do this, the Hadamard fractional differentiation matrix of these functions is obtained. In the developed method, by considering a hybrid approximation of the problem's solution using the logarithmic Chebyshev cardinal functions (for the temporal variable) and classical Chebyshev cardinal polynomials (for the spatial variable), and employing the interpolation property of these basis functions, along with the expressed derivative matrix, solving the fractional system turns into obtaining the solution of an algebraic system of equations. Two numerical examples are investigated to acknowledge the high accuracy of the introduced procedure. • A new family of basis functions called logarithmic Chebyshev cardinal functions (LCCFs) is introduced. • Hadamard fractional differentiation matrix of these functions is obtained. • The Hadamard time fractional coupled nonlinear Schrödinger–Hirota equations are introduced. • A hybrid method based on LCCFs and classical Chebyshev cardinal polynomials is developed for this system. • The accuracy of the developed method is checked by solving some examples. [ABSTRACT FROM AUTHOR]