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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]

In this work, the distributed-order time fractional Klein–Gordon–Zakharov system is introduced by substituting the second-order temporal derivative with a distributed-order fractional derivative. The Caputo fractional derivative is utilized to define this kind of distributed-order fractional derivative. A high accuracy approach based on the Chebyshev cardinal polynomials is established for this system. The proposed method turns the fractional system solution into an algebraic system solution by approximating the unknown solution via these cardinal polynomials and engaging their derivative matrices (that are obtained in this paper). Some test problems are considered to investigate the capability and accuracy of this approach. • The distributed-order time fractional Klein-Gordon-Zakharov system is defined. • The Chebyshev cardinal polynomials are employed to construct a method for this system. • Some distributed-order fractional derivative matrices for these polynomials are extracted. • The accuracy of the proposed method is shown in some numerical examples. [ABSTRACT FROM AUTHOR]

In this paper we consider multi variable orders differential equations (MVODEs) with non-local and no-singular kernel. The derivative is described in Atangana and Baleanu sense of variable order. We use the fifth-kind Chebyshev polynomials as basic functions to obtain operational matrices. We transfer the original equations to a system of algebraic equations using operational matrices and collocation method. The convergence analysis of the presented method is discussed. Few examples are presented to show the efficiency of the presented method. [ABSTRACT FROM AUTHOR]