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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]

Highlights • The index concept is generalized for categorizing system of fractional differential algebraic equations and analyzing the solvability of such systems. • A model of a simple pendulum in a Newtonian fluid is obtained and described by using nonlinear fractional differential algebraic equations. • An application in an electrical circuit is introduced and the solvability of the obtained linear system is studied. • A simple and efficient numerical method for solving a system of FDAEs with different fractional derivatives including Liouville-Caputo's definition, Caputo-Fabrizio's definition and a definition with Mittag-Leffler kernel are introduced. Abstract One of the important classes of coupled systems of algebraic, differential and fractional differential equations (CSADFDEs) is fractional differential algebraic equations (FDAEs). The main difference of such systems with other class of CSADFDEs is that their singularity remains constant in an interval. However, complete classifying and analyzing of these systems relay mainly to the concept of the index which we introduce in this paper. For a system of linear differential algebraic equations (DAEs) with constant coefficients, we observe that the solvability depends on the regularity of the corresponding pencils. However, we show that in general, similar properties of DAEs do not hold for FDAEs. In this paper, we introduce some practical applications of systems of FDAEs in physics such as a simple pendulum in a Newtonian fluid and electrical circuit containing a new practical element namely fractors. We obtain the index of introduced systems and discuss the solvability of these systems. We numerically solve the FDAEs of a pendulum in a fluid with three different fractional derivatives (Liouville–Caputo's definition, Caputo–Fabrizio's definition and with a definition with Mittag–Leffler kernel) and compare the effect of different fractional derivatives in this modeling. Finally, we solved some existing examples in research and showed the effectiveness and efficiency of the proposed numerical method. [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]