System of fractional differential algebraic equations with applications

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.