System of fractional differential algebraic equations with applications

By Babak Shiri, Dumitru Băleanu. Chaos, Solitons & Fractals, 2019

115 citations recorded by the source corpus.

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.

Subfields: Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/c715baa7d3b0270f5de7f03325644e73d2ce7e99

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Citations in the corpus

Spline collocation methods for seismic analysis of multiple degree of freedom systems with visco-elastic dampers using fractional models (2020, methodology): “One of the classical methods for solving integral or differential equations is the collocation method on piecewise polynomial spaces which is known as the spline collocation method (Baleanu and Shiri, 2018; Brunner, 2004; Shiri, 2014; Shiri and Baleanu, 2019).”
Integral inequalities for a fractional operator of a function with respect to another function with nonsingular kernel (2020, background): “After Leibniz, many famous mathematicians and physicians have studied theory and application of this model for fractional derivatives and integrals and fractional differential equations; see for detail [12, 13, 35, 55].”