Solving variable-order fractional differential algebraic equations via generalized fuzzy hyperbolic model with applicati

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Solving variable-order fractional differential algebraic equations via generalized fuzzy hyperbolic model with application in electric circuit modeling Marzieh Mortezaee1 · Mehdi Ghovatmand1 · Alireza Nazemi1

© Springer-Verlag GmbH Germany, part of Springer Nature 2020

Abstract In this paper, a new approach based on a generalized fuzzy hyperbolic model is used for the numerical solution of variableorder fractional differential algebraic equations. The fractional derivative is described in the Atangana–Baleanu sense that is a new derivative with fractional order based on the generalized Mittag–Leffler function. First, by using fuzzy solutions with adjustable parameters, the variable-order fractional differential algebraic equations are reduced to a problem consisting of solving a system of algebraic equations. For adjusting the parameters of fuzzy solutions, an unconstrained optimization problem is then considered. A learning algorithm is also presented for solving the unconstrained optimization problem. Finally, some numerical examples are given to verify the efficiency and accuracy of the proposed approach. Keywords Variable-order fractional differential algebraic equations · Fuzzy systems · Generalized fuzzy hyperbolic model · Atangana–Baleanu derivative · Convergence

1 Introduction In the last few decades, non-integer-order derivative or integral of the variable function, or fractional calculus, has been proved that is very useful in various fields such as physics, chemistry, biology, economics, engineering, signal and image processing, and control theory (see Sun et al. 2018). Due to the ability of fractional differential equations (FDEs) to provide an exact description of different nonlinear phenomena, inherent relation to various materials and processes with memory and hereditary properties and greater degrees of freedom, fractional-order modeling of many real phenomena has more advantages and consistency rather than classical integer-order mathematical modeling (see Pham Communicated by V. Loia.


Mehdi Ghovatmand [email protected] Marzieh Mortezaee [email protected] Alireza Nazemi [email protected]


Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran

et al. 2018). Fractional differentiation may be introduced in several different operators. A history of the development of fractional differential operators has been presented in Miller and Ross (1993). During the last years, researchers have introduced fractional operators that consider the order as a function of time, space or some other variables. Variable-order fractional operators describe accurately real-world problems. One can find some applications of variable-order fractional differential equations (V-OFDEs) in Solís-Pérez et al. (2018), Atangana and Alqahtani (2016), Gómez-Aguilar (2018), Ghanbari and Gómez-Aguilar (2018), Coronel-Escamilla et al. (2018). Since it is difficult to find exact solutions of fractional differential equations with constant-order or var