Analytical solutions of incommensurate fractional differential equation systems with fractional order $ 1 < \alpha, \beta < 2 $ via bivariate Mittag-Leffler functions

Abstract

<abstract><p>In this paper, we derive the explicit analytical solution of incommensurate fractional differential equation systems with fractional order $ 1 < \alpha, \beta < 2 $. The derivation is extended from a recently published paper by Huseynov et al. in <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>, which is limited for incommensurate fractional order $ 0 < \alpha, \beta < 1 $. The incommensurate fractional differential equation systems were first converted to Volterra integral equations. Then, the Mittag-Leffler function and Picard's successive approximations were used to obtain the analytical solution of incommensurate fractional order systems with $ 1 < \alpha, \beta < 2 $. The solution will be simplified via some combinatorial concepts and bivariate Mittag-Leffler function. Some special cases will be discussed, while some examples will be given at the end of this paper.</p></abstract>

Application of ARA-Residual Power Series Method in Solving Systems of Fractional Differential Equations

“…We mention that our results are obtained by using the Mathematica software package 12. As future work, we plan to solve fractional integral equations and systems of fractional partial differential equations with different orders of α using the proposed method [48][49][50].…”