Trivariate Mittag-Leffler functions used to solve multi-order systems of fractional differential equations

Linear systems of fractional differential equations have been studied from various points of view: applications to electric circuit theory, approximate solutions by numerical methods, and recently exact solutions by analytical methods. We discover here that, to obtain a fully closed-form solution in all cases, it is necessary to introduce a new type of Mittag-Leffler function involving triple series, and also to construct the associated fractional calculus operators, which we introduce and study in this paper. We then complete the rigorous analytical solutions for the aforesaid systems of fractional differential equations. As a consequence, comparing the solutions found here with the vector-matrix solutions known in the literature, we obtain explicit formulae for the elements of the 2 ? 2 matrix Mittag-Leffler function. ? 2021 Elsevier B.V. All rights reserved.

Automated summary

This study on linear systems of fractional differential equations reveals the need to introduce a new type of Mittag-Leffler function involving triple series and construct associated fractional calculus operators to obtain fully closed-form solutions. By comparing the solutions with existing vector-matrix solutions, explicit formulae for the elements of the 2×2 matrix Mittag-Leffler function are derived.