Explicit solutions for linear variable-coefficient fractional differential equations with respect to functions

TL;DR

This paper derives explicit solutions for complex fractional differential equations with variable coefficients, expressed through convergent series and multivariate Mittag-Leffler functions, extending existing formulas to broader classes.

Contribution

It provides a general method for explicit solutions of variable-coefficient fractional differential equations with respect to functions, including complex derivatives, extending prior formulas.

Findings

Solutions expressed as convergent infinite series of fractional operators

Explicit solutions in terms of multivariate Mittag-Leffler functions for constant coefficients

Extension of Luchko-Gorenflo formula to variable coefficients and complex derivatives

Abstract

Explicit solutions of differential equations of complex fractional orders with respect to functions and with continuous variable coefficients are established. The representations of solutions are given in terms of some convergent infinite series of fractional integro-differential operators, which can be widely and efficiently used for analytic and computational purposes. In the case of constant coefficients, the solution can be expressed in terms of the multivariate Mittag-Leffler functions. In particular, the obtained result extends the Luchko-Gorenflo representation formula to a general class of linear fractional differential equations with variable coefficients, to complex fractional derivatives, and to fractional derivatives with respect to a given function.