A New Representation for the Solutions of Fractional Differential Equations with Variable Coefficients

A recent development in differential equations with variable coefficients by means of fractional operators has been a method for obtaining an exact solution by infinite series involving nested fractional integral operators. This solution representation is constructive but difficult to calculate in practice. Here, we show a new representation of the solution function, as a convergent series of single fractional integrals, which is computationally simpler and which we believe will quickly prove its usefulness in future computational work for applications. In particular, for constant coefficients, the solution is given by the Mittag-Leffler function. We also show some applications in Cauchy problems involving both time-fractional and space-fractional operators and with time-dependent coefficients.