A new representation for the solutions of fractional differential equations with variable coefficients

TL;DR

This paper introduces a simplified, convergent series representation for solutions of fractional differential equations with variable coefficients, enhancing computational feasibility and practical application in complex differential problems.

Contribution

It presents a new, more practical series representation of solutions, replacing the complex nested fractional integrals with a convergent series of single fractional integrals.

Findings

Series representation is easier to compute than previous methods.

Solutions for constant coefficients involve Mittag-Leffler functions.

Applications demonstrated in fractional PDEs with time-dependent coefficients.

Abstract

A recent development in the theory of fractional differential equations with variable coefficients has been a method for obtaining an exact solution in the form of an 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 will be easier to use for computational work and applications. In the particular case of constant coefficients, the solution is given in terms of the Mittag-Leffler function. We also show some applications in Cauchy problems for partial differential equations involving both time-fractional and space-fractional operators and with time-dependent coefficients.