Prabhakar-type linear differential equations with variable coefficients

TL;DR

This paper solves linear differential equations with variable coefficients using Prabhakar-type operators with Mittag-Leffler kernels, providing explicit solutions and extending results to operators with respect to functions, including special cases with closed-form solutions.

Contribution

It introduces explicit solutions for Prabhakar-type differential equations with variable coefficients and extends these solutions to operators with respect to functions, including closed-form solutions for constant coefficients.

Findings

Explicit solutions constructed as convergent series involving Prabhakar fractional integrals

Extension of solutions to operators with respect to functions

Closed-form solutions for constant coefficient cases using multivariate Mittag-Leffler functions

Abstract

Linear differential equations with variable coefficients and Prabhakar-type operators featuring Mittag-Leffler kernels are solved. In each case, the unique solution is constructed explicitly as a convergent infinite series involving compositions of Prabhakar fractional integrals. We also extend these results to Prabhakar operators with respect to functions. As an important illustrative example, we consider the case of constant coefficients, and give the solutions in a more closed form by using multivariate Mittag-Leffler functions.