Hilfer derivatives based on the multivariate Mittag-Leffler functions and applications

Abstract

In this study, we propose a generalized definition for the Hilfer-Prabhakar derivative, namely multivariate Mittag-Leffler Hilfer derivative. This new formulation extends the existing framework by incorporating additional parameters and functions, enhancing flexibility and applicability for modelling complex systems. We explore the mathematical properties of the multivariate Mittag-Leffler Hilfer derivative, including its connection to the Hilfer-Prabhakar derivative and its behaviour under Laplace transforms. Furthermore, we demonstrate the applicability of this operator in solving classical equations in mathematical physics, such as the time-fractional heat equation, the free electron laser equation and the fractional wave equation. Additionally, we investigate numerical approximations for both the multivariate Mittag-Leffler Hilfer and Caputo-type integro-differential operators.