On complete monotonicity of solution to the fractional relaxation equation with the $n$th level fractional derivative

TL;DR

This paper derives explicit formulas for the projector and Laplace transform of the nth level fractional derivative, analyzes the solutions of the fractional relaxation equation, and establishes their complete monotonicity under certain conditions, especially for the second level case.

Contribution

It introduces explicit formulas for the nth level fractional derivative projector and Laplace transform, and studies the complete monotonicity of solutions to the fractional relaxation equation.

Findings

Solutions are completely monotone functions under certain conditions.

Solutions can be expressed as linear combinations of Mittag-Leffler functions.

Special case analysis for the second level fractional derivative.

Abstract

In this paper, we first deduce the explicit formulas for the projector of the $n$th level fractional derivative and for its Laplace transform. Then the fractional relaxation equation with the $n$th level fractional derivative is discussed. It turns out that under some conditions, the solutions to the initial-value problems for this equation are completely monotone functions that can be represented in form of the linear combinations of the Mittag-Leffler functions with some power law weights. Special attention is given to the case of the relaxation equation with the 2nd level derivative.