Twist of fractional oscillations

TL;DR

This paper investigates fractional oscillations derived from harmonic vibrations through Laplace transform, revealing their unique damped oscillatory behavior with algebraic decay and deriving their governing fractional differential equation.

Contribution

It introduces a novel approach to fractional oscillations via Laplace transform and derives their fractional differential equation, highlighting differences from classical sine and cosine functions.

Findings

Fractional oscillations exhibit finite damped oscillations with algebraic decay.

A fractional differential equation governing these oscillations is derived.

The method involves time-clock randomization of harmonic vibrations.

Abstract

Using the method of the Laplace transform, we consider fractional oscillations. They are obtained by the time-clock randomization of ordinary harmonic vibrations. In contrast to sine and cosine, the functions describing the fractional oscillations exhibit a finite number of damped oscillations with an algebraic decay. Their fractional differential equation is derived.