A new Laplace-type fractional derivative

TL;DR

This paper introduces a novel Laplace-type fractional derivative that aligns with Riemann-Liouville derivatives, offering new tools for fractional differential equations and applications in regularization.

Contribution

It proposes a new fractional derivative derived via the Laplace transform, connecting it to Riemann-Liouville derivatives and exploring its properties and generalizations.

Findings

Provides a new derivative form compatible with periodic functions

Establishes basic properties of the new derivative

Suggests applications in fractional differential equations and regularization

Abstract

In this paper, we present a new derivative via the Laplace transform. The Laplace transform leads to a natural form of the fractional derivative which is equivalent to a Riemann-Liouville derivative with fixed terminal point. We first consider a representation which interacts well with periodic functions, examine some rudimentary properties and propose a generalization. The interest for this new approach arose from recent developments in fractional differential equations involving Caputo-type derivatives and applications in regularization problems.