Theory on new fractional operators using normalization and probability tools

TL;DR

This paper introduces a new class of fractional operators with normalization and probability tools, addressing deficiencies of traditional operators, and applies them to solve linear and nonlinear equations, including epidemic models.

Contribution

It develops a novel framework for fractional calculus using normalization and probability, including inverse operators and fundamental theorems, expanding the theoretical foundation.

Findings

Constructed inverse operators for the new fractional class

Proved a fundamental theorem of calculus for these operators

Applied methods to solve the SIR epidemic model

Abstract

We show how a rescaling of fractional operators with bounded kernels may help circumvent their documented deficiencies, for example, the inconsistency at zero or the lack of inverse integral operator. On the other hand, we build a novel class of linear operators with memory effects to extend the L-fractional and the ordinary derivatives, using probability tools. A Mittag-Leffler-type function is introduced to solve linear problems, and nonlinear equations are addressed with power series, illustrating the methods for the SIR epidemic model. The inverse operator is constructed, and a fundamental theorem of calculus and an existence-and-uniqueness result for differintegral equations are proved. A conjecture on deconvolution is raised, that would permit completing the proposed theory.