Theory on new fractional operators using normalization and probability tools

Marc Jornet

Published 2024-03-10Version 1

Recently, a theory on L-fractional differential equations was developed in the work [arXiv:2403.00341]. These are based on normalizing the Caputo operator, so that alternative properties hold, such as smoothness of the solution, finite derivative at the initial instant, and units time$^{-1}$ of the system. The present paper follows that line of research. On the one hand, 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 solved with power series (as an analogue of the Cauchy-Kovalevskaya theorem), illustrating the methods for the SIR model. The inverse operator is constructed, and a fundamental theorem of calculus and an existence-and-uniqueness result for differential equations are proved. A conjecture on deconvolution is raised, that would permit completing the proposed theory.