General Fractional Calculus, Evolution Equations, and Renewal Processes

TL;DR

This paper introduces a new fractional calculus framework based on complete Bernstein functions, analyzing relaxation and diffusion equations with operators involving a kernel function, and relates solutions to renewal processes.

Contribution

It develops a novel fractional calculus theory linked to complete Bernstein functions and applies it to relaxation equations and renewal process descriptions.

Findings

Solutions are continuous and completely monotone under certain conditions.

The framework connects fractional operators to renewal processes involving inverse subordinators.

Provides a new mathematical foundation for relaxation and diffusion equations.

Abstract

We develop a kind of fractional calculus and theory of relaxation and diffusion equations associated with operators in the time variable, of the form $(Du)(t)=\frac{d}{dt}\int\limits_0^tk(t-\tau)u(\tau)\,d\tau -k(t)u(0)$ where $k$ is a nonnegative locally integrable function. Our results are based on the theory of complete Bernstein functions. The solution of the Cauchy problem for the relaxation equation $Du=-\lambda u$, $\lambda >0$, proved to be (under some conditions upon $k$) continuous on $[(0,\infty)$ and completely monotone, appears in the description by Meerschaert, Nane, and Vellaisamy of the process $N(E(t))$ as a renewal process. Here $N(t)$ is the Poisson process of intensity $\lambda$, $E(t)$ is an inverse subordinator.