Fractional boundary value problems

TL;DR

This paper investigates functionals related to fractional boundary value problems involving Caputo derivatives, linking them to fractional telegraph equations and unifying various boundary conditions like elastic and sticky cases.

Contribution

It introduces a novel analysis of additive functionals associated with FBVPs and connects them to fractional telegraph equations, providing a unified framework for different boundary conditions.

Findings

Additive functionals are related to fractional telegraph equations.

Fractional order derivatives unify elastic and sticky boundary cases.

The study advances understanding of boundary conditions in fractional processes.

Abstract

We study some functionals associated with a process driven by a fractional boundary value problem (FBVP for short). By FBVP we mean a Cauchy problem with boundary condition written in terms of a fractional equation, that is an equation involving time-fractional derivative in the sense of Caputo. We focus on lifetimes and additive functionals characterizing the boundary conditions. We show that the corresponding additive functionals are related to the fractional telegraph equations. Moreover, the fractional order of the derivative gives a unified condition including the elastic and the sticky cases among the others.