The noncommutative fractional Fourier law in bounded and unbounded domains

TL;DR

This paper develops a framework for defining and analyzing fractional powers of noncommuting vector operators related to nonhomogeneous materials, using spectral theory on the $S$-spectrum in bounded and unbounded domains.

Contribution

It introduces a novel approach to fractional diffusion problems involving noncommuting operators with variable coefficients using quaternionic spectral theory.

Findings

Established the generation of fractional powers of noncommuting vector operators.

Extended the spectral theory to handle nonconstant coefficient operators in bounded and unbounded domains.

Provided boundary condition formulations for fractional powers in different domain types.

Abstract

Using the spectral theory on the $S$-spectrum it is possible to define the fractional powers of a large class of vector operators. This possibility leads to new fractional diffusion and evolution problems that are of particular interest for nonhomogeneous materials where the Fourier law is not simply the negative gradient operator but it is a nonconstant coefficients differential operator of the form $$ T=\sum_{\ell=1}^3e_\ell a_\ell(x)\partial_{x_\ell}, \ \ \ x=(x_1,x_2,x_3)\in \bar{\Omega}, $$ where, $\Omega$ can be either a bounded or an unbounded domain in $\mathbb{R}^3$ whose boundary $\partial\Omega$ is considered suitably regular, $\bar{\Omega}$ is the closure of $\Omega$ and $e_\ell$, for $\ell=1,2,3$ are the imaginary units of the quaternions $\mathbb{H}$. The operators $T_\ell:=a_\ell(x)\partial_{x_\ell}$, for $\ell=1,2,3$, are called the components of $T$ and $a_1$, $a_2$, $a_3: \bar{\Omega} \subset\mathbb{R}^3\to \mathbb{R}$ are the coefficients of $T$. In this paper we study the generation of the fractional powers of $T$, denoted by $P_{\alpha}(T)$ for $\alpha\in(0,1)$, when the operators $T_\ell$, for $\ell=1,2,3$ do not commute among themselves. To define the fractional powers $P_{\alpha}(T)$ of $T$ we have to consider the weak formulation of a suitable boundary value problem associated with the pseudo $S$-resolvent operator of $T$. In this paper we consider two different boundary conditions. If $\Omega$ is unbounded we consider Dirichlet boundary conditions. If $\Omega$ is bounded we consider the natural Robin-type boundary conditions associated with the generation of the fractional powers of $T$.