The Noncommutative Fractional Fourier Law in Bounded and Unbounded Domains

Abstract Using the spectral theory on S -spectrum it is possible to define fractional powers of a large class vector operators. This possibility leads new diffusion and evolution problems that are particular interest for nonhomogeneous materials where Fourier law not simply negative gradient operator but nonconstant coefficients differential form $$\begin{aligned} T=\sum _{\ell =1}^3e_\ell a_\ell (x)\partial _{x_\ell }, \ x=(x_1,x_2,x_3)\in \overline{\Omega \end{aligned}$$ T = ? ? 1 3 e a ( x ) ? , 2 ? ? ¯ where, $$\Omega $$ can be either bounded or an unbounded domain in $$\mathbb {R}^3$$ R whose boundary $$\partial \Omega considered suitably regular, $$\overline{\Omega }$$ closure $$e_\ell , $$\ell =1,2,3$$ imaginary units quaternions {H}$$ H . The operators $$T_\ell :=a_\ell : called components T $$a_1$$ $$a_2$$ $$a_3: } \subset \mathbb {R}^3\rightarrow {R}$$ ? ? In this paper we study generation denoted by $$P_{\alpha }(T)$$ P ? $$\alpha \in (0,1)$$ 0 when do commute among themselves. To have consider weak formulation suitable value problem associated with pseudo -resolvent two different conditions. If Dirichlet natural Robin-type conditions represented $$\sum =1}^3a_\ell ^2(x)n_\ell (x) \partial }+a(x)I$$ n + I $$x\in I identity operator, $$a:\partial \rightarrow given function $$n=(n_1,n_2,n_3)$$ outward unit normal are, general, from Robin heat which type \sum (x)n_\ell }+b(x)I$$ b . For reason also discuss coefficient $$b:\partial so compatible physical equations.