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In the recent years, there has been a lot of interest in fractional diffusion and fractional evolution problems. The spectral theory on the S‐spectrum turned out to be an important tool to define new fractional diffusion operators stating from the Fourier law for nonhomogeneous materials. Precisely, let eℓ, eℓ=1,2,3 be orthogonal unit vectors in R3 and let Ω⊂R3 be a bounded open set with smooth boundary ∂Ω. Denoting by x_ a point in Ω, the heat equation is obtained replacing the Fourier law given by T=ax_∂xe1+bx_∂ye2+cx_∂ze3into the conservation of energy law. In this paper, we investigate the structure of the fractional powers of the vector operator T, with homogeneous Dirichlet boundary conditions. Recently, we have found sufficient conditions on the coefficients a, b, c:Ω→R such that the fractional powers of T exist in the sense of the S‐spectrum approach. In this paper, we show that under a different set of conditions on the coefficients a, b, c, the fractional powers of T have a different structure. [ABSTRACT FROM AUTHOR]

Let eℓ, for ℓ = 1,2,3, be orthogonal unit vectors in R3 and let Ω⊂R3 be a bounded open set with smooth boundary ∂Ω. Denoting by x_ a point in Ω, the heat equation, for nonhomogeneous materials, is obtained replacing the Fourier law, given by the following: T=a(x_)∂xe1+b(x_)∂ye2+c(x_)∂ze3,into the conservation of energy law, here a, b, c:Ω→R are given functions. With the S‐spectrum approach to fractional diffusion processes we determine, in a suitable way, the fractional powers of T. Then, roughly speaking, we replace the fractional powers of T into the conservation of energy law to obtain the fractional evolution equation. This method is important for nonhomogeneous materials where the Fourier law is not simply the negative gradient. In this paper, we determine under which conditions on the coefficients a, b, c:Ω→R the fractional powers of T exist in the sense of the S‐spectrum approach. More in general, this theory allows to compute the fractional powers of vector operators that arise in different fields of science and technology. This paper is devoted to researchers working in fractional diffusion and fractional evolution problems, partial differential equations, and noncommutative operator theory. [ABSTRACT FROM AUTHOR]

Recently the S -spectrum approach to fractional diffusion problems has been applied to vector operators with homogeneous Dirichlet boundary conditions. This method allows to determine the Fractional Fourier law under given boundary conditions. In this paper we consider first order linear boundary conditions and we study the case of vector operators with commuting components. Precisely, let Ω be an open bounded set in R 3 where its boundary ∂ Ω is considered suitably regular, we denote a point in Ω ¯ by x = (x 1 , x 2 , x 3) and a basis for the quaternions H will be indicated by e ℓ , for ℓ = 1 , 2 , 3. We prove that under suitable conditions on the coefficients a 1 , a 2 , a 3 : Ω ¯ ⊂ R 3 → R of the vector operator T = e 1 a 1 (x 1) ∂ x 1 + e 2 a 2 (x 2) ∂ x 2 + e 3 a 3 (x 3) ∂ x 3 , x ∈ Ω ¯ and on the coefficient a : ∂ Ω → R of the boundary operator B : B ≔ ∑ ℓ = 1 3 a ℓ 2 (x ℓ) n ℓ ∂ x ℓ + a (x) I , x ∈ ∂ Ω , where n = (n 1 , n 2 , n 3) is the outward unit normal vector to ∂ Ω , we can define the fractional powers T α , for α ∈ (0 , 1) , of T. In general the coefficients a 1 , a 2 , a 3 and a can depend on time. We omit the time dependence for the sake of simplicity but the proofs of our results can be easily extended to this more general setting considering the time as a parameter. [ABSTRACT FROM AUTHOR]