Let e ℓ , for ℓ = 1,2,3, be orthogonal unit vectors in R 3 and let Ω ⊂ R 3 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.

Colombo, F., Mongodi, S., Peloso, M., Pinton, S. (2019). Fractional powers of the noncommutative Fourier's law by the S-spectrum approach. MATHEMATICAL METHODS IN THE APPLIED SCIENCES, 42(5), 1662-1686 [10.1002/mma.5466].

### Fractional powers of the noncommutative Fourier's law by the S-spectrum approach

#### Abstract

Let e ℓ , for ℓ = 1,2,3, be orthogonal unit vectors in R 3 and let Ω ⊂ R 3 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.
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Articolo in rivista - Articolo scientifico
fractional diffusion processes; fractional Fourier's law; fractional powers of vector operators; S-spectrum; the S-spectrum approach;
English
2019
42
5
1662
1686
reserved
Colombo, F., Mongodi, S., Peloso, M., Pinton, S. (2019). Fractional powers of the noncommutative Fourier's law by the S-spectrum approach. MATHEMATICAL METHODS IN THE APPLIED SCIENCES, 42(5), 1662-1686 [10.1002/mma.5466].
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/10281/349856`