Let eℓ, for ℓ = 1,2,3, be orthogonal unit vectors in (Formula presented.) and let (Formula presented.) be a bounded open set with smooth boundary ∂Ω. Denoting by (Formula presented.) a point in Ω, the heat equation, for nonhomogeneous materials, is obtained replacing the Fourier law, given by the following: (Formula presented.) into the conservation of energy law, here a, b, (Formula presented.) 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, (Formula presented.) 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.

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

Colombo, Fabrizio;Mongodi, Samuele;Pinton, Stefano
2019-01-01

Abstract

Let eℓ, for ℓ = 1,2,3, be orthogonal unit vectors in (Formula presented.) and let (Formula presented.) be a bounded open set with smooth boundary ∂Ω. Denoting by (Formula presented.) a point in Ω, the heat equation, for nonhomogeneous materials, is obtained replacing the Fourier law, given by the following: (Formula presented.) into the conservation of energy law, here a, b, (Formula presented.) 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, (Formula presented.) 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.
2019
fractional diffusion processes; fractional Fourier's law; fractional powers of vector operators; S-spectrum; the S-spectrum approach; Mathematics (all); Engineering (all)
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11311/1078528
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