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Fractional powers of the noncommutative Fourier's law by theS‐spectrum approach
- Source :
- Mathematical Methods in the Applied Sciences. 42:1662-1686
- Publication Year :
- 2019
- Publisher :
- Wiley, 2019.
-
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.
- Subjects :
- S-spectrum
General Mathematics
fractional diffusion processe
General Engineering
fractional diffusion processes
Spectrum (topology)
Noncommutative geometry
fractional Fourier's law
symbols.namesake
Engineering (all)
Fourier transform
the S-spectrum approach
symbols
Mathematics (all)
fractional powers of vector operators
fractional powers of vector operator
Mathematical physics
Mathematics
Subjects
Details
- ISSN :
- 10991476 and 01704214
- Volume :
- 42
- Database :
- OpenAIRE
- Journal :
- Mathematical Methods in the Applied Sciences
- Accession number :
- edsair.doi.dedup.....f244615bb927699dbfcf727374bed6d3