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An Application of the S-Functional Calculus to Fractional Diffusion Processes
- Source :
- Milan Journal of Mathematics. 86:225-303
- Publication Year :
- 2018
- Publisher :
- Springer Science and Business Media LLC, 2018.
-
Abstract
- In this paper we show how the spectral theory based on the notion of S-spectrum allows us to study new classes of fractional diffusion and of fractional evolution processes. We prove new results on the quaternionic version of the $${H^\infty}$$ functional calculus and we use it to define the fractional powers of vector operators. The Fourier law for the propagation of the heat in non homogeneous materials is a vector operator of the form $$T = e_{1} a(x)\partial_{x1} + e_{2} b(x)\partial_{x2} + e_{3} c(x)\partial_{x3}$$ where $${e_{\ell}, {\ell} = 1, 2, 3}$$ are orthogonal unit vectors, a, b, c are suitable real valued functions that depend on the space variables $${x = (x_{1}, x_{2}, x_{3})}$$ and possibly also on time. In this paper we develop a general theory to define the fractional powers of quaternionic operators which contain as a particular case the operator T so we can define the non local version $${T^{\alpha}, {\rm for} \alpha \in (0, 1)}$$ , of the Fourier law defined by T. Our new mathematical tools open the way to a large class of fractional evolution problems that can be defined and studied using our spectral theory based on the S-spectrum for vector operators. This paper is devoted to researchers working on fractional diffusion and fractional evolution problems, partial differential equations, non commutative operator theory, and quaternionic analysis.
- Subjects :
- Pure mathematics
Spectral theory
Vector operator
General Mathematics
01 natural sciences
Functional calculus
Mathematics - Spectral Theory
Operator (computer programming)
Unit vector
0103 physical sciences
FOS: Mathematics
Mathematics (all)
0101 mathematics
Spectral Theory (math.SP)
Commutative property
Mathematics
fractional diffusion and fractional evolution processes
S-spectrum
010102 general mathematics
Operator theory
Quaternionic analysis
Functional Analysis (math.FA)
Mathematics - Functional Analysis
H∞ functional calculus for quaternionic operators
010307 mathematical physics
fractional powers of vector operators
Subjects
Details
- ISSN :
- 14249294 and 14249286
- Volume :
- 86
- Database :
- OpenAIRE
- Journal :
- Milan Journal of Mathematics
- Accession number :
- edsair.doi.dedup.....6ac231682632dbcd700507cfead1abb5