2020/10/09 by Fabrizio Colombo, Colombo, Fabrizio, Denis Deniz González +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Differential Equations and Boundary Problems #FOS: Mathematics #Functional Analysis (math.FA) #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2010.04688
openalex publication_date 2020/10/09 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
Using the spectral theory on the S-spectrum it is possible to define the\nfractional powers of a large class of vector operators. This possibility leads\nto new fractional diffusion and evolution problems that are of particular\ninterest for nonhomogeneous materials where the Fourier law is not simply the\nnegative gradient operator but it is a nonconstant coefficients differential\noperator of the form T=
sum
ell=13e_
ell a_
ell(x)
partialx_
ell,
\n
x=(x1,x2,x3)
in
bar
Omega, where, \Ω can be either a bounded\nor an unbounded domain in \ℝ3 whose boundary \∂\Ω is\nconsidered suitably regular, \\Ω is the closure of \Ω and\ne_\ℓ, for \ℓ=1,2,3 are the imaginary units of the quaternions\n\ℍ. The operators T_\ℓ:=a_\ℓ(x)\∂x_\ℓ, for\n\ℓ=1,2,3, are called the components of T and a1, a2, a3:\n\\Ω \⊂\ℝ3\→ \ℝ are the coefficients of T. In\nthis paper we study the generation of the fractional powers of T, denoted by\nP\α(T) for \α\∈(0,1), when the operators T_\ℓ, for\n\ℓ=1,2,3 do not commute among themselves. To define the fractional powers\nP\α(T) of T we have to consider the weak formulation of a suitable\nboundary value problem associated with the pseudo S-resolvent operator of\nT. In this paper we consider two different boundary conditions. If \Ω\nis unbounded we consider Dirichlet boundary conditions. If \Ω is bounded\nwe consider the natural Robin-type boundary conditions associated with the\ngeneration of the fractional powers of T.\n