Fractional vector analysis based on invariance requirements (critique of coordinate approaches)

A Note on Gradient/Fractional One-Dimensional Elasticity and Viscoelasticity

“…In addition to questions related to 3D equilibrium and compatibility requirements, which do not arise in a non-trivial manner in the present 1D case, one also needs to address the issue of frame indifference, i.e., "the invariance requirements under translations and rotations", as a reviewer indicated by pointing, for example, to reference [25]. For a more general discussion on fractional aspects and invariance considerations in continuum mechanics formulations, the reader can consult a most interesting recent reference [26]. Another pertinent issue is the question of identifying the length scale in relation to three-dimensional micro/nano structures in various materials, which still remains an open problem to collectively elaborate upon.…”

“…For a constant applied stress σ = σ 0 (the same as in Section 2), the integration of Equation (26) gives…”

“…For the remainder of this Section, we discuss a fractional extension of the classical Kelvin-Voigt model given by Equation (26). By replacing the integer order derivative ε = ∂ε/∂t with its time fractional counterpart, ∂ β ε/∂t β , where β is a non-integer number 0 < β < 1, we first derive the corresponding creep equation associated with the fractional index β, i.e., the fractional counterpart of Equation ( 27).…”

On an Anisotropic Fractional Stefan-Type Problem with Dirichlet Boundary Conditions

“…(1.2) with the operator L 1 A , where D 1 = D. The operator L s A can be viewed as an anisotopic generalisation of the fractional Laplacian. Indeed, following the works of Silhavy [47], Shieh-Spector [45]- [46] and Comi-Stefani [17]- [18], the Riesz fractional s-gradient (D s ) and the s-divergence (D s •) are defined in integral form for sufficiently regular functions u and vector fields φ, respectively, by . Then, in the distributional sense, it is well-known (see for instance, [45])…”

A note on Riemann-Liouville fractional Sobolev spaces

“…This is indeed the one dimensional case of the new notion of fractional gradient on the whole R, suitable also for a meaningful extension to R n , for all n ≥ 1, which has been investigated in some recent papers, as [13,14,31,32,33,34], with the aim of generalizing classical vector calculus rules to the fractional setting, and thus providing a way to define weakly fractionally differentiable functions.…”