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

The paper discusses the fractional operators Delta.a, diva, (- )a/2, where a is a real number, the order of the operator. A frequently encountered definition of the fractional gradient uses an orthogonal basis e1,..., en in the physical space V and one-dimensional "partial" fractional derivatives Da.i f of a function f to lay down the formula Delta a f (x) = Da.1 f (x)e1 + center dot center dot center dot + Da.n f ( x)en. will be shown that this definition is wrong: unlike the classical case a = 1, it depends on the chosen basis, i.e.,. a f does not transform as a vector under rotations. The same objection applies to similarly constructed fractional divergence and laplacean. The paper presents a novel approach to the operators of fractional vector analysis based on elementary requirements, viz., translational invariance, rotational invariance, homogeneity of degree a. R under isotropic scaling; certain weak requirement of continuity. Using methods of the theory of homogeneous distributions the paper proves that these requirements determine the fractional operators uniquely to within a multiplication by a scalar factor; derives explicit formulas for these operators. For (- )a/2 we recover the standard formulas for the fractional laplacean. For the fractional gradient, the requirements lead to Delta f (x) =............. mu a lim{.0 |h|= h f (x + h) |h|n+a+1 dh if 0 = a < 1,. f (x) if a = 1, mu a Rn h( f (x + h) - f (x)-. f ( x) center dot h) |h|n+a+1 dh if 1 < a = 2, x. Rn, where mu a is a normalization factor to be determined below from extra additional requirements. (The general case-8 < a < 8 is treated in Sect. 4.) The paper then proceeds to prove some basic properties of the fractional operators, such as, e.g., the identity diva (Delta beta f) = -(- )(a+ ss)/2 f, which generalizes the classical case div(. f) = Delta f.