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

Abstract

The paper discusses the fractional operators

$$\begin{aligned} \nabla ^\alpha , \quad {\mathrm{div}}^\alpha , \quad (-\,\Delta )^{\alpha /2}, \end{aligned}$$

where $\alpha$ is a real number, the order of the operator. A frequently encountered definition of the fractional gradient uses an orthogonal basis $e_1,\dots ,e_n$ in the physical space V and one-dimensional “partial” fractional derivatives $D^\alpha _{\xi _i}\,f$ of a function f to lay down the formula

$$\begin{aligned} \nabla ^\alpha f(x) = D^\alpha _{\xi _1}\,f(x)e_1 + \dots + D^\alpha _{\xi _n}\,f(x)e_n. \end{aligned}$$

It will be shown that this definition is wrong: unlike the classical case$\alpha =1,$it depends on the chosen basis, i.e.,$\nabla ^\alpha f$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 $\alpha \in \mathbf{R}$ 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 $(-\,\Delta )^{\alpha /2}$ we recover the standard formulas for the fractional laplacean. For the fractional gradient, the requirements lead to

$$\begin{aligned} \nabla ^\alpha \,f(x) = \left\{ {\begin{array}{ll} \mu _\alpha \lim \limits _{\epsilon \downarrow 0} \int _{|h|\ge \epsilon } \frac{hf(x+h)}{|h|^{n+\alpha +1}} \mathrm{d}h &{}\quad \hbox {if} \quad 0\le \alpha<1,\\ \nabla f(x) &{}\quad \hbox {if} \quad \alpha =1,\\ \mu _\alpha \int _{\mathbf{R}^n} \frac{h\big ( f(x+h)-f(x)-\nabla f(x)\cdot h \big )}{|h|^{n+\alpha +1}} \mathrm{d}h &{}\quad \hbox {if} \quad 1<\alpha \le 2, \end{array}}\right. \end{aligned}$$

$x\in \mathbf{R}^n,$ where $\mu _\alpha$ is a normalization factor to be determined below from extra additional requirements. (The general case $-\,\infty<\alpha <\infty$ is treated in Sect. 4.) The paper then proceeds to prove some basic properties of the fractional operators, such as, e.g., the identity

$$\begin{aligned} {\mathrm{div}}^\alpha (\nabla ^\beta f) = -\, (-\,\Delta )^{(\alpha +\beta )/2}f , \end{aligned}$$

which generalizes the classical case ${\mathrm{div}}(\nabla f) = \Delta f$.