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 \( {\mathrm{div}}(\nabla f) = \Delta f\).

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Acknowledgements

This research was supported by RVO 67985840. The author thanks Daniel Spector, Brian Seguin and Giovanni Comi for their remarks on the preceding versions of the paper.

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Correspondence to M. Šilhavý.

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In memory of Walter Noll.

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Šilhavý, M. Fractional vector analysis based on invariance requirements (critique of coordinate approaches). Continuum Mech. Thermodyn. 32, 207–228 (2020). https://doi.org/10.1007/s00161-019-00797-9

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Keywords

  • Fractional gradient, divergence, and laplacean
  • Translation invariance
  • Rotation invariance
  • Positive homogeneity of degree \(\alpha \)
  • Fractional vector identities
  • Distributions