Skip to main content
Log in

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

  • Original Article
  • Published:
Continuum Mechanics and Thermodynamics Aims and scope Submit manuscript


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

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. Aksoy, H.G.: Wave propagation in heterogeneous media with local and nonlocal material behavior. J. Elast. 122, 1–25 (2016)

    MathSciNet  MATH  Google Scholar 

  2. Ben Adda, F.: Geometric interpretation of the fractional derivative. J. Fract. Calcul. 11, 21–52 (1997)

    MathSciNet  MATH  Google Scholar 

  3. Ben Adda, F.: Geometric interpretation of the differentiability and gradient of real order. Comptes Rendus de l’Academie des Sciences. Sciences I: Mathematics 1326, 931–934 (1998)

    MathSciNet  MATH  Google Scholar 

  4. Ben Adda, F.: The differentiability in the fractional calculus. Comptes Rendus de l’Academie des Sciences. Sciences I: Mathematics 326, 787–790 (1998)

    ADS  MathSciNet  MATH  Google Scholar 

  5. Ben Adda, F.: The differentiability in fractional calculus. Nonlinear Anal. 47, 5423–5428 (2001)

    MathSciNet  MATH  Google Scholar 

  6. Bucur, C., Valdinoci, E.: Nonlocal Diffusion and Applications. Springer, Cham (2016)

    MATH  Google Scholar 

  7. Caffarelli, L., Vázquez, J.L.: Nonlinear porous medium flow with fractional potential pressure. Arch. Ration. Mech. Anal. 202, 537–565 (2011)

    MathSciNet  MATH  Google Scholar 

  8. D’Ovidio, M., Garra, R.: Multidimensional fractional advection-dispersion equations and related stochastic processes. Electron. J. Probab. 19, 1–31 (2014)

    MathSciNet  MATH  Google Scholar 

  9. Drapaca, C.S., Sivaloganathan, S.: A fractional model of continuum mechanics. J. Elast. 107, 105–123 (2012)

    MathSciNet  MATH  Google Scholar 

  10. Engheta, N.: Fractional curl operator in electromagnetics. Microwave Opt. Technol. Lett. 17, 86–91 (1998)

    Google Scholar 

  11. Estrada, R., Kanwal, R.P.: Asymptotic Analysis. Birkhäuser, Boston (1994)

    MATH  Google Scholar 

  12. Gel’fand, I.M., Shapiro, Z.Y.: Homogeneous functions and their applications (in Russian). Uspekhi Mat. Nauk 10, 3–70 (1955)

    Google Scholar 

  13. Gel’fand, I.M., Shilov, G.E.: Generalized Functions I. Properties and Operations. Academic Press, New York (1964)

    MATH  Google Scholar 

  14. Gel’fand, I.M., Shilov, G.E.: Generalized Functions II. Spaces of Fundamental and Generalized Functions. Academic Press, New York (1968)

    MATH  Google Scholar 

  15. Hörmander, L.: The Analysis of Partial Differential Operators I. Distribution Theory and Fourier Analysis, 2nd edn. Springer, Berlin (1990)

    MATH  Google Scholar 

  16. Horváth, J.: On some composition formulas. Proc. Am. Math. Soc. 10, 433–437 (1959)

    MathSciNet  MATH  Google Scholar 

  17. Horváth, J.: Composition of hypersingular integral operators. Appl. Anal. 7, 171–190 (1978)

    MathSciNet  MATH  Google Scholar 

  18. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  19. Kwaśnicki, M.: Ten equivalent definitions of the fractional Laplace operator. Frac. Calc. Appl. Anal. 20, 7–51 (2017)

    MathSciNet  MATH  Google Scholar 

  20. Landkof, N.S.: Foundations of Modern Potential Theory. Springer, Berlin (1972)

    MATH  Google Scholar 

  21. Lemoine, C.: Fourier transforms of homogeneous distribution. Ann. Scuola Normale Superiore di Pisa, Classe di Scienze 3e série 26, 117–149 (1972)

    MathSciNet  MATH  Google Scholar 

  22. Martínez, C., Sanz, M.: The Theory of Fractional Powers of Operators. Elsevier, Amsterdam (2001)

    MATH  Google Scholar 

  23. Martínez, C., Sanz, M., Periago, F.: Distributional fractional powers of the Laplacean. Riesz potentials. Stud. Math. 135, 253–271 (1999)

    MathSciNet  MATH  Google Scholar 

  24. Meerschaert, M.M., Benson, D.A., Baeumer, B.: Multidimensional advection and fractional dispersion. Phys. Rev. E 59, 5026–5028 (1999)

    ADS  Google Scholar 

  25. Meerschaert, M.M., Mortensen, J., Scheffler, H.P.: Vector Grünwald formula for fractional derivatives. Fract. Calc. Appl. Anal. 7, 61–81 (2004)

    MathSciNet  MATH  Google Scholar 

  26. Meerschaert, M.M., Mortensen, J., Wheatcraft, S.W.: Fractional vector calculus for fractional advection–dispersion. Phys. A 367, 181–190 (2006)

    Google Scholar 

  27. Miller, K., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  28. Oldham, K.B., Spanier, J.: The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press, New York (1974)

    MATH  Google Scholar 

  29. Ortigueira, M.D., Laleg-Kirati, T.-M., Tenreiro Machado, J.A.: Riesz potential versus fractional Laplacian. J. Stat. Mech. Theory Exp. 2014, P09032 (2014)

    MathSciNet  Google Scholar 

  30. Ortigueira, M.D., Rivero, M., Trujillo, J.J.: From a generalised Helmholtz decomposition theorem to fractional Maxwell equations. Commun. Nonlinear Sci. Numer. Simul. 22, 1036–1049 (2015)

    ADS  MathSciNet  MATH  Google Scholar 

  31. Ortner, N.: On some contributions of John Horváth to the theory of distributions. J. Math. Anal. Appl. 297, 353–383 (2004)

    MathSciNet  MATH  Google Scholar 

  32. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  33. Pozrikidis, C.: The Fractional Laplacian. CRC Press, Boca Raton (2016)

    MATH  Google Scholar 

  34. Riesz, M.: L’intégrale de Riemann-Liouville et le probleme de Cauchy pour l’équation des ondes. BulL Soc. math. France. 67, 153–170 (1939)

    MathSciNet  MATH  Google Scholar 

  35. Rudin, W.: Functional Analysis. McGraw-Hill, New York (1973)

    MATH  Google Scholar 

  36. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Gordon and Breach Science Publishers, Amsterdam (1993)

    MATH  Google Scholar 

  37. Schikorra, A., Shieh, T.-T., Spector, D.: \(L^p\)-theory for fractional gradient PDE with VMO coefficients. Rendiconti della Accademia dei Lincei 26, 433–443 (2015)

    MathSciNet  MATH  Google Scholar 

  38. Schwartz, L.: Théorie des Distributions. Herrman, Paris (1966)

    MATH  Google Scholar 

  39. Shieh, T.-T., Spector, D.E.: On a new class of fractional partial differential equations. Adv. Calc. Var. 8, 321–336 (2015)

    MathSciNet  MATH  Google Scholar 

  40. Shieh, T.-T., Spector, D.E.: On a new class of fractional partial differential equations II. Adv. Calc. Var. 11, 289–307 (2018)

    MathSciNet  MATH  Google Scholar 

  41. Tarasov, V.E.: Fractional generalization of gradient systems. Lett. Math. Phys. 73, 49–58 (2005)

    ADS  MathSciNet  MATH  Google Scholar 

  42. Tarasov, V.E.: Fractional vector calculus and fractional Maxwell’s equations. Ann. Phys. 323, 2756–2778 (2008)

    ADS  MathSciNet  MATH  Google Scholar 

  43. Tarasov, V.E.: Fractional Dynamics. Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, Heidelberg (2010)

    MATH  Google Scholar 

Download references


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.

Author information

Authors and Affiliations


Corresponding author

Correspondence to M. Šilhavý.

Additional information

Communicated by Andreas Öchsner.

In memory of Walter Noll.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Šilhavý, M. Fractional vector analysis based on invariance requirements (critique of coordinate approaches). Continuum Mech. Thermodyn. 32, 207–228 (2020).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: