# 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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## References

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

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

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

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)

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

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

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

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)

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

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

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)

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

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

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

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

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

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

## 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.

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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• DOI: https://doi.org/10.1007/s00161-019-00797-9

### Keywords

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