Home New Results from Old Investigation: A Note on Fractional M-Dimensional Differential Operators. The Fractional Laplacian
Article
Licensed
Unlicensed Requires Authentication

New Results from Old Investigation: A Note on Fractional M-Dimensional Differential Operators. The Fractional Laplacian

  • Humberto Prado EMAIL logo , Margarita Rivero , Juan J. Trujillo and M. Pilar Velasco
Published/Copyright: March 13, 2015

Abstract

The non local fractional Laplacian plays a relevant role when modeling the dynamics of many processes through complex media. From 1933 to 1949, within the framework of potential theory, the Hungarian mathematician Marcel Riesz discovered the well known Riesz potential operators, a generalization of the Riemann-Liouville fractional integral to dimension higher than one. The scope of this note is to highlight that in the above mentioned works, Riesz also gave the necessary tools to introduce several new definitions of the generalized coupled fractional Laplacian which can be applied to much wider domains of functions than those given in the literature, which are based in both the theory of fractional power of operators or in certain hyper-singular integrals. Moreover, we will introduce the corresponding fractional hyperbolic differential operator also called fractional Lorentzian Laplacian.

References

[1] D. Baleanu, J. Tenreiro Machado, A. C. J. Luo (Ed.), Fractional Dynamics and Control. Springer, New York (2011).10.1007/978-1-4614-0457-6Search in Google Scholar

[2] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates. Annal. l’Inst. Henri Poincare C 31, No 1 (2014), 23-53.Search in Google Scholar

[3] L. Caffarelli and A. Figalli, Regularity of solutions to the parabolic fractional obstacle problema. J. Reine Angew. Math. 680 (2013), 191-233.Search in Google Scholar

[4] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian. Comm. Part. Diff. Equat. 32, No 8 (2007), 1245-1260.Search in Google Scholar

[5] L. Caffarelli, F. Soria, J. L. Vázquez, Regularity of solutions of the fractional porous medium flow. J. Eur. Math. Soc. 15, No 5 (2013), 1701-1746.Search in Google Scholar

[6] L. Caffarelli and J. L. Vázquez, Nonlinear porous medium flow with fractional potential pressure. Arch. Rat. Mech. Anal. 202, No 2 (2011), 537-565.Search in Google Scholar

[7] Q. Du, M. D. Gunzburger, R. Lehoucq, and K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws. Math. Model. Meth. Appl. Sci. 23, No 3 (2013), 493-540.Search in Google Scholar

[8] W. Feller, On a generalization of Macel Riesz potentials and the semigroups, generated by them. Communications du seminaire mathematique de universite de Lund, Tome suppl. dédiéà M. Riesz 21 (1952), 72-81.Search in Google Scholar

[9] A. Le. Méhauté, J. Tenreiro Machado, J. C. Trigeassou, J. Sabatier (Ed.), Fractional Differentiation and its Applications. Ubooks Verlag, Neusäss (2005).Search in Google Scholar

[10] J. Tenreiro. Machado, V. Kiryakova, F. Mainardi, Recent history of fractional calculus. Comm. Nonlin. Sci. Num. Sim. 16, No 3 (2011), 1140-1153.Search in Google Scholar

[11] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Imperial College Press, London (2010).10.1142/p614Search in Google Scholar

[12] C. Martínez and M. A. Sanz, The Theory of Fractional Powers of Operators. Elsevier, Amsterdam (2001).Search in Google Scholar

[13] A. C. McBride, Fractional Calculus and Integral Transforms of Generalized Functions. Pitman Press, San Francisco (1979).Search in Google Scholar

[14] M. M. Meerschaert, J. Mortensen, S. W. Wheatcraft, Fractional vector calculus for fractional advection-dispersion. Physica A 367, No 15 (2006), 181-190.Search in Google Scholar

[15] R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep. 339, No 1 (2000), 1-77.Search in Google Scholar

[16] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006).Search in Google Scholar

[17] V. S. Kiryakova, Generalized Fractional Calculus and Applications. Longman & J. Wiley, Harlow & New York (1994).Search in Google Scholar

[18] M. Riesz, L’intégrale de Riemann-Liouville et le problème de Cauchy. Acta Math. 81 (1949), 1-223.Search in Google Scholar

[19] P. Rusev, I. Dimovski, V. Kiryakova (Eds.), Transform Methods & Special Functions, Varna’96 (Proc. 2nd International Workshop, with Special Session on FC and Open Problems in FC Round Table). Institute of Mathematics and Informatics (IMI - BAS), Sofia (1998).Search in Google Scholar

[20] J. Sabatier, O. P. Agrawal, J. Tenreiro Machado (Eds.), Advances in Factional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht (2007).10.1007/978-1-4020-6042-7Search in Google Scholar

[21] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon (1993).Search in Google Scholar

[22] S. Samko, Hypersingular Integrals and Their Applications. Taylor and Francis, New York (2002). 10.1201/9781482264968Search in Google Scholar

[23] L. Silvestre, Regularity of the obstacle problem for a fractional power of the laplace operator. Comm. Pure Appl. Math. 60, No 1 (2007), 67-112.Search in Google Scholar

[24] V. E. Tarasov, Fractional vector calculus and fractional Maxwell’s equations. Annals Phys. 323, No 11 (2008), 2756-2778.Search in Google Scholar

[25] V. E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, Heidelberg (2011).Search in Google Scholar

[26] V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers, Vol. I-II. Springer, publ. jointly with Higher Education Press, Bejing and Heidelberg (2013).10.1007/978-3-642-33911-0Search in Google Scholar

[27] D. Valerio, J. J. Trujillo, M. Rivero, J. Tenreiro Machado, and D. Baleanu, Fractional Calculus: A survey of useful formulas. Eur. Phys. J. St. 222, No 8 (2013), 1825-1844.Search in Google Scholar

[28] J. L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators. Discrete Contin. Dyn. Syst. (2014), To appear; ArXiv:1401.3640.Search in Google Scholar

[29] G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics. Oxford University Press, Oxford (2008). Search in Google Scholar

Received: 2014-7-31
Published Online: 2015-3-13
Published in Print: 2015-4-1

© 2015 Diogenes Co., Sofia

Downloaded on 16.11.2025 from https://www.degruyterbrill.com/document/doi/10.1515/fca-2015-0020/html
Scroll to top button