SPECIAL ISSUE PAPER
Two-sided and regularised Riesz-Feller derivatives
Manuel Duarte Ortigueira,
Corresponding Author
Manuel Duarte Ortigueira
Centre of Technology and Systems-UNINOVA and Department of Electrical Engineering, NOVA School of Science and Technology of NOVA University of Lisbon, Lisbon, Portugal
Correspondence
Manuel Duarte Ortigueira, Centre of Technology and Systems-UNINOVA and Department of Electrical Engineering, NOVA School of Science and Technology of NOVA University of Lisbon, Lisbon, Portugal.
Email: [email protected]
Communicated by: D. Zeidan
Search for more papers by this authorManuel Duarte Ortigueira,
Corresponding Author
Manuel Duarte Ortigueira
Centre of Technology and Systems-UNINOVA and Department of Electrical Engineering, NOVA School of Science and Technology of NOVA University of Lisbon, Lisbon, Portugal
Correspondence
Manuel Duarte Ortigueira, Centre of Technology and Systems-UNINOVA and Department of Electrical Engineering, NOVA School of Science and Technology of NOVA University of Lisbon, Lisbon, Portugal.
Email: [email protected]
Communicated by: D. Zeidan
Search for more papers by this authorPresent Address: Campus da FCT, Quinta da Torre, 2829–516 Caparica, Portugal
Abstract
The two-sided derivatives, the Riesz-Feller potentials, and their interrelations are studied in this paper. A general integral formulation for two-sided derivatives and anti-derivatives is introduced. The integer order cases are studied. Regularised Riesz-Feller derivatives are proposed.
CONFLICT OF INTEREST
The author declare no potential conflict of interests.
REFERENCES
- 1Ortigueira MD, Machado JT. Fractional derivatives: the perspective of system theory. Mathematics. 2019; 7(2): 150.
- 2Riesz M. L'integral de Riemann-Liouville et le probleme de Cauchy. Acta Math. 1949; 81(1): 1-222.
- 3Kilbas AA, Srivastava HM, Trujillo JJ. Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204. Amsterdam: Elsevier; 2006.
10.1016/S0304-0208(06)80001-0 Google Scholar
- 4Samko SG, Kilbas AA, Marichev OI. Fractional Integrals and Derivatives: Theory and Applications. Amsterdam: Gordon and Breach Science Publishers; 1993.
- 5Feller W. On a generalization of Marcel Riesz potentials and the semigroups, generated by them. Communications du Seminaire Mathematique de Universite de Lund. Lund, Sweden: 1952: 72-81.
- 6Ortigueira MD, Machado JT. Which derivative? Fractal Fract. 2017; 1(1): 3.
- 7Pozrikidis C. The Fractional Laplacian. Boca Raton, USA: CRC Press, Taylor & Francis Group; 2016.
10.1201/b19666 Google Scholar
- 8Luchko Y, Gorenflo R. Scale-invariant solutions of a partial differential equation of fractional order. Fract Calc Appl Anal. 1998; 1(1): 63-78.
- 9Mainardi F, Gorenflo R. On Mittag-Leffler-type functions in fractional evolution processes. J Comput Appl Math. 2000; 118(1–2): 283-299.
- 10Metzler R, Nonnenmacher TF. Space- and time-fractional diffusion and wave equations, fractional fokker–planck equations, and physical motivation. Chem Phys. 2002; 284(1): 67-90.
- 11Mainardi F. Applications of integral transforms in fractional diffusion processes. Integral Transform Spec Funct. 2004; 15(6): 477-484.
- 12Baeumer B, Meerschaert MM, Nane E. Space–time duality for fractional diffusion. J Appl Probab. 2009; 46(4): 1100-1115.
- 13Laskin N. Fractional quantum mechanics and lévy path integrals. Phys Lett A. 2000; 268(4): 298-305.
- 14Laskin N. Fractional schrödinger equation. Phys Rev E. 2002; 66:056108.
- 15Naber M. Time fractional schrödinger equation. J Math Phys. 2004; 45(8): 3339-3352.
- 16Dong J, Xu M. Space–time fractional schrödinger equation with time-independent potentials. J Math Anal Appl. 2008; 344(2): 1005-1017.
- 17Al-Saqabi B, Boyadjiev L, Luchko Y. Comments on employing the riesz-feller derivative in the schrödinger equation. Eur Phys J Spec Top. 2013; 222(8): 1779-1794.
- 18Baqer S, Boyadjiev L. Fractional Schrödinger equation with zero and linear potentials. Fract Calc Appl Anal. 2016; 19(04): 973-988.
- 19Baqer S, Boyadjiev L. On the space-time fractional schrödinger equation with time independent potentials. In: CM da Fonseca, D Van Huynh, S Kirkland, VK Tuan, eds. A Panorama of Mathematics: Pure and Applied, Contemporary Mathematics, vol. 658. Safat, Kuwait: American Mathematical Society; 2016: 81-90.
10.1090/conm/658/13121 Google Scholar
- 20Laskin N. Time fractional quantum mechanics. Chaos, Solitons Fractals. 2017; 102: 16-28.
- 21Luchko Y. Anomalous Diffusion: Models, Their Analysis, and Interpretation. In: AA Rogosin Koroleva, ed. Advances in Applied Analysis. Basel: Birkhäuser; 2012: 115-145.
10.1007/978-3-0348-0417-2_3 Google Scholar
- 22Ortigueira MD. Riesz potential operators and inverses via fractional centred derivatives. Int J Math Math Sci. 2006; 2006(Article ID 48391): 12.
- 23Ortigueira MD. Fractional central differences and derivatives. J Vib Control. 2008; 14(9-10): 1255-1266.
- 24Ortigueira MD, Trujillo JJ. A unified approach to fractional derivatives. Commun Nonlinear Sci Numer Simul. 2012; 17(12): 5151-5157.
- 25Roberts MJ. Signals and Systems: Analysis Using Transform Methods and Matlab. New York: McGraw-Hill; 2003.
- 26Gel'fand IM, Shilov GE. Generalized Functions: Properties and Operations. New York: Academic Press; 1964.
- 27Mainardi F, Gorenflo R. Fractional calculus and special functions. Lect Notes Math Phys. 2000: 1-62. www.fracalmo.org
- 28Gorenflo R, Mainardi F. Some recent advances in theory and simulation of fractional diffusion processes. J Comput Appl Math. 2009; 229(2): 400-415. Special Issue: Analysis and Numerical Approximation of Singular Problems.
- 29Dong J, Xu M. Some solutions to the space fractional schrödinger equation using momentum representation method. J Math Phys. 2007; 48(7):072105.
- 30Bayin S. Definition of the Riesz derivative and its application to space fractional quantum mechanics. J Math Phys. 2016; 57(12): 123501.
- 31Liemert A, Kienle A. Fractional schrödinger equation in the presence of the linear potential. Mathematics. 2016; 4(2): 31.
- 32Ortigueira MD. Fractional Calculus for Scientists and Engineers, Lecture Notes in Electrical Engineering. Berlin Heidelberg: Springer; 2011.
10.1007/978-94-007-0747-4 Google Scholar
- 33Proakis JG, Manolakis DG. Digital Signal Processing: Principles, Algorithms, and Applications. New Jersey: Prentice–Hall; 1996.
- 34Ortigueira MD, Magin RL, Trujillo JJ, Velasco MP. A real regularised fractional derivative. Signal Image Video Process. 2012; 6(3): 351-358.
