On a connection between the discrete fractional Laplacian and superdiffusion. (English) Zbl 1381.35215
Summary: We relate the fractional powers of the discrete Laplacian with a standard time-fractional derivative in the sense of Liouville by encoding the iterative nature of the discrete operator through a time-fractional memory term.
MSC:
| 35R11 | Fractional partial differential equations |
Keywords:
discrete Laplacian; superdiffusion; heat semigroup; fractional Laplacian; modified Bessel functionsReferences:
| [1] | Ciaurri, Ó.; Gillespie, T. A.; Roncal, L.; Torrea, J. L.; Varona, J. L., Harmonic analysis associated with a discrete Laplacian, J. Anal. Math. (2015), in press. arXiv:1401.2091 |
| [2] | Kulish, V. V.; Lage, J. L., Application of fractional calculus to fluid mechanics, J. Fluids Eng., 124, 803-806 (2002) |
| [3] | Keyantuo, V.; Lizama, C., On a connection between powers of operators and fractional Cauchy problems, J. Evol. Equ., 12, 245-265 (2012) · Zbl 1336.47046 |
| [4] | Allouba, H., Time-fractional and memoryful \(\Delta^{2^k}\) SIEs on \(R_+ \times R^d\): how far can we push white noise?, Illinois J. Math., 57, 919-963 (2013) · Zbl 1323.35224 |
| [5] | Baeumer, B.; Meerschaert, M. M.; Nane, E., Brownian subordinators and fractional Cauchy problems, Trans. Amer. Math. Soc., 361, 3915-3930 (2009) · Zbl 1186.60079 |
| [6] | Stinga, P. R.; Torrea, J. L., Extension problem and Harnack’s inequality for some fractional operators, Comm. Partial Differential Equations, 35, 2092-2122 (2010) · Zbl 1209.26013 |
| [7] | Grünbaum, F. A.; Iliev, P., Heat kernel expansions on the integers, Math. Phys. Anal. Geom., 5, 183-200 (2002) · Zbl 0996.35077 |
| [8] | Prudnikov, A. P.; Brychkov, A. Y.; Marichev, O. I., Integrals and Series. Vol. 1. Elementary Functions (1986), Gordon and Breach Science Publishers: Gordon and Breach Science Publishers New York · Zbl 0733.00004 |
| [9] | Ortigueira, M. D., Riesz potentials operators and inverses via centred derivatives, Int. J. Math. Math. Sci. (2006), Article ID 48391 · Zbl 1122.26007 |
| [10] | Ortigueira, M. D., Fractional central differences and derivatives, J. Vib. Control, 14, 1255-1266 (2008) · Zbl 1229.26015 |
| [11] | Ortigueira, M. D.; Trujillo, J. J., A unified approach to fractional derivatives, Commun. Nonlinear Sci. Numer. Simul., 17, 5151-5157 (2012) · Zbl 1263.35217 |
| [12] | Kilbas, A. A.; Srivastava, H. R.; Trujillo, J. J., (Theory and Applications of Fractional Differential Equations. Theory and Applications of Fractional Differential Equations, North-Holland Math. Studies, vol. 44 (2006), Elsevier) · Zbl 1092.45003 |
| [13] | Landkof, N. S., Foundations of modern potential theory, (Die Grundlehren der Mathematischen Wissenschaften, Band 180 (1972), Springer-Verlag: Springer-Verlag New York) · Zbl 0253.31001 |
| [15] | Meerschaert, M. M.; Sikorskii, A., (Stochastic Models for Fractional Calculus. Stochastic Models for Fractional Calculus, De Gruyter Studies in Mathematics, vol. 43 (2012), De Gruyter: De Gruyter Berlin) · Zbl 1247.60003 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
