Some applications of generalized fractional derivatives
DOI:
https://doi.org/10.17721/1812-5409.2022/2.3Keywords:
generalized convolution–type derivatives, Bernstein functions, subordinators, eigenfunctions, differential equations with generalized fractional derivatives, random initial conditions, random fields on the sphereAbstract
The paper presents a concise summary of main properties of generalized fractional derivatives, so-called convolution type derivatives with respect to Bernstein functions. Applications are considered to modeling time dependent random fields on the sphere as solutions to partial differential equations with the generalized fractional derivative in time and random initial condition.
Pages of the article in the issue: 28 - 34
Language of the article: English
References
ALRAWASHDEH, M.S., KELLY, J.F., MEERSCHAERT, M.M., SCHEFFLER, H.-P. (2017) Applications of inverse tempered stable subordinators. Comput. Math. Appl., Vol. 73, no. 6, p. 892–905.
BEGHIN, L., GAJDA, J. (2020) Tempered relaxation equation and related generalized stable processes. Fract. Calc. Appl. Anal., Vol. 23(5), p. 1248–1273.
BUCHAK, K., SAKHNO, L. (2019) On the governing equations for Poisson and Skellam processes time-changed by inverse subordinators. Theor. Probab. Math. Stat., Vol. 98, p. 91–104.
CHEN, Z.-Q. (2017) Time fractional equations and probabilistic representation. Chaos, Solitons & Fractals, Vol. 102, p. 168–174.
D’OVIDIO, M. (2014) Coordinates changed random fields on the sphere. J. Stat. Phys., Vol. 154, p. 1153–1176 .
D’OVIDIO, M., LEONENKO, N., ORSINGHER, E. (2016) Fractional spherical random fields. Stat. Probab. Lett., Vol. 116, p. 146–156.
D’OVIDIO, M., ORSINGHER, E., SAKHNO, L. (2022) Models of space-time random fields on the sphere Modern Stoch. Theory Appl., Vol. 9, Issue 2, p. 139–156.
KOCHUBEI, A.N.(2011) General fractional calculus, evolution equations, and renewal processes. Integral Equ. Oper. Theory. Vol. 71, no. 4, p. 583–600.
MARINUCCI, D., PECCATI, G. (2011) Random Fields on the Sphere: Representation, Limit Theorems and Cosmological Applications. Cambridge University Press, 356 p.
MEERSCHAERT, M.M., NANE, E., VELLAISAMY, P. (2013) Transient anomalous sub-diffusion on bounded domains. Proc. Amer. Math. Soc. Vol. 141(2), p. 699–710.
MEERSCHAERT, M.M., TOALDO, B.(2019) Relaxation patterns and semi-Markov dynamics. Stoch. Proc. Appl. Vol. 129, Issue 8, p. 2850–2879.
TOALDO, B.(2015) Convolution-type derivatives, hitting-times of subordinators and time-changed C0-semigroups. Potential Analysis. Vol. 42, p. 115–140.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2022 The Author(-s)
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
