University of Oulu

Kemppainen, J. Positivity of the fundamental solution for fractional diffusion and wave equations. Math Meth Appl Sci. 2021; 44: 2468– 2486. https://doi.org/10.1002/mma.5974

Positivity of the fundamental solution for fractional diffusion and wave equations

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Author: Kemppainen, Jukka1
Organizations: 1Applied and Computational Mathematics, University of Oulu, PO Box 8000, Oulu 90014, Finland
Format: article
Version: accepted version
Access: open
Online Access: PDF Full Text (PDF, 0.5 MB)
Persistent link: http://urn.fi/urn:nbn:fi-fe2021051930607
Language: English
Published: John Wiley & Sons, 2021
Publish Date: 2020-11-05
Description:

Abstract

We study the question of positivity of the fundamental solution for fractional diffusion and wave equations of the form, which may be of fractional order both in space and time. We give a complete characterization for the positivity of the fundamental solution in terms of the order of the time derivative α ∈ (0,2), the order of the spatial derivative β ∈ (0,2], and the spatial dimension d. It turns out that the fundamental solution fails to be positive for all α ∈ (1,2) and either β ∈ (0,2] and d ≥ 2 or β < α and d = 1, whereas in the other cases, it remains positive. The proof is based on delicate properties of the Fox H-functions and the Mittag-Leffler functions.

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Series: Mathematical methods in the applied sciences
ISSN: 0170-4214
ISSN-E: 1099-1476
ISSN-L: 0170-4214
Volume: 44
Issue: 3
Pages: 2468 - 2486
DOI: 10.1002/mma.5974
OADOI: https://oadoi.org/10.1002/mma.5974
Type of Publication: A1 Journal article – refereed
Field of Science: 111 Mathematics
Subjects:
Copyright information: © 2019 John Wiley & Sons, Ltd. This is the peer reviewed version of the following article: Kemppainen, J. Positivity of the fundamental solution for fractional diffusion and wave equations. Math Meth Appl Sci. 2021; 44: 2468– 2486. https://doi.org/10.1002/mma.5974, which has been published in final form at https://doi.org/10.1002/mma.5974. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.