Positivity of the fundamental solution for fractional diffusion and wave equations
Kemppainen, Jukka (2019-11-05)
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
© 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.
https://rightsstatements.org/vocab/InC/1.0/
https://urn.fi/URN:NBN:fi-fe2021051930607
Tiivistelmä
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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