Premium
Positivity of the fundamental solution for fractional diffusion and wave equations
Author(s) -
Kemppainen Jukka
Publication year - 2021
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5974
Subject(s) - mathematics , fractional calculus , order (exchange) , space (punctuation) , dimension (graph theory) , mittag leffler function , mathematical analysis , wave equation , diffusion , derivative (finance) , pure mathematics , physics , quantum mechanics , linguistics , philosophy , finance , financial economics , economics
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.