Next Article in Journal
Inequalities Pertaining Fractional Approach through Exponentially Convex Functions
Previous Article in Journal
A Criterion for Subfamilies of Multivalent Functions of Reciprocal Order with Respect to Symmetric Points
Open AccessArticle

# Green’s Function Estimates for Time-Fractional Evolution Equations

by Ifan Johnston 1,*,† and Vassili Kolokoltsov 2,†
1
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
2
Department of Statistics, University of Warwick, Coventry CV4 7AL, UK
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Fractal Fract 2019, 3(2), 36; https://doi.org/10.3390/fractalfract3020036
Received: 3 June 2019 / Revised: 18 June 2019 / Accepted: 20 June 2019 / Published: 25 June 2019
We look at estimates for the Green’s function of time-fractional evolution equations of the form $D 0 + ∗ ν u = L u$ , where $D 0 + ∗ ν$ is a Caputo-type time-fractional derivative, depending on a Lévy kernel $ν$ with variable coefficients, which is comparable to $y − 1 − β$ for $β ∈ ( 0 , 1 )$ , and L is an operator acting on the spatial variable. First, we obtain global two-sided estimates for the Green’s function of $D 0 β u = L u$ in the case that L is a second order elliptic operator in divergence form. Secondly, we obtain global upper bounds for the Green’s function of $D 0 β u = Ψ ( − i ∇ ) u$ where $Ψ$ is a pseudo-differential operator with constant coefficients that is homogeneous of order $α$ . Thirdly, we obtain local two-sided estimates for the Green’s function of $D 0 β u = L u$ where L is a more general non-degenerate second order elliptic operator. Finally we look at the case of stable-like operator, extending the second result from a constant coefficient to variable coefficients. In each case, we also estimate the spatial derivatives of the Green’s functions. To obtain these bounds we use a particular form of the Mittag-Leffler functions, which allow us to use directly known estimates for the Green’s functions associated with L and $Ψ$ , as well as estimates for stable densities. These estimates then allow us to estimate the solutions to a wide class of problems of the form $D 0 ( ν , t ) u = L u$ , where $D ( ν , t )$ is a Caputo-type operator with variable coefficients. View Full-Text
MDPI and ACS Style

Johnston, I.; Kolokoltsov, V. Green’s Function Estimates for Time-Fractional Evolution Equations. Fractal Fract 2019, 3, 36.