We prove that the (nonlocal) Marchaud fractional derivative in R can be obtained from a parabolic extension problem with an extra (positive) variable as the operator that maps the heat conduction equation to the Neumann condition. Some properties of the fractional derivative are deduced from those of the local operator. In particular, we prove a Harnack inequality for Marchaud-stationary functions.
An extension problem for the fractional derivative defined by Marchaud / C. Bucur, F. Ferrari. - In: FRACTIONAL CALCULUS & APPLIED ANALYSIS. - ISSN 1311-0454. - 19:4(2016 Aug), pp. 867-887. [10.1515/fca-2016-0047]
An extension problem for the fractional derivative defined by Marchaud
C. BucurPrimo
;
2016
Abstract
We prove that the (nonlocal) Marchaud fractional derivative in R can be obtained from a parabolic extension problem with an extra (positive) variable as the operator that maps the heat conduction equation to the Neumann condition. Some properties of the fractional derivative are deduced from those of the local operator. In particular, we prove a Harnack inequality for Marchaud-stationary functions.
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