Nonlinear Dynamics

, Volume 71, Issue 4, pp 653–662

Fractional integration and differentiation of variable order: an overview

Review

DOI: 10.1007/s11071-012-0485-0

Cite this article as:
Samko, S. Nonlinear Dyn (2013) 71: 653. doi:10.1007/s11071-012-0485-0

Abstract

We give an overview of a selection of studies on fractional operations of integration and differentiation of variable order, when this order may vary from point to point. We touch on both the Euclidean setting and also the general setting within the framework of quasimetric measure spaces.

Keywords

Fractional integralsRiesz potentialsLaplaciansVariable ordersQuasimetric measure spaces

Notation

n

is the n-dimensional Euclidean space, \(|x|=\sqrt {x_{1}^{2}+\cdots+ x_{n}^{2}}\);

\(\mathbb{S}^{n-1}\)

is the unit sphere in ℝn centered at the origin, \(|\mathbb{S}^{n-1}|\) is its surface area;

Δ

is the Laplace operator;

(X,ϱ,μ)

denotes a quasimetric measure space with quasidistance ϱ and measure μ;

δ(x,Ω)=infyΩϱ(x,y)

is the distance of a point xX to a set ΩX.

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.University of AlgarveFaroPortugal