Boundary problems for the fractional and tempered fractional operators

Weihua Deng, Buyang Li, Wenyi Tian, Pingwen Zhang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

39 Citations (Scopus)

Abstract

To characterize the Brownian motion in a bounded domain Ω, it is well known that the boundary conditions of the classical diffusion equation just rely on the given information of the solution along the boundary of a domain; in contrast, for the Lévy flights or tempered Lévy flights in a bounded domain, the boundary conditions involve the information of a solution in the complementary set of Ω, i.e., Rn\Ω, with the potential reason that paths of the corresponding stochastic process are discontinuous. Guided by probability intuitions and the stochastic perspectives of anomalous diffusion, we show the reasonable ways, ensuring the clear physical meaning and well-posedness of the partial differential equations (PDEs), of specifying “boundary” conditions for space fractional PDEs modeling the anomalous diffusion. Some properties of the operators are discussed, and the well-posednesses of the PDEs with generalized boundary conditions are proved.
Original languageEnglish
Pages (from-to)125-149
Number of pages25
JournalMultiscale Modeling and Simulation
Volume16
Issue number1
DOIs
Publication statusPublished - 1 Jan 2018

Keywords

  • Generalized boundary conditions
  • Lévy flight
  • Tempered Lévy flight
  • Well-posedness

ASJC Scopus subject areas

  • Chemistry(all)
  • Modelling and Simulation
  • Ecological Modelling
  • Physics and Astronomy(all)
  • Computer Science Applications

Cite this