Analysis and numerics of traveling waves for asymmetric fractional reaction-diffusion equations

Franz Achleitner, Christian Kuehn

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

We consider a scalar reaction-diffusion equation in one spatial dimension with bistable nonlinearity and a nonlocal space-fractional diffusion operator of Riesz-Feller type. We present our analytical results on the existence, uniqueness (up to translations) and sta-bility of a traveling wave solution connecting two stable homogeneous steady states. Moreover, we review numerical methods for the case of reaction-diffusion equations with fractional Laplacian and discuss possible extensions to our reaction-diffusion equations with Riesz-Feller operators. In particular, we present a direct method using integral op- erator discretization in combination with projection boundary conditions to visualize our analytical results about traveling waves.

Original languageEnglish
JournalCommunications in Applied and Industrial Mathematics
Volume6
Issue number2
DOIs
StatePublished - 2015
Externally publishedYes

Keywords

  • Allen-cahn type equation
  • Comparison principle
  • Fractional derivative
  • Nagumo equation
  • Nonlocal diffusion
  • Projection boundary conditions
  • Quadrature
  • Real ginzburg-landau equation
  • Riesz-feller operator
  • Traveling wave

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