A finite difference method for fractional diffusion equations with Neumann boundary conditions

Abstract

A finite difference numerical method is investigated for fractional order diffusion problems in one space dimension. The basis of the mathematical model and the numerical approximation is an appropriate extension of the initial values, which incorporates homogeneous Dirichlet or Neumann type boundary conditions. The wellposedness of the obtained initial value problem is proved and it is pointed out that each extension is compatible with the original boundary conditions. Accordingly, a finite difference scheme is constructed for the Neumann problem using the shifted Grünwald-Letnikov approximation of the fractional order derivatives, which is based on infinite many basis points. The corresponding matrix is expressed in a closed form and the convergence of an appropriate implicit Euler scheme is proved.

Models of space-fractional diffusion: A critical review

“…Remark: We still think that non-local analysis in [20] is a proper tool for modeling anomalous diffusion but the zero extension in (8) does not correspond to the Dirichlet boundary conditions. Instead, in the one-dimensional case, one should apply the extension in [5].…”

“…In many real situations, one should use Neumann boundary conditions in the models. A corresponding analysis can be found in [4] and [5].…”

Törtrendű diffúzió modelljei és szimulációja

“…Ha például zárt térben (kémcső, lombik, tartály) végbemenő anyagtranszportot modellezünk, akkor annak határán mindig homogén Neumannféle peremfeltételt kell vennünk. Az egydimenziós esetben sikerült a homogén feltételt a modellbe beépíteni [13]…”

Finite element approximation of fractional order elliptic boundary value problems

“…The favor of this approach is not only that we can deal with complex domains but also that in the corresponding bilinear forms the homogeneous boundary conditions can be included in a natural way. The choice of an appropriate boundary condition -which corresponds to the real-life phenomenon to simulate and leads to a well-posed problem -is not obvious [20], [21] due to the non-local nature [9] of the fractional Laplacian.…”