English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

An essential part of any boundary value problem is the domain on which the problem is defined. The domain is often given by scanning or another digital image technique with limited resolution. This leads to significant uncertainty in the domain definition. The paper focuses on the impact of the uncertainty in the domain on the Neumann boundary value problem (NBVP). It studies a scalar NBVP defined on a sequence of domains. The sequence is supposed to converge in the set sense to a limit domain. Then the respective sequence of NBVP solutions is examined. First, it is shown that the classical variational formulation is not suitable for this type of problem as even a simple NBVP on a disk approximated by a pixel domain differs much from the solution on the original disk with smooth boundary. A new definition of the NBVP is introduced to avoid this difficulty by means of reformulated natural boundary conditions. Then the convergence of solutions of the NBVP is demonstrated. The uniqueness of the limit solution, however, depends on the stability property of the limit domain. Finally, estimates of the difference between two NBVP solutions on two different but close domains are given.

Effects of uncertainties in the domain on the solution of Neumann boundary value problems in two spatial dimensions