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 Tools annual

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 monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

This work presents a probabilistic scheme for solving semilinear nonlocaldiffusion equations with volume constraints and integrable kernels. Thenonlocal model of interest is defined by a time-dependent semilinear partialintegro-differential equation (PIDE), in which the integro-differentialoperator consists of both local convection-diffusion and nonlocal diffusionoperators. Our numerical scheme is based on the direct approximation of thenonlinear Feynman-Kac formula that establishes a link between nonlinear PIDEsand stochastic differential equations. The exploitation of the Feynman-Kacrepresentation successfully avoids solving dense linear systems arising fromnonlocality operators. Compared with existing stochastic approaches, our methodcan achieve first-order convergence after balancing the temporal and spatialdiscretization errors, which is a significant improvement of existingprobabilistic/stochastic methods for nonlocal diffusion problems. Erroranalysis of our numerical scheme is established. The effectiveness of ourapproach is shown in two numerical examples. The first example considers athree-dimensional nonlocal diffusion equation to numerically verify the erroranalysis results. The second example presents a physics problem motivated bythe study of heat transport in magnetically confined fusion plasmas.

A probabilistic scheme for semilinear nonlocal diffusion equations with volume constraints