2025/09/19 by Pereira, Antonio Luiz
#37L05 #45H05 #45J05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.2509.16041
The nonlocal diffusion equation with continuous kernel K(x,y, with ∫R K(y,x) d y = 1 has been proposed as a model for some evolution process with diffusion, including population models. However, in general, we don't have ∫Ω K(y,x) d y = 1, as expected from its interpretation as a probability density. In this note, we propose a modification of the kernel, based on the idea of `reflection' at the boundary, familiar in one dimensional problems. We show that a similar construction is possible in higher dimensions, with the new kernel satisfying the above integral equality and being also symmetric in some special cases.