A Note on the formulation of the Neumann boundary condition for a nonlocal problem

Antonio Luiz Pereira

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arXiv · 2509.16041

A Note on the formulation of the Neumann boundary condition for a nonlocal problem

Antonio Luiz Pereira

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Abstract

The nonlocal diffusion equation with continuous kernel $K(x,y$, with $ \int_{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 $ \int_{\Omega} 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.

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Antonio Luiz Pereira. 2025-09-19. A Note on the formulation of the Neumann boundary condition for a nonlocal problem. https://arxiv.org/abs/2509.16041

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