A Note on the formulation of the Neumann boundary condition for a nonlocal problem
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911163670855680 |
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| author | Pereira, Antonio Luiz |
| author_facet | Pereira, Antonio Luiz |
| contents | 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_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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_16041 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Note on the formulation of the Neumann boundary condition for a nonlocal problem Pereira, Antonio Luiz Classical Analysis and ODEs 45J05, 45H05, 37L05 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_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. |
| title | A Note on the formulation of the Neumann boundary condition for a nonlocal problem |
| topic | Classical Analysis and ODEs 45J05, 45H05, 37L05 |
| url | https://arxiv.org/abs/2509.16041 |