Lattice Green/Neumann function for the 2D Laplacian operator defined on square lattice on cylinders, tori and other geometries with some applications - Archive ouverte HAL
Article Dans Une Revue Journal of Physics A: Mathematical and Theoretical Année : 2023

Lattice Green/Neumann function for the 2D Laplacian operator defined on square lattice on cylinders, tori and other geometries with some applications

Résumé

Abstract The lattice Green function for the discrete planar Laplacian defined on regular square lattice wrapped around cylinders and tori are rigorously defined and obtained in an exact analytic form. The method of images well-known in potential theory is implemented to derive for many other geometries with free boundaries (semi-infinite or finite cylinders and strips, rectangle) the related exact lattice Green-Neumann functions needed to readily solve discrete Neumann problems or, via a Neumann-to-Dirichlet mapping, discrete Dirichlet problems for these flat square lattices. Some applications are thus proposed as explicit expressions of two-point resistances for related resistor networks, and some probability-based characteristics regarding the associated P\`{o}lya's random walks.
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hal-04081635 , version 1 (25-04-2023)

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Malik Mamode. Lattice Green/Neumann function for the 2D Laplacian operator defined on square lattice on cylinders, tori and other geometries with some applications. Journal of Physics A: Mathematical and Theoretical, In press, ⟨10.1088/1751-8121/accfd5⟩. ⟨hal-04081635⟩
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