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Cohen, Yossi; Rothman, Daniel H.

Exact solution for the Poisson field in a semi-infinite strip. (English) Zbl 1404.35145

Proc. R. Soc. Lond., A, Math. Phys. Eng. Sci. 473, No. 2200, Article ID 20160908, 11 p. (2017).
Summary: The Poisson equation is associated with many physical processes. Yet exact analytic solutions for the two-dimensional Poisson field are scarce. Here we derive an analytic solution for the Poisson equation with constant forcing in a semi-infinite strip. We provide a method that can be used to solve the field in other intricate geometries. We show that the Poisson flux reveals an inverse square-root singularity at a tip of a slit, and identify a characteristic length scale in which a small perturbation, in a form of a new slit, is screened by the field. We suggest that this length scale expresses itself as a characteristic spacing between tips in real Poisson networks that grow in response to fluxes at tips.

Cited in 1 Document

MSC:

35J25 Boundary value problems for second-order elliptic equations
35C05 Solutions to PDEs in closed form
30C20 Conformal mappings of special domains
30E25 Boundary value problems in the complex plane

Keywords:

Poisson equation; Laplacian growth; diffusion field; analytic solution; ramified network; golden ratio
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References:

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