Exact solution for the Poisson field in a semi-infinite strip

Abstract

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.

The Saint‐Venant type isoperimetric inequalities for assessing saturated water storage in lacunary shallow perched aquifers

“…Physically, accretion results in groundwater mounds whose summits hydraulically command the regional discharge zones (the curve G in Figure 1, see e.g., [31]). Analytical and numerical solutions to these BVPs are used in groundwater hydrology, agricultural and geotechncial engineering, geomorphology, among others (see e.g., [20,[25][26][27][32][33][34][35][36][37]). If wells are pumped in D, the mound is drained causing a near-well drawdown of the water table.…”