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

Area of Science:

• Physics
• Applied Mathematics
• Computational Science

Background:

• The Poisson equation governs numerous physical phenomena, but exact analytical solutions for 2D fields are rare.
• Solving the Poisson equation is essential for modeling diverse physical processes.

Purpose of the Study:

• To derive an exact analytical solution for the 2D Poisson equation in a semi-infinite strip with constant forcing.
• To develop a generalizable method for solving the Poisson equation in complex geometries.
• To investigate the behavior of the Poisson flux and identify characteristic length scales.

Main Methods:

• Derivation of an analytical solution for the Poisson equation in a semi-infinite strip geometry.
• Analysis of the Poisson flux to identify singularities and characteristic length scales.
• Mathematical modeling of field screening effects from perturbations (new slits).

Main Results:

• An exact analytical solution for the specified 2D Poisson problem was successfully derived.
• An inverse square-root singularity in the Poisson flux was identified at the tip of a slit.
• A characteristic length scale was determined, defining the screening range of perturbations within the field.

Conclusions:

• The derived method offers a pathway to solve the Poisson equation in intricate geometries.
• The identified length scale provides insights into the structure and growth of real-world Poisson networks.
• This work contributes to a deeper understanding of fields governed by the Poisson equation and their physical implications.