The Solution of the Modified Helmholtz Equation in a Wedge and an Application to Diffusion-Limited Coalescence

TL;DR

This paper derives a general solution to the modified Helmholtz equation in a wedge-shaped domain and applies it to explicitly solve the steady-state diffusion-limited coalescence process with a trap-source at the origin.

Contribution

It provides a new explicit solution to the modified Helmholtz equation in a wedge and applies it to a specific diffusion-limited coalescence problem.

Findings

Explicit steady-state solution for diffusion-limited coalescence with a trap-source.

General solution of the modified Helmholtz equation in a wedge geometry.

Application of mathematical solution to a physical diffusion process.

Abstract

The general solution of the modified Helmholtz equation, q_{xx}(x,y)+q_{yy}(x,y)-4b^2q(x,y)=0, in the wedge 0 < x < y < infinity, is presented. This solution is used to find the explicit steady-state of diffusion-limited coalescence, A+A<-->A, on the half-line, with a trap-source at the origin.