A Note on the Axisymmetric Diffusion equation

TL;DR

This paper derives an explicit, rapidly converging series solution for the axisymmetric diffusion equation using Mellin inversion and residue calculus, facilitating efficient computation of the solution.

Contribution

It introduces a novel series representation of the axisymmetric diffusion equation solution via Mellin inversion and residue calculus, improving computational efficiency.

Findings

Derived a Mellin inversion formula for the solution

Represented the solution as a rapidly converging series

Provided a method for efficient computation

Abstract

We consider the explicit solution to the axisymmetric diffusion equation. We recast the solution in the form of a Mellin inversion formula, and outline a method to compute a formula for $u(r,t)$ as a series using the Cauchy residue theorem. As a consequence, we are able to represent the solution to the axisymmetric diffusion equation as rapidly converging series.