Revisiting Diffusion: Self-similar Solutions and the $t^{-1/2}$ Decay in Initial and Initial-Boundary Value Problems

TL;DR

This paper explores the solutions of the diffusion equation, demonstrating that the classic $t^{-1/2}$ decay law is not universal and can be altered through specific initial and boundary conditions, with implications for understanding decay behaviors.

Contribution

It introduces novel self-similar solutions to the diffusion equation, showing that the $t^{-1/2}$ decay law is not necessary and can be replaced by other decay rates under certain conditions.

Findings

The $t^{-1/2}$ decay law is not mandatory for diffusion solutions.

Different decay rates can be engineered through initial and boundary conditions.

The dominant decay mode corresponds to finite-mass initial data.

Abstract

The diffusion equation is a universal and standard textbook model for partial differential equations (PDEs). In this work, we revisit its solutions, seeking, in particular, self-similar profiles. This problem connects to the classical theory of special functions and, more specifically, to the Hermite as well as the Kummer hypergeometric functions. Reconstructing the solution of the original diffusion model from novel self-similar solutions of the associated self-similar PDE, we infer that the $t^{-1/2}$ decay law of the diffusion amplitude is {\it not necessary}. In particular, it is possible to engineer setups of {\it both} the Cauchy problem and the initial-boundary value problem in which the solution decays at a {\it different rate}. Nevertheless, we observe that the $t^{-1/2}$ rate corresponds to the dominant decay mode among integrable initial data, i.e., ones corresponding to finite mass. Hence, unless the projection to such a mode is eliminated, generically this decay will be the slowest one observed. In initial-boundary value problems, an additional issue that arises is whether the boundary data are \textit{consonant} with the initial data; namely, whether the boundary data agree at all times with the solution of the Cauchy problem associated with the same initial data, when this solution is evaluated at the boundary of the domain. In that case, the power law dictated by the solution of the Cauchy problem will be selected. On the other hand, in the non-consonant cases a decomposition of the problem into a self-similar and a non-self-similar one is seen to be beneficial in obtaining a systematic understanding of the resulting solution.