General self-similar solutions of diffusion equation and related constructions

TL;DR

This paper explores a family of self-similar solutions to the diffusion equation, including less-known solutions, to better understand the long-time decay behaviors in various diffusive systems.

Contribution

It introduces new solutions derived from trial functions, expanding the understanding of long-term decay rates in diffusion processes.

Findings

Presented a family of solutions for infinite horizon cases

Identified different decay rates for non-equilibrium systems

Extended the set of known solutions to the diffusion equation

Abstract

Transport phenomena plays an important role in science and technology. In the wide variety of applications both advection and diffusion may appear. Regarding diffusion, for long times, different type of decay rates are possible for different non-equilibrium systems. After summarizing the existing solutions of the regular diffusion equation, we present not so well known solution derived from three different trial functions, as a key point we present a family of solutions for the case of infinite horizon. By this we tried to make a step toward understanding the different long time decays for different diffusive systems.