# An exact solution to a Stefan problem with variable thermal conductivity and a Robin boundary condition

## TL;DR

This paper derives an exact similarity solution for a Stefan problem with temperature-dependent thermal conductivity and Robin boundary condition, introducing a generalized error function and analyzing its properties.

## Contribution

It provides a novel exact solution framework for a Stefan problem with variable thermal conductivity and Robin boundary condition, including properties and approximations of the generalized error function.

## Key findings

- Existence of a unique, bounded, analytic solution for small parameter values.
- The generalized modified error function is concave and increasing.
- Explicit approximations for the generalized error function are proposed.

## Abstract

In this article it is proved the existence of similarity solutions for a one-phase Stefan problem with temperature-dependent thermal conductivity and a Robin condition at the fixed face. The temperature distribution is obtained through a generalized modified error function which is defined as the solution to a nonlinear ordinary differential problem of second order. It is proved that the latter has a unique non-negative bounded analytic solution when the parameter on which it depends assumes small positive values. Moreover, it is shown that the generalized modified error function is concave and increasing, and explicit approximations are proposed for it. Relation between the Stefan problem considered in this article with those with either constant thermal conductivity or a temperature boundary condition is also analysed.