# Approximation of the modified error function

## TL;DR

This paper develops explicit approximations for the modified error function related to heat transfer problems, demonstrating convergence to the classical error function as a key parameter approaches zero.

## Contribution

It introduces a novel approximation method for the modified error function using power series in the parameter, applicable to phase-change heat transfer models.

## Key findings

- Accurate approximations involving error and exponential functions.
- Modified error function converges to the classical error function as delta approaches zero.
- Characterizes the properties of the modified error function for small positive delta.

## Abstract

In this article, we obtain explicit approximations of the modified error function introduced in Cho, Sunderland. Journal of Heat Transfer 96-2 (1974), 214-217, as part of a Stefan problem with a temperature-dependent thermal conductivity. This function depends on a parameter $\delta$, which is related to the thermal conductivity in the original phase-change process. We propose a method to obtain approximations, which is based on the assumption that the modified error function admits a power series representation in $\delta$. Accurate approximations are obtained through functions involving error and exponential functions only. For the special case in which $\delta$ assumes small positive values, we show that the modified error function presents some characteristic features of the classical error function, such as monotony, concavity, and boundedness. Moreover, we prove that the modified error function converges to the classical one when $\delta$ goes to zero.