Existence and uniqueness of the modified error function

TL;DR

This paper proves the existence and uniqueness of a non-negative analytic solution to a nonlinear second-order differential problem defining the modified error function, for small positive parameter values.

Contribution

It establishes the first rigorous proof of existence and uniqueness for the modified error function as introduced in earlier heat transfer research.

Findings

Unique non-negative analytic solution exists for small positive parameters.

The solution's existence is proven using nonlinear differential equation methods.

The work extends understanding of the modified error function in heat transfer models.

Abstract

This article is devoted to prove the existence and uniqueness of solution to the non-linear second order differential problem through which is defined the modified error function introduced in Cho-Sunderland, J. Heat Transfer, 96-2:214-217, 1974. We prove here that there exists a unique non-negative analytic solution for small positive values of the parameter on which the problem depends.