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][215][216][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.Key words Modified error function, error function, phase change problem, temperature-dependent thermal conductivity, nonlinear second order ordinary differential equation.2000 MSC 35R35, 80A22, 34B15, 34B08.

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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 δ, 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 δ. Accurate approximations are obtained through functions involving error and exponential functions only. For the special case in which δ 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 δ goes to zero.