Approximation of the modified error function

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 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.

Auxiliary functions in the study of Stefan-like problems with variable thermal properties

“…The method described in [1] had already been considered by Cho and Sunderland in [2] for the analogous Stefan problem with constant heat capacity, corresponding to γ = 0 in (1.1). The modified error function for the case when γ = 0 and δ > 0 was studied by the authors in [3,4] (see also [5]), where existence and uniqueness in the space of bounded analytic functions were proven and explicit approximations were provided (see [6] for improved approximations). In particular, this paper extends the existence and uniqueness result in [3] for the case δ < 0 (γ = 0).…”

A Note on Corrections in Approximation of the Modified Error Function

“…To exhibit the usefulness of the results derived here we have compared the correction terms ϕn(x) in (1.4) with the results obtained by Ceretani et al [18] in Fig.1. It is observed that the lack of monotonicity of successive corrections of MEF present in [18] disappears.…”

“…of MEF was overlooked. In their studies [17,18] and references therein, Tarzia and his collaborators investigated the mathematical aspects of MEF as mentioned above.…”

“…The authors in [18] obtained first and second order corrections ϕ1(x) and ϕ2(x) involved in the approximations Ψ δ,1 (x) and Ψ δ,2 (x) of the MEF for δ > −1. But it is observed that Ψ δ,1 (x) appears to be better approximation than Ψ δ,2 (x), which is not desirable.…”

“…With these expressions it is observed that the order of magnitude of the corrections decreases order by order, thus resolves the apparent problem of monotonicity of the successive correction terms that is necessary for the convergence of the series in (1.1). Hence, the inconsonance which arises in [18] has been dispelled.…”