The solution is obtained and validated by an existence and uniqueness theorem for the following nonlinear boundary value problemwhich was proposed in 1974 by Cho and Sunderland to represent a Stefan problem with a nonlinear temperature-dependent thermal conductivity on the semi-infinite line (0, ∞). The modified error function of two parameters 𝜑 𝛿,𝛾 is introduced to represent the solution of the problem above, and some properties of the function are established. This generalizes the results obtained in earlier studies.

Approximate solutions are obtained in implicit forms for the following general form of the nonlinear Stefan problem ddx(1+δ1yp)dydx+2x(1+δ2yp)dydx=4Steβ(x),0<x<λ, with y(0)=1,y(λ)=0, where λ>0 is a solution to the nonlinear equation y′(λ)=−2λSte, where δi>−1,i=1,2,p>0, and Ste is the Stefan number, which represents a phase-change problem with a nonlinear temperature-dependent thermal parameters (i.e., thermal conductivity and specific heat) on (0,λ).

In a recent research, the authors established an approximation to the modified error function (MEF) 𝜑 𝛿 for a small positive thermal conductivity coefficient 𝛿 > 0, leaving the problem open for the general case −1 < 𝛿 < ∞. The approximation represents an approximate solution to the Stephan problem with linear thermal conductivity in a semi-infinite body, which converges to the classical MEF as 𝛿 → 0 + . In this paper, a new approximation to the MEF is found for −1 < 𝛿 < ∞ and converges to the classical MEF as 𝛿 → 0 ± . A comparative analysis shows that our proposed approximation gives a better estimate than the one in this recent reasearch by Ceratani et al.

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