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.

An analytical solution for a Stefan problem with variable latent heat, Int. J. Heat Mass Transfer 47 (2004) 5387-5390], and Lorenzo-Trueba and Voller [J. Lorenzo-Trueba, V.R. Voller, Analytical and numerical solution of a generalized Stefan problem exhibiting two moving boundaries with application to ocean delta formation, J. Math. Anal. Appl. 366 (2010) 538-549], a model associated with the formation of sedimentary ocean deltas is studied through a one-phase Stefan-like problem with variable latent heat. Motivated by these works, we consider a two-phase Stefan problem with variable latent of fusion and initial temperature, and constant heat flux boundary conditions. We obtain the sufficient condition on the data in order to have an explicit solution of a similarity type of the corresponding free boundary problem for a semi-infinite material. Moreover, the explicit solution given in the first quoted paper can be recovered for a particular case by taking a null heat flux condition at the infinity.