Existence and uniqueness of the modified error function

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][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.

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

“…Theorem 2.1 improves the analogous result given by the authors in [3] for the case when δ > 0 and γ = 0. In fact, the result given in [3] holds true provided that δ > 0 satisfies δ 2 (1 + δ) 3/2 (3 + δ)(1 + (1 + δ) 3/2 ) < 1, whereas condition (2.1) in Theorem 2.1 just requires δ(1 + δ) 3/2 (3 + δ) < 1. Note that condition (2.1) is sufficient but not necessary, therefore we show the plots of the modified error function for parameters included and not included in the set A.…”

A note on the existence and uniqueness solutions of the modified error function

“…Figures 1 and 2 have been drawn to show the approximate solutions. Also, errors analysis ER i (x), i = 1, 2 for y i,p (x), i = 1, 2 are considered, where the residuals ER i (x), i = 1, 2 with different values of are plotted in Figures 3 and 4 for small and large values, respectively, of x by substituting y i,p (x), i = 1, 2 for y(x) in the differential equation of Problem (1), which shows that the present solutions are highly accurate.…”

“…In Ceretani et al, the authors proved a result on the existence and uniqueness theorem of the modified error function introduced by Cho and Sunderland, which is a solution of the following nonlinear boundary value problem: where δ ≥ − 1.…”

“…Firstly, the authors introduced a similar linear problem to obtain the results they wanted: where and It is shown in theorem 2.1 that the function is a solution to Problem .…”

“…where ≥ − 1. Firstly, the authors introduced a similar linear problem to obtain the results they wanted 1 :…”