On the existence, uniqueness, and new analytic approximate solution of the modified error function in two‐phase Stefan problems

Abstract

This paper provides a new proof of the existence and uniqueness of the solution for a nonlinear boundary value problem {right leftarray(1+δy)y′′+2x(1+γy)y′array=0,0<x<∞,arrayy(0)=0,y(∞)array=1, which describes the study of two‐phase Stefan problems on the semi‐infinite line [0, ∞). This result considerably extends the analysis of a recent work. A highly accurate analytic approximate solution of this problem is also provided via the Adomian decomposition method.

On the approximation of the modified error function

“…A similar result can be found in Bougoffa 4 . Also, the general case $$$ \delta &amp;gt;-1 $$$ was proved in Bougoffa et al 11 …”

A Stefan problem with nonlinear thermal conductivity

“…The analysis of the result obtained in the preceding sections reveals two important observations regarding the nature and behavior of the solution $$$ y $$$ to Problem (). The solution () of Problem () can be viewed as a generalization of the modified error function established in Bougoffa et al 4 for $$$ \delta &gt;-1 $$$ and in Ceretani et al 3 when $$$ \delta &gt;0 $$$. Problem () represents Stefan's problem with a nonlinear thermal conductivity, and when $$$ n&#x0003D;1 $$$, the thermal conductivity becomes quadratic of the form $$$ K(y)&#x0003D;\gamma {y}&#x0005E;2&#x0002B;\delta y&#x0002B;1 $$$.…”

“…1,2 This has motivated many researchers to develop existence and uniqueness theorems for the solutions of this problem which has been ever since the subject of many research papers; see, for example previous research. [3][4][5][6][7][8][9][10][11][12][13][14] The authors in Ceretani et al 3 proved the existence and uniqueness of the modified error function for small values of 𝛿 > 0, and the general case 𝛿 > −1 was proved in Bougoffa et al 4 The purpose of this paper is to provide an existence and uniqueness theorem for the solution of (1.4) that was proposed in Cho and Sunderland, 1 which represents a Stefan problem with a nonlinear thermal conductivity of the form (1 + 𝛿𝑦 + 𝛾𝑦 2 ) n , where 𝛿 > −1 and 𝛾 > −1. To the best of our knowledge, the question of existence and uniqueness of the solution of Problem (1.4) has not been answered since proposing the problem in 1974.…”

“…No existence and uniqueness theorems have been established in earlier studies 1,2 . This has motivated many researchers to develop existence and uniqueness theorems for the solutions of this problem which has been ever since the subject of many research papers; see, for example previous research 3–14 . The authors in Ceretani et al 3 proved the existence and uniqueness of the modified error function for small values of $$$ \delta &gt;0 $$$, and the general case $$$ \delta &gt;-1 $$$ was proved in Bougoffa et al 4 …”

“…The solution (3.1) of Problem(1.4) can be viewed as a generalization of the modified error function established in Bougoffa et al4 for 𝛿 > −1 and in Ceretani et al3 when 𝛿 > 0. Problem (1.4) represents Stefan's problem with a nonlinear thermal conductivity, and when n = 1, the thermal conductivity becomes quadratic of the form K(𝑦) = 𝛾𝑦 2 + 𝛿𝑦 + 1.…”