An Analysis of the One-Phase Stefan Problem with Variable Thermal Coefficients of Order p

Abstract

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).

A Nonclassical Stefan Problem with Nonlinear Thermal Parameters of General Order and Heat Source Term

“…As noted in [31], this function helps understand the absorption between dried and frozen regions and facilitates the analytical solution of the problem. Several papers have investigated the solution of the Stefan problem for heat sources with constant thermal parameters [32][33][34][35] and temperature-dependent thermal parameters [36][37][38][39]. The existence of solutions has been established, and explicit solutions have been obtained for particular cases.…”

“…( 4)- (5) and Pr. ( 4)-(6) when β(x) = e −kx 2 , k > 0, which was adopted in [39]. The paper is organized as follows: In Section 2, we present a preliminary analysis of the homogeneous problem.…”

“…For the given BVP (39), there are functions A(x, v), B(x, v) > 0, which are bounded when v varies in a bounded set [−1, 0] such that…”

“…If g(x, v, p) is monotone increasing in v for each fixed x ∈ [0, ∞) and p ∈ R. Then, the boundary problem Pr. (39) has at most one solution v in C 2 [0, ∞).…”