Exact Solutions for Nonclassical Stefan Problems

Abstract

We consider one-phase nonclassical unidimensional Stefan problems for a source function 𝐹 which depends on the heat flux, or the temperature on the fixed face 𝑥=0. In the first case, we assume a temperature boundary condition, and in the second case we assume a heat flux boundary condition or a convective boundary condition at the fixed face. Exact solutions of a similarity type are obtained in all cases.

1. Introduction

The one-phase Stefan problem for a semi-infinite material is a free boundary problem for the classical heat equation which requires the determination of the temperature distribution 𝑢 of the liquid phase (melting problem) or the solid phase (solidification problem) and the evolution of the free boundary 𝑥=𝑠(𝑡). Phase change problems appear frequently in industrial processes and other problems of technological interest [1–4].

Nonclassical heat conduction problem for a semi-infinite material was studied in [5–11]. A problem of this type is the following:(i)𝑢_𝑡−𝑢_𝑥𝑥=−𝐹(𝑊(𝑡),𝑡),𝑥>0,𝑡>0,(ii)𝑢(0,𝑡)=𝑓(𝑡),𝑡>0,(iii)𝑢(𝑥,0)=ℎ(𝑥),𝑥>0, (1.1) where functions 𝑓=𝑓(𝑡) andℎ=ℎ(𝑥) are continuous real functions, and 𝐹 is a given function of two variables. A particular and interesting case is the following:𝜆𝐹(𝑊(𝑡),𝑡)=⁰√𝑡𝑊(𝑡)(𝜆₀>0), (1.2) where 𝑊=𝑊(𝑡) represents the heat flux on the boundary 𝑥=0, that is 𝑊(𝑡)=𝑢_𝑥(0,𝑡).Problems of the types (1.1) and (1.2) can be thought of by modelling of a system of temperature regulation in isotropic mediums [10, 11], with a nonuniform source term which provides a cooling or heating effect depending upon the properties of 𝐹 related to the course of the heat flux (or the temperature in other cases) at the boundary 𝑥=0[10].

In the particular case of a bounded domain, a class of problems, when the heat source is uniform and belongs to a given multivalued function from ℝ into itself, was studied in [8] regarding existence, uniqueness, and asymptotic behavior. Moreover, in [5] conditions are given on the nonlinearity of the source term 𝐹so as to accelerate the convergence of the solution to the steady-state solution. Other references on the subject are in [7, 12, 13].

Nonclassical free boundary problems of the Stefan type were recently studied in [14–16] from a theoretical point of view by using an equivalent formulation through a system of second kind Volterra integral equations [17–19]. A large bibliography on free boundary problems for the heat equation was given in [20].

In this paper, firstly we consider a free boundary problem which consists in determining the temperature 𝑢=𝑢(𝑥,𝑡) and the free boundary 𝑥=𝑠(𝑡) such that the following conditions are satisfied:𝜌𝑐𝑢_𝑡−𝑘𝑢_𝑥𝑥=−𝛾𝐹(𝑊(𝑡),𝑡),0<𝑥<𝑠(𝑡),𝑡>0,𝑢(0,𝑡)=𝑓>0,𝑡>0,𝑢(𝑠(𝑡),𝑡)=0,𝑡>0,𝑘𝑢_𝑥(𝑠(𝑡),𝑡)=−𝜌𝑙𝑠(̇𝑡),𝑡>0,𝑠(0)=0, (1.3)(1.4)(1.5)(1.6)(1.7) where the thermal coefficients 𝑘,𝜌,𝑐,𝑙,𝛾>0, the boundary temperature 𝑓>0, and the control function 𝐹 depend on the evolution of the heat flux at the boundary 𝑥=0as follows:𝑊(𝑡)=𝑢_𝑥(0,𝑡),𝐹(𝑊(𝑡),𝑡)=𝐹(𝑢_𝑥𝜆(0,𝑡),𝑡)=⁰√𝑡𝑢_𝑥(0,𝑡), (1.8) where 𝜆₀>0 is a given constant.The existence and the uniqueness of the solution of a general free boundary problem of the type (1.3)–(1.8) was given recently in [14, 15]. Moreover, we consider other two free boundary problems which consist in determining the temperature 𝑢=𝑢(𝑥,𝑡) and the free boundary 𝑥=𝑠(𝑡) such that (1.3), (1.5), (1.6), and (1.7) are satisfied, and in these cases the control function 𝐹 depends on the evolution of the temperature at the boundary 𝑥=0as follows:𝜆𝑊(𝑡)=𝑢(0,𝑡),𝐹(𝑊(𝑡),𝑡)=𝐹(𝑢(0,𝑡),𝑡)=⁰𝑡𝑢(0,𝑡),𝜆₀>0. (1.9) In this case, a heat flux boundary condition𝑘𝑢_𝑥(0,𝑡)=−𝑞₀√𝑡>0,𝑡>0 (1.10) or a convective boundary condition𝑘𝑢_𝑥𝑞(0,𝑡)=⁰√𝑡(𝑢(0,𝑡)−𝑓)>0,𝑡>0 (1.11) can be considered at the fixed face 𝑥=0in order to obtain the corresponding explicit solutions.

The plan of this paper is the following. In Section 2, we show an explicit solution of a similarity type for the nonclassical one-phase Stefan problem (1.3)–(1.7) for a control function 𝐹 given by (1.8).

In Sections 3 and 4, we obtain sufficient conditions on data in order to have a similarity type solution to the problems (1.3), (1.5), (1.6), and (1.7), where the control function 𝐹 is given by (1.9) (instead of (1.8)) and we take into account the heat flux condition (1.10) or the convective condition (1.11) at the fixed face 𝑥=0, respectively.

The restrictions on data we have obtained for these two free boundary problems with a heat flux boundary condition (1.10) or a convective boundary condition (1.11) at the fixed face 𝑥=0 can be interpreted in the same way as we have obtained in the classical Stefan problem with the same boundary conditions in [21, 22] in order to have an instantaneous phase-change problem (see, e.g., sufficient condition 𝜆₀<𝜌𝑐/2𝛾 in Theorems 3.2 and 4.1).

2. Explicit Solution to a One-Phase Stefan Problem for a Nonclassical Heat Equation with Control Function of the Type 𝐹(𝑢_𝑥(0,𝑡),𝑡)=(𝜆₀/√𝑡)𝑢_𝑥(0,𝑡) and a Temperature Condition at the Fixed Boundary

We consider the following free boundary problem for a semi-infinite material given by the following conditions:𝜌𝑐𝑢_𝑡−𝑘𝑢_𝑥𝑥=−𝛾𝐹(𝑢_𝑥(0,𝑡),𝑡),0<𝑥<𝑠(𝑡),𝑡>0,𝑢(0,𝑡)=𝑓>0,𝑡>0,𝑢(𝑠(𝑡),𝑡)=0,𝑡>0,𝑘𝑢_𝑥(𝑠(𝑡),𝑡)=−𝜌𝑙𝑠(̇𝑡),𝑡>0,𝑠(0)=0, (2.1) where the thermal coefficients 𝑘,𝜌,𝑐,𝑙,𝛾are positive and the control function 𝐹, which depends on the evolution of the heat flux at the extremum 𝑥=0, is given by (1.8).

In order to obtain an explicit solution of a similarity type, we define𝑥Φ(𝜂)=𝑢(𝑥,𝑡),𝜂=2𝑎√𝑡, (2.2) where 𝑎²=𝑘/𝜌𝑐 is the diffusion coefficient of the phase change material. The problem (2.1) and (1.8) becomeΦ^′′(𝜂)+2𝜂Φ^′(𝜂)=2𝜆Φ^′(0),0<𝜂<𝜂₀,Φ(0)=𝑓,Φ(𝜂₀Φ)=0,_′(𝜂₀)=−2𝑙𝑐𝜂₀, (2.3)(2.4)(2.5)(2.6) where the dimensionless parameter 𝜆is defined by𝜆=𝛾𝜆₀𝜌𝑐𝑎>0, (2.7) and the free boundary 𝑠(𝑡) must be of the type𝑠(𝑡)=2𝑎𝜂₀√𝑡, (2.8) where 𝜂₀ is an unknown parameter to be determined later. The general solution of the differential equation (2.3) is given byΦ(𝜂)=𝐶₂+𝐶₁[√𝜋2erf(𝜂)+2𝜆^𝜂₀𝑓₁(𝑧)𝑑𝑧], (2.9) where 𝐶₁ and𝐶₂are arbitrary constants, and2erf(𝑥)=√𝜋^𝑥₀exp(−𝑧²)𝑑𝑧,𝑓₁(𝑥)=exp(−𝑥²)^𝑥₀exp(𝑟²)𝑑𝑟 (2.10) are the error function and the Dawson's integral (see [23, page 298] and [24, page 43]), respectively.

After some elementary computations, from (2.3), (2.4), and (2.5) we obtainΦ(𝜂)=𝑓[1−𝐸(𝜂,𝜆)𝐸(𝜂₀,𝜆)],0<𝜂<𝜂₀, (2.11) where𝐸(𝑥,𝜆)=erf(𝑥)+4𝜆√𝜋^𝑥₀𝑓₁(𝑟)𝑑𝑟. (2.12)

Taking into account condition (2.6), the unknown parameter 𝜂₀=𝜂₀(𝜆,Ste)must be the solution of the following equation:Ste√𝜋[exp(−𝑥²)+2𝜆𝑓₁(𝑥)]=𝑥[erf(𝑥)+4𝜆√𝜋^𝑥₀𝑓₁(𝑧)𝑑𝑧],𝑥>0, (2.13) where Ste=𝑓𝑐/𝑙>0 is the Stefan's number. Equation (2.13) is equivalent to the following one:𝑊₁(𝑥)=2𝜆𝑊₂(𝑥),𝑥>0, (2.14) where the real functions 𝑊₁and 𝑊₂ are defined by𝑊₁(𝑥)=Steexp(−𝑥²)−√𝑊𝜋𝑥erf(𝑥),₂(𝑥)=2𝑥^𝑥₀𝑓₁(𝑟)𝑑𝑟−Ste𝑓₁(𝑥). (2.15)(2.16)

Remark 2.1. If 𝜆=0 (i.e.,𝜆₀=0), then the problem (2.1) and (1.8) represented the classical Lamé-Clapeyron problem [25]. In this case, there exists a unique solution 𝜂₀₀of (2.17) (equivalent to (2.13))given by 𝐹₀(𝑥)=Ste√𝜋,𝑥>0, (2.17) where 𝐹₀(𝑥)=𝑥erf(𝑥)exp(𝑥²), (2.18) and the explicit solution is given by [2, 23]: 𝑢(𝑥,𝑡)=𝑓[1−erf(𝜂)erf(𝜂₀₀)𝑥],0<𝜂=2𝑎√𝑡<𝜂₀₀,𝑠(𝑡)=2𝑎𝜂₀₀√𝑡. (2.19)

In order to solve (2.14), we will study firstly the behavior of function 𝑓₁. We obtain some preliminary properties.

Lemma 2.2. The Dawson's integral satisfies the following properties:(i)𝑓₁(0)=0,(ii)𝑓₁(+∞)=0,(iii)𝑓^′₁(𝑥)=1−2𝑥𝑓₁(𝑥)=>0if0<𝑥<𝑥₁,=0if𝑥=𝑥₁,<0if𝑥>𝑥₁, (2.20) where 𝑥₁≃0.924,𝑓₁(𝑥₁)≃0.541,(iv)𝑓₁^′′(𝑥)=−2[1+𝑓₁(𝑥)(1−2𝑥²)]=<0if0<𝑥<𝑥₂,=0if𝑥=𝑥₂,>0if𝑥>𝑥₂, (2.21) where 𝑥₂≃1.502,𝑓₁(𝑥₂)≃0.428,(v)lim_𝑥→+∞2𝑥𝑓₁(𝑥)=1.

Proof. The properties (i)–(iv) have been proved in [23, page 298] (see also [24, pages 42–45])
(v) By the L'Hopital Theorem, we have lim_𝑥→+∞2𝑥𝑓₁(𝑥)=lim_𝑥→+∞2𝑥∫^𝑥₀exp(𝑟²)𝑑𝑟exp(𝑥²)=lim_𝑥→+∞∫^𝑥₀exp(𝑟²)𝑑𝑟+𝑥exp(𝑥²)𝑥exp(𝑥²)=lim_𝑥→+∞∫(1+^𝑥0exp(𝑟²)𝑑𝑟𝑥exp(𝑥²))=lim_𝑥→+∞𝑓(1+¹(𝑥)𝑥)=1, (2.22) then (v) holds.

Next, we define the following auxiliary functions:𝜑₁(𝑥)=^𝑥₀𝑓₁(𝑟)𝑑𝑟,𝜑₂(𝑥)=𝑥𝜑₁(𝑥)=𝑥^𝑥₀𝑓₁𝜑(𝑟)𝑑𝑟,₃(𝑥)=𝑥𝑓₁(𝑥),𝜑₄(𝑥)=𝑥(2𝑥𝑓₁(𝑥)−1)=−𝑥𝑓^′₁𝜑(𝑥),₅(𝑥)=𝑓₁(𝑥)−𝑥𝑓^′₁(𝑥),𝜑₆(𝑥)=Ste−2(1+Ste)𝑥𝑓₁(𝑥). (2.23)

We have the following results.

Lemma 2.3. (a) Function 𝜑₁satisfies the following properties:(i)𝜑₁(0)=0,(ii)𝜑^′₁(𝑥)=𝑓₁(𝑥),(iii)𝜑^′₁(0⁺)=0,(iv)𝜑₁(+∞)=+∞,(v)𝜑₁^′′(𝑥)=𝑓^′₁(𝑥)=1−2𝑥𝑓₁(𝑥)=>0if0<𝑥<𝑥₁,=0if𝑥=𝑥₁,<0if𝑥>𝑥₁, (2.24)(vi)lim_𝑥→+∞(𝜑₁(𝑥)/log(𝑥))=1/2,(vii)lim_𝑥→+∞𝜑₁(𝑥)𝑓^′₁(𝑥)=0.(b) Function 𝜑₄satisfies the following properties:(i)𝜑₄(0⁺)=0⁻,(ii)𝜑^′₄(𝑥)=−1+4𝑥𝑓₁(𝑥)−2𝑥²(2𝑥𝑓₁(𝑥)−1),(iii)𝜑₄(+∞)=0⁺,(iv)𝜑^′₄(0⁺)=−1,(v)𝜑^′₄(+∞)=0⁺,(vi)𝜑₄(𝑥)=0⇔𝑥=𝑥₁(themaximumpointof𝑓₁),(vii)𝜑^′₄(𝑥₁)=1.(c) Function 𝜑₃satisfies the following properties:(i)𝜑₃(0⁺)=0,(ii)𝜑₃(+∞)=1/2,(iii)𝜑^′₃(𝑥)=𝑓₁(𝑥)+𝑥(1−2𝑥𝑓₁(𝑥)),(iv)𝜑^′₃(0⁺)=0,(v)𝜑^′₃(+∞)=0,(vi)𝜑₃(𝑥₁)=𝑥₁𝑓₁(𝑥₁)≃0.4999,(vii)𝜑₃(𝑥₂)=𝑥₂𝑓₁(𝑥₂)≃0.64.(d) Function 𝜑₂satisfies the following properties:(i)𝜑₂(0⁺)=0,(ii)𝜑₂(+∞)=+∞,(iii)𝜑^′₂(𝑥)=𝜑₁(𝑥)+𝑥𝑓₁(𝑥)>0,for all 𝑥>0,(iv)𝜑^′₂(0⁺)=0,(v)𝜑^′₂(+∞)=+∞,(vi)𝜑₂^′′(𝑥)=2𝑓₁(𝑥)−𝑥(2𝑥𝑓₁(𝑥)−1),(vii)𝜑₂^′′(+∞)=0,(viii)𝜑₂^′′(0⁺)=0.(e) Function 𝜑₅satisfies the following properties:(i)𝜑₅(0⁺)=0,(ii)𝜑₅(+∞)=0⁺,(iii)𝜑^′₅(𝑥)=−𝑥𝑓₁^′′(𝑥)=>0if0<𝑥<𝑥₂,=0if𝑥=𝑥₂,<0if𝑥>𝑥₂, (2.25)(iv)𝜑₅(𝑥)>0,for all 𝑥>0.(f) Function 𝜑₆satisfies the following properties:(i)𝜑₆(0⁺)=Ste>0,(ii)𝜑₆(+∞)=−1,(iii)𝜑^′₆(𝑥)=−2(1+Ste)𝜑^′₃(𝑥),(iv)𝜑^′₆(0⁺)=0,(v)𝜑^′₆(+∞)=0,(vi)𝜑₆(𝑥₁)=𝑥₁𝑓₁(𝑥₁)≃0.4999,(vii)𝜑₆(𝑥₂)=𝑥₂𝑓₁(𝑥₂)≃0.64.

Proof. (a) Taking into account properties of 𝑓₁, we have 𝜑^′₁(𝑥)=𝑓₁(𝑥)>0,∀𝑥>0,𝜑^′₁(0)=𝑓₁(0)=0, (2.26) and (v) holds. If we consider Lemma 2.2(v), we get 𝜑₁(+∞)=+∞ and we have lim_𝑥→+∞𝜑₁(𝑥)log(𝑥)=lim_𝑥→+∞𝑥𝑓₁1(𝑥)=2, (2.27) then (iv) and (vi) hold.
To prove (vii), we consider 𝜑₁(𝑥)𝑓^′₁(𝑥)=(^𝑥₀𝑓₁(𝑟)𝑑𝑟)𝑓^′₁(𝑥)=𝑓₁(𝑐)𝑥𝑓^′₁(𝑥), (2.28) where 𝑐=𝑐(𝑥)∈(0,𝑥). Then lim_𝑥→+∞𝜑₁(𝑥)𝑓^′₁(𝑥)=0 because lim_𝑥→+∞𝑥𝑓^′₁(𝑥)=0 and 𝑓₁is a bounded function.
(b) From the definition of 𝜑₄, we obtain (i) and (ii). To prove (iii), we have𝜑₄(+∞)=lim_𝑥→+∞𝑥(2𝑥𝑓₁(𝑥)−1)=lim_𝑥→+∞2𝑥𝑓₁(𝑥)−11/𝑥=lim_𝑥→+∞2[𝑓₁(𝑥)+𝑥(1−2𝑥𝑓₁(𝑥))]1/𝑥²=2lim_𝑥→+∞[𝑥²𝑓₁(𝑥)+𝑥²𝑥(1−2𝑥𝑓₁(𝑥))], (2.29) then lim_𝑥→+∞𝑥(2𝑥𝑓₁(𝑥)−1)=2lim_𝑥→+∞[𝑥²𝑥(2𝑥𝑓₁(𝑥)−1)−𝑥²𝑓₁(𝑥)]. (2.30) If we suppose that lim_𝑥→+∞𝑥(2𝑥𝑓₁(𝑥)−1)=𝐿>0, (2.31) we get 𝐿=2lim_𝑥→+∞[𝑥²𝑥(2𝑥𝑓₁(𝑥)−1)−𝑥²𝑓₁(𝑥)]=+∞, (2.32) which is a contradiction.If we suppose thatlim_𝑥→+∞𝑥(2𝑥𝑓₁(𝑥)−1)=+∞, (2.33) then 𝜑^′₄(+∞)=lim_𝑥→+∞−1+4𝑥𝑓₁(𝑥)−2𝑥²(2𝑥𝑓₁(𝑥)−1)=−∞, (2.34) which is also a contradiction. Therefore, lim_𝑥→+∞𝑥(2𝑥𝑓₁(𝑥)−1)=0and (iii) hold.
Taking into account (ii), we have 𝜑^′₄(𝑥)=−1+4𝑥𝑓₁(𝑥)−2𝑥²(2𝑥𝑓₁(𝑥)−1), then𝜑^′₄(0)=−1 and if we consider (iii) we have𝜑^′₄(+∞)=0⁺. From properties of 𝑓₁, we have 𝜑₄(𝑥)=0⇔2𝑥𝑓₁(𝑥)−1=0⇔𝑓^′₁(𝑥)=0⇔𝑥=𝑥₁, (2.35) and (vi) holds. Taking into account 𝑓^′₁(𝑥)=1−2𝑥𝑓₁(𝑥)=0, we get 𝜑^′₄(𝑥₁)=1.
(c) From Lemmas 2.2 and 2.3(b) we get (i)–(vii).
(d) We have 𝜑₂(𝑥)=𝑥𝜑₁(𝑥)=𝑥∫^𝑥₀𝑓₁(𝑟)𝑑𝑟, then from (a) and (b)(iii) we get (i)–(vi).
(e) As we have 𝜑₅(𝑥)=𝑓₁(𝑥)−𝑥𝑓^′₁(𝑥)=𝑓₁(𝑥)+𝜑₄(𝑥), then by using the properties of 𝑓₁and (b) we obtain the properties of 𝜑₅.
(f) We have𝜑₆(𝑥)=Ste−2(1+Ste)𝑥𝑓₁(𝑥)=Ste−2(1+Ste)𝜑₃(𝑥), and from the properties of 𝜑₃,we obtain (i)–(v).

Corollary 2.4. One has(i)lim_𝑥→+∞𝑥²[2𝑥𝑓₁(𝑥)−1]=1/2,(ii)lim_𝑥→+∞𝑥[𝑥²(2𝑥𝑓₁(𝑥)−1)−𝑥𝑓₁(𝑥)]=0.

Now, we are in conditions to enunciate properties of functions 𝑊₁and 𝑊₂ in order to study after (2.14).

Lemma 2.5. The functions 𝑊₁(𝑥)and 𝑊₂(𝑥), defined by (2.15) and (2.16), respectively, satisfy the following properties.(a)Properties of function 𝑊₁:(i)𝑊₁(0)=Ste,(ii)𝑊₁(+∞)=−∞,(iii)lim_𝑥→+∞(𝑊₁(𝑥)/𝑥)=−√𝜋,(iv)lim_𝑥→+∞(𝑊₁(𝑥)+√𝜋𝑥)=0,(v)𝑊^′₁(𝑥)<0,for all 𝑥>0,(vi)𝑊₁(𝜂₀₀)=0, where 𝜂₀₀ is the unique solution of (2.17),(vii)𝑊₁^′′(𝑥)=<0if0<𝑥<𝑥₀,=0if𝑥=𝑥₀,<0if𝑥>𝑥₀, (2.36) where 𝑥₀=3+2Ste4(1+Ste), (2.37)(viii)𝑊₁^′′(0⁺)=−2(3+2Ste)<0.(b) Properties of function 𝑊₂∶(i)𝑊₂(0)=0,(ii)𝑊₂(+∞)=+∞,(iii) there exists a unique 𝑥₄>0such that 𝑊₂(𝑥₄)=0,(iv)𝑊^′₂(𝑥)=2∫^𝑥₀𝑓₁(𝑟)𝑑𝑟+2𝑥𝑓₁(𝑥)(1+Ste)−Ste,(v) there exists a unique 𝑥₃>0such that𝑊^′₂(𝑥₃)=0and 𝑊₂(𝑥₃)<0,(vi)𝑊^′₂(0⁺)=−Ste<0,(vii)𝑊^′₂(+∞)=+∞,(viii)𝑊₂^′′(𝑥)=2(1+Ste)𝑥+2𝑓₁(𝑥)[2+Ste−2(1+Ste)𝑥²],(ix)𝑊₂^′′(0⁺)=0,(x)𝑊₂(𝜂₀₀)<0.

Proof. (a) Taking into account the definition of the function 𝑊₁,we get (i) and (ii).
(iii) We have lim_𝑥→+∞𝑊₁(𝑥)𝑥=lim_𝑥→+∞[Steexp(−𝑥²)𝑥−√𝜋erf(𝑥)]=−√𝜋. (2.38)
(iv) We have lim_𝑥→+∞(𝑊₁(𝑥)+√𝜋𝑥)=lim_𝑥→+∞(Steexp(−𝑥²)−√𝜋𝑥erf(𝑥)+√𝜋𝑥)=lim_𝑥→+∞(Steexp(−𝑥²)+√𝜋𝑥erf𝑐(𝑥))=lim_𝑥→+∞(Steexp(−𝑥²)+𝑄(𝑥)exp(−𝑥²))=lim_𝑥→+∞exp(−𝑥²)(Ste+𝑄(𝑥))=0, (2.39) where 𝑄 is the function defined by 𝑄(𝑥)=√𝜋𝑥exp(𝑥²)erf𝑐(𝑥),erf𝑐(𝑥)=1−erf(𝑥), (2.40) which satisfies the following properties: 𝑄(0)=0,𝑄(+∞)=1,𝑄^′(𝑥)>0,∀𝑥>0. (2.41)
(v) We have 𝑊^′₁(𝑥)=−√𝜋erf(𝑥)−2𝑥exp(−𝑥²)[Ste+1]<0,∀𝑥>0. (2.42)
(vi) Taking into account (i), (iii), and (v), we get that there exists a unique zero of 𝑊₁ which is given by𝜂₀₀,the unique solution of (2.17).
(vii)We have 𝑊₁^′′(𝑥)=−2exp(−𝑥²)[3+2Ste−4(1+Ste)𝑥²], (2.43) then 𝑊₁^′′(𝑥)=0⇔4(1+Ste)𝑥²=3+2Ste⇔𝑥=𝑥₀=3+2Ste.4(1+Ste) (2.44)
Since sign(𝑊₁^′′(𝑥))=sign(4(1+Ste)𝑥²−3−2Ste),then we obtain (vii).
(b) Taking into account Lemmas 2.2 and 2.3, we have (i) and (ii).
We can write 𝑊^′₂(𝑥)=2^𝑥₀𝑓₁(𝑟)𝑑𝑟+2𝑥𝑓₁(𝑥)(1+Ste)−Ste=2𝜑₁(𝑥)−𝜑₆(𝑥), (2.45) then 𝑊^′₂(0⁺)=−Ste,𝑊^′₂(+∞)=+∞and 𝑊₂^′′(𝑥)=2𝜑^′₁(𝑥)−𝜑^′₆(𝑥) satisfies 𝑊₂^′′(0⁺)=0.Then (iv), (vi), (vii), (viii), and (ix) hold.
We have 𝑊₂(𝑥)=0⇔2𝜑₂(𝑥)=Ste𝑓₁(𝑥), (2.46) then taking into account the properties of 𝜑₂and𝑓₁, we get that there exists a unique 𝑥₄>0such that 𝑊₂(𝑥)=0,𝑥>0. (2.47) Moreover, we have 𝑊₂(𝑥)==0if𝑥=0,<0if0<𝑥<𝑥₄,=0if𝑥=𝑥₄,>0if𝑥>𝑥₄. (2.48) In the same way, we have 𝑊^′₂(𝑥)=0⇔2𝜑₁(𝑥)=𝜑₆(𝑥). (2.49) Then, if we consider the properties of the functions 𝜑₁and𝜑₂, we have that there exists a unique 𝑥₃such that 𝑊^′₂(𝑥₃)=0. Moreover, 𝑊₂(𝑥₃)=−2𝑥²₃𝑓₁(𝑥₃)−Ste𝜑₅(𝑥₃)<0 and then (v) holds.
To prove (x), we take into account that 𝑊₂(𝑥)=2𝑥^𝑥₀𝑓₁(𝑟)𝑑𝑟−Ste𝑓₁=√(𝑥)𝜋𝑥erf(𝑥)𝐹(𝑥)−√𝜋𝑥^𝑥₀erf(𝑟)exp(𝑟²)𝑑𝑟−Steexp(−𝑥²=√)𝐹(𝑥)𝜋exp(−𝑥²)[𝐹₀(𝑥)−Ste√𝜋]𝐹(𝑥)−√𝜋𝑥^𝑥₀erf(𝑟)exp(𝑟²)𝑑𝑟, (2.50) where 𝐹(𝑥)=∫^𝑥₀exp(𝑟²)𝑑𝑟 and𝐹₀ was defined in (2.18). Then by using (2.17), we have 𝑊₂(𝜂₀₀)=−√𝜋𝜂₀₀^𝜂00₀erf(𝑟)exp(𝑟²)𝑑𝑟<0. (2.51)

Lemma 2.6. For each 𝜆>0, there exists a unique solution 𝜂₀ of (2.14). This solution 𝜂₀=𝜂₀(𝜆) satisfies the following properties: (i)𝜂₀(0⁺)=𝜂₀₀,(ii)𝜂₀(+∞)=𝑥₄,(iii)𝜂₀=𝜂₀(𝜆)isanincreasingfunctionon𝜆, (2.52) where 𝜂₀₀ and 𝑥₄ are the unique solution of (2.17) and (2.47), respectively.

Proof. Taking into account Lemma 2.5, we get that there exists a unique solution 𝜂₀ of (2.14). Let 0<𝜆₁<𝜆₂ be given, taking into account properties of function 𝑊₂, we obtain that the real functions 𝑍₁and 𝑍₂ defined by 𝑍₁(𝑥)=2𝜆₁𝑊₂(𝑥),𝑍₂(𝑥)=2𝜆₂𝑊₂(𝑥) (2.53) satisfy the following properties: 𝑍₂(𝑥)<𝑍₁(𝑥)if0<𝑥<𝑥₄,𝑍₂(𝑥)=𝑍₁(𝑥)if𝑥=𝑥₄,𝑍₂(𝑥)>𝑍₁(𝑥)if𝑥>𝑥₄. (2.54) Then 𝜂₀(𝜆₁)<𝜂₀(𝜆₂), where 𝜂₀(𝜆_𝑖)is the solution of equation 𝑍_𝑖(𝑥)=𝑊₁(𝑥),𝑖=1,2.Therefore, 𝜂₀=𝜂₀(𝜆)is an increasing function on 𝜆. Moreover, we obtain 𝜂₀₀<𝜂₀(𝜆)<𝑥₄because 𝑊₂(𝜂₀₀)<0.

Then, we have proved the following result.

Theorem 2.7. For each 𝜆>0, the free boundary problem (2.1), where 𝐹 is defined by (1.8), has a unique similarity solution of the type 𝑢(𝑥,𝑡,𝜆)=𝑓[1−𝐸(𝑛,𝜆)𝐸(𝜂₀𝑥(𝜆),𝜆)],0<𝜂=2𝑎√𝑡<𝜂₀(𝜆),𝑠(𝑡,𝜆)=2𝑎𝜂₀(𝜆)√𝑡, (2.55) where 𝐸(𝜂,𝜆)=erf(𝜂)+4𝜆√𝜋^𝜂₀𝑓₁(𝑟)𝑑𝑟 (2.56) and 𝜂₀=𝜂₀(𝜆) is the unique solution of (2.14) with 𝜂₀₀<𝜂₀(𝜆)<𝑥₄.

3. Explicit Solution to a One-Phase Stefan Problem for a Nonclassical Heat Equation with Control Function of the Type 𝐹(𝑢(0,𝑡),𝑡)=(𝜆₀/𝑡)𝑢(0,𝑡) and a Heat Flux Condition at the Fixed Face

In this section, the free boundary problem consists in determining the temperature 𝑢=𝑢(𝑥,𝑡) and the free boundary 𝑥=𝑠(𝑡) with a control function 𝐹 which depends on the evolution of the temperature at the extremum 𝑥=0given by the following conditions:𝜌𝑐𝑢_𝑡−𝑘𝑢_𝑥𝑥=−𝛾𝐹(𝑢(0,𝑡),𝑡),0<𝑥<𝑠(𝑡),𝑡>0,𝑘𝑢_𝑥(0,𝑡)=−𝑞₀√𝑡>0,𝑡>0,𝑢(𝑠(𝑡),𝑡)=0,𝑡>0,𝑘𝑢_𝑥(𝑠(𝑡),𝑡)=−𝜌𝑙𝑠(̇𝑡),𝑡>0,𝑠(0)=0, (3.1) where the coefficient 𝑞₀>0 characterizes the heat flux on the 𝑥=0 [21] and the control function 𝐹is given by (1.9).

In order to obtain an explicit solution of a similarity type, we define the same transformation given by (2.2). The problem (3.1) and (1.9) are equivalent to the following one:Φ^′′(𝜂)+2𝜂Φ^′(𝜂)=ΛΦ(0),0<𝜂<𝜇₀,Φ^′(0)=−𝑞^∗₀,Φ(𝜇₀Φ)=0,_′(𝜇₀)=−2𝑙𝑐𝜇₀, (3.2)(3.3)(3.4)(3.5) where the dimensionless parameters Λand 𝑞^∗₀ are defined byΛ=4𝛾𝜆₀𝜌𝑐>0,𝑞^∗₀=2𝑎𝑞₀𝑘,𝑠(𝑡)=2𝑎𝜇₀√𝑡 (3.6)(3.7) is the free boundary, where 𝜇₀ is an unknown parameter to be determined.

From (3.2), (3.3), and (3.4), we obtain the similarity solution𝑞Φ(𝜂)=^∗0√𝜋2𝐺(𝜇₀,Λ)[erf(𝜇₀)𝐺(𝜂,Λ)−erf(𝜂)𝐺(𝜇₀,Λ)],0<𝜂<𝜇₀, (3.8) where𝐺(𝑥,Λ)=1+Λ^𝑥₀𝑓₁(𝑟)𝑑𝑟=1+Λ𝜑₁(𝑥), (3.9) and 𝑓₁ is the Dawson's integral and 𝜑₁ is given by (2.23).

By condition (3.5), the unknown parameter 𝜇₀=𝜇₀(Λ,𝑙,𝑐,𝑞^∗₀)must be solution of the following equation:Λerf(𝑥)𝑓₁2(𝑥)=√𝜋𝐺(𝑥,Λ)[exp(−𝑥²)−2𝑙𝑐𝑞^∗₀𝑥],𝑥>0, (3.10) which is equivalent to the following one:𝐻₂(𝑥)=𝐻₃(𝑥),𝑥>0, (3.11) where the real functions 𝐻₂and 𝐻₃ are defined by𝐻₂(𝑥)=Λerf(𝑥)𝑓₁𝐻(𝑥),₃2(𝑥)=√𝜋𝐺(𝑥,Λ)𝐻₁𝐻(𝑥),₁(𝑥)=[exp(−𝑥²)−2𝑙𝑐𝑞^∗₀𝑥]. (3.12)(3.13)(3.14)

Remark 3.1. If Λ=0(i.e., 𝜆₀=0), we have the solution 𝑞Φ(𝜂)=^∗0√𝜋2[erf(𝜇₀₀)−erf(𝜂)],0<𝜂<𝜇₀₀, (3.15) where 𝜇₀₀is the unique solution of the following equation: exp(−𝑥²)=2𝑙𝑐𝑞^∗₀𝑥. (3.16)

In order to solve (3.11), we consider properties of Dawson's integral, error function, and some auxiliary functions, and then we obtain the following result.

Theorem 3.2. For each 𝜆₀<𝜌𝑐/2𝛾, the free boundary problem (3.1), where 𝐹 is defined by (1.9), has a unique similarity solution of the type 𝑢(𝑥,𝑡,𝜆₀𝑞)=⁰𝑎√𝜋𝑘𝐺(𝜇₀(𝜆₀),4𝛾𝜆₀𝑥/𝜌𝑐)[erf(2𝑎√𝑡)𝐺(𝜇₀(𝜆₀),4𝛾𝜆₀)𝜌𝑐−erf(𝜇₀(𝜆₀𝑥))𝐺(2𝑎√𝑡,4𝛾𝜆₀𝑥𝜌𝑐)],0<2𝑎√𝑡<𝜇₀(𝜆₀),𝑡>0,𝑠(𝑡,𝜆₀)=2𝑎𝜇₀(𝜆₀)√𝑡,𝑡>0, (3.17) where 𝜇₀=𝜇₀(𝜆₀) is the unique solution of (3.11), 0<𝜇₀(𝜆₀)<𝜇₀₀.

Proof. We follow a similar method developed in Theorem 2.7.

4. Explicit Solution to a One-Phase Stefan Problem for a Nonclassical Heat Equation with Control Function of the Type 𝐹(𝑢(0,𝑡),𝑡)=(𝜆₀/𝑡)𝑢(0,𝑡) and a Convective Condition at the Fixed Face

In this section, we consider a similar problem to the one given in Section 3 for a convective boundary condition [22, 26] on the fixed face given by𝜌𝑐𝑢_𝑡−𝑘𝑢_𝑥𝑥=−𝛾𝐹(𝑢(0,𝑡),𝑡),0<𝑥<𝑠(𝑡),𝑡>0,𝑘𝑢_𝑥ℎ(0,𝑡)=⁰√𝑡(𝑢(0,𝑡)−𝑓)>0,𝑡>0,𝑢(𝑠(𝑡),𝑡)=0,𝑡>0,𝑘𝑢_𝑥(𝑠(𝑡),𝑡)=−𝜌𝑙𝑠(̇𝑡),𝑡>0,𝑠(0)=0, (4.1) where 𝐹is defined by (1.9) and ℎ₀ characterizes the heat transfer coefficients [22, 26]. To solve this problem, we consider again a similarity type solution given by (2.2). Then, the problem (4.1) and (1.9) are equivalent to the following one:Φ^′′(𝜂)+2𝜂Φ^′(𝜂)=ΛΦ(0),0<𝜂<𝜇₀,Φ^′(0)=ℎ^∗₀(Φ(0)−𝑓),ℎ^∗₀=2𝑎ℎ₀𝑘,Φ(𝜇₀Φ)=0,_′(𝜇₀)=−2𝑙𝑐𝜇₀, (4.2)(4.3)(4.4)(4.5) where the dimensionless parameter Λis defined by (3.6) and𝑠(𝑡)=2𝑎𝜇₀√𝑡 (4.6) is the free boundary, where 𝜇₀ is an unknown parameter to be determined. We obtain the solutionℎΦ(𝜂)=^∗0𝑓√𝜋2[erf(𝜇₀)𝐺(𝜂,Λ)−erf(𝜂)𝐺(𝜇₀,Λ)]𝐺(𝜇₀,Λ)+(ℎ^∗₀√𝜋/2)erf(𝜇₀),0<𝜂<𝜇₀, (4.7) where 𝐺(𝑥,Λ) is given by (3.9).Taking into account the condition (4.5), the unknown parameter 𝜇₀=𝜇₀(Λ,𝑙,𝑐,ℎ^∗₀)must be the solution of the following equation:Λerf(𝑥)𝑓₁2(𝑥)+2Steerf(𝑥)𝑥=√𝜋𝐺(𝑥,Λ)[exp(−𝑥²2)−ℎ^∗₀Ste𝑥],𝑥>0, (4.8) which is equivalent to𝐻^∗₂(𝑥)=𝐻^∗₃(𝑥),𝑥>0, (4.9) where𝐻^∗₂(𝑥)=𝐻₂2(𝑥)+𝐻Steerf(𝑥)𝑥,𝑥>0,_∗32(𝑥)=√𝜋𝐺(𝑥,Λ)[exp(−𝑥²2)−ℎ^∗₀Ste𝑥],𝑥>0, (4.10) and the function 𝐻₂ is defined by (3.12).

Similarly to the previous cases, we can enunciate the following result.

Theorem 4.1. (a) For each Λ<2 (𝜆₀<𝜌𝑐/2𝛾), the free boundary problem (4.1), where 𝐹 is defined by (1.9), has a unique similarity solution given by 𝑢(𝑥,𝑡,𝜆₀)=−ℎ₀𝑎𝑓√𝜋𝑘[erf(𝑥/2𝑎√𝑡)𝐺(𝜇₀(𝜆₀),4𝛾𝜆₀/𝜌𝑐)(ℎ₀𝑎𝑓√𝜋/𝑘)erf(𝜇₀(𝜆₀))+𝐺(𝜇₀(𝜆₀),4𝛾𝜆₀−/𝜌𝑐)erf(𝜇₀(𝜆₀))𝐺(𝑥/2𝑎√𝑡,4𝛾𝜆₀/𝜌𝑐)(ℎ₀𝑎𝑓√𝜋/𝑘)erf(𝜇₀(𝜆₀))+𝐺(𝜇₀(𝜆₀),4𝛾𝜆₀𝑥/𝜌𝑐)],0<2𝑎√𝑡<𝜇₀(𝜆₀),𝑡>0,𝑠(𝑡,𝜆₀)=2𝑎𝜇₀(𝜆₀)√𝑡,𝑡>0, (4.11) where 𝜇₀=𝜇₀(𝜆₀) is the unique solution of (4.9).
(b) Let 𝑀(𝑥)=Λ𝑓₁(𝑥)and 𝑁(𝑥)=2𝑥𝐺(𝑥,Λ) be, there exists a unique solution 𝑥^∗>0 of the equation 𝑀(𝑥)=𝑁(𝑥).
For each Λ>2(𝜆₀>𝜌𝑐/2𝛾) such that 𝑀(𝛼(Λ))−𝑁(𝛼(Λ))<2/ℎ^∗₀Ste, where 0<𝛼(Λ)<𝑥^∗ satisfies 𝑀^′(𝛼(Λ))−𝑁^′(𝛼(Λ))=0,there exists a unique similarity solution to the free boundary problem (3.1), where 𝐹 is defined by (1.9). The solution is given by (4.11).

Acknowledgments

This paper has been partially sponsored by Projects PIP no. 0460 from CONICET-UA (Rosario, Argentina) and Fondo de Ayuda a la Investigacion de la Universidad Austral (Argentina). The authors would like to thank the three referees for their constructive comments which improved the readability of the paper.