Exact Solutions for Nonclassical Stefan Problems

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 casewe 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.


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: where functions f f t and h h x are continuous real functions, and F is a given function of two variables.A particular and interesting case is the following: International Journal of Differential Equations 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 F so 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: 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 0 as follows: where λ 0 > 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 0 as follows: In this case, a heat flux boundary condition or a convective boundary condition can be considered at the fixed face x 0 in 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 λ 0 < ρc/2γ in Theorems 3.2 and 4.1 .

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 λ 0 / √ 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: where the thermal coefficients k, ρ, c, l, γ are 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 define where a 2 k/ρc is the diffusion coefficient of the phase change material.The problem 2.1 and 1.8 become where the dimensionless parameter λ is defined by and the free boundary s t must be of the type where η 0 is an unknown parameter to be determined later.The general solution of the differential equation 2.3 is given by where C 1 and C 2 are arbitrary constants, and 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 obtain where Taking into account condition 2.6 , the unknown parameter η 0 η 0 λ, Ste must be the solution of the following equation: where Ste fc/l > 0 is the Stefan's number.Equation 2.13 is equivalent to the following one: where the real functions W 1 and W 2 are defined by Remark 2.1.If λ 0 i.e., λ 0 0 , then the problem 2.1 and 1.8 represented the classical Lamé-Clapeyron problem 25 .In this case, there exists a unique solution η 00 of 2.17 equivalent to 2.13 given by where and the explicit solution is given by 2, 23 :

2.19
In order to solve 2.14 , we will study firstly the behavior of function f 1 .We obtain some preliminary properties.

2.21
where 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 Next, we define the following auxiliary functions:

2.23
We have the following results.

Lemma 2.3.
a Function ϕ satisfies the following properties: b Function ϕ satisfies the following properties: c Function ϕ satisfies the following properties: d Function ϕ satisfies the following properties: International Journal of Differential Equations e Function ϕ 5 satisfies the following properties: 2.25 iv ϕ 5 x > 0, for all x > 0.
f Function ϕ 6 satisfies the following properties: Proof. a Taking into account properties of f 1 , we have and v holds.If we consider Lemma 2.2 v , we get ϕ 1 ∞ ∞ and we have lim then iv and vi hold.
To prove vii , we consider b From the definition of ϕ 4 , we obtain i and ii .To prove iii , we have

2.29
International Journal of Differential Equations

2.30
If we suppose that lim which is a contradiction.If we suppose that lim which is also a contradiction.Therefore, lim x → ∞ x 2xf 1 x − 1 0 and iii hold.Taking into account ii , we have ϕ 4 x −1 4xf 1 x − 2x 2 2xf 1 x − 1 , then ϕ 4 0 −1 and if we consider iii we have ϕ 4 ∞ 0 .From properties of f 1 , we have Now, we are in conditions to enunciate properties of functions W 1 and W 2 in order to study after 2.14 .
Lemma 2.5.The functions W 1 x and W 2 x , defined by 2.15 and 2.16 , respectively, satisfy the following properties.

International Journal of Differential Equations a Properties of function W
vi W 1 η 00 0, where η 00 is the unique solution of 2.17 , vii

2.36
where iii there exists a unique Proof. a Taking into account the definition of the function W 1 , we get i and ii .
iii We have lim where Q is the function defined by which satisfies the following properties: vi Taking into account i , iii , and v , we get that there exists a unique zero of W 1 which is given by η 00 ,the unique solution of 2.17 .
vii We have Since sign W 1 x sign 4 1 Ste x 2 − 3 − 2Ste ,then we obtain vii .b Taking into account Lemmas 2.2 and 2.3, we have i and ii .We can write x satisfies W 2 0 0. Then iv , vi , vii , viii , and ix hold.

International Journal of Differential Equations
We have then taking into account the properties of ϕ 2 and f 1 , we get that there exists a unique x 4 > 0 such that Moreover, we have

2.48
In the same way, we have

2.49
Then, if we consider the properties of the functions ϕ 1 and ϕ 2 , we have that there exists a unique x 3 such that W 2 x 3 0.Moreover, W 2 x 3 −2x 2 3 f 1 x 3 − Ste ϕ 5 x 3 < 0 and then v holds.
To prove x , we take into account that

2.50
where F x x 0 exp r 2 dr and F 0 was defined in 2.18 .Then by using 2.17 , we have International Journal of Differential Equations 13 Lemma 2.6.For each λ > 0, there exists a unique solution η 0 of 2.14 .This solution η 0 η 0 λ satisfies the following properties: iii η 0 η 0 λ is an increasing function on λ,

2.52
where η 00 and x 4 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 η 0 of 2.14 .Let 0 < λ 1 < λ 2 be given, taking into account properties of function W 2 , we obtain that the real functions Z 1 and Z 2 defined by satisfy the following properties:
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

14
International Journal of Differential Equations

Explicit Solution to a One-Phase Stefan Problem for a Nonclassical Heat Equation with Control Function of
the Type F u 0, t , t λ 0 /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 0 given by the following conditions: where the coefficient q 0 > 0 characterizes the heat flux on the x 0 21 and the control function F is 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: where the dimensionless parameters Λ and q * 0 are defined by is the free boundary, where μ 0 is an unknown parameter to be determined.From 3.2 , 3.3 , and 3.4 , we obtain the similarity solution International Journal of Differential Equations 15 where and f 1 is the Dawson's integral and ϕ 1 is given by 2.23 .By condition 3.5 , the unknown parameter μ 0 μ 0 Λ, l, c, q * 0 must be solution of the following equation: which is equivalent to the following one: where the real functions H 2 and H 3 are defined by

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

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 by ρcu t − ku xx −γF u 0, t , t , 0 < x < s t , t > 0, where F is defined by 1.9 and h 0 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: where the dimensionless parameter Λ is defined by 3.6 and is the free boundary, where μ 0 is an unknown parameter to be determined.We obtain the solution where G x, Λ is given by 3.9 .Taking into account the condition 4.5 , the unknown parameter μ 0 μ 0 Λ, l, c, h * 0 must be the solution of the following equation: where μ 0 μ 0 λ 0 is the unique solution of 4.9 .
International Journal of Differential Equations b Let M x Λf 1 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 λ 0 > ρc/2γ such that M α Λ −N α Λ < 2/h * 0 Ste, 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 .
then by using the properties of f 1 and b we obtain the properties of ϕ 5 .fWehaveϕ6 x Ste−2 1 Ste xf 1 x Ste−2 1 Ste ϕ 3x , and from the properties of ϕ 3 ,we obtain i -v .
x 0 f 1 r dr, then from a and b iii we get i -vi .eAs we haveϕ 5 x f 1 x − xf 1 x f 1 x ϕ 4 x , H 2 is defined by 3.12 .Similarly to the previous cases, we can enunciate the following result.
Theorem 4.1.a For each Λ < 2 (λ 0 < ρc/2γ , the free boundary problem 4.1 , where F is defined by 1.9 , has a unique similarity solution given by