EXISTENCE AND UNIQUENESS OF THE P-GENERALIZED MODIFIED ERROR FUNCTION

. In this article, we deﬁne a p-generalized modiﬁed error function as the solution to a non-linear ordinary diﬀerential equation of second order, with a Robin type boundary condition at x = 0. We prove existence and uniqueness of a non-negative C ∞ solution by using a ﬁxed point argument. We show that the p-generalized modiﬁed error function converges to the p-modiﬁed error function deﬁned as the solution to a similar problem with a Dirichlet boundary condition. In both problems, for p = 1, the generalized modiﬁed error function and the modiﬁed error function are recovered. In addition, we analyze the existence and uniqueness of solution to a problem with a Neumann boundary condition.


Introduction
Ceratani et al. [5] studied a fusion Stefan problem with variable thermal conductivity and a Robin boundary condition at the fixed face x = 0.They studied ρc ∂T ∂t = ∂ ∂x k(T ) ∂T ∂x , 0 < x < s(t), t > 0, (1.1) T (s(t), t) = T f , t > 0, ( k (T (s(t), t)) ∂T ∂x (s(t), t) = −ρl ṡ(t), t > 0, (1.4) where the unknown functions are the temperature T and the free boundary s separating both phases.The parameters ρ > 0 (density), l > 0 (latent heat per unit mass), T f (phase-change temperature), T 0 > T f (bulk temperature), h > 0 (coefficient that characterizes the heat transfer at x = 0), and c (specific heat) are all known constants.Problem (1.1)-(1.5) is a phase-change problem known in the literature as a Stefan problem.It corresponds to the melting of a semi-infinite material which is initially solid at the phase-change temperature T f .As T 0 > T f , a phase-change interface x = s(t), t > 0 is beginning at t = 0 with the initial position s(0) = 0.Then, the temperature of the liquid phase is T = T (x, t) defined in the domain 0 < x < s(t), t > 0, and the temperature of the solid phase is T = 0 defined in the domain x > s(t), t > 0.
In [6], the thermal conductivity k is defined as where δ is a given positive constant and k 0 is the reference thermal conductivity.
The existence of a solution to (1.1)-(1.5)when the thermal conductivity k(T ) is defined by (1.6) has been proved through the existence of what the authors in [5] called a generalized modified error function (GME), which is defined as the solution to the ordinary differential where The solution to (1.1)-(1.5) is given as a function of the solution of (1.7) through the similarity variable x/(2 √ α 0 t), see [5,6,12].More explanations are given in [1,9,14].
Motivated by [10] we define a generalized thermal conductivity as Then the existence of a solution to (1.1)-(1.5)with k given by (1.10) will be studied through the p-generalized modified error function (p-GME) which we define as the solution to the nonlinear differential problem (1 + δy p (0))y (0) − γy(0) = 0, (1.11b) Note that when p = 1, we recover the problem studied in [4,5] and originally defined in [6,12].Others studies for p = 1 can be found in [2,13].In that sense, the p-GME function constitutes a mathematical generalization of the GME function.
With the purpose of proving existence and uniqueness of the p-GME function, i.e. a solution to (1.11), we define a convenient contracting mapping, in Section 2. In Section 3, we study the asymptotic behavior of the p-GME function when γ → ∞.We will show that this function converges to the solution of an ordinary differential equation that arises by changing the Robin condition at x = 0 [3] by a Dirichlet condition.Finally, in Section 4 we change the Robin condition by a Neumann condition in a solidification process and analyze the existence and uniqueness of a new ordinary differential problem.In conclusion, the aim of this paper is to prove existence and uniqueness of a solution to three ordinary differential problems that have been motivated by Stefan problems.This is done imposing different boundary conditions at the fixed face x = 0: Robin, Dirichlet and Neumann conditions.

Existence and uniqueness of the p-GME function
Let us define X = {h : R + 0 → R : h is a bounded and continuous real-valued function}, (2.1) We remark that K is a non-empty closed convex and bounded subset of the Banach space X with the norm [8, page 152], [11, page 132].
In this section we prove existence and uniqueness of the p-GME function (problem (1.11)) by using the Banach fixed point theorem.First, we show that the ordinary differential problem (1.11) becomes equivalent to an integral equation.We consider that γ is a parameter for problem (1.11), and in Section 3 we will study the asymptotic behavior when γ → ∞.
Theorem 2.1.Let δ ≥ 0, γ > 0, p ≥ 1.For each γ > 0, the function y γ ∈ K is a solution to problem (1.11) if and only if y γ is a fixed point to the operator T γ : K → K given by with (2.4) Proof.Notice first that for each y = y γ ∈ K we can easily obtain from where it follows that Taking into account (2.6), T γ (y) is a continuous function, since y ∈ X.Also, according to (2.1)-(2.3)and (2.6), T γ (y) ∈ K.
Through the substitution v = y , the ordinary differential equation (1.7a) is equivalent to from where we obtain Then, condition (1.7b) is satisfied if and only if c 0 = γy(0).In addition, from (1.7c) we obtain Therefore, y is a solution to problem (1.11) if and only if y is a fixed point of the operator T γ , i.e. y(x) = T γ (y)(x) for all x ≥ 0. Conversely, if y is a fixed point of the operator T γ we obtain immediately that (1.7c) is verified, and y(0) is given by (2.7).Then, by differentiation (1.7a) and (1.7b) hold, and then y is a solution of (1.11).
Remark 2.2.The notation y γ , T γ is adopted to emphasize the dependence of the solution to (1.11) on γ, although it also depends on p and δ.This fact is going to facilitate the subsequent analysis of the asymptotic behavior of y γ when γ → ∞, to be presented in Section 3.
By Theorem 2.1, we will focus on proving that T γ is a contracting mapping on K.For that purpose, we need the following lemmas.
The above lemma follows immediately from the fact that g γ is an increasing function, g γ (0) = 0 and lim x→∞ g γ (x) = +∞.Now, we are able to formulate the following result.
Theorem 2.5.Let γ > 0 and p ≥ 1.The problem (1.11) has a unique solution y γ ∈ K if and only if 0 ≤ δ < δ γ , where δ γ is given by Lemma 2.4.Moreover, y γ is a C ∞ function in R + with the following properties: (2.13) Proof.Let y 1 , y 2 ∈ K and x ≥ 0. Taking into account Lemma 2.3, we have Then from Lemma 2.4, if 0 ≤ δ < δ γ it follows that T γ is a contracting mapping what allows to apply the Banach fixed point theorem.Therefore, the problem (1.11) has a unique non-negative continuous solution.Moreover, by differentiation and easy computation the solution is a C ∞ function in R + with the useful properties (2.13).

Asymptotic behavior of p-GME function when γ → ∞
In this section if we consider the Stefan problem (1.1)-(1.5)and we change the Robin condition (1.2) by a Dirichlet condition.
we obtain the ordinary differential problem [(1 + δy p (x))y (x)] + 2xy (x) = 0, 0 < x < +∞, (3.2a) ) For the special case p = 1, the solution to this problem is called modified error function (ME) and was considered in [2,4,5,6,12].In [4] the existence and uniqueness of the ME function was proved.Moreover, if it is considered δ = 0, the classical error function defined by arises as a solution.
In a similar way to the above section we can analyze the existence and uniqueness of the p-modified error function (p-ME), which is defined as the solution to problem (3.2) and constitutes a generalization of the ME function.Now, let us define where X is given by (2.1).We remark that K * is a non-empty closed convex and bounded subset of the Banach space X.We will show that the ordinary differential problem (3.2) becomes equivalent to an integral equation.
Theorem 3.1.Let δ ≥ 0, p ≥ 1.Then the function y * ∈ K * is a solution to (3.2) if and only if y * is a fixed point of the operator T * : K * → K * given by: with f h defined by (2.4).
Proof.In a similar way as in the proof of Theorem 2.1, the operator T * is well defined and it is easy to see that To prove that the operator T * is a contracting mapping on K * , we enunciate the following lemmas which proofs are analogous to Lemma 2.3 and Lemma 2.4.
Then there exists a unique δ * > 0 such that g * (δ * ) = 1. Theorem Then from Lemma 3.3, if 0 ≤ δ < δ * it follows that T * is a contracting mapping what allows to apply the Banach fixed point theorem.Therefore, the problem (3.2) has a unique non-negative continuous solution which is also a C ∞ function by simple differentiation in R + .
Proof.First let us note that if 0 ≤ δ < min{ δ, δ γ }, then as δ γ < δ * , we obtain that y γ and y * are well defined because of Theorems 2.5 and 3.4.Then for x ≥ 0 we have The above inequalities are obtained by applying Lemma 2.3, and they lead to with C defined by (3.5).Finally, the desired convergence and order of convergence in Theorem 3.7 are obtained by noting that if 0 ≤ δ < δ, then 0 ≤ C(δ) < 1 because of Lemma 3.6.

Existence and uniqueness considering a Neumann condition
In this section we consider a solidification Stefan problem with a Neumann condition at the fixed face x = 0, given by T (s(t), t) = T f , t > 0, ( k(T (s(t), t)) ∂T ∂x (s(t), t) = ρl ṡ(t), t > 0, (4.4) where the unknown functions are the temperature T and the free boundary s separating both phases.The parameters ρ > 0 (density), l > 0 (latent heat per unit mass), T f (phase-change temperature), q 0 > 0 (characterizes the heat flux on the fixed face x = 0 of the face-change material which can be determined experimentally) and c > 0 (specific heat) are all known constants.In this case, the thermal conductivity k is defined as where δ is a given positive constant and k 0 is the reference thermal conductivity.
In a similar way as in previous sections, this Stefan problem leads us to the study the ordinary differential problem (1 + δy p (0))y (0) = γ * , (4.7b) where In a similar way to the above sections we can state the following results: . Then y γ * ∈ K is a solution to (4.7) if and only if y γ * is a fixed point of the operator T γ * : K → K given by with f h defined by (2.4) and K given by (2.2).
In a similar way as in the proof of Theorem 2. Since g is an increasing function such that g(0) = 0 and g(+∞) = +∞, there exists a unique δ γ * > 0 with g(δ γ * ) = 1.Then, if 0 ≤ δ < δ γ * it follows that T γ * is a contracting mapping what allows to apply the Banach fixed point theorem.Therefore, the problem (4.7) has a unique non-negative continuous solution which is also a C ∞ function.

Conclusion.
In this article, the ordinary differential problems studied in [4,5] have been generalized by defining what we call the p-GME function and the p-ME function corresponding to the case when a Robin or Dirichlet boundary condition are imposed at x = 0, respectively.In both problems, existence and uniqueness of C ∞ solution has been proved by defining convenient contracting mappings.In addition it has been studied the behavior of the p-GME function when the coefficient γ that characterizes the Robin condition goes to infinity, obtaining its convergence to the p-ME function with an order of convergence of the type 1/γ when γ → ∞.Finally, existence and uniqueness of a solution to a solidification problem with a Neumann condition has been studied.