Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 35, pp. 1–11.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

EXISTENCE AND UNIQUENESS OF THE P-GENERALIZED

MODIFIED ERROR FUNCTION

JULIETA BOLLATI, JOSÉ A. SEMITIEL, MARÍA F. NATALE, DOMINGO A. TARZIA

Communicated by Jesus Ildefonso Diaz

Abstract. 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.

1. Introduction

Ceratani et al. [5] studied a fusion Stefan problem with variable thermal con-
ductivity 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)

k(T (0, t))
∂T

∂x
(0, t) =

h√
t
[T (0, t)− T

0
], t > 0, (1.2)

T (s(t), t) = T
f
, t > 0, (1.3)

k (T (s(t), t))
∂T

∂x
(s(t), t) = −ρlṡ(t), t > 0, (1.4)

s(0) = 0, (1.5)

where the unknown functions are the temperature T and the free boundary s sep-
arating both phases. The parameters ρ > 0 (density), l > 0 (latent heat per unit
mass), Tf (phase-change temperature), T0 > Tf (bulk temperature), h > 0 (coef-
ficient 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 Tf . As T0 > Tf , a phase-change interface

2010 Mathematics Subject Classification. 34A34, 47H10, 33E30, 34A12, 35R35.
Key words and phrases. Modified error function; generalized modified error function;

nonlinear ordinary differential equation; Banach fixed point theorem; Stefan problem.
c©2020 Texas State University.

Submitted April 19, 2019. Published April 18, 2020.

1



2 J. BOLLATI, J. A. SEMITIEL, M. F. NATALE, D. A. TARZIA EJDE-2020/35

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

k(T ) = k0

(
1 + δ

( T − Tf
T0 − Tf

))
, (1.6)

where δ is a given positive constant and k0 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

[(1 + δy(x))y′(x)]′ + 2xy′(x) = 0, 0 < x < +∞, (1.7a)(
1 + δy(0)

)
y′(0)− γy(0) = 0, (1.7b)

y(+∞) = 1, (1.7c)

where

γ = 2 Bi, Bi =
h
√
α0

k0
(generalized Biot number), (1.8)

α0 =
k0

ρc
(thermal diffusivity). (1.9)

The solution to (1.1)–(1.5) is given as a function of the solution of (1.7) through
the similarity variable x/(2

√
α0t), see [5, 6, 12]. More explanations are given in

[1, 9, 14].
Motivated by [10] we define a generalized thermal conductivity as

k(T ) = k0

(
1 + δ

( T − Tf
T0 − Tf

)p)
, p ≥ 1. (1.10)

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 + δyp(x))y′(x)]′ + 2xy′(x) = 0, 0 < x < +∞, (1.11a)

(1 + δyp(0))y′(0)− γy(0) = 0, (1.11b)

y(+∞) = 1 . (1.11c)

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



EJDE-2020/35 P-GENERALIZED MODIFIED ERROR FUNCTION 3

have been motivated by Stefan problems. This is done imposing different boundary
conditions at the fixed face x = 0: Robin, Dirichlet and Neumann conditions.

2. 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)

K = {h ∈ X : ‖h‖∞ ≤ 1, 0 ≤ h, h(+∞) = 1}. (2.2)

We remark that K is a non-empty closed convex and bounded subset of the
Banach space X with the norm

‖h‖∞ = sup
x∈R+

0

|h(x)| <∞;

see [7, page 2487], [8, page 152], [11, page 132].
In this section we prove existence and uniqueness of the p-GME function (prob-

lem (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 yγ ∈ K
is a solution to problem (1.11) if and only if yγ is a fixed point to the operator
Tγ : K → K given by

Tγ(h)(x) =
1 + γ

∫ x
0
fh(η)dη

1 + γ
∫∞

0
fh(η)dη

, x ≥ 0, (2.3)

with

fh(x) =
1

Ψh(x)
exp

(
− 2

∫ x

0

ξ

Ψh(ξ)
dξ
)
, Ψh(x) = 1 + δhp(x). (2.4)

Proof. Notice first that for each y = yγ ∈ K we can easily obtain

exp(−η2)

1 + δ
≤ fy(η) ≤ exp

(
− η2

1 + δ

)
, (2.5)

from where it follows that

0 <
γ
√
π

2(1 + δ)
< 1 + γ

∫ ∞
0

fy(η)dη ≤ 1 +
γ
√

1 + δ
√
π

2
. (2.6)

Taking into account (2.6), Tγ(y) is a continuous function, since y ∈ X. Also,
according to (2.1)–(2.3) and (2.6), Tγ(y) ∈ K.

Through the substitution v = y′, the ordinary differential equation (1.7a) is
equivalent to

−
Ψ′y(x) + 2x

Ψy(x)
=
v′(x)

v(x)
,

from where we obtain

y(x) = y(0) + c0

∫ x

0

fy(η)dη.

Then, condition (1.7b) is satisfied if and only if c0 = γy(0). In addition, from (1.7c)
we obtain

y(0) =
(

1 + γ

∫ ∞
0

fy(η)dη
)−1

. (2.7)



4 J. BOLLATI, J. A. SEMITIEL, M. F. NATALE, D. A. TARZIA EJDE-2020/35

Therefore, y is a solution to problem (1.11) if and only if y is a fixed point of the
operator Tγ , i.e. y(x) = Tγ(y)(x) for all x ≥ 0. Conversely, if y is a fixed point of
the operator Tγ 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 yγ , Tγ 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 yγ when γ → ∞,
to be presented in Section 3.

By Theorem 2.1, we will focus on proving that Tγ is a contracting mapping on
K. For that purpose, we need the following lemmas.

Lemma 2.3. Let y1, y2 ∈ K, δ ≥ 0, γ > 0, p ≥ 1 and x ≥ 0. Then, the following
estimates hold: √

π

2(1 + δ)
≤
∣∣ ∫ ∞

0

fy1(η)dη
∣∣ ≤ √1 + δ

√
π

2
, (2.8)∣∣ 1

Ψy1(η)
− 1

Ψy2(η)

∣∣ ≤ δp‖y1 − y2‖∞, (2.9)

∣∣ exp
(∫ η

0

−2ξ

Ψy1(ξ)
dξ
)
− exp

(∫ η

0

−2ξ

Ψy2(ξ)
dξ
)∣∣ ≤ 2δpη2

exp( η2

1+δ )
‖y1 − y2‖∞, (2.10)∫ x

0

|fy1(η)− fy2(η)|dη ≤
√
π

2
δp
√

1 + δ(2 + δ)‖y1 − y2‖∞, (2.11)∣∣ 1

1 + γ
∫∞

0
fy1(η)dη

− 1

1 + γ
∫∞

0
fy2(η)dη

∣∣
≤ 2(1 + δ)5/2

γ
√
π

δp(2 + δ)‖y1 − y2‖∞
(2.12)

Proof. We follow the method was developed in [4].
Inequality (2.8) follows from integrating (2.5) in (0,+∞). For inequality (2.9)

we note that from the Mean Value Theorem applied to the function r(x) = xp and
the fact that 1 ≤ Ψy(x) ≤ 1 + δ for all y ∈ K, we obtain∣∣ 1

Ψy1(η)
− 1

Ψy2(η)

∣∣ ≤ δ|yp2(η)− yp1(η)
∣∣ ≤ δp‖y2 − y1‖∞ .

For inequality (2.10), applying the Mean Value Theorem to r(x) = exp(−2x) we
have ∣∣∣ exp

(∫ η

0

−2ξ

Ψy1(ξ)
dξ
)
− exp

(∫ η

0

−2ξ

Ψy2(ξ)
dξ
)∣∣∣

≤ 2 exp
(
− η2

1 + δ

)∫ η

0

∣∣ ξ

Ψy1(ξ)
− ξ

Ψy2(ξ)

∣∣dξ
≤ 2 exp

(
− η2

1 + δ

)
η

∫ η

0

∣∣ 1

Ψy1(ξ)
− 1

Ψy2(ξ)

∣∣dξ .
Taking into account (2.9) we obtain the corresponding estimate. For inequality
(2.11), from items (2.9) and (2.10) we obtain∫ x

0

∣∣fy1(η)− fy2(η)
∣∣dη



EJDE-2020/35 P-GENERALIZED MODIFIED ERROR FUNCTION 5

≤
∫ x

0

{∣∣∣fy1(η)−
exp(−2

∫ x
0

ξ
Ψy2 (ξ)dξ)

Ψy1(η)

∣∣∣+
∣∣∣exp(−2

∫ x
0

ξ
Ψy2 (ξ)dξ)

Ψy1(η)
− fy2(η)

∣∣∣}dη

≤
∫ x

0

{ 1

Ψy1(η)

∣∣∣ exp
(∫ η

0

−2ξ

Ψy1(ξ)
dξ
)
− exp

(∫ η

0

−2ξ

Ψy2(ξ)
dξ
)∣∣∣

+ exp
(∫ η

0

−2ξ

Ψy2(ξ)
dξ
)∣∣∣ 1

Ψy1(η)
− 1

Ψy2(η)

∣∣∣}dη

≤ ‖y1 − y2‖∞δp
∫ x

0

exp
( −η2

1 + δ

)
(2η2 + 1)dη

= ‖y1 − y2‖∞δp
√

1 + δ
[√π

2
(2 + δ) erf

( x√
1 + δ

)
− x
√

1 + δ exp
( −x2

1 + δ

)]
≤
√
π

2
δp
√

1 + δ(2 + δ)‖y1 − y2‖∞.

Inequality (2.12) follows immediately by using (2.6) and (2.11). �

Lemma 2.4. Let γ > 0, p ≥ 1 and

gγ(x) = xp(1 + x)3/2
[
(2 + x)

(
1 + (1 + x)3/2

)
+

2

γ
√
π

(1 + x)
]
, x ≥ 0 .

Then there exist a unique δγ > 0 such that gγ (δγ) = 1.

The above lemma follows immediately from the fact that gγ is an increasing
function, gγ(0) = 0 and limx→∞ gγ(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
yγ ∈ K if and only if 0 ≤ δ < δγ , where δγ is given by Lemma 2.4. Moreover, yγ
is a C∞ function in R+ with the following properties:

y′γ(x) > 0, y′′γ (x) < 0, ∀x ≥ 0. (2.13)

Proof. Let y1, y2 ∈ K and x ≥ 0. Taking into account Lemma 2.3, we have

|Tγ(y1)(x)− Tγ(y2)(x)|

≤
∣∣∣ 1 + γ

∫ x
0
fy1(η)dη

1 + γ
∫∞

0
fy1(η)dη

−
1 + γ

∫ x
0
fy2(η)dη

1 + γ
∫∞

0
fy1(η)dη

∣∣∣
+
∣∣∣ 1 + γ

∫ x
0
fy2(η)dη

1 + γ
∫∞

0
fy1(η)dη

−
1 + γ

∫ x
0
fy2(η)dη

1 + γ
∫∞

0
fy2(η)dη

∣∣∣
≤
γ
∫ x

0
|fy1(η)− fy2(η)|dη∣∣1 + γ
∫∞

0
fy1(η)dη

∣∣
+
∣∣∣1 + γ

∫ x

0

fy2(η)dη
∣∣∣∣∣∣ 1

1 + γ
∫∞

0
fy1(η)dη

− 1

1 + γ
∫∞

0
fy2(η)dη

∣∣∣
≤ gγ(δ)‖y1 − y2‖∞.

Then from Lemma 2.4, if 0 ≤ δ < δγ it follows that Tγ 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). �



6 J. BOLLATI, J. A. SEMITIEL, M. F. NATALE, D. A. TARZIA EJDE-2020/35

3. 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.

T (0, t) = T0 > 0, (3.1)

we obtain the ordinary differential problem

[(1 + δyp(x))y′(x)]′ + 2xy′(x) = 0, 0 < x < +∞, (3.2a)

y(0) = 0, (3.2b)

y(+∞) = 1 . (3.2c)

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

erf(x) =
2√
π

∫ x

0

exp(−z2)dz, x > 0, (3.3)

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

K∗ = {h ∈ X : ‖h‖∞ ≤ 1, 0 ≤ h, h(0) = 0, h(+∞) = 1},
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:

T ∗(h)(x) =

∫ x
0
fh(η)dη∫∞

0
fh(η)dη

, x ≥ 0, (3.4)

with fh 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

y∗(x) = y∗(0) + c∗0

∫ x

0

f∗y (η)dη,

with y∗(0) = 0 and c∗0 =
( ∫∞

0
fh(η)dη

)−1
. Then, using (3.2b) and (3.2c), we obtain

(3.4). Therefore, y∗ is a solution to (3.2) if and only if y∗ is a fixed point of the
operator T ∗. �

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.

Lemma 3.2. Let y∗1 , y
∗
2 ∈ K∗, δ ≥ 0, p ≥ 1 and x ≥ 0. Then∣∣∣ 1∫∞

0
fy∗1 (η)dη

− 1∫∞
0
fy∗2 (η)dη

∣∣∣ ≤ 2(1 + δ)5/2

√
π

δp(2 + δ)‖y∗1 − y∗2‖∞.



EJDE-2020/35 P-GENERALIZED MODIFIED ERROR FUNCTION 7

Lemma 3.3. Let p ≥ 1 and

g∗(x) = xp(1 + x)3/2(2 + x)
(
1 + (1 + x)3/2

)
, x ≥ 0 .

Then there exists a unique δ∗ > 0 such that g∗ (δ∗) = 1.

Theorem 3.4. Problem (3.2) has a unique solution y∗ ∈ K if and only if 0 ≤ δ <
δ∗, where δ∗ is given by Lemma 3.3. Moreover, y∗ is a C∞ function in R+.

Proof. Let y∗1 , y∗2 ∈ K∗ and x ≥ 0. Taking into account Lemmas 2.3 and 3.2 we
obtain

|T ∗(y∗1)(x)− T ∗(y∗2)(x)|

≤
∣∣∣ ∫ x0 fy∗1 (η)dη∫∞

0
fy∗1 (η)dη

−
∫ x

0
fy∗2 (η)dη∫∞

0
fy∗1 (η)dη

∣∣∣+
∣∣∣ ∫ x0 fy∗2 (η)dη∫∞

0
fy∗1 (η)dη

−
∫ x

0
fy∗2 (η)dη∫∞

0
fy∗2 (η)dη

∣∣∣
≤
∫ x

0
|fy∗1 (η)− fy∗2 (η)|dη
|
∫∞

0
fy∗1 (η)dη|

+
∣∣∣ ∫ x

0

fy∗2 (η)dη
∣∣∣∣∣∣ 1∫∞

0
fy∗1 (η)dη

− 1∫∞
0
fy∗2 (η)dη

∣∣∣
≤ g∗(δ∗)‖y∗1 − y∗2‖∞.

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+. �

To problem (1.11), we impose a Robin boundary condition characterized by the
coefficient γ > 0 at x = 0. This condition constitutes a generalization of the
Dirichlet condition, in the sense that taking the limit when γ → ∞ in condition
(1.7b), we obtain condition (3.2b). Now, we show that the solution to problem
(1.11) converges to the solution to problem (3.2) when γ → ∞. For this purpose,
first, we need the following lemmas which proofs are immediate.

Lemma 3.5. For every p ≥ 1, when γ →∞, the following convergence results hold

(a) Tγ(h)(x)→ T ∗(h)(x) for every h ∈ K and x ≥ 0.
(b) gγ(x)→ g∗(x) for every x ≥ 0.
(c) δγ → δ∗.

In addition gγ(x) ≥ g∗(x) and δγ < δ∗ for all x ≥ 0, γ > 0.

Lemma 3.6. Let p ≥ 1 and

C(x) = 2xp(1 + x)3(2 + x), x ≥ 0. (3.5)

Then there exists a unique δ̂ > 0 such that C(δ̂) = 1.

Theorem 3.7. Let p ≥ 1 and 0 ≤ δ < min{δ̂, δγ}. Then ‖yγ − y∗‖∞ → 0 when
γ →∞. Furthermore, the order of convergence is 1/γ when γ →∞.

Proof. First let us note that if 0 ≤ δ < min{δ̂, δγ}, then as δγ < δ∗, we obtain that
yγ and y∗ are well defined because of Theorems 2.5 and 3.4. Then for x ≥ 0 we
have

|yγ(x)− y∗(x)|

=
∣∣∣ (1 + γ

∫ x
0
fyγ (η)dη

)( ∫∞
0
fy∗(η)dη

)
−
( ∫ x

0
fy∗(η)dη

)(
1 + γ

∫∞
0
fyγ (η)dη

)(
1 + γ

∫∞
0
fyγ (η)dη

)( ∫∞
0
fy∗(η)dη

) ∣∣∣



8 J. BOLLATI, J. A. SEMITIEL, M. F. NATALE, D. A. TARZIA EJDE-2020/35

≤
∣∣∣[ ∫ ∞

0

fy∗(η)dη + γ
( ∫ x

0

fyγ (η)dη
)( ∫ ∞

0

fy∗(η)dη
)
−
∫ x

0

fy∗(η)dη

− γ
( ∫ x

0

fy∗(η)dη
)( ∫ ∞

0

fyγ (η)dη
)]/[(

1 + γ

∫ ∞
0

fyγ (η)dη
)( ∫ ∞

0

fy∗(η)dη
)]∣∣∣

=
∣∣∣[ ∫ ∞

x

fy∗(η)dη + γ
( ∫ x

0

fyγ (η)dη
)( ∫ ∞

0

fy∗(η)dη
)

− γ
( ∫ x

0

fy∗(η)dη
)( ∫ ∞

0

fyγ (η)dη
)]/[(

1 + γ

∫ ∞
0

fyγ (η)dη
)( ∫ ∞

0

fy∗(η)dη
)]∣∣∣

≤
∣∣∣[ ∫ ∞

0

fy∗(η)dη + γ
( ∫ x

0

fyγ (η)dη
)( ∫ ∞

0

fy∗(η)dη −
∫ ∞

0

fyγ (η)dη
)

+ γ
( ∫ ∞

0

fyγ (η)dη −
∫ x

0

fy∗(η)dη
)( ∫ ∞

0

fyγ (η)dη
)]

÷
[(

1 + γ

∫ ∞
0

fyγ (η)dη
)( ∫ ∞

0

fy∗(η)dη
)]∣∣∣

≤
[√

1 + δ

√
π

2
+ γ
√

1 + δ

√
π

2

( ∫ ∞
0

|fy∗(η)− fyγ (η)|dη
)

+ γ
√

1 + δ

√
π

2

( ∫ x

0

|fy∗(η)− fyγ (η)|dη
)]/[ γ

√
π

2(1 + δ)

√
π

2(1 + δ)

]
≤
√

1 + δ
√
π

2 + 2γ
√

1 + δ
√
π

2

∫∞
0
|fy∗(η)− fyγ (η)|dη

γπ
4(1+δ)2

≤ 4(1 + δ)2

γπ

(√
1 + δ

√
π

2
+ γ

π

4
δp(1 + δ)(2 + δ)||yγ − y∗||∞

)
≤ 2(1 + δ)5/2

γπ
+ 2(1 + δ)3δp(2 + δ)‖yγ − y∗‖∞ .

The above inequalities are obtained by applying Lemma 2.3, and they lead to

(1− C(δ))‖yγ − y∗‖∞ ≤
1

γ

(2(1 + δ)5/2

√
π

)
,

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 < δ̂, then 0 ≤ C(δ) < 1 because
of Lemma 3.6. �

4. Existence and uniqueness considering a Neumann condition

In this section we consider a solidification Stefan problem with a Neumann con-
dition at the fixed face x = 0, given by

ρc
∂T

∂t
=

∂

∂x

(
k(T )

∂T

∂x

)
, 0 < x < s(t), t > 0, (4.1)

k(T (0, t))
∂T

∂x
(0, t) =

q0√
t
, t > 0, (4.2)

T (s(t), t) = T
f
, t > 0, (4.3)

k(T (s(t), t))
∂T

∂x
(s(t), t) = ρlṡ(t), t > 0, (4.4)

s(0) = 0, (4.5)



EJDE-2020/35 P-GENERALIZED MODIFIED ERROR FUNCTION 9

where the unknown functions are the temperature T and the free boundary s sep-
arating both phases. The parameters ρ > 0 (density), l > 0 (latent heat per unit
mass), Tf (phase-change temperature), q0 > 0 (characterizes the heat flux on the
fixed face x = 0 of the face-change material which can be determined experimen-
tally) and c > 0 (specific heat) are all known constants. In this case, the thermal
conductivity k is defined as

k(T ) = k0

(
1 + δ

( T
Tf

)p)
, p ≥ 1, (4.6)

where δ is a given positive constant and k0 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 + δyp(x))y′(x)]′ + 2xy′(x) = 0, 0 < x < +∞, (4.7a)

(1 + δyp(0))y′(0) = γ∗, (4.7b)

y(+∞) = 1 , (4.7c)

where

γ∗ = 2 Bi∗ with Bi∗ =
q0
√
α0

k0Tf
. (4.8)

In a similar way to the above sections we can state the following results:

Theorem 4.1. Let δ ≥ 0, p ≥ 1 and 0 < γ∗ ≤ 2√
π(1+δ)

. Then yγ∗ ∈ K is a

solution to (4.7) if and only if yγ∗ is a fixed point of the operator Tγ∗ : K → K
given by

Tγ∗(h)(x) = 1− γ∗
∫ +∞

x

fh(η)dη, x ≥ 0, (4.9)

with fh defined by (2.4) and K given by (2.2).

Proof. Given yγ∗ ∈ K and taking into account (2.5), we obtain

0 <
γ∗ erfc(x)

1 + δ
≤ γ∗

∫ ∞
x

fy(η)dη <
γ∗
√

1 + δ
√
π

2
≤ 1. (4.10)

Note that from (4.9) we have that Tγ∗(yγ∗) is an analytic function, since yγ∗ ∈ X.
Also, according to (4.9) and (4.10), Tγ∗(yγ∗) ∈ K.

In a similar way as in the proof of Theorem 2.1, yγ∗ is a solution to (4.7) if and
only if yγ∗ is a fixed point of the operator Tγ∗ . �

Theorem 4.2. Let p ≥ 1, δ > 0 and 0 < γ∗ ≤ 2√
π(1+δ)

. Then (4.7) has a unique

C∞ solution yγ∗ ∈ K if and only if δ < δγ∗ where δγ∗ is the unique solution to the
equation g(x) = 1, with

g(x) = x
p√
π

[
(1 + x)

(√
1 + x exp(−1

4
) +
√
π
)

+
√
π
]
.

Proof. Let y1γ∗ , y2γ∗ ∈ K and x ≥ 0. Taking into account (2.9) and (2.10) we
obtain

|Tγ∗(y1γ∗ )(x)− Tγ∗(y2γ∗ )(x)|

≤ ‖y1 − y2‖∞δpγ∗
[
(1 + δ)3/2

( x√
1 + δ

exp
(
− x2

1 + δ

)
+

√
π

2

)
+
√

1 + δ

√
π

2

]



10 J. BOLLATI, J. A. SEMITIEL, M. F. NATALE, D. A. TARZIA EJDE-2020/35

≤ δ p√
π

[
(1 + δ)

(√
1 + δ exp(−1

4
) +
√
π
)

+
√
π
]
‖y1γ∗ − y2γ∗ ‖∞

≤ g(δ)‖y1γ∗ − y2γ∗‖∞.

Since g is an increasing function such that g(0) = 0 and g(+∞) = +∞, there exists
a unique δγ∗ > 0 with g(δγ∗) = 1.

Then, if 0 ≤ δ < δγ∗ it follows that Tγ∗ 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 ad-
dition 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.

Acknowledgments. The author gratefully acknowledges the helpful suggestions
from the reviewer to improve this article. This work was partially sponsored by the
European Union’s Horizon 2020 research, by the innovation programme under the
Marie Sklodowska-Curie Grant Agreement No. 823731 CONMECH, by the Project
PIP No. 0275 from CONICET - Univ. Austral, Rosario, Argentina, and by the
ANPCyT PICTO Austral No. 0090.

References

[1] Alexiades, V.; Solomon, A. D.; Mathematical Modelling of Melting and Freezing Processes.
Hemisphere-Taylor, Francis, Washington, 1993.

[2] Bougoffa, L.; A note on the existence and uniqueness solutions of the modified error function.

Mathematical Methods in Applied Science, 2018 (2018), 1–9.
[3] Carslaw, H. S.; Jaeger, J. C.; Conduction of Heat in Solids. Clarendon Press, Oxford, 1959.

[4] Ceretani, A. N.; Salva, N. N.; Tarzia, D. A.; Existence and uniqueness of the modified error

function. Applied Mathematics Letters, 70 (2017), 14-17.
[5] Ceretani, A. N.; Salva, N. N.; Tarzia, D. A.; An exact solution to a Stefan problem with

variable thermal conductivity and a Robin boundary condition, Nonlinear Analysis: Real

World Applications 40 (2018), 243-259.
[6] Cho, S. H.; Sunderland, J., E.; Phase-change problems with temperature-dependent thermal

conductivity. Journal of Heat Transfer, 96 (1974), 214-217.

[7] Dhage, B. C.; Global attractivity results for nonlinear functional integral equations via a
Krasnoselskii type fixed point theorem, Nonlinear Analysis, 70 (2009), 2485-2493.

[8] Dhage, B. C.; Some characterization of nonlinear first order differential equations of un-
bounded intervals, Differential Equations and Applications, 2 (2010), 151-162.

[9] Gupta, S. C.; The classical Stefan problem. Basic concepts, modelling and analysis. Elsevier,

Amsterdam, 2003.
[10] Kumar, A. Rajeev; A moving boundary problem with variable specific heat and thermal

conductivity, Journal of King Saud University Sci., 32 (2020), 384-389.
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132-140.
[12] Oliver D. L. R.; Sunderland, J. E.; A phase-change problem with temperature-dependent

thermal conductivity and specific heat. International Journal of Heat Mass Transfer, 30
(1987), 2657-2661.



EJDE-2020/35 P-GENERALIZED MODIFIED ERROR FUNCTION 11

[13] Salva, N. N.; Tarzia, D. A.; A sensitivity analysis for the determination of unknown thermal

coefficients through a phase-change process with temperature-dependent thermal conductiv-

ity. International Communication in Heat and Mass Transfer, 38 (2011), 418-424.
[14] Tarzia, D. A.; Explicit and approximated solutions for heat and mass transfer problem with

a moving interface, Chapter 20, in Advanced Topics in Mass Trasnfer, M. El-Amin (Ed.),

InTech Open Access Publisher, Rijeka, (2011), 439-484.

Julieta Bollati
CONICET, Argentina.

Depto. Matemática, FCE, Univ. Austral, Paraguay 1950, S2000FZF Rosario, Argentina

Email address: jbollati@austral.edu.ar

José A. Semitiel

Depto. Matemática, FCE, Univ. Austral, Paraguay 1950, S2000FZF Rosario, Argentina
Email address: jsemitiel@austral.edu.ar

Maŕıa F. Natale
Depto. Matemática, FCE, Univ. Austral, Paraguay 1950, S2000FZF Rosario, Argentina

Email address: fnatale@austral.edu.ar

Domingo A. Tarzia

CONICET, Argentina.

Depto. Matemática, FCE, Univ. Austral, Paraguay 1950, S2000FZF Rosario, Argentina
Email address: dtarzia@austral.edu.ar


	1. Introduction
	2. Existence and uniqueness of the p-GME function
	3. Asymptotic behavior of p-GME function when 
	4. Existence and uniqueness considering a Neumann condition
	Conclusion
	Acknowledgments

	References