doi:10.1155/2010/868059
Research Article
Exact Solutions for Nonclassical Stefan Problems
Adriana C. Briozzo and Domingo A. Tarzia
Departamento de Matem´atica-CONICET, FCE, Universidad Austral, Paraguay 1950, S2000FZF Rosario, Argentina
Correspondence should be addressed to Domingo A. Tarzia,[email protected] Received 17 February 2010; Revised 17 June 2010; Accepted 3 August 2010
Academic Editor: Maurizio Grasselli
Copyrightq2010 A. C. Briozzo and D. A. Tarzia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider one-phase nonclassical unidimensional Stefan problems for a source functionFwhich depends on the heat flux, or the temperature on the fixed facex0. 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 st.Phase change problems appear frequently in industrial processes and other problems of technological interest1–4.
Nonclassical heat conduction problem for a semi-infinite material was studied in5–
11. A problem of this type is the following:
i ut−uxx−FWt, t, x >0, t >0, ii u0, t ft, t >0,
iii ux,0 hx, x >0,
1.1
where functionsfftand hhxare continuous real functions, andFis a given function of two variables. A particular and interesting case is the following:
FWt, t √λ0
tWt λ0>0, 1.2
where W Wt represents the heat flux on the boundary x 0, that is Wt ux0, t.Problems of the types1.1and1.2can be thought of by modelling of a system of temperature regulation in isotropic mediums10,11, with a nonuniform source term which provides a cooling or heating effect depending upon the properties ofFrelated to the course of the heat fluxor the temperature in other casesat the boundaryx010.
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 fromRinto itself, was studied in 8 regarding existence, uniqueness, and asymptotic behavior. Moreover, in5conditions are given on the nonlinearity of the source termF so as to accelerate the convergence of the solution to the steady-state solution. Other references on the subject are in7,12,13.
Nonclassical free boundary problems of the Stefan type were recently studied in14–
16from a theoretical point of view by using an equivalent formulation through a system of second kind Volterra integral equations17–19. A large bibliography on free boundary problems for the heat equation was given in20.
In this paper, firstly we consider a free boundary problem which consists in determining the temperature u ux, t and the free boundary x st such that the following conditions are satisfied:
ρcut−kuxx −γFWt, t, 0< x < st, t >0, 1.3
u0, t f >0, t >0, 1.4
ust, t 0, t >0, 1.5
kuxst, t −ρlst,˙ t >0, 1.6
s0 0, 1.7
where the thermal coefficientsk, ρ, c, l, γ >0,the boundary temperaturef >0,and the control functionFdepend on the evolution of the heat flux at the boundaryx0 as follows:
Wt ux0, t, FWt, t Fux0, t, t √λ0
tux0, t, 1.8 whereλ0>0 is a given constant.The existence and the uniqueness of the solution of a general free boundary problem of the type1.3–1.8was given recently in14,15. Moreover, we consider other two free boundary problems which consist in determining the temperature uux, tand the free boundaryxstsuch that1.3,1.5,1.6, and1.7are satisfied, and in these cases the control functionF depends on the evolution of the temperature at the boundaryx0 as follows:
Wt u0, t, FWt, t Fu0, t, t λ0
t u0, t, λ0>0. 1.9 In this case, a heat flux boundary condition
kux0, t −q0
√t >0, t >0 1.10
or a convective boundary condition
kux0, t q0
√t
u0, t−f
>0, t >0 1.11
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. InSection 2, we show an explicit solution of a similarity type for the nonclassical one-phase Stefan problem1.3–1.7for a control function Fgiven by1.8.
In Sections3and4, 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 functionF is given by1.9 instead of1.8and we take into account the heat flux condition1.10or the convective condition1.11at the fixed facex0, respectively.
The restrictions on data we have obtained for these two free boundary problems with a heat flux boundary condition1.10or a convective boundary condition1.11at the fixed facex 0 can be interpreted in the same way as we have obtained in the classical Stefan problem with the same boundary conditions in21,22in order to have an instantaneous phase-change problemsee, e.g., sufficient conditionλ0< ρc/2γin Theorems3.2and4.1.
2. Explicit Solution to a One-Phase Stefan Problem for a Nonclassical Heat Equation with Control Function of the Type Fu
x0 , t, t λ
0/ √
tu
x0 , 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:
ρcut−kuxx−γFux0, t, t, 0< x < st, t >0, u0, t f >0, t >0,
ust, t 0, t >0, kuxst, t −ρlst,˙ t >0,
s0 0,
2.1
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 extremumx0,is given by1.8.
In order to obtain an explicit solution of a similarity type, we define
Φ η
ux, t, η x
2a√
t, 2.2
wherea2 k/ρcis the diffusion coefficient of the phase change material. The problem2.1 and1.8become
Φ η
2ηΦ η
2λΦ0, 0< η < η0, 2.3
Φ0 f, 2.4 Φ
η0
0, 2.5
Φ η0
−2l
cη0, 2.6
where the dimensionless parameterλis defined by
λ γλ0
ρca >0, 2.7
and the free boundarystmust be of the type st 2aη0
√t, 2.8
where η0 is an unknown parameter to be determined later. The general solution of the differential equation2.3is given by
Φ η
C2C1
√ π 2 erf
η 2λ
η
0
f1zdz
, 2.9
whereC1and C2 are arbitrary constants, and
erfx 2
√π x
0
exp
−z2
dz, f1x exp
−x2 x
0
exp 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, from2.3,2.4, and2.5we obtain
Φ η
f 1− E η, λ E
η0, λ
, 0< η < η0, 2.11
where
Ex, λ erfx 4λ
√π x
0
f1rdr. 2.12
Taking into account condition2.6, the unknown parameterη0 η0λ,Stemust be the solution of the following equation:
√Ste π
exp
−x2
2λf1x x
erfx 4λ
√π x
0
f1zdz
, x >0, 2.13
where Ste fc/l > 0 is the Stefan’s number. Equation2.13is equivalent to the following one:
W1x 2λW2x, x >0, 2.14
where the real functionsW1 andW2are defined by
W1x Ste exp
−x2
−√
πxerfx, 2.15
W2x 2x x
0
f1rdr−Stef1x. 2.16
Remark 2.1. Ifλ 0i.e.,λ0 0, then the problem2.1and1.8represented the classical Lam´e-Clapeyron problem 25. In this case, there exists a unique solution η00 of 2.17 equivalent to2.13given by
F0x Ste
√π, x >0, 2.17
where
F0x xerfxexp x2
, 2.18
and the explicit solution is given by2,23:
ux, t f 1− erf η erf
η00
, 0< η x 2a√
t < η00, st 2aη00
√t.
2.19
In order to solve2.14, we will study firstly the behavior of functionf1.We obtain some preliminary properties.
Lemma 2.2. The Dawson’s integral satisfies the following properties:
if10 0, iif1∞ 0, iii
f1x 1−2xf1x
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
>0 if 0< x < x1, 0 ifxx1,
<0 ifx > x1,
2.20
wherex10.924, f1x10.541, iv
f1x −2
1f1x
1−2x2
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
<0 if 0< x < x2, 0 ifxx2,
>0 ifx > x2,
2.21
wherex21.502, f1x20.428, vlimx→∞2xf1x 1.
Proof. The propertiesi–ivhave been proved in23, page 298 see also24, pages 42–45 vBy the L’Hopital Theorem, we have
xlim→∞2xf1x lim
x→∞
2xx
0 exp r2
dr expx2 lim
x→∞
x
0 exp r2
drxexp x2 xexpx2 lim
x→∞
1
x
0 exp r2
dr xexpx2
lim
x→∞
1f1x x
1,
2.22
thenvholds.
Next, we define the following auxiliary functions:
ϕ1x x
0
f1rdr, ϕ2x xϕ1x x x
0
f1rdr, ϕ3x xf1x, ϕ4x x
2xf1x−1
−xf1x, ϕ5x f1x−xf1x, ϕ6x Ste−21Stexf1x.
2.23
We have the following results.
Lemma 2.3.
aFunctionϕ1 satisfies the following properties:
iϕ10 0, iiϕ1x f1x, iiiϕ10 0, ivϕ1∞ ∞,
v
ϕ1x f1x 1−2xf1x
⎧⎪
⎪⎨
⎪⎪
⎩
>0 if 0< x < x1, 0 ifxx1,
<0 ifx > x1,
2.24
vilimx→∞ϕ1x/logx 1/2, viilimx→∞ϕ1xf1x 0.
bFunctionϕ4 satisfies the following properties:
iϕ40 0−,
iiϕ4x −14xf1x−2x22xf1x−1, iiiϕ4∞ 0,
ivϕ40 −1, vϕ4∞ 0,
viϕ4x 0⇔xx1 the maximum point of f1, viiϕ4x1 1.
cFunctionϕ3 satisfies the following properties:
iϕ30 0, iiϕ3∞ 1/2,
iiiϕ3x f1x x1−2xf1x, ivϕ30 0,
vϕ3∞ 0,
viϕ3x1 x1f1x10.4999, viiϕ3x2 x2f1x20.64.
dFunctionϕ2 satisfies the following properties:
iϕ20 0, iiϕ2∞ ∞,
iiiϕ2x ϕ1x xf1x>0,for allx >0, ivϕ20 0,
vϕ2∞ ∞,
viϕ2x 2f1x−x2xf1x−1, viiϕ2∞ 0,
viiiϕ20 0.
eFunctionϕ5 satisfies the following properties:
iϕ50 0, iiϕ5∞ 0, iii
ϕ5x −xf1x
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
>0 if 0< x < x2, 0 ifxx2,
<0 ifx > x2,
2.25
ivϕ5x>0,for allx >0.
fFunctionϕ6 satisfies the following properties:
iϕ60 Ste>0, iiϕ6∞ −1,
iiiϕ6x −21Steϕ3x, ivϕ60 0,
vϕ6∞ 0,
viϕ6x1 x1f1x10.4999, viiϕ6x2 x2f1x20.64.
Proof. aTaking into account properties off1, we have
ϕ1x f1x>0, ∀x >0, ϕ10 f10 0, 2.26 andvholds. If we considerLemma 2.2v, we getϕ1∞ ∞and we have
xlim→∞
ϕ1x
logx lim
x→∞xf1x 1
2, 2.27
thenivandvihold.
To provevii, we consider ϕ1xf1x
x
0
f1rdr
f1x f1cxf1x, 2.28
whereccx∈0, x.Then limx→∞ϕ1xf1x 0 because limx→∞xf1x 0 andf1 is a bounded function.
bFrom the definition ofϕ4,we obtainiandii. To proveiii, we have
ϕ4∞ lim
x→∞x
2xf1x−1 lim
x→∞
2xf1x−1 1/x lim
x→∞2
f1x x
1−2xf1x
1/x2 2 lim
x→∞
x2f1x x2x
1−2xf1x ,
2.29
then
xlim→∞x
2xf1x−1
2 lim
x→∞
x2x
2xf1x−1
−x2f1x
. 2.30
If we suppose that
xlim→∞x
2xf1x−1
L >0, 2.31
we get
L2 lim
x→∞
x2x
2xf1x−1
−x2f1x
∞, 2.32
which is a contradiction. If we suppose that
xlim→∞x
2xf1x−1
∞, 2.33
then
ϕ4∞ lim
x→∞−14xf1x−2x2
2xf1x−1
−∞, 2.34
which is also a contradiction. Therefore, limx→∞x2xf1x−1 0 andiiihold.
Taking into accountii, we haveϕ4x −14xf1x−2x22xf1x−1,then ϕ40
−1 and if we consideriiiwe haveϕ4∞ 0.From properties off1,we have
ϕ4x 0⇐⇒2xf1x−10⇐⇒f1x 0⇐⇒xx1, 2.35 andviholds. Taking into accountf1x 1−2xf1x 0,we getϕ4x1 1.
cFrom Lemmas2.2and2.3bwe geti–vii.
dWe haveϕ2x xϕ1x xx
0 f1rdr,then fromaandbiiiwe geti–vi.
eAs we haveϕ5x f1x−xf1x f1x ϕ4x,then by using the properties of f1 andbwe obtain the properties ofϕ5.
fWe haveϕ6x Ste−21Stexf1x Ste−21Steϕ3x,and from the properties ofϕ3,we obtaini–v.
Corollary 2.4. One has
ilimx→∞x22xf1x−1 1/2,
iilimx→∞xx22xf1x−1−xf1x 0.
Now, we are in conditions to enunciate properties of functionsW1 andW2in order to study after2.14.
Lemma 2.5. The functionsW1xandW2x,defined by2.15and2.16, respectively, satisfy the following properties.
aProperties of functionW1: iW10 Ste, iiW1∞ −∞,
iiilimx→∞W1x/x −√ π, ivlimx→∞W1x √
πx 0, vW1x<0,for allx >0,
viW1η00 0,whereη00is the unique solution of 2.17, vii
W1x
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
<0 if 0< x < x0, 0 ifxx0,
<0 ifx > x0,
2.36
where
x0
32 Ste
41Ste, 2.37
viiiW10 −232 Ste<0.
bProperties of functionW2: iW20 0,
iiW2∞ ∞,
iiithere exists a uniquex4>0 such thatW2x4 0, ivW2x 2x
0f1rdr2xf1x1Ste−Ste,
vthere exists a uniquex3>0 such that W2x3 0 andW2x3<0, viW20 −Ste<0,
viiW2∞ ∞,
viiiW2x 21Stex2f1x2Ste−21Stex2, ixW20 0,
xW2η00<0.
Proof. aTaking into account the definition of the functionW1,we getiandii.
iiiWe have
xlim→∞
W1x
x lim
x→∞ Steexp
−x2
x −√
πerfx
−√
π. 2.38
ivWe have
xlim→∞
W1x √ πx
lim
x→∞
Ste exp
−x2
−√
πxerfx √ πx lim
x→∞
Ste exp
−x2 √
πxerfcx lim
x→∞
Ste exp
−x2
Qxexp
−x2
lim
x→∞exp
−x2
SteQx 0,
2.39
whereQis the function defined by
Qx √
πxexp x2
erfcx, erfcx 1−erfx, 2.40
which satisfies the following properties:
Q0 0, Q∞ 1, Qx>0, ∀x >0. 2.41 vWe have
W1x −√
πerfx−2xexp
−x2
Ste1<0, ∀x >0. 2.42 viTaking into accounti,iii, andv, we get that there exists a unique zero ofW1
which is given byη00,the unique solution of2.17.
viiWe have
W1x −2 exp
−x2
32Ste−41Stex2
, 2.43
then
W1x 0⇐⇒41Stex2 32Ste⇐⇒xx0
32Ste
41Ste. 2.44
Since signW1x sign41Stex2−3−2Ste,then we obtainvii.
bTaking into account Lemmas2.2and2.3, we haveiandii.
We can write W2x 2
x
0
f1rdr2xf1x1Ste−Ste2ϕ1x−ϕ6x, 2.45
thenW20 −Ste, W2∞ ∞ andW2x 2ϕ1x−ϕ6xsatisfiesW20 0.Then iv,vi,vii,viii, andixhold.
We have
W2x 0⇐⇒2ϕ2x Stef1x, 2.46
then taking into account the properties ofϕ2 andf1, we get that there exists a uniquex4 >
0 such that
W2x 0, x >0. 2.47
Moreover, we have
W2x
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎩
0 ifx0,
<0 if 0< x < x4, 0 ifxx4,
>0 ifx > x4.
2.48
In the same way, we have
W2x 0⇐⇒2ϕ1x ϕ6x. 2.49
Then, if we consider the properties of the functions ϕ1 andϕ2, we have that there exists a uniquex3 such thatW2x3 0.Moreover,W2x3 −2x23f1x3−Steϕ5x3<0 and thenv holds.
To provex,we take into account that
W2x 2x x
0
f1rdr−Stef1x √
πxerfxFx−√ πx
x
0
erfrexp r2
dr−Ste exp
−x2 Fx
√ πexp
−x2
F0x− Ste
√π
Fx−√ πx
x
0
erfrexp r2
dr,
2.50
whereFx x
0expr2drand F0was defined in2.18. Then by using2.17, we have
W2
η00
−√ πη00
η00
0
erfrexp r2
dr <0. 2.51
Lemma 2.6. For eachλ > 0,there exists a unique solutionη0of 2.14. This solutionη0 η0λ satisfies the following properties:
i η00 η00, ii η0∞ x4,
iii η0 η0λis an increasing function onλ,
2.52
whereη00andx4are the unique solution of 2.17and2.47, respectively.
Proof. Taking into accountLemma 2.5, we get that there exists a unique solutionη0of2.14.
Let 0 < λ1 < λ2 be given, taking into account properties of functionW2,we obtain that the real functionsZ1 andZ2defined by
Z1x 2λ1W2x, Z2x 2λ2W2x 2.53
satisfy the following properties:
Z2x< Z1x if 0< x < x4, Z2x Z1x if xx4, Z2x> Z1x if x > x4.
2.54
Then η0λ1 < η0λ2, where η0λiis the solution of equation Zix W1x, i 1,2.Therefore,η0 η0λis an increasing function onλ.Moreover, we obtainη00 < η0λ <
x4 becauseW2η00<0.
Then, we have proved the following result.
Theorem 2.7. For eachλ > 0,the free boundary problem2.1, whereF is defined by1.8, has a unique similarity solution of the type
ux, t, λ f 1− En, λ E
η0λ, λ
, 0< η x 2a√
t < η0λ, st, λ 2aη0λ√
t,
2.55
where
E η, λ
erf η
√4λ π
η
0
f1rdr 2.56
andη0η0λis the unique solution of 2.14withη00 < η0λ< x4.
3. Explicit Solution to a One-Phase Stefan Problem for a Nonclassical Heat Equation with Control Function of the Type Fu0 , t, t λ
0/tu0 , t and
a Heat Flux Condition at the Fixed Face
In this section, the free boundary problem consists in determining the temperatureuux, t and the free boundaryxstwith a control functionFwhich depends on the evolution of the temperature at the extremumx0 given by the following conditions:
ρcut−kuxx −γFu0, t, t, 0< x < st, t >0, kux0, t −q0
√t >0, t >0, ust, t 0, t >0, kuxst, t −ρlst,˙ t >0,
s0 0,
3.1
where the coefficientq0 > 0 characterizes the heat flux on the x 0 21and the control functionF is given by1.9.
In order to obtain an explicit solution of a similarity type, we define the same transformation given by2.2. The problem3.1and1.9are equivalent to the following one:
Φ η
2ηΦ η
ΛΦ0, 0< η < μ0, 3.2 Φ0 −q0∗, 3.3
Φ μ0
0, 3.4
Φ μ0
−2l
cμ0, 3.5
where the dimensionless parametersΛ andq0∗are defined by
Λ 4γλ0
ρc >0, q0∗ 2aq0
k , 3.6
st 2aμ0
√t 3.7
is the free boundary, whereμ0is an unknown parameter to be determined.
From3.2,3.3, and3.4, we obtain the similarity solution
Φ η
q∗0√ π 2G
μ0,Λ erf
μ0
G η,Λ
−erf η
G μ0,Λ
, 0< η < μ0, 3.8
where
Gx,Λ 1 Λ x
0
f1rdr 1 Λϕ1x, 3.9
andf1is the Dawson’s integral andϕ1is given by2.23.
By condition3.5, the unknown parameterμ0μ0Λ, l, c, q∗0must be solution of the following equation:
Λerfxf1x 2
√πGx,Λ exp
−x2
− 2l cq∗0x
, x >0, 3.10
which is equivalent to the following one:
H2x H3x, x >0, 3.11
where the real functionsH2 andH3are defined by
H2x Λerfxf1x, 3.12
H3x 2
√πGx,ΛH1x, 3.13
H1x exp
−x2
− 2l cq∗0x
. 3.14
Remark 3.1. IfΛ 0i.e.,λ00, we have the solution
Φ η
q∗0√ π 2
erf μ00
−erf η
, 0< η < μ00, 3.15
whereμ00 is the unique solution of the following equation:
exp
−x2 2l
cq∗0x. 3.16
In order to solve3.11, we consider properties of Dawson’s integral, error function, and some auxiliary functions, and then we obtain the following result.
Theorem 3.2. For eachλ0< ρc/2γ, the free boundary problem3.1, whereFis defined by1.9, has a unique similarity solution of the type
ux, t, λ0 q0a√ π kG
μ0λ0,4γλ0/ρc
erf x
2a√ t
G
μ0λ0,4γλ0
ρc
−erf μ0λ0
G x
2a√ t,4γλ0
ρc
, 0< x
2a√
t < μ0λ0, t >0, st, λ0 2aμ0λ0√
t, t >0,
3.17
whereμ0μ0λ0is the unique solution of 3.11, 0< μ0λ0< μ00. Proof. We follow a similar method developed inTheorem 2.7.
4. Explicit Solution to a One-Phase Stefan Problem for a Nonclassical Heat Equation with Control Function of the Type Fu0 , t, t λ
0/tu0 , t and
a Convective Condition at the Fixed Face
In this section, we consider a similar problem to the one given inSection 3for a convective boundary condition22,26on the fixed face given by
ρcut−kuxx −γFu0, t, t, 0< x < st, t >0, kux0, t √h0
t
u0, t−f
>0, t >0, ust, t 0, t >0,
kuxst, t −ρlst,˙ t >0, s0 0,
4.1
whereF is defined by1.9andh0characterizes the heat transfer coefficients22,26. To solve this problem, we consider again a similarity type solution given by2.2. Then, the problem 4.1and1.9are equivalent to the following one:
Φ η
2ηΦ η
ΛΦ0, 0< η < μ0, 4.2
Φ0 h∗0
Φ0−f
, h∗0 2ah0
k , 4.3
Φ μ0
0, 4.4
Φ μ0
−2l
cμ0, 4.5
where the dimensionless parameterΛis defined by3.6and st 2aμ0
√t 4.6
is the free boundary, whereμ0 is an unknown parameter to be determined. We obtain the solution
Φ η
h∗0f√ π 2
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 Gx,Λ is given by 3.9. Taking into account the condition 4.5, the unknown parameterμ0μ0Λ, l, c, h∗0must be the solution of the following equation:
Λerfxf1x 2
Steerfxx 2
√πGx,Λ exp
−x2
− 2 h∗0Stex
, x >0, 4.8
which is equivalent to
H2∗x H3∗x, x >0, 4.9
where
H2∗x H2x 2
Steerfxx, x >0, H3∗x 2
√πGx,Λ exp
−x2
− 2 h∗0Stex
, x >0,
4.10
and the functionH2is defined by3.12.
Similarly to the previous cases, we can enunciate the following result.
Theorem 4.1. aFor eachΛ<2 (λ0< ρc/2γ, the free boundary problem4.1, whereFis defined by1.9, has a unique similarity solution given by
ux, t, λ0
−h0af√ π k
⎡
⎢⎣ erf x/2a√
t G
μ0λ0,4γλ0/ρc h0af√
π/k erf
μ0λ0 G
μ0λ0,4γλ0/ρc
− erf μ0λ0
G x/2a√
t,4γλ0/ρc h0af√
π/k erf
μ0λ0 G
μ0λ0,4γλ0/ρc
⎤
⎥⎦,
0< x 2a√
t < μ0λ0, t >0, st, λ0 2aμ0λ0√
t, t >0,
4.11
whereμ0μ0λ0is the unique solution of 4.9.
bLetMx Λf1xandNx 2xGx,Λbe, there exists a unique solutionx∗>0 of the equationMx Nx.
For eachΛ>2λ0> ρc/2γsuch thatMαΛ−NαΛ<2/h∗0Ste, where 0< αΛ< x∗ satisfiesMαΛ−NαΛ 0,there exists a unique similarity solution to the free boundary problem3.1, whereFis defined by1.9. The solution is given by4.11.
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.
References
1 V. Alexiades and A. D. Solomon, Mathematical Modeling of Melting and Freezing Processes, Hemisphere—Taylor & Francis, Washington, DC, USA, 1983.
2 J. Crank, Free and Moving Boundary Problems, Oxford Science Publications, The Clarendon Press Oxford University Press, New York, NY, USA, 1984.
3 A. Fasano, Mathematical Models of Some Diffusive Processes with Free Boundaries, vol. 11 of MAT serie A:
Conferencias, Seminarios Y Trabajos de Matematica, 2005.
4 V. J. Lunardini, Heat Transfer with Freezing and Thawing, Elsevier, Amsterdam, The Netherlands, 1991.
5 L. R. Berrone, D. A. Tarzia, and L. T. Villa, “Asymptotic behaviour of a non-classical heat conduction problem for a semi-infinite material,” Mathematical Methods in the Applied Sciences, vol. 23, no. 13, pp.
1161–1177, 2000.
6 J. R. Cannon and H.-M. Yin, “A class of nonlinear nonclassical parabolic equations,” Journal of Differential Equations, vol. 79, no. 2, pp. 266–288, 1989.
7 K. Glashoffand J. Sprekels, “An application of Glicksberg’s theorem to set-valued integral equations arising in the theory of thermostats,” SIAM Journal on Mathematical Analysis, vol. 12, no. 3, pp. 477–486, 1981.
8 N. Kenmochi and M. Primicerio, “One-dimensional heat conduction with a class of automatic heat- source controls,” IMA Journal of Applied Mathematics, vol. 40, no. 3, pp. 205–216, 1988.
9 M. N. Koleva, “Numerical solution of heat-conduction problems on a semi-infinite strip with nonlinear localized flow source,” in Proceedings of the International Conference “Pioneers of Bulgarian Mathematics”, Sofia, Bulgaria, July 2006.
10 D. A. Tarzia and L. T. Villa, “Some nonlinear heat conduction problems for a semi-infinite strip with a non-uniform heat source,” Revista de la Uni´on Matem´atica Argentina, vol. 41, no. 1, pp. 99–114, 1998.
11 L. T. Villa, “Problemas de control para una ecuaci ´on unidimensional del calor,” Revista de la Uni´on Matem´atica Argentina, vol. 32, no. 1, pp. 163–169, 1986.
12 K. Glashoffand J. Sprekels, “The regulation of temperature by thermostats and set-valued integral equations,” Journal of Integral Equations, vol. 4, no. 2, pp. 95–112, 1982.
13 N. Kenmochi, “Heat conduction with a class of automatic heat source controls,” in Free Boundary Problems: Theory and Applications,, vol. 186 of Pitman Research Notes in Mathematics Series, pp. 471–474, 1990.
14 A. C. Briozzo and D. A. Tarzia, “Existence and uniqueness for one-phase Stefan problems of non- classical heat equations with temperature boundary condition at a fixed face,” Electronic Journal of Differential Equations, vol. 2006, no. 21, pp. 1–16, 2006.
15 A. C. Briozzo and D. A. Tarzia, “A one-phase Stefan problem for a non-classical heat equation with a heat flux condition on the fixed face,” Applied Mathematics and Computation, vol. 182, no. 1, pp. 809–
819, 2006.
16 D. A. Tarzia, “A Stefan problem for a non-classical heat equation,” in VI Seminar on Free Boundary Value Problems and their Applications, vol. 3 of MAT serie A: Conferencias, Seminarios y Trabajos de Matematica, pp. 21–26, 2001.
17 A. Friedman, “Free boundary problems for parabolic equations. I. Melting of solids,” Journal of Mathematics and Mechanics, vol. 8, pp. 499–517, 1959.
18 L. I. Rubenstein, The Stefan Problem, Translations of Mathematical Monographs, American Mathemat- ical Society, Providence, RI, USA, 1971.
19 B. Sherman, “A free boundary problem for the heat equation with prescribed flux at both fixed face and melting interface,” Quarterly of Applied Mathematics, vol. 25, pp. 53–63, 1967.
20 D. A. Tarzia, A Bibliography on Moving-Free Boundary Problems for the Heat-Diffusion Equation. The Stefan and Related Problems, vol. 2 of MAT serie A: Conferencias, Seminarios Y Trabajos de Matematica, 2000,with 5869 titles on the subject, http://web.austral.edu.ar/descargas/facultad-cienciasEmpresariales/
mat/Tarzia-MAT-SerieA-22000.pdf.
21 D. A. Tarzia, “An inequality for the coefficientσof the free boundaryst 2σ√
tof the Neumann solution for the two-phase Stefan problem,” Quarterly of Applied Mathematics, vol. 39, no. 4, pp. 491–
497, 1981.
22 D. A. Tarzia, An Explicit Solution for a Two-Phase Unidimensional Stefan Problem with a Convective Boundary Condition at the Fixed Face, vol. 8 of MAT serie A: Conferencias, Seminarios Y Trabajos de Matematica, 2004.
23 M. Abramowitz and I. E. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Washington, DC, USA, 1972.
24 A. G. Petrova, D. A. Tarzia, and C. V. Turner, “The one-phase supercooled Stefan problem with temperature boundary condition,” Advances in Mathematical Sciences and Applications, vol. 4, no. 1, pp. 35–50, 1994.
25 G. Lam´e and B. P. Clapeyron, “Memoire sur la solidification par refroidissement d’un globe liquide,”
Annales Chimie Physique, vol. 47, pp. 250–256, 1831.
26 S. M. Zubair and M. A. Chaudhry, “Exact solutions of solid-liquid phase-change heat transfer when subjected to convective boundary conditions,” Heat and Mass Transfer, vol. 30, no. 2, pp. 77–81, 1994.
