1
Revista de la Union Matematica Argentina Volumen 4 1 , Nro. 4 , 20 0 0 , 1-8
INTEGRAL EQUATION FOR A STEFAN PROBLEM WITH MANY PHASES AND A SINGULAR SOURCE AN
Pr'esentado por: Ra/ael Panzone
Julio E. BOUILLET (t) - Domingo A. TARZIA (*) (t) Dept.o Matermttica
and CONIC ET, FCEyN, Univ.
Aires and lAM (CONICET),
de Buenos
Ciudad Universitaria, Pah. 1, (l428) Buenos ' Ai res , Argentina.
(*) Depto Matematica and CONlCET, FCE, Univ. Austral, Para g u ay 1950, (2000) Rosario, A rgen tina.
E-mail:
[email protected]
Au thor
to
whom
correspondence should be
the
addressed.
ABSTRACT.
We
di s c l l ss
the self- sim i lar solutions 0 ( x, t)
E(O)t-A(O)xx=t.B(1)), 1» 'Ve
E
that E
assume
(0) = A (0) = 0
nd
a
and
A are
m on
(temperature) 0, A'(O) 2:
or sink depending of
(x/Ji:)/t
..
otone
a
sufficicnt
o
( ll di
=
E (0)
Moreover,
of
the problem
E(O(x,O))=O,x>O .
being
cont.inllous, with
conservation
0+ A
Math.
ord er to
PHRASES: Stefall
equa.tions, Singular sou rce .
AMS Subject Classification:
35
R 3 5,
ACKNOWLEDGEMENTS.
This paper
80
A
problem,
12 (1993),
2 2, 35
h ...
=
(1)}/ t reprcs/,·nl.s
It
solution ) .
boundary
123
- 142,
for
t.he
singular
source
particular
one
0 is studied where IH'Cl'ssary alld
aract e ri ze the
ch
Free
�our('e o r
represent.s energy per unit volume at level
(0) 0), E (0) = 0 and A (0) in
of t.hermal
mat.erial with a �self-similar"
0 is t.he thermal conductivity and 13
tion s were given
....d
(x / y'i)
the sign of the function B. We ge n e rali zc results obtaiued in:
unique sol u t ion (a generalized Lamc-Cll1peyron KEY WORDS
0
increasing functions, A
semi-infinite
(i) J .L. Menaldi - D.A. Tarzia, Comp. Appl.
phase case E (0)
=
This eq u ation can describe the
energy in a heat c.onduction process for sink term of the t.ype R
0 (1])
O (O , t)= C > O , t > O ;
0;
A = E (0+) >O.
=
SOllrce
problem,
tt'cm B which provides a
Self-similar solutions, Integral
C15
been partially sponsored by
national project #221 "Aplicaciones de P"oblemas de Front.era Lib,"e".
CONJCET
(Argentina) through the
J ulio E. Bouillet and
2
(ii) B
Bouillet, IMA Preprints # 230, Univ.
J.E.
==
°
for the two-phase Stefan problem,
source term.
We ob tain for
Minnesota March
(
we
and
obtain
the inverse function T] = ." (0)
Domingo
A. Tarzia
1986), for the particular case
some new results in connec tion to the
an i ntegral
equation equivalent to the above
problem and we prove that for certain hypothese over data there exists
at
least a solu tion of the
corresponding integral equation.
1. INTRODUCTION In
this paper,
with
we
the i nitial
discuss the self-similar solutio ns
and
of
(temperature)
conduction process
the type
medium
heating, In
is
n(x/Ji)/t.
0, A/(O) � °
or sink depe ndi ng of porous
as
the
sign of the fu n c tion
[4].
nece ss a ry and sufficient
which provides
E(O) repr esen t.s
B.
A
case
E (0) = 0 +).
energy per unit volume at level
B (1])/ t rep r s n ts a singular source e e
study of
sublimation-dehydration within
as
(0) 0),
E
a unique solution
(0) = °
(a genera li zed Lame-Clapeyron
to the sou r ce
a
that associated with m ic r ow ave
a nd
A
condi tions are gi ven in order to ch aract eriz e
we follow the analysis presented in [1]
resuHs in connec tion
and
a res u lt of volumetric heating, such
presen ted in
In this paper
for a semi-infini te material with a "self-similar" source or Moreover,
is the thermal conducti vi ty
[3] the particul ar one-p hase
where
x> O.
and A are monotone increasing func tions, A bein g continuous, with 0 and ,\ = E (0+) >O. This equation can describe the conservation of thermal
energy in a heat term
E(O(x,O»=O,
t>o
E
vVe assume that
sink
of the equation
boundary condition given by:
() (0, t) = C > 0,
E(0) = A(0) =
0 (x, t) = 0 (T]) = () (xl vi)
sol uti on
for the case B ==
(0) = 0 is st.udied
the
source term B
[2] ) .
0, and we obtain some
new
term.
n. SELF-SIMILAR SOLUTION
We consider the following problem for the function 0 = 0 (r/),
where '/
variable:
(1)
-h(E(O»' - (A (0) )
(2)
0 (0) = C > 0,
"
=
B(1]) ,
E(O(+oo»=O,
'1 >
= x/ Ji
0,
is the simi l ari t y
An integral equation for a Stefan problem with many phases and ..... .
3
d where the prime denotes . d ", . If
we define the function
h('7) = (A (0('7)))' +hE(O(",)),
(3)'
the differential equation (1) c/!-n be written as h' (11)
(4)
=
! E (0 ('7))
-
13 ('7).
(1) it is easy to sec that bot.h A (0 ('7») and h ('7) defined by (3f are absolutely continuous (A C) functions of 0 > 0, and equation (4) is satisfied a distributional version of equation
From
almost everywhere ( recall that no asumptions on regularity are made on the monotone functions
E
an d A). We shall assume that side derivatives always exist at 1)> 0 in order to avoid
technical details which are not essential and will obscure this exposition (cf.
[1). Observe that if thermal energy is suplied to the medium x > 0 ( initially at 8 = 0) by (i ) heat conduction from x = 0, and (ii ) the source B �
1)
�
0 for t > 0,
therefore it is natural to expect that 8 ( x, t ) be zero for unit volume at x> 0 grows from
E (0 ( x, 0)
=
0 < t < t (x)
0 according to the law
while
local energy per
the
(5)
E (0+) . E (8 (x, t» = E (0 (1))
and remains less than The notation
A
=
mind a multivalued graph for
E,
is inadequate in this case B � 0, and we
where E (0)
=
[0 , A). This
fact
is
seen
ill
the
clearly have in
integral equ ation
(6)-(7) whose fixed point gives a solution to our problem. Lemma
1. We
obtain
the jot/owing
the sign of th.e source / sink term
(iJ
(ii)
If
for the solution o
f lite
problem (1). (2)
:::; 0, V 1) � 0, then 8 = () (1) is a non increasing function of 11. B (1) � 0, V 1) � 0, and the solution verifies the' condit!oll 0' (0+) < 0,
B (1)
If
B.
r�s1llt
non increasing functioll of 1)
•
Proof. Case (i ) . If this were not. the case, there would exist
0 :::; 1)1
< '12 and
according to
then 0
=
0
(11)
i.�
a
D >O. s ll ch that
4
Julio E .. Boumet amI DOJ}lingo
Employing (3). (4)
and
[A (D)'+!I] E(DH'
lE(D)
=
we obtain by
A .. Tarzia
substraction
, {(.AoO)'+h[E(O(ll))-E(D))} �![E(O(IJ})-E(D») - B(7/)
integrating in
(1]1 .1]2) gives
1]2 ) + A o � ( 00)' (1]2 -(A 0 0)' (1]1 ) �! [E(O(I]» - E (D)] dl] > 0,
J
1]1
a
contradiction.
Case (ii).
Again by contradiction, if 0
and 0$1]1 O.
and
[E(0('1»-E(D»)d1/-
1]1
we can
be
112
decreasi ng
Ratisfies the following property
Theorem 2. Assllme that
(6)
0 (1])
0(1])
there
'12
0$(AoO)'(1]2-)-(AoO)'(1]1 +)�! a
monoto ne
Oe(o,C).
(0)
wc
have the
An integral equation for a
'
S te fan problem with
many phases and
.. . ..
'
5
(7)
Proof.
We have
__
h ('I» )' 'I ( 'I -
hence
(A 00)' ('I)
'I
B
('I),
that is (A 00)' ('1) Assuming
and
for the moment that
h
�?)
'I2
->
B('I) '1 - . --
0 when
'I -> +00,
we have:
hence
(Ao�'('I)_!E(O)= J (Ao:t(S)ds+ JB�s)dS .
00
00
'I
'I
'
whence i.e.
( 6 ) , (7).
B � 0).
From
(4)
it
( '1)/ '1-> 0
follows
We 'now prove that h
when
h('1)=h(0+)+
'1-+ + 00
(under
'I
'I d 1 J E(O(s» s- J
o
0
the
hypothesis
B(s)ds ,
.r B (s) ds
