Nonlinear Two-Phase Stefan Problem
K. Ivaz
1,*and N. Aliyev
21
Department of Mathematics, University of Tabriz, Tabriz, Islamic Republic of Iran
2
Department of Applied Mathematics and Economical Cybernetics, Baku State University, 370148 Baku, Azerbaijan
Abstract
In this paper we consider a nonlinear two-phase Stefan problem in
one-dimensional space. The problem is mapped into a nonlinear Volterra integral
equation for the free boundary.
Keywords: Free boundary problems; Integral equation
*
Corresponding author, Tel.: +98(411)3342102, Fax: +98(411)3342102, E-mail: [email protected]
Introduction
Processes related to solidification (or melting) are important in many engineering applications such as freezing of foodstuff, casting of alloys, preservation of human blood, solar energy storage and many others. The study on the moving boundary problems involving solidification (or melting) has become a highly popular subject recently. These problems are inherently nonlinear due to the presence of moving boundary solution.
One and two phase Stefan problems for the linear heat equation have been the subject of many studies in the past [1,2]. Indeed these problems have a great physical relevance since they provide a mathematical model for the processes of phase changes [3,4].
The boundary between the two phases is a free boundary its motion has to be determined as part of the solution.
In [5,6] exact solution were found in parametric form for a class of Stefan problems in nonlinear heat conduction. More recently the previous analysis [7-13] was extended to nonlinear diffusion models.
It is the aim of this paper to analyse a two-phase Stefan problem for the nonlinear heat equation. Such an equation arises as a model of heat conduction in
solidification [14,15].
In the next section we show that the two-phase Stefan problem for the nonlinear heat equation reduces to a nonlinear Volterra integral equation.
Statement of the Problem
Let
}
{
( , ) 0 ( ), ,l x t x γ t t t
Ω = < < >
{( , ) ( ) 1, 0}
s x t γ t x t
Ω = < < >
be the liquid and solid domains, respectively. We start our analysis with the following system of nonlinear heat equations:
(2.0.1) l [ ( l ) l ] 0 , ( , ) l ,
u u
a u x t
t x x
∂ − ∂ ∂ = ∈ Ω
∂ ∂ ∂
(2.0.2) us [ (a us ) us ] 0 , ( , )x t s,
t x x
∂ − ∂ ∂ = ∈ Ω
∂ ∂ ∂
(2.0.3)
0
( , 0) ( ) , 0 ( 0) ,
l l
u x =u x ≤ ≤ =x b γ
(2.0.4)
0
( , 0) ( ) , 1,
s s
(2.0.5) ul ( 0, )t =f1( ) ,t t ≥0 ,
(2.0.6) us (1, )t =f2( ) ,t t ≥0 ,
(2.0.7) ( ( ), )ul γ t t =us ( ( ), )γ t t =0 ,t ≥0 ,
(2.0.8) ( ) ( )
( , ) ( , )
( ), 0,
s l
x t x t
u x t u x t
x x
t t
γ γ
γ
= =
∂ ∂
−
∂ ∂
′
= − ≥
where
0 0
( l ) , ( s ) , l , s ,
a u a u u u f1 and f2 are
assumed to be known and ul ( , ) ,x t us ( , )x t and the free boundary ( )γ t are unknown and must be determined. We also suppose that
(2.0.9) (0)a =0. If we set
(2.0.10)
0
0
( ( , )) ( ( ) )
[ ( ( , )) ( ( ))]
( ) ( , ( , )),
l l
l l
l l l
a u x t a u x
a u x t a u x
x A x u x t
α
=
+ −
= +
(2.0.11)
0
0
( ( , )) ( ( ) )
[ ( ( , )) ( ( ))]
( ) ( , ( , )),
s s
s s
s s s
a u x t a u x
a u x t a u x
x A x u x t
α
=
+ −
= +
it is clear that, we can write
(2.0.12)
{
}
( ) ( ) ( ) ,
( , ) ( ) 0 ,
l x s x x
x t x t t
α α α
γ
= =
∈ Γ ≡ = <
(2.0.13) ( ( ), 0)Al γ t =As ( ( ), 0)γ t = −α γ( ( )).t
We also suppose that
(2.0.14) αl (0)=a f( 1( 0)) ,αs (1)=a f( 2( 0)) ,
then we can write (2,0,1)-(2,0,2) as follow:
(2.0.15)
[ ( ) ]
[ ( , ) ], ( , ) ,
l l
l l
l
l l l
u u
lu x
t x x
u
A x u x t
x x
α
∂ ∂ ∂
≡ −
∂ ∂ ∂
∂ ∂
= ∈ Ω
∂ ∂
(2.0.16)
[ ( ) ]
[ ( , ) ], ( , ) ,
s s
s s
s
s s s
u u
lu x
t x x
u
A x u x t
x x
α
∂ ∂ ∂
≡ −
∂ ∂ ∂
∂ ∂
= ∈ Ω
∂ ∂
We also define
(lu v, )≡(lul ,vl ) (+ lus,vs )
( , )
( , ) l
l
l l
u x t
v x t d t
Ω
∂
≡ Ω
∂
∫
( , )
( , )
s s
s s
u x t
v x t d t
Ω
∂
+ Ω
∂
∫
[ ( ) ] ( , )
l
l
l l l
u
x v x t d
x α x
Ω
∂ ∂
− Ω
∂ ∂
∫
[ ( ) ] ( , )
s
s
s s s
u
x v x t d
x α x
Ω
∂ ∂
− Ω
∂ ∂
∫
( , )
[ ( , ) ] ( , )
l
l
l l l l
u x t
A x u v x t d
x x
Ω
∂ ∂
= Ω
∂ ∂
∫
( , )
[ ( , ) ] ( , )
s
s
s s s s
u x t
A x u v x t d
x x
Ω
∂ ∂
+ Ω
∂ ∂
∫
Now, integrating by parts, we obtain
( , ) ( , ) cos( , )
l l l l
u x t v x t ν t d
∂Ω ∂Ω
∫
( , ) ( , )
s
s
l l
v x t
u x t d
t
Ω
∂
− Ω
∂
∫
( , ) ( , ) cos( , )
ls s s s
u x t v x t ν t d
∂Ω
+
∫
∂Ω( , ) ( , )
s
s
s s
v x t
u x t d
t
Ω
∂
− Ω
∂
∫
( , )
[ ( ) ( , )
l
l
l l
u x t
x v x t
x
α
∂Ω
∂ −
∂
∫
( , )
( , ) ( ) l ]cos( , )
l l l
v x t
u x t x x d
x
α ∂ ν
− ∂Ω
∂
( , )
( , ) [ ( ) ]
l
l
l l l
v x t
u x t x d
x α x
Ω
∂ ∂
− Ω
∂ ∂
∫
( , )
[ ( ) ( , )
s
s
s s
u x t
x v x t
x
α
∂Ω
∂ −
∂
( , )
( , ) ( ) s ]cos( , )
s s s
v x t
u x t x x d
x
α ∂ ν
− ∂Ω
∂
( , )
( , ) [ ( ) ]
s
s
s s s
v x t
u x t x d
x α x
Ω
∂ ∂
− Ω
∂ ∂
∫
( , )
( , ) ( , ) cos( , )
l
l
l l l l
u x t
A x u v x t x d
x ν
∂Ω
∂
= ∂Ω
∂
∫
( , ) ( , ) ( , )
l
l l
l l l
u x t v x t
A x u d
x x
Ω
∂ ∂
− Ω
∂ ∂
∫
( , )
( , ) ( , ) cos( , )
s
s
s s s s
u x t
A x u v x t x d
x ν
∂Ω
∂
+ ∂Ω
∂
∫
( , ) ( , )
( , ) .
s
s s
s s s
u x t v x t
A x u d
x x
Ω
∂ ∂
− Ω
∂ ∂
∫
By choosing
( ( ), ) ( ( ), ) ( ( ), ),
s l
v γ t t =v γ t t =v γ t t
and by (2.0.3)-(2.0.8) we obtain
0
0
( ) ( , 0) b
l l
u x v x dx
−
∫
0
( ( ), ) ( ( ), ) ( )
l l
u γ t t v γ t t γ t dt
∞
′
+
∫
( )
0
limt ul ( , )x t vl ( , )x t dx
γ ∞
→∞
+
∫
0
( ( ), ) ( ( ), ) ( )
s s
u γ t t v γ t t γ t dt
∞
′
+
∫
0
1
( ) ( , 0)
s s
b
u x v x dx
−
∫
1
( )
limt us ( , )x t vs( , )x t dx
γ →∞
∞
+
∫
0 0
( , )
[ l ( 0) l x l ( 0. )
u x t
v t
x
α
∞
=
∂ +
∂
∫
0
( , )
(0, ) (0) l ]
l l x
v x t
u t dt
x
α ∂ =
−
∂
( ) 0
( , )
[ l ( ( ))t ul x t x t vl ( ( ), )t t
x γ
α γ γ
∞
=
∂ −
∂
∫
( )
( , )
( ( ), ) ( ( )) l ]
l l x t
v x t
u t t t dt
x γ
γ α γ ∂ =
−
∂
( ) 0
( , )
[ s ( ( )) s x t s ( ( ), )
u x t
t v t t
x γ
α γ γ
∞
=
∂ +
∂
∫
( )
( , )
( ( ), ) ( ( )) s ]
s s x t
v x t
u t t t dt
x γ
γ α γ ∂ =
−
∂
1 0
( , )
[ s (1) s x s (1, )
u x t
v t
x
α
∞
=
∂ −
∂
∫
1
( , )
(1, ) (1) s ]
s s x
v x t
u t dt
x
α ∂ =
−
∂
0 0
( , )
( 0, ( 0, )) l ( 0, )
l l x l
u x t
A u t v t dt
x
∞
=
∂ +
∂
∫
0
[As ( ( ),γ t ul ( ( ), )γ t t
∞
−
∫
( )
( , )
( ( ), ) s
x t l
u x t
v t t dt
x =γ γ
∂ ∂
0
[As ( ( ),γ t us ( ( ), ))γ t t
∞
+
∫
( )
( , )
( ( ), ) s
x t s
u x t
v t t dt
x =γ γ
∂ ∂
1 0
( , )
[ s (1, s (1, )) s x s (1, )
u x t
A u t v t dt
x
∞
=
∂ −
∂
∫
( , ) ( , ) ( , )
l
l l
l l l
u x t v x t
A x u d
x x
Ω
∂ ∂
+ Ω
∂ ∂
∫
( , ) ( , ) ( , )
s
s
s s s
u x t v x t
A x u d
x x
Ω
∂ ∂
+ Ω
∂ ∂
∫
( , ) ( , )[
l l
v x t u x t
t
Ω
∂ =
∂
∫
( , )
( l ( ) l )] l
v x t
x d
x α x
∂ ∂
+ Ω
( , ) ( , )[
s
s s
v x t u x t
t
Ω
∂ +
∂
∫
( , ) ( s ( )x vs x t )]d s
x α x
∂ ∂
+ Ω
∂ ∂
(2.0.17) We note that
( )
0
limt ul ( , )x t vl ( , )x t dx
γ ∞
→∞
∫
1
( )
limt us ( , )x t vs ( , )x t dx 0
γ →∞
∞
+
∫
=( ) 0
( , )
( ( )) l x t l ( ( ), )
u x t
t v t t dt
x γ
α γ γ
∞
=
∂ −
∂
∫
( ) 0
( , )
( ( )) s ( ( ), )
s x t s
u x t
t v t t dt
x γ
α γ γ
∞
=
∂ +
∂
∫
0
( ( )) ( ( ), )t v t t ( )t dt
α γ γ γ
∞
′
= −
∫
We now suppose that Vl ( , ;x ξ t −τ) and
( , ; )
s
V x ξ t−τ are the fundamental solutions of the following problem in x direction, respectively
(2.0.18)
( , ; )
( , ; )
[ ( ) ]
( ) ( ) , ( , ) l
l l
l
V x t t
V x t x
x x
x t x t
ξ τ
ξ τ
α
δ ξ δ τ
∂ −
∂
∂ −
∂ +
∂ ∂
= − − ∈ Ω
(2.0.19)
( , ; )
( , ; )
[ ( ) ]
( ) ( ) , ( , ) s
s s
s
V x t t
V x t x
x x
x t x t
ξ τ
ξ τ
α
δ ξ δ τ
∂ −
∂
∂ −
∂ +
∂ ∂
= − − ∈ Ω
By using (2,0,16) and (2,0,1)-(2,0,6) we find that first important relation for the solution of two-phase Stefan problem for the nonlinear heat equation is given by:
0
0
( ) ( , ; )
b
l l
u x V x ξ τ dx
−
∫
−0
1
( ) ( , ; )
s s
b
u x V x ξ τ dx
−
∫
−0 0
( , )
[ l ( 0) l x l ( 0, ; )
u x t
V t
x
α ξ τ
∞
=
∂
+ −
∂
∫
0
( , ; )
(0, ) (0) l ]
l l x
V x t t
u t dt
x
ξ τ
α ∂ − =
−
∂
0
( ( ))t Vl ( ( ), ;t t ) ( )t dt
α γ γ ξ τ γ
∞
′
−
∫
−1 0
( , )
[ s (1) s x s (1, ; )
u x t
V t
x
α ξ τ
∞
=
∂
− −
∂
∫
1
( , ; )
(1, ) (1) s ]
s l x
V x t
u t dt
x
ξ τ
α ∂ − =
−
∂
1 0
0
( , )
( 0, ( )) l ( 0, ; )
l x l
u x t
A f t V t dt
x ξ τ
∞
=
∂
+ −
∂
∫
( ) 0
( , )
( ( ), 0) l ( ( ), ; )
l x t l
u x t
A t V t t dt
x γ
γ γ ξ τ
∞
=
∂
− −
∂
∫
( ) 0
( , )
[ s ( ( ), 0) s x t s ( ( ), ; )
u x t
A t V t t dt
x γ
γ γ ξ τ
∞
=
∂
+ −
∂
∫
2 1
0
( , )
[ s (1, ( )) s x s (1, ; )
u x t
A f t V t t dt
x ξ τ
∞
=
∂
− −
∂
∫
1
( , ) ( , ; )
( , ( , )) l
l l
l l l
u x t V x t
A x u x t d
x x
ξ τ
Ω
∂ ∂ −
+ Ω
∂ ∂
∫
( , ) ( , ; )
( , ( , )) s s
s s s
s
u x t V x t
A x u x t d
x x
ξ τ
Ω
∂ ∂ −
+ Ω
∂ ∂
∫
(2.0.20)
( , ) ( , ) ,
1
( , ) ( , ) ,
2
( , ) ( , ) ,
1
( , ) ( , ) .
2
l l
l l
s s
s s
u
u
u
u
ξ τ ξ τ
ξ τ ξ τ
ξ τ ξ τ
ξ τ ξ τ
⎧ ∈ Ω
⎪ ⎪
∈ ∂Ω ⎪
⎪ = ⎨
∈ Ω ⎪
⎪ ⎪
Therefore, by (2,0,20) we can compute the ul and s
u at the domains and boundaries of domains. It is clear that the values of ul and us at the domains and boundaries of domains are dependent to the following parameters
0 1
( , ) ( , )
, s ,
l
x x
u x t u x t
x = x =
∂ ∂
∂ ∂
( , ) ( , )
( , ), ( , ), l , s ,
l s
u x t u x t
u x t u x t
x x
∂ ∂
∂ ∂
( , ) l ( , ) s.
for x t ∈ Ω and x t ∈ Ω
We now try to obtain the following
0 1
( , ) ( , )
, s ,
l
x x
u x t u x t
x = x =
∂ ∂
∂ ∂
For this purpose, we do similar work to [16]-[19] as follows:
( , ) ( l , l ) ( s , s )
V V
V
lu lu lu
x x x
∂ ∂
∂
≡ +
∂ ∂ ∂
( , ) ( , ; )
l
l l
l
u x t V x t d
t x
ξ τ
Ω
∂ ∂ −
≡ Ω
∂ ∂
∫
( , ) ( , ; )
s
s s
s
u x t V x t d
t x
ξ τ
Ω
∂ ∂ −
+ Ω
∂ ∂
∫
( , ) ( , ; )
[ ( ) ]
l
l l
l l
u x t V x t
x d
x x x
ξ τ
α
Ω
∂ ∂ −
∂
− Ω
∂ ∂ ∂
∫
( , ) ( , ; )
[ s ( ) s ] s s
s
u x t V x t
x d
x x x
ξ τ
α
Ω
∂ ∂ −
∂
− Ω
∂ ∂ ∂
∫
( , )
[ ( , ) ]
l
l
l l
u x t A x u
x x
Ω
∂ ∂
=
∂ ∂
∫
( , ; )
l
l
V x t d x
ξ τ
∂ −
Ω ∂
( , )
[ ( , ) ]
s
s
s s
u x t A x u
x x
Ω
∂ ∂
+
∂ ∂
∫
( , ; )
s
s
V x t d x
ξ τ
∂ −
Ω
∂ .
We now try the derivatives of functions ( , ), ( , ), ( , ; )
l s l
u x t u x t V x t t−τ and Vs ( , ;x t t−τ) in the domains that do not appear higher than 1 on the boundaries ∂Ωl and ∂Ωs and do not appear higher than 2 in the domains.
For this purpose, we do as follows:
( , ; )
( , ) cos( , )
l
l
l l
V x t
u x t t d
x
ξ τ
ν
∂Ω
∂ −
∂Ω ∂
∫
2
( , ; )
( , ) s
l
l l
V x t
u x t d
x t
ξ τ
Ω
∂ −
− Ω
∂ ∂
∫
( , ; )
( , ) cos( , )
s
s
s s
V x t
u x t t d
x
ξ τ
ν
∂Ω
∂ −
+ ∂Ω
∂
∫
2
( , ; )
( , ) s
s s
s
V x t
u x t d
x t
ξ τ
Ω
∂ −
− Ω
∂ ∂
∫
( , )
[ ( ) ]
l
l l
u x t x
x
α
∂Ω
∂ −
∂
∫
( , ; )
cos( , ) l
l
V x t
x d x
ξ τ
ν
∂ −
∂Ω ∂
1
2
2
( , ) ( , ; )
( ) l
l l
l l
u x t V x t
x d
x x
ξ τ
α
Ω
∂ ∂ −
+ Ω
∂ ∂
∫
( , )
[ ( ) ]
s
s s
u x t x
x
α
∂Ω
∂ −
∂
∫
( , ; )
cos( , ) s
s
V x t
x d x
ξ τ
ν
∂ −
∂Ω ∂
2
2
( , ) ( , ; )
( ) s
s s
s s
u x t V x t
x d
x x
ξ τ
α
Ω
∂ ∂ −
+ Ω
∂ ∂
∫
( , )
[ ( , ( , )) ]
l
l
l l
u x t A x u x t
t
∂Ω
∂ =
∂
∫
( , ; )
cos( , ) l
l
V x t
x d x
ξ τ
ν
∂ −
∂Ω ∂
( , )
[ ( , ( , )) ]
s
l
l l
u x t A x u x t
x
Ω
∂ −
∂
2
2
( , ; )
l
l
V x t d x
ξ τ
∂ −
Ω ∂
( , )
[ ( , ( , )) ]
s
s
s s
u x t A x u x t
t
∂Ω
∂ +
∂
∫
( , ; )
cos( , ) s
s
V x t
x d x
ξ τ
ν
∂ −
∂Ω ∂
( , )
[ ( , ( , )) ]
s
s
s s
u x t A x u x t
x
Ω
∂ −
∂
∫
2
2
( , ; )
s
s
V x t d x
ξ τ
∂ −
Ω ∂
And integration by parts, we obtain
( , ; )
( , ) cos( , )
l
l
l l
V x t
u x t t d
x
ξ τ
ν
∂Ω
∂ −
∂Ω ∂
∫
( , ; )
( , ) cos( , )
l
l
l l
V x t
u x t x d
t
ξ τ
ν
∂Ω
∂ −
− ∂Ω
∂
∫
1
( , ) ( , ; )
l l
l
u x t V x t d
x t
ξ τ
Ω
∂ ∂ −
+ Ω
∂ ∂
∫
( , ; )
( , ) cos( , )
s
s
s s
V x t
u x t t d
x
ξ τ
ν
∂Ω
∂ −
+ ∂Ω
∂
∫
( , ; )
( , ) s cos( , )
s s
s
V x t
u x t x d
t
ξ τ
ν
∂Ω
∂ −
− ∂Ω
∂
∫
( , ) ( , ; )
s
s s
s
u x t V x t d
x t
ξ τ
Ω
∂ ∂ −
+ Ω
∂ ∂
∫
( , ) ( )
l
l l
u x t x
x
α
∂Ω
∂ −
∂
∫
( , ; )
cos( , ) l
l
V x t
x d x
ξ τ
ν
∂ −
∂Ω ∂
1
2
2
( , ) ( , ; )
[ ( )
l l
l
u x t V x t
x
x x
ξ τ
α
Ω
∂ ∂ −
+
∂ ∂
∫
( , ; )
( )
( , ; )
( ) ]
l l
l
l l
V x t x
x
V x t
x d
x
ξ τ
α
ξ τ
α
∂ −
′ +
∂
∂ −
′
− Ω
∂
( , ) ( , ; )
( ) s s
s s
u x t V x t x
x x
ξ τ
α
∂Ω
∂ ∂ −
−
∂ ∂
∫
cos( , )ν x d∂Ωs
2
2
( , ) ( , ; )
[ ( ) s
s s
s
u x t V x t
s
x x
ξ τ
α
Ω
∂ ∂ −
+
∂ ∂
∫
( , ; )
( )
( , ; )
( ) ]
s s
s
s s
V x t x
x
V x t
x d
x
ξ τ
α
ξ τ
α
∂ −
′ +
∂
∂ −
′
− Ω
∂
( , ) ( , ; )
( , ( , )) l
l l
l l
u x t V x t A x u x t
t x
ξ τ
∂Ω
∂ ∂ −
=
∂ ∂
∫
cos( , )ν x d∂Ωl
( , ) ( , ( , ))
[ ( ) ( )
( ) s
l l l
l
u x t A x u x t
x t
x α x δ ξ δ τ
Ω
∂
− − −
∂
∫
( , ; ) ( , ; )
( ) ]
l l
l l
V x t V x t
x d
t x
ξ τ ξ τ
α
∂ − ∂ −
′
− − Ω
∂ ∂
( , ) ( , ; )
( , ( , )) s s
s s
s
u x t V x t A x u x t
t x
ξ τ
∂Ω
∂ ∂ −
+
∂ ∂
∫
cos( , )ν x d∂Ωs
( , ) ( , ( , ))
[ ( ) ( )
( ) s
s s s
s
u x t A x u x t
x t
x α x δ ξ δ τ
Ω
∂
− − −
∂
∫
( , ; ) ( , ; )
( ) ]
s s
s s
V x t V x t
x d
t x
ξ τ ξ τ
α
∂ − ∂ −
′
− − Ω
∂ ∂
By using of (2.0.3)-(2.0.7), (2.0.17)-(2.0.18) and some computations, we can write
( )
0
( , ; )
limt l ( , ) l
V x t
u x t dx
x
γ ∞ ξ τ
→∞
∂ −
∂
∫
1
( )
( , ; )
limt s ( , ) s 0,
V x t
u x t dx
x
γ
ξ τ
→∞ ∞
∂ −
+ =
∂
∫
( ) 0
( , ; )
[ l ( , ) l ] x t ( ) 0 ,
V x t
u x t t dt
x γ
ξ τ
γ
∞
=
∂ −
′ =
∂
( ) 0
( , ; )
[ ( , ) l ] 0,
l x t
V x t
u x t dt
t γ
ξ τ
∞
=
∂ −
= ∂
∫
( ) 0
( , ; )
[ s ( , ) s ] x t ( ) 0 ,
V x t
u x t t dt
x γ
ξ τ
γ
∞
=
∂ −
′ =
∂
∫
( ) 0
( , ; )
[ s ( , ) s ] x t 0 ,
V x t
u x t dt
t γ
ξ τ
∞
=
∂ −
− =
∂
∫
( ) 0
( , ) ( , ; )
( ( ))[ l l ]
l x t
u x t V x t
t dt
x x γ
ξ τ
α γ
∞
=
∂ ∂ −
∂ ∂
∫
( ) 0
( , ) ( , ; )
( ( ))[ s s ]
s x t
u x t V x t
t dt
x x γ
ξ τ
α γ
∞
=
∂ ∂ −
−
∂ ∂
∫
( ) 0
( , ; )
( ( )) x t ( ) ,
V x t
t t dt
x γ
ξ τ
α γ γ
∞
=
∂ −
′ =
∂
∫
and
0
[Al ( ,x ul ( , ))x t
∞
∫
( )
( , ) ( , ; )
]
l l
x t
u x t V x t
dt
t x γ
ξ τ
=
∂ ∂ −
∂ ∂
0
[As ( ,x us ( , ))x t
∞
−
∫
( )
( , ) ( , ; )
]
s s
x t
u x t V x t
dt
x x γ
ξ τ
=
∂ ∂ −
∂ ∂
( ) 0
( , ; )
( ( )) x t ( )
V x t
t t dt
x γ
ξ τ
α γ γ
∞
=
∂ −
′ = −
∂
∫
A applying above relation, we now obtain a similar relation to (2.0.20) for computing the values of unknown function at the boundary of domain. For this purpose we do as follows:
0 0
0
( , ; )
[ ( ) ]
b
l
l t
V x t
u x dx
x
ξ τ
=
∂ −
∂
∫
1 0
0
( , ; )
[ ( ) l ] x
V x t
f t dt
t
ξ τ
∞
=
∂ −
−
∂
∫
0 0
0
( , ; )
[ ( ) ]
b
s
s t
V x t
u x dx
x
ξ τ
=
∂ −
+
∂
∫
2 1
0
( , ; )
[f ( )t Vs x t ] x dt t
ξ τ
∞
=
∂ −
−
∂
∫
0 0
0
( , ) ( , ; )
[ l ( 0) l x l ] x
u x t V x t
dt
x x
ξ τ
α
∞
= =
∂ ∂ −
−
∂ ∂
∫
1 1
0
( , ) ( , ; )
[ s (1) s x s ] x
u x t V x t
dt
x x
ξ τ
α
∞
= −
∂ ∂ −
+
∂ ∂
∫
0
[Al ( ,x ul ( , ))x t
∞
−
∫
0
( , ) ( , ; )
]
l l
x
u x t V x t
dt
x x
ξ τ
=
∂ ∂ −
∂ ∂
0
[As ( ,x us ( , ))x t
∞
+
∫
1
( , ) ( , ; )
]
s s
x
u x t V x t
dt
x x
ξ τ
=
∂ ∂ −
∂ ∂
( , ) ( , ; )
( ) l
l l
l l
u x t V x t
x d
x x
ξ τ
α
Ω
∂ ∂ −
′
+ Ω
∂ ∂
∫
( , ) ( , ; )
( ) s
s s
s s
u x t V x t
x d
x x
ξ τ
α
Ω
∂ ∂ −
′
+ Ω
∂ ∂
∫
( , ) ( , ; )
[ l
l l
u x t V x t
x t
ξ τ
Ω
∂ ∂ −
+
∂ ∂
∫
( , ; ) ( , ( , ))
( ) ]
( )
l l l
l l
l
V x t A x u x t
x d
x x
ξ τ
α
α
∂ −
′
+ Ω
∂
( , ) ( , ; )
[ s
s s
u x t V x t
x t
ξ τ
Ω
∂ ∂ −
+
∂ ∂
∫
( , ; ) ( , ( , ))
( ) ]
( )
s s s
s s
s
V x t A x u x t
x d
x x
ξ τ
α
α
∂ −
′
+ Ω
∂
(2.0.21)
( ( , )) ( , )
( , ) ,
( )
( ( , )) ( , ) 1
( , ) ,
2 ( )
( ( , )) ( , )
( , ) ,
( )
( ( , )) ( , ) 1
( , ) .
2 ( )
l l
l l
l l
l l
s s
s s
s s
s s
a u u
a u u
a u u
a u u
ξ τ ξ τ
ξ τ
α ξ ξ
ξ τ ξ τ
ξ τ
α ξ ξ
ξ τ ξ τ
ξ τ
α ξ ξ
ξ τ ξ τ
ξ τ
α ξ ξ
∂ ⎧
∈ Ω
⎪ ∂
⎪
⎪ ∂
⎪ ∈ ∂Ω
⎪ ∂
⎪ = ⎨
∂ ⎪
∈ Ω
⎪ ∂
⎪
⎪ ∂
⎪ ∈ ∂Ω
⎪ ∂
We thus proved:
Theorem. Let the function a u( ) in problem (2.0.1)-(2.0.8) that satisfies (2.0.9), (2.0.14) and
( , ; ) , ( , ; )
l s
V x ξ t−τ V x ξ t−τ be the fundamental solution of (2.0.17) and (2.0.18), respectively. Then the solution of problem (2.0.1)-(2.0.8) satisfies (2.0.20) and (2.0.21).
By using (2.0.20), we can obtain
0
( ( , )) ( , ) 1
[ ]
2 ( )
l l
l
a u u
ξ
ξ τ ξ τ
α ξ ξ =
∂ ∂
as follows:
0
( ( , )) ( , ) 1
[ ]
2 ( )
l l
l
a u u
ξ
ξ τ ξ τ
α ξ ξ =
∂ ∂
0 0 , 0
0
( , ; )
[ ( ) ]
b
l
l t
V x t
u x dx
x ξ
ξ τ
= =
∂ −
=
∂
∫
1 0 , 0
0
( , ; )
[ ( ) l ] x
V x t
f t dt
t ξ
ξ τ
∞
= =
∂ −
−
∂
∫
0
1
0 , 0
( , ; )
[ s ( ) s ] t
b
V x t
u x dx
x ξ
ξ τ
= =
∂ −
+
∂
∫
2 1 , 0
0
( , ; )
[ ( ) s ] x
V x t
f t dt
t ξ
ξ τ
∞
= =
∂ −
−
∂
∫
0
[Al ( ,x ul ( , ))x t
∞
−
∫
0 , 0
( , ) ( , ; )
]
l l
x
u x t V x t
dt
t x ξ
ξ τ
= =
∂ ∂ −
∂ ∂
0
[As ( ,x us( , ))x t
∞
+
∫
1 , 0
( , ) ( , ; )
]
s s
x
u x t V x t
dt
x x ξ
ξ τ
= =
∂ ∂ −
∂ ∂
0
( , ) ( , ; )
( ) l
l l
l l
u x t V x t
x d
x x ξ
ξ τ
α =
Ω
∂ ∂ −
′
+ Ω
∂ ∂
∫
0
( , ) ( , ; )
( )
s s
s s
s
u x t V x t
x d
x x ξ
ξ τ
α =
Ω
∂ ∂ −
′
+ Ω
∂ ∂
∫
( , ) ( , ; )
[ l
l l
u x t V x t
x t
ξ τ
Ω
∂ ∂ −
+
∂ ∂
∫
0
( , ; ) ( , ( , ))
( ) ]
( )
l l l
l l
l
V x t A x u x t
x d
x ξ x
ξ τ
α
α
=
∂ −
′
+ Ω
∂
( , ) ( , ; )
[ s
s s
u x t V x t
x t
ξ τ
Ω
∂ ∂ −
+
∂ ∂
∫
0
( , ; ) ( , ( , ))
( ) ]
( )
s s s
s s
s
V x t A x u x t
x d
x ξ x
ξ τ
α
α
=
∂ −
′
+ Ω
∂
(2.0.22)
In similar fusion we obtain 1[ ( ( , ))
2 ( )
s
s
a u ξ τ
α ξ
1
( , ) ] s
u
ξ
ξ τ
ξ =
∂
∂ and call it (2.0.23).
Now, considering (2.0.21) and (2.0.22) we obtain a system of Fredholm integral equation for the
0
( , ) l
u
ξ
ξ τ
ξ =
∂
∂ and 1
( , ) s
u
ξ
ξ τ
ξ =
∂
∂ and call it (2.0.24)
and (2.0.25), respectively.
We also obtain some similar relation for
0
( , ) l
u
ξ
ξ τ
ξ =
∂
∂ and 1
( , ) s
u
ξ
ξ τ
ξ =
∂
∂ in domains Ωl and s
Ω .
Thus by (2.0.19)-(2.0.25) we will obtain a system of nonlinear Fredholm integral equation of the second kind for ul and us. Hence, by substitution of ul and us in (2.0.8), we will obtain a nonlinear Fredholm integral equation for the free boundary ( )γ t .
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