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Online Publication Date: 12 December, 2011
Publisher: Asian Economic and Social Society
Solution Of A System Of Two Partial Differential Equations Of The
Second Order Using Two Series
Hayder Jabbar Abood (Department of Mathematics, College of
Education, Babylon University, Babylon,Iraq)
Citation: Hayder Jabbar Abood (2011): “ Solution Of A System Of Two Partial Differential Equations Of The
Second Order Using Two Series ” Journal of Asian Scientific Research , Vol.1, No.8,pp.407-418.
Journal of Asian Scientific Research, 1 (8), pp.407-418
407
Introduction
Levenshtam (2009), considered systems of ordinary differential equations of the first order whose terms oscillate
with a high frequency ω.For the problem on periodic solutions, the author justifies the averaging method and
establishes a posteriori bounds for the error of partial sums of the complete asymptotic expansion for the solution.
Asymptotic expansions of periodic solutions of second –and third- order equations and formal asymptotics of such
solutions in the case of equations of arbitrary order were constructed in Abood1,2
(2004). The first-order asymptotic
form was obtained and proved for the solution of a system of two partial differential equations with small
parameters in the derivatives for the regular part, two boundary-layer parts and corner boundary part in Vasil'eva
and Butuzob (1990). In Levenshtam and Abood (2005), an algorithm of asymptotic integration of the initial-
boundary-value problem for the heat-conduction equation with minor terms (nonlinear sources of heat) in a thin rod
of thickness1/2 oscillating in time with frequency
1was proposed. In the present paper, we consider a
system of partial differential equations of the second order and solve this system with the aid of two series and some
conditions. Our paper continues the line of research initiated in Levenshtam and Abood (2005). Examples of
applications of systems of second order partial differential equations in modelling can be found in elasticity in
Nerantzaki and Kandilas (2008) and packed-bed electrode in Xiao-Ying Qin and Yan-Ping Sun (2009).
Statement Of The Problem
We study a system of second order partial differential equations with initial-boundary-value conditions
2
1 1 120
2
2 2 220
, , , , ,
, , , , ,
i
i
i
i
i
i
u u ub x a x t u f u x y t
t y x
v v vb x a x t v f u x y t
t y x
(2.1)
in the domain ( , , ) (0 1) (0 1) (0 ).x y t x y t T
The initial boundary conditions are
Solution Of A System Of Two Partial Differential Equations Of The Second
Order Using Two Series
Abstract
Author (s)
Hayder Jabbar Abood
Department of Mathematics, College of
Education, Babylon University,
Babylon-Iraq.
E-mail: drhayder_jabbar@yahoo.com
Key Words: Partial differential
equations 1; System 2; Asymptotic 3.
For a system of two partial differential equations of second order, we obtain and justify two
asymptotics solutions in the form of two series with respect to the small parameter . We have
proven the solution is unique and uniform in the domain , and, further, each the asymptotics
approximations are within 1nO .
Solution Of A System Of Two Partial Differential Equations…..
408
0 0,1 0 0,1
0,1 0,1
| 0, | 0, | 0, | 0,
0, 0.
t x t x
y y
u u v v
u v
y y
(2.2)
where is a small parameter, and the functions 1( , , , , ),f u x y t and 2( , , , , )f u x y t are continuous and infinitely
differentiable with respect to each of their arguments. The construction is made under the following conditions.
I. The functions , ,i ib x a x t and , 1,2if i have continuous derivatives of order
2n .
II. We can assume that 2 1b . If 2 2b b x , then making the change of the variable 1
2
0
x
z b d , and we
shall assume that 1 0b b x and 2 1b .
Algorithm For Construction Of The Asymptotics
We seek an asymptotics expansion of the solution of problem (2.1) and (2.2) as the following two series in the
powers of in the form
0
0
( , , , ) [ ( , , ) ( , , ) ( , , )],
( , , , ) [ ( , , ) ( , , ) ( , , )],
i
i i ii
i
i i ii
u x y t u x y t p u x y q u y t
v x y t v x y t p v x y q v y t
(3.1)
where iu and iv are coefficients of the regular part of the asymptotic, , ,i i ip u p v q u and ip v are boundary-layer
functions describing boundary layers near the initial instant of time 0t and the ends of the rod 0x and 1x .
The boundary-layer variables are
t
, and
x
.
Regular Parts Of The Asymptotics u And v
We substitute series (3.1) into equations (2.1) - (2.2) and the coefficients of the same powers of in the left and
right-hand sides of the obtained relations are equated. If the variable is assumed to be independent of t and
functions depending on ε are represented by the corresponding asymptotic series then problem (2.1) and (2.2)
becomes
Journal of Asian Scientific Research, 1 (8), pp.407-418
409
2
1 1 120
2
2 2 220
, 0 0,1 , 0 0,1
0,1 0,1
, , , , ,
, , , , ,
| 0, | 0, | 0, | 0,
0, 0.
i
i
i
i
i
i
t x t x
y y
u u ub x a x t u f u x y t
t y x
v v vb x a x t v f v x y t
t y x
u u v v
u v
y y
(4.1)
The following problems for the regular coefficients 0u and 0v are obtained:
01 0 10 0
02 0 20 0
0 0
0 0
0,1 0,1
, , , , ,
, , , , ,
0, , 0, , ,0 0,
0, 0,y y
ub x a x t u f u x y t
x
va x t v f v x y t
x
u y t v x y
u v
y y
(4.2)
By direct integration, it can be seen [Viik (2010), Viika and Room (2008), Tokovyy and Chien-Ching Ma,(2009),
Toshiki, Son Shin, Murakami and Ngoc(2007)] system (4.2) is equivalent to system of integral equations
1 1
0 1 0 10
0
, exp , , , , ,x x
u x t b p a p t dp b v t f y t d
(4.3)
0 2 0 20
0
, exp , , , , .t t
s
v x t a x p dp u x s f x y t ds
(4.4)
Substituting (4.4) in (4.3), we arrive at an integral equation with respect to 0, :u x t
0 0 0 0
0 0
, , , , , , ,x t
u x t G x t s u s d ds g x t
where 0 , , ,G x t s and
0( , )g x t are known functions. The solution of this integral equation can be expressed in terms
of the resolvent 0, , ,x t s of the Kernel 0
, , ,G x t s (as in [Viik (2010), Viika and Room (2008), Tokovyy
and Chien-Ching Ma,(2009), Toshiki, Son Shin, Murakami and Ngoc(2007)])
0 0 0 0
0 0
, , , , , , .x t
u x t g x t x t s g s d ds
After that, the function 0,v x t can be determined by (4.4).
The system of equations for 1
u and 1
v is
Solution Of A System Of Two Partial Differential Equations…..
410
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
11 1 10 0 0 11 0
2
0 0
2
12 1 20 0 0 21 0
2
0 0
2
1 1 1 1
1 1
0,1 0,1
, , , , , ,
,
, , , , , ,
,
0, , 0, , , , ,0 , ,0 ,
0, 0.y y
ub x a x t u x t f u x y t u f u x y t
x u
u u
t y
va x t v x t f v x y t u f v x y t
t v
u v
y x
u y t q u y t v x y p v x y
u v
y y= =
¶ ¶= + + -
¶ ¶
¶ ¶-
¶ ¶
¶ ¶= + + -
¶ ¶
¶ ¶-
¶ ¶
= - = -
¶ ¶= =
¶ ¶
(4.5)
From system (4.5) and direct integration, we have the system of integral equations
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
1 1
1 1 2 1
0
2
0 010 0 0 11 0 2
, exp , [ , ,
, , , , ] ,
x x
u x t b p a p t dp b a t v t
u uf u x y t u f u x y t d
u t y
s
s s s
s
- -æ ö
÷ç ÷ç= +÷ç ÷ç ÷è ø
¶ ¶¶+ - -
¶ ¶ ¶
ò ò (4.6)
( ) ( ) ( ) ( )
( ) ( )
1 2 1 1
0
2
0 020 0 0 21 0 2
, exp , [ , ,
, , , , ] .
t t
s
v x t a x p dp a x s u x s
u vf v x y t v f v x y t ds
v y x
æ ö÷ç ÷ç= +÷ç ÷ç ÷è ø
¶ ¶¶+ + - -¶ ¶ ¶
ò ò (4.7)
Substituting (4.7) in (4.6), we arrive at the integral equation with respect to 1, :u x t
1 1 1 1
0 0
, , , , , , , , ,x t
u x t G x y t s u s d ds g x y t
where 1 , , , ,G x y t s and
1( , , )g x y t are known functions. The solution of this integral equation can be expressed in
terms of the resolvent 1, , , ,x y t s of the Kernel 1
, , , ,G x y t s [Viik (2010), Viika and Room (2008),
Tokovyy and Chien-Ching Ma,(2009), Toshiki, Son Shin, Murakami and Ngoc(2007)] and we have
1 1 1 1
0 0
, , , , , , , ,x t
u x y t g x y t x y s g s d ds . After that, the function 1, ,v x y t can be determined by
(4.7).
For regular terms nu and nv we obtain the problem
Journal of Asian Scientific Research, 1 (8), pp.407-418
411
( ) ( ) ( ) ( )
( ) ( )
1 0
0,1
, , , , ,
0, , 0, , , 0.
nn n
nn n
y
ub x a x t u x t u x y t
x
uu y t q u y t
y=
¶= + ¡
¶
¶= - =
¶
(4.8)
( ) ( ) ( )
( ) ( )
2 0
0,1
, , , , ,
, ,0 , ,0 , 0.
nn n
nn n
y
va x t v x t v x y t
t
vv x y p v x y
y=
¶= + ¡
¶
¶= - =
¶
(4.9)
where ( ) ( ) ( )2
1 10 1 0 2 2
0
1, , , , ,
!
kn
n nn k kk
k
u uu x y t f u x y t u x t
k u t y
- -
=
¶ ¶¶¡ = - -
¶ ¶ ¶å and
( ) ( ) ( )2
0 00 2 0 2 2
0
1, , , , ,
!
kn
n k kkk
u vv x y t f v x y t v x t
k u y x=
¶ ¶¶¡ = - -
¶ ¶ ¶å . This problem is quite similar to
problem (4.5) and can be solved by reduction to a system of integral equations.
Boundary Parts Of The Asymptotics pu And pv
We construct the following group of coefficients of two series (3.1) that is the boundary-layer parts
, , ,pu x y and , , ,pv x y . We consider problem (2.1) in a neighbourhood of the upper boundary 0t of
the domain and perform the change of variables t ; we obtain the system
2
1 12
2
2 22
, , , , ,
, , , , , ,
pu pu pub x a x t pu f pu x y
y x
pv pv pva x t pv f pv x y
y x
(5.1)
0,1 0,1
, ,0 , ,0 , , ,0 , ,0 , 0, 0.y y
dpu dpvpu x y u x y pv x y v x y
dy dy
The boundary-layer functions are0
p u and0
p v . In this case0
0p v .
For 0p u we obtain the problem
0 01 0
0 0 0
0 0
0,1 0,1
,0 , 0 , 0,
0, , 0, , ,0 , ,0 ,
0, 0.y y
p u p ub x a x p u x X
x
p u y p u x y u x y
dp u dp v
dy dy
(5.2)
By direct integration, equation (5.2) has a classical solution
Solution Of A System Of Two Partial Differential Equations…..
412
1 1
0 1
0 0
( ),0 exp ( ),0 ,0,
0, ,
u x a x s dsp u x
where 1
0
x
x b d and 1 z is the function inverse to z x .
The system of equations for 1p u and 1p v is
1 11 1 1
1 1 1
1
0,1
, ,
, ,0 , ,0 , 0, , 0,
0,y
p u p ub x a x t p u x
x
p u x y u x y p u y
dp u
dy
12 1 1
1 1
10,1
, , ,
, ,0 , ,0
0.y
p va x t p v x
x
p v x y v x y
dp v
dy
Here 2
011 0 10 0 2
, ,0 , , , ,0,0p ua
x x p u x f p u x yt y
and
2
2 11 0 20 0 2
, ,0 , , , ,0,0a p v
x x p v x f p v x yt y
is smooth. In the
same way we obtain
2 0
1 2 0
,0 , , 0,0 ,
0, .
x
a x p v x s ds xp v a x p v x s ds
x
1 1
1 1
0
11 1
( ),0 exp ( ),0
( ),0 , 0
0, .
u x a x s
p u x s ds x
x
Journal of Asian Scientific Research, 1 (8), pp.407-418
413
The function np u can be defined as the solution of the problems
1 , , , ,n nn n
p u p ub x a x t p u x y
x
(5.3)
0,1
, ,0 , ,0 , 0, , 0,
0,
n n n
n
y
p u x y u x y p u y
dp u
dy
(5.4)
where
11 1 0
0
2
11 0 2
0
1, , [ ,0 , , , ,0,0
!
, , ,0,0 ] ,
nk k
n k k
k
k
nkk
ax y x p u x f p u x y
k t
p uf p u x y
p u y
since (5.3) is a partial differential equation of the first order and first degree. Here the smooth function , ,n x y
is known. Using the initial and boundary conditions (5.4), we obtain
1 1
1
0
1
( ),0 exp ( ),0
( ),0 , 0
0, .
n
n n
u x a x s
p u x s ds x
x
For the function np v , it is the solution of the problem
0,1
, , ,
, ,0 , ,0
0,
nn
n
y
p vx y
p v x y v x y
dpv
dy
where
22 1 1 0
0
2
1 11 0 2
0
1, , , [ ,0 , , , ,0,0
!
, , ,0,0 ] ,
nk k
n n k k
k
k
n nkk
ax y a x t p v x p v x f p v x y
k t
p v p vf p v x y
p v x y
Solution Of A System Of Two Partial Differential Equations…..
414
that satisfies the condition , , 0,np x y where , ,n x y is a known continuous function vanishing
for x and with partial derivative making jump on the characteristic line x . Integrating, we obtain
, , , 0 ,, ,
0, ,
n
xn
x y r dr xp v x y
x
where the function , ,np v x y is smooth everywhere.
Boundary Parts Of The Asymptotics qu And qv
Now, we find the coefficients , ,iq u y t and , ,iq v y t . Consider problem (2.1) and the conditions (2.2) in a
neighbourhood of the left boundary of the domain and perform the change of variable1x . We obtain the
system
2
1 120
2
2 22
1 10 , , , , ,
!
, , , , , , , , .
k k
k
qu qu qub a t qu f qu y t
t y k
qv qv qva t qv y t f qu y t
t y
For 0 , ,q u y t we obtain the equation 00 0.q u
b
From this equation, taking into account the standard condition for boundary parts at infinity, that
means 0 , 0q u t , and so we have 0 , , 0q u y t .
Now, for 0 , ,q v y t we have the problem
0 02 0
0 0 0
0
0,1
0, , , , 0, 0 ,
0, , 0, , , , ,0 0,
0y
q v q va t q v y t t T
t
q v y t v y t q v y
q v
y
Since this is a partial differential of first order and first degree, it’s solution have the form
0 2
0 0
0, , exp 0, , , 0, ,
0, .
v y t a y s t ds t Tq v y t
t
Journal of Asian Scientific Research, 1 (8), pp.407-418
415
It’s possible to find 1q u and 1q v in the same way. Finally, we will continue to find nq u and nq v .
For nq u we have the equation
0 , , ,nn
q ub y t
(6.1)
where
11 1 0
0
2
1 11 0 2
0
1, , [ 0, , ,0, , ,0
!
, , ,0,0 ] ,
nk k
n k k
k
k
n nkk
ay t y t q u f q u y t
k t
q u q uf q u x y
q u t y
is known and continuous everywhere.
The function , , 0n y t for t (is evident from our findings in the above), so we can seek this function
in the form
, , , , for .n ny t z y t t t (6.2)
The function ,n t is smooth everywhere. By the condition , , 0nq u y t and integrating (6.1), we have
1 0 , , 0
, ,
0, .
n
n t
b z y t s s t ds t Tq u y t
t
We can define the part , ,nq v y t as the solution of the problem
, , ,n nn
q v q vy t
t
(6.3)
0,1
0, , 0, , , , ,0 0,
0
n n n
n
y
q v y t v y t q v y
q v
y
(6.4)
here
Solution Of A System Of Two Partial Differential Equations…..
416
22 1 1 0
0
2
1 11 0 2
0
1, , 0, , [ ,0 0, , ,0, , ,0
!
,0, , ,0 ] .
nk k
n n k k
k
k
n nkk
ay t a y t p v x p v y t f p v y t
k t
p v p vf p v y t
p v x y
Also
, ,n y t is a known continuous function that satisfies the condition , , 0,np v y t vanishing for t
and with partial derivative making jump on the characteristic line t . Solving (6.3) and using (6.4), we obtain
0
0, , exp 0, , , 0, ,
0, .
n n
n
v y t y s t ds t Tq v y t
t
where the function , ,nq v y t is smooth everywhere.
7. JUSTIFICATION OF THE ASYMOTOTICS
By , , ,nU x y t and , , ,nV x y t we denote the n -th
partial sums of the series (3.1).
Theorem. The solution , , , , , , ,u x y t v x y t of problem (2.1) and (2.2) admits the asymptotic solution
1 1( , , , ) ( , , , ) , ( , , , ) ( , , , )n n
n nu x y t U x y t O v x y t V x y t O , (7.1)
uniformly in the domain 0 1 0 1 0x y t T .
Proof. We set 1 1n
u U w
and 1 2n
v V w
. Substituting this in (1.1) and (2.1) we obtain the following problem for
the remainder terms 1
w and 2
w
2
1 1 1
1 1 1 12
2
2 2 2
2 2 2 22
1 0 1 0,1 2 0 2 0,1
1 2
0,1 0,1
, , , , ,
, , , , ,
| 0, | 0, | 0, | 0,
0, 0.
t x t x
y y
w w wb x a x t w x y t
t y x
w w wb x a x t w x y t
t y x
w w w w
w w
y y
Obviously, the inhomogeneous terms 1
and
2
can be estimated as 1, , , n
ix y t O uniformly in . We
shall prove that the same estimate is valid for 1
w and2
w . In view of the equalities
1
1 1 1
1
2
,
,
n
n n n n
n
n
u U u U U U w O
v V w O
this will directly imply the assertion of the theorem.
Journal of Asian Scientific Research, 1 (8), pp.407-418
417
We make the change of the variables 1,2
k x y t
i iw re i
, where k const >0. We obtain the following system of
equations for 1
r and2
r :
2
1 1 1
1 1 1 12
2
2 2 2
2 2 2 22
ˆ, , , , ,
ˆ, , , , ,
r r rb x c x t r x y t
t y x
r r rb x c x t r x y t
t y x
(7.2)
with the homogeneous boundary conditions, as in the case of , 1,2i
r i . Here
( ) 1
1 1 2 2ˆ, , , 1 , ( ).k x t n
i ic a x t k b x c a x t k e O
Assume that
1r has a maximum at
a point 1 1 1,x t of and
2r has a maximum at a point 2 2 2
,x t , and assume that 1 1 2 2r r . We shall
consider the first equation of (7.2) at1
, (If 1 1 2 2r r , then we shall consider the first equation of (7.2)
at2
). We rewrite this equation as
2
1 1 1
1 1 1 12
ˆ, , , , ,r r r
b x c x t r x y tt y x
(7.3)
Assume that the function1
r is negative and has a minimum at1
. (The case of a positive maximum can be
considered in the same way.) Then 1 0r
t
and 1 0
r
x
at this point (the inequality sign is possible only if
1 lies on
the boundary of ). Hence the left-hand side of (7.3) is negative at 1
and is not larger than 1 1r , while the right-
hand side is of order1n .
Consequently, 1
1 1 1max , nr r x t O
.Since 1 1 2 2
r r , it follows
that 1
2 2 2max , nr r x t O
.Therefore 1, n
ir x t O uniformly in . Hence we also have
1k x t n
i iw r e O
uniformly in G .
The proof of Theorem is complete.
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