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(1)

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)

(2)

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 thickness

 



1/2oscillating 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 1

2

0 2

2 2 2

2

0

,

, , ,

,

, , ,

,

i i i

u

b x

a x t u

f

u x y t

t

y

x

v

b

x

a

x t v

f

u x y t

t

y

x



   

















_



_











_



_



_

_







_



_



(2.1)

in the domain

( , , )

x y t

        

(0

x

1) (0

y

1) (0

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: [email protected]

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

 

n1

(3)

408

0 0,1 0 0,1

0,1 0,1

|

0,

|

0,

|

0,

|

0,

0.

t x t x

y y

u

v

u

v

y

   

 





_



_



(2.2)

where



is a small parameter, and the functions

f u x y t

₁

( , , , , ),



and

f u x y t

₂

( , , , , )



are continuous and infinitely differentiable with respect to each of their arguments. The construction is made under the following conditions.

I. The functions

b x a x t

_i

   

,

_i

,

and

f i

_i

,



1, 2

have continuous derivatives of order



n



2 

.

II. We can assume that

b

₂



1

. If

b

₂



b

₂

 

x

, then making the change of the variable ₂1

 

0

x

z





b



 

d

, and we

shall assume that

b

₁



b x

 



0

and

b

₂



1

. 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

( , , , )

[ ( , , )

( , , )

( , , )],

( , , , )

[ ( , , )

( , , )

( , , )],

i

i i i

i

i i i

i

u x y t

p u x y

q u

y t

v x y t

p v x y

q v

y t













   

 



 



(3.1)

where

u

_i and

v

_i are coefficients of the regular part of the asymptotic,

p u p v q u

_i

,

_i

,

_i and

p v

_i are boundary-layer functions describing boundary layers near the initial instant of time

t



0

and the ends of the rod

x



0

and

x



1

. The boundary-layer variables are

t







, and



x





.

Regular Parts Of The Asymptotics

u

And

v

(4)

409

 





 





2

1 1 1

2

0 2

2 2 2

2

0

, 0 0,1 , 0 0,1

0,1 0,1

,

, , ,

,

, , ,

,

|

0,

|

0, |

0,

|

0,

0.

i i i

t x t x

y y

u

b x

a x t u

f

u x y t

t

y

x

v

b

x

a

x t v

f

v x y t

t

y

x

u

v

u

v

y

 



   

   

 





_





_



_

_



_



_













_



_







_



_





_



_





(4.1)

The following problems for the regular coefficients

u

₀and

v

₀are obtained:

 





 













0

1 0 10 0

0

2 0 20 0

0 0

0,1 _0,1

,

, , ,

,

, , ,

,

0, ,

0,

, , 0

0,

y _y

u

b x

a x t u

f

u x y t

x

v

a

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 tou₀

 

x t, :

0

 

0



 

0



0

 

0 0

, , , , , , ,

x t

u x t 

 

G x t s u  s d ds g x t

whereG x t₀



, , , s



andg x t₀( , ) are known functions. The solution of this integral equation can be expressed in terms of the resolvent ₀



x t, , , s



of the Kernel G x t₀



, , , 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 functionv₀

 

x t, can be determined by (4.4).

(5)

410

( )

(

) (

)

(

)

(

)

(

) (

)

(

)

(

)

(

)

(

)

(

)

(

)

1

1 1 10 0 0 11 0

2

0 0

2 1

2 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.

y _y

u

b x

a x t u x t

f

u x y t u

f

u x y t

x

u

t

y

v

a 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 0

10 0 0 11 0 2

,

exp

,

[

,

, ,

]

,

x x

u x t

b

p a p t dp b

a

t v

t

u

f

u x y t u

f

u x y t

d

u

t

y

s

-

-æ

_ö÷

ç

_÷

ç

=

_ç

_÷

+

ç

÷

è

ø

¶

+

-

-¶

¶

ò

(4.6)

(

)

(

)

(

) (

)

(

)

(

)

1 2 1 1

0

2

0 0

20 0 0 21 0 2

,

exp

,

[

,

, ,

] .

t t

s

v

x t

a x p dp

a x s u x s

u

v

f

v x y t v

f

v x y t

ds

v

y

x

æ

_ö÷

ç

_÷

ç

=

_ç

_÷

+

ç

÷

è

ø

¶

+

-

-¶

¶

ò

(4.7)

Substituting (4.7) in (4.6), we arrive at the integral equation with respect to u x t₁

 

, :

 



  





1 1 1 1

0 0

, , , , , , , , ,

x t

u x t 

 

G x y t  s u  s d ds g x y t

whereG x y t₁



, , , , s



andg x y t₁( , , ) 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 G x y t₁



, , , , 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 functionv x y t₁



, ,



can be determined by

(4.7).

(6)

411

( )

(

) (

)

(

)

(

)

(

)

1 0 0,1

,

, ,

,

0, ,

,

0.

n n n n n n y

u

b x

a x t u

x t

u x y t

x

u

y t

q u

y t

y

₌

¶

=

+ ¡

¶

= -

=

¶

(4.8)

(

) (

)

(

)

(

)

(

)

2 0 0,1

,

, ,

,

, , 0

, , 0 ,

0.

n n n n n n y

v

a x t v

x t

v x y t

t

v

x y

p v x y

y

₌

¶

=

+ ¡

¶

= -

=

¶

(4.9)

where

(

)

(

) (

)

2

1 1

0 1 0 2 2

0

1 , ,

, ,

,

!

k n n n

n k k k

k

u

u x y t

f

u x y t u

x t

k

u

t

y

- -=

¶

¡

=

-

-¶

¶

å

and

(

)

(

) (

)

2 0 0

0 2 0 2 2

0

1 , ,

, ,

,

!

k n

n k k k

k

u

v

v 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  andpv x y



, , , 



. We consider problem (2.1) in a neighbourhood of the upper boundary



t0



of the domain  and perform the change of variablest



; we obtain the system

 





 





2 1 1 2 2 2 2 2

,

, , ,

,

, , ,

,

pu

b x

a x t pu

f

pu x y

y

x

pv

a

x t pv

f

pv x y

y

x



 





 





_



_



_

_











_



_







_



_

(5.1)

















0,1 0,1

, , 0

, , 0 ,

, , 0

, , 0 ,

0,

0.

y y

dpu

dpv

pu x y

u x y

pv x y

v x y

dy

_

dy

_

 



The boundary-layer functions arep u₀ andp v₀ . In this casep v₀ 0.

For

p u

₀ we obtain the problem

 













0 0

1 0

0 0 0

0 0

0,1 0,1

, 0

,

0,

0, ,

0,

, , 0

, , 0 ,

0,

0.

y y

p u

b x

a x

p u

x

X

x

p u

y

p u x y

u

x y

dp u

dp v

dy



 



_



_

_{ }

_





 



(5.2)

(7)

412

 



 





 



1 1

0 1

0 0

(

), 0 exp

(

), 0

, 0

,

0,

,

u

x

a

x

s

ds

p u x



 



 



 



 

 









 

 







 

_

_



_





where

 

1

 

0

x

b

d



_



 



and



1

 

z

is the function inverse to

z





 

x

.

The system of equations for

p u

₁ and

p v

₁ is

 













1 1

1 1 1

1 0,1

,

, , 0

, , 0 ,

0, ,

0,

y

p u

b x

a x t p u

x

p u x y

u x y

p u

y

dp u

dy







_



_

_{ }



 



 









1

2 1 1

1 1

1 0,1

,

, , 0

0.

y

p v

a

x t p v

x

p v x y

v x y

dp v

dy









 



 



Here

 





2 0 1

1

,

, 0

0

,

10 0

, , , 0, 0

2

p u

a

x

p u x

f

p u x y

t

y



















and

 

2

 





2 1

1

,

, 0

0

,

20 0

, , , 0, 0

2

a

p v

x

p v x

f

p v x y

t

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

x

p v

a

x

p v x s ds

x

 



 





 





_



_





 







 



 





 

1 1

0 1

1 1

(

), 0 exp

(

), 0

(

), 0

, 0

0,

.

u

x

a

x

s

p u

x

s

ds

x



 



 



 



 

 











 























 

 



_



(8)

413 The function

p u

_n can be defined as the solution of the problems

n

 

n ₁

 

,



, ,



,

n n

p u

b x

a x t p u

x y

x









 



(5.3)













0,1

, , 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





 









1

1 1 0

0

2 1

1 0 2

0

1 , ,

[

, 0

,

, , , 0, 0

!

, , , 0, 0 ]

,

n

k k

n k k

k k n k k

a

x y

x

p u x

f

p u x y

k

t

p u

f

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



 



 



 



 

  









 























 

 



_





For the function

p v

_n , it is the solution of the problem













0,1

, ,

,

, , 0

0,

n n n y

p v

x y

p v x y

v x y

dpv

dy







_{ }



 



where





 









2

2 1 1 0

0

2

1 1

1 0 2

0

1 , ,

,

[

, 0

,

, , , 0, 0

!

, , , 0, 0 ]

,

n

k k

n n k k

k k n n k k

a

x y

a

x t p v

x

p v x

f

p v x y

k

t

p v

f

p v x y

p v

x

y

(9)

414 that satisfies the condition

p

_n



x y

, ,

 



0,

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 x n

x y r dr

x

p v x y

x





 



 





 



 



_





where the function

p v x y

_n



, ,





is smooth everywhere.

Boundary Parts Of The Asymptotics

qu

And

qv

Now, we find the coefficients

q u

_i





, ,

y t



and

q v

_i





, ,

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 variable





x



1. We obtain the system

 











 







2

1 1

2

0 2

2 2

2

1

0 ,

,

, , ,

!

,

, , ,

,

, , ,

.

k k

k

qu

b

a

t qu

f qu

y t

t

y

k

qv

a

t qv

y t

f

qu

y t

t

y

























 





_





_







_

_



_







_











_



_



_

_





For

q u

₀





, ,

y t



we obtain the equation

b

 

0 q u

0

0. 







From this equation, taking into account the standard condition for boundary parts at infinity, that

means

q u

₀

 



,

t



0

, and so we have

q u

₀





, ,

y t





0

. Now, for

q v

₀





, ,

y t



we have the problem

  















0 0

2 0

0 0 0

0 0,1

0,

, ,

,

0, 0

,

0, ,

,

, , 0

0,

0

y

q v

a

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

T

q v

y t

t











 

  





_



_

 

_

_



_



(10)

415 It’s possible to find

q u

₁ and

q v

₁ in the same way. Finally, we will continue to find

q u

_n and

q v

_n .

For

q u

_n we have the equation

b

 

0 q u

n _n





, ,

y t



,





 



(6.1)

where

















1

1 1 0

0

2

1 1

1 0 2

0

1 , ,

[

0, ,

, 0, , , 0

!

, , , 0, 0 ]

,

n

k k

n k k

k

n n

k k

a

y t

y t q u

f

q u

y t

k

t

q u

f

q u x y

q u

t

y



_



 











_





is known and continuous everywhere.

The function



n





, ,

y t





0

for





t

(is evident from our findings in the above), so we can seek this function

in the form



_n





, ,

y t

  



z y t





  



_n



,

t

, for





t

.

(6.2) The function



_n

 



,

t

is smooth everywhere. By the condition

q u

_n





, ,

y t





0

and integrating (6.1), we have





    

  

1

0 ,

,

0 , ,

0,

.

n

n t

b

z y

t

s

s t ds

t

T

q u

y t

t







 

  



 



_





We can define the part

q v

_n





, ,

y t



as the solution of the problem

n n



, ,



,

n

q v

y t

t







 



(6.3)













0,1

0, ,

,

, , 0

0,

0

n n n

n

y

q v

y t

v

y t

q v

y

q v

y





 





_



(6.4)

(11)

416









 













2

2 1 1 0

0

2

1 1

1 0 2

0

1 , ,

0, ,

[

, 0

0, ,

, 0, , , 0

!

, 0, , , 0 ]

.

n

k k

n n k k

k k

n n

k k

a

y t

a

y t p v

x

p v

y t

f

p v

y t

k

t

p v

f

p v

y t

p v

x

y



_



 

















Also



, ,



n



y t



is a known continuous function that satisfies the condition

p v

_n





, ,

y t





0,

vanishing for





t

and with partial derivative making jump on the characteristic line





t

. Solving (6.3) and using (6.4), we obtain



, ,





0, ,



exp

₀



0, ,



, 0

0,

.

n n

n

v

y t

y s t

ds

t

T

q v

y t

t













 

  





_



_

 

_

_



_





where the function

q v

_n





, ,

y t



is smooth everywhere.

7. JUSTIFICATION OF THE ASYMOTOTICS

By

U

n



x y t

, , ,





and

V

n



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 _{( , , , )} _{( , , , )}

 

n1 _, _{( , , , )} _{( , , , )}

 

n1

n n

u x y t U x y t O   v x y t  V x y t  O   , (7.1)

uniformly in the domain  



0       x 1

 

0 y 1

 

0 t T



.

Proof. We set uU_n₁w₁and vVn1w2. Substituting this in (1.1) and (2.1) we obtain the following problem for

the remainder terms w₁and w₂

 





 





2

1 1 1

1 1 1 1

2

2 2 2

2 2 2 2

2

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 w

b x a x t w x y t

t y x

w w w

b 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

i x y t  O 



  uniformly in. We

shall prove that the same estimate is valid for w₁andw₂. 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

 

  



      

  

(12)

417 We make the change of the variables k x y t 



1, 2



i i

w re   i , where kconst >0. We obtain the following system of equations for r₁andr₂:

 





 





2

1 1 1

1 1 1 1

2

2 2 2

2 2 2 2

2

ˆ

, , , , ,

ˆ

, , , , ,

r r r

b x c x t r x y t

t y x

r r r

b x c x t r x y t

t y x

 

   

    

_ _  _

 

 

 _  _  _ _{ }

 

 _  _

(7.2)

with the homogeneous boundary conditions, as in the case ofr ii, 1, 2. Here

 



 



  



( ) 1

1 1 , , 2 2 , 1 , ˆ ( ).

k x t n

i i

c a x t k b x  c a x t k     e  O  Assume that r₁ has a maximum at a point ₁



x t₁,₁



of  and r₂ has a maximum at a point₂



x t₂, ₂



, and assume thatr₁

 

 ₁ r₂

 

₂ . We shall

consider the first equation of (7.2) at₁, (Ifr₁

 

 ₁ r₂

 

₂ , then we shall consider the first equation of (7.2) at₂). We rewrite this equation as

 





2

1 1 1

1 1 1 1

2

ˆ

, , , , ,

r r r

b x c x t r x y t

t y x

_  _     

  

  (7.3)

Assume that the functionr₁ is negative and has a minimum at₁. (The case of a positive maximum can be

considered in the same way.) Then r1 ₀

t

 

 and

1 ₀

r x

 

 at this point (the inequality sign is possible only if 1lies on

the boundary of). Hence the left-hand side of (7.3) is negative at ₁and is not larger thanr₁

 

₁ , while the right-hand side is of order



n1

.

Consequently,

 

1

1 1 max 1 ,

n

r r x t O







   .Since r₁

 

 ₁ r₂

 

₂ , it follows

that

 

1

2 2 max 2 ,

n

r   _ r x t O



 .Therefore

 

1

, n

i

r x t O   uniformly in. Hence we also have

 

1

k x t n

i i

w r e  O   uniformly inG. The proof of Theorem is complete.

References

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Abood2, H. J. (2004) Asymptotic Integration Problem of Periodic Solutions of Ordinary Differential Equation of Third Order with Rapidly Oscillating Terms,” Dep. in VINITI, no. 1357-B2004, 32 pages, Moscow, Russian, 05.08.2004.

Levenshtam, V. B. (2009) “Kantorovich Theorem and Justification of Asymptotics of Solutions of Differential Equations with Large High-Frequency Summands” Journal of Mathematical Sciences Vol.158, No.2, pp.228-238.

Levenshtam, V. B. and Abood, H. J. (2005) “Asymptotic integration of the problem on heat distribution in a thin rod with rapidly varying sources of heat” Journal of Mathematical Sciences, Vol.129, No.1, pp. 3626- 3634.

(13)

418 Toshiki Naito; Jong Son Shin; Satoru Murakami and Pham Huu Anh Ngoc(2007) “Characterizations of linear Volterra integral equations with nonnegative kernels” J. Math. Anal. Appl., Vol.335, pp.298–313.

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Room,

 

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