online publication date: 12 december, 2011 publisher ... vol.1, no.8, pp.407-418..pdf ·...

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: [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 1nO .

mailto:[email protected]

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


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


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


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


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


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


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


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


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.


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.

References

Abood1, H. J. (2004) Asymptotic Integration Problem of Periodic Solutions of Ordinary Differential Equations

Containing a Large High-Frequency Terms, Dep. in VINITI, no., 338-B2004, 28 Pages, Moscow, Russian,

26.02.2004.

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.

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Vol. 215, pp. 270–275.

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3850–3859.

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