Uniform Difference Scheme on the Singularly Perturbed System

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Uniform Difference Scheme on the Singularly

Perturbed System

Ilhame G. Amiraliyeva

Department of Mathematics, Faculty of Science, Sinop University, Sinop, Turkey Email: [email protected]

Received July 12, 2012; revised August 12, 2012; accepted August 19, 2012

ABSTRACT

This paper is concerned with the numerical solution for singular perturbation system of two coupled second ordinary differential equations with initial and boundary conditions, respectively. Fitted finite difference scheme on a uniform mesh, whose solution converges pointwise independently of the singular perturbation parameter is constructed and ana-lyzed.

Keywords: Singular Perturbation; Linear System; Difference Scheme; Uniform Convergence

1. Introduction

We consider the following singularly perturbed initial/ boundary value problem for the linear system of ordinary differential equations in the interval

 

0,1 :

 

   

 

1 1 1

1 1

:

,

L u u x a x u x b x u x

c x v x f x

  

  

  (1)

 

   

 

2 2

:

,

L u v x a x v x

c x u x f x



 



  

  (2)

 

1

0 , 0 ,

u A u B





  (3)

 

0 2, 1

 

v A v B₂, (4) where 0 is a small parameter

,

0,

a x

A1, A2, B1,

B2, are given constants. The functions i

 

i0,

 

,

i

c x f xi

 

are given functions sat- isfying certain regularity conditions which are specified whenever necessary.



i1, 2 ,



b x₁

 

The above type initial/boundary value problems arise in many areas of mechanics and physics [1,2].

Differential equations with a small parameter  mul-tiplying the highest order derivative terms are said to be singularly perturbed and normally boundary layers occur in their solutions. The numerical analysis of singular perturbation cases has always been far from trivial be-cause of the boundary layer behavior of the solution. Such problems undergo rapid changes within very thin layers near the boundary or inside the problem domain. It is well known that standard numerical methods for solv-ing such problems are unstable and fail to give accurate

results when the perturbation parameter  is small. Therefore, it is important to develop suitable numerical methods to these problems, whose accuracy does not depend on the parameter value , i.e. methods that are

-uniformly convergent. These include fitted finite dif-ference methods, finite element methods using special elements such as exponential elements, and methods which use a priori refined or special non-uniform grids which condense in the boundary layers in a special man-ner. The various approaches to the design and analysis of appropriate numerical methods for singularly perturbed differential equations can be found in [3-8] (see also ref-erences cited in them).

In this present paper, we analyze the numerical solu-tion of the initial/boundary problems (1)-(4). The nu-merical method presented here comprises a fitted differ-ence scheme on an uniform mesh. Fitted operator method is widly used to construct and analyse uniform difference methods, especially for a linear differential problems (see, e.g., [4-7]). In the Section 2, we state some important properties of the exact solution. The derivation of the difference scheme and uniform convergence analysis have been given in Section 3. Uniform convergence is proved in the discrete maximum norm. The approach to the construction of the discrete problem and the error analysis for the approximate solution are similar to those in [8,9].

Difference schemes for singularly perturbed systems with another type of initial/boundary conditions was in-vestigated in [9-14].

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2. Analytical Results

Here we give useful asymptotic estimates of the exact solution of (1.1)-(1.4), that are needed in later sections.

Lemma 2.1 Under the





1



1



1 2 c1 c2 exp b 1 1



     

  

  (5)

the problem (1.1)-(1.4) has a unique solution, which satisfies

,

u _C (6)

 



1



1

1 exp , 0

u x C  x x l



 

  _   _  

  , (7)

,

v_C (8)

 



 







 





1 2

2

1 2

2

1 exp 0

exp 1 1 ,

v x C x a

x a

 

 



    





   _



(9)

where _{ }

 

0,1 max

g _  g x for any continuous function

 

g x .

Proof. Consider the iterative process

 

_{ }

 

_{ }

 

_{ }

 

_{ }

 

_{ }

 

_{ }

1

1 1 1

2 2 2

1 1

2 2

, ,

0 , 0 ,

0 , 1 ,

n n

L u f x c x v

L v f x c x u

B

u A u

v A v B







  



  

 

 

 

  



(10)

where  0

 



_0,1 is an arbitrary function.

v x C



First we prove that for the solution of initial-value problem of the type

         

 

,

u x a x u x b x u x F x

     

 

0 ,

 

0 B

u A u





 

the following estimates hold

 

_exp



1



1 1



_,

u x _ b_ A B F _ (11)

 

















1

1 1

1 1 1

exp

exp 1_exp

exp 1 .

u x b b A

b b B

x B

b b F



 

  

  



 

 

 

  

 

 



 

 

 _  _ _

(12)

To prove (2.7), after some manipulations we have

 





 

1 1 1

0

1 1

0 d

d ,

x

u x A B b u F

A B F b u

    

   

  

 

 

 

   

   



hence

 

0

d ,

x

u x AB



u  

with





1_,

A A B F _ 

1 _.

B b_

From here by virtue of integral inequality it follows that

 

B ,

u x A e 

which leads to (2.7). Now we prove (2.8). Clearly

 

B _exp









1_.

u x  x u b F 

    

    

Then by using (2.7) we get

 









































1 1

2 1

1 1 1

exp exp

exp

exp exp

exp 1 ,

B

u x x b A b

F b B b

b A b

1

x b b

b b F

   



  

 

    

  

 

 

  

  

 

 

 

 

  

  

   

 



  

 

B

which arrive at (2.8).

Further, note that, by virtue of maximum principle the problem of the form

   

 

2 , 0

L vF x F x C ,1

 

0 , 1

 

v A v B

admits the estimate

1 _.

v  A B F _ (13)

Denoting

 

_{ }

 

_{ }

 

_{ }

 

_{ }

 

_{ }

 

_{ }

1

,

n n n

x u x u x x v x v x

 

 

 

from (1.1)-(1.4) and (2.6) we have

   

 

_{ }

 

_{ }

 

_{ }

1

1 1

1

2 2

, ,

0 0 0

0 1

n n

n

n n

L c L c

 

  

 

 

 

 



 

 



 



 ,

0.

 

(3)

Next, applying here (2.7), (2.8), (2.9) we arrive at

  1



1



 1

1 exp 1 ,

n n

b c

    

 

 

  1 1



1



1

1 1 exp 1 1 ,

n n

b b c

     

   

  

  _  _

   1

2 .

n n

c



 



 

  

2



Therefore

   

1 1

1 3

2

, , ,

n n

  

  



 



 



 



(14)

with









1 1

1 1 1

1

2 2 2

1 1 1

3 1 1 1

exp ,

,

exp 1 .

b c c

b b c



  

  

   

 

 





  

 

 

 

 _  _ _

m (2.10) we have

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Fro

   1

_

_

1  1

1 2 1 2 ,

n

n n

        

    (16)

 

_

_

2  1

1 2 1 ,

n n

     

  (17)

 

_

_

2  1

3 1 2 .

n n

     



 

 (18)

From (2.12), (2.13), (2.14) follows that the se uniformly converges on

quences

 

 

n _,

 

 n _,

 

 n

u u v x

 

0,1

for . Replacing (2.6) by appropri

ons we conclude that for the limit functions are the solution of (1.1)-(1.4).

Now the using (2.7) and (2.8) with the function

n 

integral equati

ate system of

n 

 

1

   

1

 

F x c x v x f

bounds

x

  yield the following stability

















1 1 1

1 1

e

exp

u

b 

  



 1

1

f _

xp exp

exp

b A b B

c b v

  

 

  

  

 

  



 (19)





















1 1

1 1 1

1 1

1 1 1

1 1 1 1

1 1 1

1 1 1 1

exp

exp 1

exp

exp 1

exp 1 .

u b b A

b b B

x B

b b f

b b c



 

  

  



 

 

 

  

  

  

  

  

 

 

 _  _

 

 _  _ v_

(20)

Next from (2.9), with F x

 

c2

   

x u x  f2

 

x

it follows that

1 1

2 2 2 2 2 2 .

v_ A  B    c _ u _ f _ (21) From (2.13) and (2.15) obviously



























1 1

1 1 1 1

1 1 1

1 1 1 2 2 2 2

1 exp

exp exp

exp

u b A

b B b f

c b A B f

 

   

  

 

 

   

1

  

  

  

 

 

  















2

1

2 2 2 2

1 1 1

1 1 1 exp 1 1

f c

b A b B



  

  









1 1

1 exp b 1 f1 .

 

1

2 2

1

exp

v_   A B _ _

  

 



Using the last relation in (2.14) we obtain

 

 



   

 





































1

1 1

1 1 1

1 1 1 1

1 1 1

1 1

1 2 2 2 2

1 1 1 1

1 1 1 1 1 1

exp

1 exp

exp 1

1

exp exp .

x

u b b A

b b B e B

b b f

b b

c A B f

b B b f

 



 



  

 

   



 



 

 

  

  

  

 

 

 

   

  

 

 

 

 _  _

 

 _  _

   

 

The last three inequalities show the validity of (2.2)- (2.4). Now we prove (2.5). Since

1 1

2 c2 exp b 1 A1



  

 



 

   

 

2 2

2 2 ,

L v c x u x

c x u x f x C

 

 

 



  

which leads to (2.5), which completes the proof.

3. The Difference Scheme and Convergence

Now we construct the difference scheme and investigate it. In what follows, we denote by  the uniform mesh in

 

0,1 :



xi ih i, 1, 2, ,N 1, 1



    h N

and





x 0,l



    . Before describing our n erical method, we introduce some notation for the mesh

func-um tions. For any mesh function g x

 

, we use

 













, 1 ,

x i i i

g g g h

 

, 1 ,

2 ,

x i i i

g g g h





1 1

,

2

1 1

,

, ₀

2 ,

max .

i i

x i

i i i

xx i

i d _{i N}

,

g g x



 

g g g g h

g g g

 

  _{ }

  

 





 

(4)

On  we propose the following difference scheme for approximating (1.1)-(1.4):

 1

: ,

h

U  U   c V  f

1 , 1 1 1

,

i i xx i i i i i i i x i

a U bU (22)

 2

2 : , 2 2 ,

h

i i xx i i i i i V   V a V c U 

 f₂_i (23)

     

0 1

1

2 1 2

,0 1,0 1,0 1 0 1,0 1,0 1,0

,

1

x

U A

h h h

U a   B b   f

  

 



  

 _ _ _  

  

(24)

2 ,

  

(25)

0 2 N 2

and

, ,

V A V B

 1 1



 2  1



1

1 1 1 1 coth ,

h

2 2 2

i i

i i i i

a h h a

a

  

  



    _

   

 1 1 1

1 1

exp i 1 ,

i i

a h

 





  

 _ _ _ _  

 



 2 1 1

1

1 exp i ,

i

a h

 





  

 _ _ __ 

 

 2 2 2

2 2

. 4 sinh

2

i i

i

h a

a h

 





 

 

 

For solving of the (3.1)-(3.4) we give the following iterative procedure:

i

2i

   1

1 1 ,

n n

h

i i i

U c V  

 f₁

   

2 2 ,

n n

h

i i i

V c U  f



 

0 1,

n

U A

  1,0  2 1 1 ₁  2  2

,0 1 1,0 0, 1,0 1,0 1,0

A h n

x

a h h

U   B b _  f 

 

 

   

 _ _{ }_   _



 

2 1

   

0 2, ,

n n

N

V A V B

where (0)

i

V is arbitrary. Lemma 3.1

  1 1  1  0

, ,

, 1

n n

d d

U U   V V





 

  

 (26)

  1  1  0

,

, 1

n n

d

V V  V V



 

  

 ,d

(27)

where

, .

d

 implies the discrete maximum norm on h;

 and 1 are defined by (2.1) and (2.11) appro-

priately.

Proof. Denoting      11,

n n n

i Ui Ui

 



       1

1

n n n

i Vi Vi

 



 

we will have

(28)

i

   1

1 1 1 ,

n n

h i

c

 _  



   

2 2 ,

n n

h

i ci



  

 (29)

   

0 ,0 0,

n n

x

   (30)

   

0 0.

n n

N

   (31) (3.

From 7)-(3.10), it is not difficult to get

 n j n1_,

h c

 _  



1

i i i

i

where





1 1

4 exp 4 b

  





and thereby

   

1 1 ,

, ,

1 1

1 _, 1

, , .

n n

d

d d

n n

d _d _d

jh c

c

  

   

 

 

 

 _ _



 

In similar manner we also obtain

     

2 , 2

, ,

2

.

n n

d

d c d





  

 

    ,

n d





e, Henc

   1

, , , 1, 2,

n n

d d

n

   

    

tly, Consequen

  1  1

, , ,

n n

d d

   

  

  2  1

1

, , .

n n

d d

    

  

From this follows that the _ n _,_{ } n ₀_to

hence the sequences

n ,

 

Ui ,

 

Vi

n i

are

 n i

U U and the Cauchy

quences and converg

i

se-ent: lim

 

lim n

nVi V . The lim

e (3

Now we prove (3.5), (3.6). We have

it functions Ui, Vi will be

solution of schem .1)-(3.4).

   

       

   





   





   

,

1 1 2

, ,

1

,

1 0

2 3 1

2

1 0

, ...

1

. 1

n m n d

n m n m n m n m

d d

n n

d

n m n m n

d

U U U

U U

V V

V

V V







      

 





    





   

  

 

 



 



1 _,

1 0

1 1

1 _,

1 0

1 1

...

1

d

n m m

d m

n

V

V V

  

   

  





 

 

  

  

 

 ,

1

d n



 

U U

(5)

The limit case for leads to (3.5). The ine-quality (3.6) is being proved analoguosly.

Lemma 3.2 The solution of the difference problem

(3.1)-(3.4) satisfies

m 





1

,0 1

,

1

2

1 , 1 1

2

1 x N i

d

i

d

U U

f

c A B

                   _   __   



h f

32) (





1



1

2 2 2 2

, ,

1

2 2 , 1

1 1 . d d N N i d i

V A B f

c U h f

                          



 (33)

where  h1_coth h1 _, 2  2 

Proof. Using the est

   p1, 2, , N1.

imates for the difference equations

1

h i i U F



and

with conditions (3.3) and (3.4) appropriately, which is being obtained analoguosly as in differential case, after

setting i

2

h i i V G



1 1

i i i

F  f c v and _i we will

get i 2i 2i

G  f c u

,0 1 1

, , , ,

1

N

x i

d d d d

i U    U  c  V  h_

 

 _   _





f  (34)

1

2 2 2 2

, ,

1

2 c2 ,d U ,d



 

 

 (35)

d d

V _  A  B  f _

The using each of these into another immediately leads .11) and (3.12).

For the truncation errors



to (3

Lemma 3.3

 

1i 1i 1 i 1i i ,

R  f L u x c v x

 

2

 

,

h i

R  f L v x c u x

h

2i 2i 2 i i

 

   

_{ }

1 2 1 10 2 2 1

1 1 10 10 ,0

1 0

0 0 x

h r a

h h

B b A f u

            _ _     _   _  

the following estimates hold

 

1 1

1 1 d d ,

i i

i

x

R Ch h u x x h v x x

 



 _  _

 _   _





(36)

x x

x

 



 

_ _



1 1



2 1 _, _, ,

i i

i _x _x

R Ch  u x

     

 

1 0 1 10 1 d 1 0 x x Ch r u h a        _  _  _ _{ } _    



x x . (38)

Proof. We may write

 

   

 

     

   

 

   

     

1 1 1 1 1 1 1 1

1 1 1 1

1

1 1 1

1 1 1 1 1 1 1 1 1 1 1 d d d d , i i i i i i i i h

i i i i i

x i x i i x x

i i i

x x i i x x i i

R f u x c v x

h u x c x v x f x x

h a x a x u x x x

h b x u x b x u x

h f x f x x x

h c x v x c x v x x x

                      _   _     _  _    _  _    _  _   _



 

 

x dx x (39) 1 i x 1 1 i i x 

_

_

 

   

 

     

 

   

 

1 1 1 1 1 1 1 1

2 2 2 2

1

2 2 2

1 2 2 1 2 2 1 2 2 d d d d , i i i i i i i i h

i i i i i

x

i i

x x

i i i

x x

i i i

x x

i i i i

x

R f v x c u x

h v c x u x f x x

h a x a x v x x x

h f x f x x x

h c x u x c x u x x

                                _   _    _  _    _  _    _  _



  (40) x x

 

       

 

_{ }

 

   

 

_{ }

   

 

_{ }

 

_{ }

1 0 1 0 1 0 1 0 2 1 10 2 0 2

1 1 0

2

1 0

2

1 1 0

1

1 0

0 0 d

0 d

d

0 d ,

x x x x x x x x r h a

b x u x b u x x

a x a u x x x

c x v x x xx

f x f x x

(6)

 

_{ }

  

 

_{ }

  

 





1 1

1

1 1

1

2

1

1, i1 ,

1_, _,

1 1

, ,

1 0,

i i

i

i i

i

a x x x

i a x h i i

a x x x

i i a x h i i

i e

x x x x

e

x x x x x

e

x x

 



 







 

 







 _ _{ }



 _



 

_   

 

 

   

x_

 

_{ }

  



 

_{ }

  



 





2 1

1

1 2

2 1

2

1 2

1 1

sinh

, ,

sinh

, sinh

0, , ,

i

i i

i

i i

i

i i x

a x x x

,

i

i i

x x x x

a x h

a x x x

x x

a x h

x x x



 

  

 





 

 _

  

 

x x





_   



 _

   

 

1

2 1

2 d

2 tanh

i

x

i

i i

x i

h a x

h x x

a x h



  .







 

 

_ _ 







We note that i 1 ,

 2

i

 and i 1 ,

 2

i

 are the solutions of following problem respectively: s

 

x a2

   

x x 0,xi 1 x xi

  

    

 

xi 1 0,

 

xi 1,

 _   

 

x a2

   

x x 0,xi x xi1

  _

    

 

xi 1,

 

xi 1 0.

   _ 

The relations (3.18)-(3.20), by using also the above properties of 

 

x and 

 

x leads immediately to (3.15)-(3.17).

Theorem 3.1 Let

 

,

k

a x bk

 

x , ck

 

x ,

 

 

1 0,1 ,

f x C f₂

 

x C1

 

0,1 , 1,



k1, 2 .



(3.1)-(3.4)

Then con-the solution

verges uniformly in

of the difference problem

 to the solution of (1.1)-(1.4

hen fo ) with

r the errors of the app

,



r

are a ating errors

a 3.2, we obtain:

rate O h

 

.

Proof. Let z1iUiui, z2i Vi vi. T

roximate so



0,1, 2, , N

lution



we have

1, 2;

ki

z k i

1 1 1 2 1, 1, 2, , 1

h

i i i i

z c z R i  

 N

2 2 2 1 2, 1, 2, , 1

h

i i i i

z c z R i  N



10 0, 1 ,0x 0,

z  z 

20 2N 0,

z z 

where pproxim from Lemma

3.3. Using Lemm

1i, 2i, 0

R R r

1 , 2 , 0 1 2 ,

1 i

d d d

i



  



 

By virtue that of (3.15)-(3.17) all terms in right-hand side of this

, .

z z C_ r h



R  R _

inequality have the rate and hence the

pr mediately.

4. Numerical Exampl

Consider the particular problem with

N

 

O h oof follows im

e

2

1 2, 1 2, 1 sinπ , 1 ,

x

a  x b e c  x f x

1

2 , 2 2, 2 1,

x

a e c x f  x 1,

1 1 1, 2 1, 2 0.

A B  A   B 

The i ss is chosen as

 0 ₁

V x

nitial gue

i  i

and stopping criterion is

   1 ₁₀5

U U 

   

n n 

1 ₁₀

n n

V V   5.

ental rates of convergence We calculate an experim



1, 2



k

p k using double mesh method as follows [4,5]:





,h _ln ,h , 2h _ln

k k k

p  e e 2,

where

, ,

1h max i h i ,

i

e  U U, 2h

,h _max ,h , 2h _.

i

e  V V

The convergence is uniform, i.e., rate of convergence

independenty of perturbation parameter. Some obtained va

2 i

i

lues for





, 2 4 6

max 0.5,10 ,1 ,10

h h

k k

p p

 

  

  0

are listed in the table

h 0.1 0.05 0.025

1, 2

p p 1.61, 1.58 1.32, 1.26 1.01, 0.99

5. Conclusion

The singularly perturbed initial-boundary value problem for a linear second order differential system is considered. To solve this problem, an exponentially fitted difference scheme on a uniform mesh is presented. First order con-ve

(7)

ith theoretical values.

6. Acknowledgements

the anonymous referees for his omments and suggestions which helped improve the quality of manuscrip

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y, “Singular Perturbations Methods for agreement w

The author is grateful to c

t.

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