A note on the numerical-integration method for solving singular perturbation problems

A Note on the Numerical-Integration Method for Solving Singular Perturbation

Problems

Y. N. Reddy Department of Mathematics Visz;esuaraya Regional College of Engineering Nagpur 440011,

Transmitted

India

by John Casti

ABSXUCT The numerical-integration method recently proposed by Reddy for solving linear, singularly perturbed two-point boundary-value problems is investigated. The method was derived by approximating the original second-order problem with a first-order problem with a deviating argument, and then using Simpson’s rule. It is shown here that,

in the limit

one-sided

1.

as the deviating

difference

approximation

argument

tends

to zero,

to the original

problem

the method

converges

in conservation

to a

form.

INTRODUCTION

Recently, a numerical-integration method for solving linear, singularly perturbed two-point boundary-value problems with a boundary layer on the left end of the interval was presented by Reddy [I]. The method was derived by replacing the original second-order problem with an approximate firstorder problem with a deviating argument, and then using Simpson’s rule for finite differences to obtain effkiently a three-term recurrence relationship. The method is iterative on the deviating argument. In this note, we examine the proposed method and show that, in the limiting case as the deviating argument tends to zero, it converges to a one-sided finite-difference approximation scheme for the original secondorder problem in conservation form.

APPLIED MATHEMATICS AND COMPUTATZON 43:175-i80

0 Elsevier Science Publishing Co., Inc., 1991 655 Avenue of the Americas, New York, NY 10010

175

(1991)

0096-3003/91/$03.50

176 2.

Y. N. REDDY THE

METHOD

To recapitulate

the method,

we consider

with

O
y(0)

the following

=(Y and y(1)

singular problem:

=/3,

(2)

where E is a small positive parameter (0 < E +Z 1); cy, /3 are given constants; a(x), b(x), and f(x) are assumed to be sufficiently continuously differentiable functions in [O,l]; and b(x) < 0, a(x) > M > 0 on [O,l], where M is some positive constant. Under these conditions, (l>-(2) has a unique solution y(x) which, in general, displays a boundary layer of width O(E) at x = 0 for small values of E. Let 6 be a small positive deviating (retarded) argument, 0 < 6 -=x 1. By using Taylor series expansion in the neighborhood of the point, we have

y'(x

- Sy"(x).

-S) - y’(x)

(3)

Consequently, the equation (1) is replaced by the following first-order differential equation of neutral type with a small deviating argument:

Y’(X)

=

P(X) Y'(X - 6) + Y(X) Y(X) + 4x)

(4)

for 6
P(X)= EfSU(X) q(x)= ?-(x) =

’

-G(x) E +

Su(x) ’

Sf(x) E+sSa(x)

(5)

(6) (7)

The three-term recurrence relationship is then obtained by integrating the equation (4) over the interval [xi, xi+ r] using Simpson’s rule, together with

Singular

Perturbation

Problems

177

the approximations

y( Xi - 8) -

YCxi+l/2

(8)

-S)-Y(xj+,/,)-hY(Xi+*)+~Y(Xi),

Ytxi+l/2)

N

Y(‘,)

6

+ Y(‘i+l> 2

(9) ’

+ ,[!/‘(q)

-

Y’<%,,>l)

(10)

and is given by Eiyi_l-Fiyi+Giy,+l=H,

(11)

for i =l , 2 ,...> N - 1, where

Fi =

(15)

Y. N. REDDY

178 and pi = p(x,), pi = p’(xi>, oi = q(xj), ri = r(xi), Yi = l/N. The process is to be repeated for different choices argument, satisfying the condition 0 < 6 -CC1) until the not differ materially from iteration to iteration. Several ments have been presented to substantiate these results.

y(x,), xi = ih, h = of 6 (the deviating solution profiles do numerical experiFor details, refer to

Reddy [l]. 3.

THE

ANALYSIS

We now establish the behavior of the difference scheme (11) in the limit as the deviating argument tends to zero. In particular, we show that the difference scheme (11) converges to a one-sided finite-difference approximation to the original second-order We start by observing

lim

Y'(X)

-

problem

in conservation

form.

that

P(X)Y'(X - 6) =

y,,(x)

a(x)y'(x)

+

6

6-O

’

E

lim

4x1 -=_-

4x1

6

E

6-O

4x1 -=-

lim

f(x)

6

s-0

’

E

.

From these results and with the relation o(r)

Y’(X)

it is clear that the equation

= Mr) (4,

Y(X)l’-

al(r)

Y(X)

after division by 6, converges

in the limit as

6 + 0 to the equation y,,(x)

+ [a(r)

y(r)]’

+ [b(r)-

o’(r)lY(r) E

E which is nothing but the equation it can easily be seen that shmOP(r) lim 6-O

_ f(r)

I-p(r) 6

(1) written in conservation

,‘imop’( r) = 0.

= I,

I-

u(r) E

’

E

’

(16)

form. Further,

Singular

Perturbation

179

Problems

and that

-

,im -= P’(X) s-0 6

Using

these

difference

1.

results,

scheme

4

Im -=-, s-0 6

EU’(x)

--

Eo[E+SU(X)]2

it can

easily

(11), divided

be

shown

by 6, converge

a’(x)

=

that

E

the

.

coefficients

of the

to the limits

1 h

G. 1 --! = - + h ,340 6 lim

where ai = a(~,>, ai = a’(~~>, bi = b(xi), and fi = jIxi>. Thus, the three-term recurrence relationship (II), after dividing h, converges, as 6 + 0, to the following difference equation (after

by 6 and rearrang-

ing the terms):

Yi-1

-2Yi + h”

YjCl

+

“i+!Ui+l

-

“iYi

eh

(17)

180

Y. N. FtEDDY

Using

(Yi+l+ 2

Yi-le2Yi

Yi>

+ Yi+l

h2

= yi+ 1,2, the equation

+

(17) becomes

ai+15i+I - aiYi

ch

Clearly the equation (18) is a one-sided finite-difference approximation to the original differential equation (11, written in the conservation form (16). Thus, it is established that, in the limiting case as the deviating argument tends to zero, the difference scheme (11) converges to a one-sided finite-difference scheme (18) for the original second-order problem in conservation form. REFERENCE 1

Y. N. Reddy, A numerical integration method for solving singular perturbation problems, Appl. Math. Cornput., 37 (1990) 83-95.