A solution of the discrepancy occurs due to using the fitted mesh approach rather than to the fitted...

Applied Mathematics and Computation 181 (2006) 756–766

www.elsevier.com/locate/amc

A solution of the discrepancy occurs due to using the fittedmesh approach rather than to the fitted operator

for solving singularly perturbed differential equations

K.K. Sharma a,*, Aditya Kaushik b

a Department of Mathematics, Panjab University Chandigarh, Chandigarh 160 014, Indiab Department of Mathematics, Kurukshetra University, Kurukshetra 136 119, India

Abstract

The solution of the boundary value problems for singularly perturbed differential equations, i.e., where the highestorder derivative is multiplied by a small parameter, exhibits layer behavior. The classical numerical schemes to solve suchtypes of the boundary value problems do not give satisfactory result when the perturbation parameter is sufficiently small.To resolve this difficulty, there are mainly two approaches, namely, fitted operator and fitted mesh. Both the scheme areuniformly convergent, i.e., their convergence is independent of the small perturbation parameter.

It is justified to adopt the two approach rather than the classical numerical schemes to solve the boundary value prob-lems for singularly perturbed differential equations. Now if we compare the two approaches, one thing is common thatboth the approaches give the parameter-uniform schemes which is the primary requirement in construction of the numer-ical scheme to solve such type of problem. Secondly, one desire a higher order numerical scheme to approximate the solu-tion of a problem. As far as order of convergence is concerned, the numerical scheme based on fitted operator approach isbetter than the numerical scheme constructed using fitted mesh approach.

The researchers who adopted the fitted mesh rather than fitted operator approach to solve a singularly perturbed prob-lem faced the question due to the loss of order of convergence, most of them justified it by quoting the simplicity of themethod and there are some non-linear problems for which a parameter uniform scheme cannot be constructed based onfitted operator approach while for the same problem, a parameter uniform scheme is constructed based on fitted meshmethod. Now question remains unanswered in the case of linear problem. In this article, we replied to this question bygiving an example of linear problem for which one cannot construct a parameter uniform scheme based on fitted operatorapproach while for the same problem a parameter uniform numerical scheme based fitted mesh approach has been con-structed [M.K. Kadalbajoo, K.K. Sharma, e uniform fitted mesh method for singularly perturbed differential differenceequations with mixed type of shifts with layer behavior, Int. J. Comput. Math. 81 (2004) 49–62]. A theoretical reasonbehind the inability in construction of a parameter uniform scheme using fitted operator approach is revealed. In supportof the predicted theory, a number of numerical experiments are carried out.� 2006 Elsevier Inc. All rights reserved.

Keywords: Fitted operator; Fitted mesh; Singularly perturbed; Boundary value problem; Ordinary differential equation

0096-3003/$ - see front matter � 2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2006.02.009

* Corresponding author.E-mail address: [email protected] (K.K. Sharma).

mailto:[email protected]

K.K. Sharma, A. Kaushik / Applied Mathematics and Computation 181 (2006) 756–766 757

1. Introduction

The classical numerical methods usually do not behave uniformly well for each value of the singular per-turbation parameter e and in particular give unsatisfactory results when the singular perturbation parameter eis small. To overcome this problem, there are mainly two approaches based on the fitted operator [5] and thefitted mesh [7]. The first approach involves replacing the standard finite difference operator by a finite differ-ence operator which reflects the singularly perturbed nature of the differential operator. Such difference oper-ators are referred to in general as fitted finite difference operators and the numerical methods with fitted finitedifference operators on a uniform mesh are called fitted operator methods. In most of the cases, the fitted finitedifference operator is used at all points of the mesh. However, in [2], Farrell proved that in some cases thefitted finite difference operator can be used on the mesh points in the boundary layer region and in the restof the domain a standard finite difference operator is used. The fitted operator methods on uniform meshesare thoroughly investigated and applied successfully to singularly perturbed boundary value problems for lin-ear ordinary differential equations in [1,8].

The second approach comprises a special type of piecewise uniform mesh condensed in the boundary layerregions to reflect the singularly perturbed nature of the solution and a standard finite difference operator. In[9], Shishkin introduced such types of meshes. The numerical method comprising a standard finite differenceoperator on a piecewise uniform mesh is referred to as the fitted mesh method. The first numerical results usinga fitted mesh method were presented by Miller et al. [6]. But when one adopt the second approach to solve asingularly perturbed differential equation rather than first, it results the loss of order of convergence. Theresearchers who adopted the second approach rather than first one to solve a singularly perturbed problemfaced the questions due to loss of order convergence of the scheme in comparison to first one.

The objective of this paper is to provide a solution to a discrepancy occurring between the two approachesfor constructing parameter uniform method to solve singularly perturbed linear differential equation. Pertain-ing to our goal, we consider a model problem for a class of singularly perturbed differential difference equa-tions of the reaction–diffusion type with small delay as well as advance. During my doctoral program, we cameacross with this model problem which is motivated by a certain biological problem, in which the stochasticactivity of neurons is under consideration. When the numerical study of such type of boundary value problemsis considered, one encounters with two difficulties, one is because of occurrence of the difference terms alongwith the differential terms and another one is due to presence of singular perturbation parameter.

To overcome the first difficulty, we took Taylor approximations for the difference arguments which con-verted the problem to a boundary value problem for a singularly perturbed differential equation. Then we con-structed a numerical scheme based on standard finite difference method and established the error estimate forthe numerical scheme. But the error estimate was not independent of singular perturbation parameter, i.e., thescheme works nicely till the mesh size is smaller than the perturbation parameter but as soon as the conditionis violated, the convergence of method destroys for detail one can see [4].

Thus the second problem still remains, as we have discussed above that there are two approaches to playwith singular perturbation parameter, first we tried to construct a parameter uniform numerical scheme usingfitted operator approach but all efforts went into vein. Then we went for second approach, i.e., fitted mesh andfinally got succeeded to construct a parameter uniform numerical scheme based on fitted mesh approach toapproximate the solution of such type of boundary value problem [5]. But till then we were not able to findout a theoretical reason for the inability to construct a parameter uniform numerical scheme based on fittedoperator approach, so we faced the questions why you adopted fitted mesh approach rather than fitted oper-ator. We also defended in the same way as earlier, the researcher who adopted the fitted mesh approach to dealwith singular perturbation parameter that there are some problems for which no parameter uniform numericalscheme can be constructed on uniform mesh using fitted operator approach while for the same problems, aparameter uniform numerical scheme is constructed based on fitted mesh approach [7]. Farrell et al. came withsomehow a strong reply that they presented a singularly perturbed quasilinear parabolic partial differentialequation for which they constructed parameter uniform scheme and computationally showed the numericalscheme based on fitted operator approach does not converge uniformly with respect to perturbation parameter[3] though they did not give any theoretical reason behind it. Also this example is a non-linear partial differ-ential, it remains what about the linear case. This motivated us to find out a theoretical reason behind the

758 K.K. Sharma, A. Kaushik / Applied Mathematics and Computation 181 (2006) 756–766

inability in construction a parameter uniform numerical scheme for the model problem which is stated in Sec-tion 2.

In this paper, we consider a similar problem with a little change in the coefficient of highest order derivativeterm with the problem which is considered in the paper [4,5], to avoid the square root sign every time in theestimates as well proofs of lemmas and theorems. Some important results are just stated for the proof one cansee [4,5].

2. Statement of the problem

Here, we consider the boundary value problem for the singularly perturbed differential difference equationwith small delay as well as advance with layer behavior

e2y00 þ aðxÞyðx� dÞ þ xðxÞyðxÞ þ bðxÞyðxþ gÞ ¼ f ðxÞ ð2:1Þ

on X = (0,1), subject to the interval conditions

yðxÞ ¼ /ðxÞ on � d 6 x 6 0; ð2:2aÞyðxÞ ¼ cðxÞ on 1 6 x 6 1þ g; ð2:2bÞ

where e is the singular perturbation parameter, 0 < e� 1, d and g are the delay and advance parameters,respectively, 0 < d = o(e2) and 0 < g = o(e2). a(x), b(x), f(x), /(x) and w(x) are smooth functions. For a func-tion y(x) to be a smooth solution to the problem (2.1) and (2.2), it must satisfy Eq. (2.1) with the given bound-ary conditions, be continuous on X;¼ ½0; 1� and continuously differentiable on X = (0, 1).

However, for the delay (d) and advance (g) equal to zero, the solution of the boundary value problem exhib-its layer behavior or oscillatory behavior depending on the sign of (a(x) + b(x) + x(x)), i.e., according to(a(x) + b(x) + x(x)) < 0 or (a(x) + b(x) + x(x)) > 0 on [0,1], respectively. Here, we consider the problemwhose solution exhibits only boundary layer behavior, i.e., (a(x) + b(x) + x(x)) 6 �h < 0 for all x 2 [0,1],where h is a positive constant. The novelty of this class of boundary value problems is the existence of delayas well advance.

3. Numerical analysis for the stated problem

Now, we start the numerical treatment of the problem (2.1) and (2.2). The solution of the problems (2.1)and (2.2) is sufficiently differentiable and the delay as well advance are very small, therefore we use Taylorseries approximation for the terms containing delay and advance. Upon using Taylor approximations forthe difference arguments in the original boundary value problem (2.1) and (2.2), it reduces to

e2z00ðxÞ þ ðbðxÞg� aðxÞdÞz0ðxÞ þ ðaðxÞ þ bðxÞ þ xðxÞÞzðxÞ ¼ f ðxÞ; ð3:1Þzð0Þ ¼ /0; /0 ¼ /ð0Þ; ð3:2aÞzð1Þ ¼ c1; c1 ¼ cð1Þ; ð3:2bÞ

which differ from the original problems (2.1) and (2.2), by terms of O(d2z00,g2z00). Here, the delay as well ad-vance are assumed to be sufficiently small, i.e., are of o(e2), so that the solution z of the problems (3.1) and(3.2) will provide a good approximation to the solution y of the problems (2.1) and (2.2). So to obtain anapproximate solution of the boundary value problems (2.1) and (2.2), it is sufficient to approximate the solu-tion of the boundary value problems (3.1) and (3.2). Thus in the latter part of the paper, we will deal theboundary value problems (3.1) and (3.2).

The differential operator Le corresponding to the problems (3.1) and (3.2) for any smooth functionw 2 C2ðXÞ is defined as

LewðxÞ ¼ e2w00ðxÞ þ ðbðxÞg� aðxÞdÞw0ðxÞ þ ðaðxÞ þ bðxÞ þ xðxÞÞwðxÞ.
Minimum principle. Assume w 2 C2ðXÞ is a smooth function satisfying w(0) P 0, w(1) P 0 and Lew(x) 6 0 forall x 2 X. Then w(x) P 0 for all x 2 X.


Lemma 3.1. Let z(x) be the solution of the problems (3.1) and (3.2), then we have

kzk 6 h�1kf k þmaxðj/0j; jc1jÞ.

3.1. Standard finite difference method

3.1.1. Numerical scheme

To replace the continuous problems (3.1) and (3.2) by discrete approximation, we place a uniform mesh ofsize h = 1/N on the interval [0,1]. Denote the values of the given functions at the mesh points by

xi ¼ ih; zi ¼ zðxiÞ; ai ¼ aðxiÞ; bi ¼ bðxiÞ; wi ¼ xðxiÞ; f i ¼ f ðxiÞ; i ¼ 0; 1; . . . ;N .

Replace z00 and z 0 by central and forward finite difference approximations in Eq. (3.1), respectively

LN1 zi ¼ f ðxiÞ; ð3:3Þ

z0 ¼ /0; /0 ¼ /ð0Þ; ð3:4aÞzN ¼ c1; c1 ¼ cð1Þ; ð3:4bÞ

where the discrete operator LN1 is defined as LN

1 zi ¼ e2DþD�zi þ ðbðxiÞg� aðxiÞdÞDþzi þ ðaðxiÞ þ bðxiÞþxðxiÞÞzi;DþD�zi ¼ ðzi�1 � 2zi þ ziþ1Þ=h2;Dþzi ¼ ðziþ1 � ziÞ=h and D�zi = (zi � zi�1)/h, which on simplificationgives a three point difference scheme

LN1 zi ¼ Eizi�1 � F izi þ Giziþ1 ¼ H i; ð3:5Þ

where

Ei ¼ e2=h2;

F i ¼ 2e2=h2 þ ðbig� aidÞ=h� ðai þ bi þ wiÞ;Gi ¼ e2=h2 þ ðbig� aidÞ=h;

H i ¼ fi; i ¼ 1; 2; . . . ;N � 1.

ð3:6Þ

Theorem 3.2. Under the assumptions that (b(x)g � a(x)d) P M > 0 and (a(x) + b(x) + w(x)) 6 �h <0 "x 2 [0,1], the solution to the system of the difference equations (3.5) together with the given boundary

conditions exists, is unique and satisfies

kykh;1 6 C�1kf kh;1 þ ðk/kh;1 þ kckh;1Þ; ð3:7Þ

where C is a constant independent of h and e and C = M or kbg � adkh,1. kÆkh,1 is the discrete l1-norm, given by

kxkh;1 ¼ max06i6N

jxij.

Corollary 3.3. Under the conditions for Theorem 3.2, the error ei = y(xi) � yi between the solution y(x) of the

continuous problems (3.1) and (3.2) and the solution yi of the discretized problem (3.5) with boundary conditions,

satisfies the estimate

kekh;1 6 M�1kT k; ð3:8Þ

where

jT ij 6 maxxi�16x6xiþ1

h2jðbðxÞg� aðxÞdÞky00ðxÞj

� �þ max

xi�16x6xiþ1

h2

6jbðxÞg� aðxÞdky000ðxÞj

� �

þ maxxi�16x6xiþ1

h2

24f2e2 þ hjðbðxÞg� aðxÞdÞjgjyivðxÞj

� �. ð3:9Þ


3.2. Fitted operator finite difference method


In this section, we will use the fitted operator method to obtain the approximate solution of the problems(3.1) and (3.2). After discretization of the problems (3.1) and (3.2) using fitted operator finite differencescheme, we obtain

LN2 zi ¼ f ðxiÞ; ð3:10Þ

i = 1, . . . ,N � 1, with the boundary conditions

z0 ¼ /0; ð3:11aÞzN ¼ c1; ð3:11bÞ

where the discrete operator LN2 is defined as

LN2 zi ¼ e2#iDþD�zi þ ðgbðxiÞ � daðxiÞÞDþzi þ ðaðxiÞ þ bðxiÞ þ xðxiÞÞzi;

#i ¼qðxiÞ½1� expð�qðxiÞÞ�

4ðsinhðqðxiÞ=2ÞÞ2ð3:12Þ

is a fitting parameter with

qðxiÞ ¼ hðbðxiÞg� aðxiÞdÞ=e2. ð3:13Þ
Calculation of the fitting parameter. To compute the fitting parameter, we shall first prove the followinglemma.
Lemma 3.4. Let z ¼ z0 þ u0 be the zeroth order asymptotic approximation to the solution, where z0 represents

the zeroth order approximate outer solution (i.e., the solution of the reduced problem) and u0 represents the zeroth

order approximate solution in the boundary layer region. Also we assume that the scheme (3.10) and (3.11) isuniformly convergent. Then for a fixed positive integer n

limh!0

zðnhÞ ¼ z0ð0Þ þ ð/0 � z0ð0ÞÞ expð�nðgbð0Þ � dað0ÞÞsÞ. ð3:14Þ

Proof. We have

jLeðzðxÞ � zðxÞÞj 6 jLezðxÞ � Lez0ðxÞj þ jLeu0ðxÞj¼ jf ðxÞ � e2z000ðxÞ � ðgbðxÞ � daðxÞÞz00ðxÞ � ðaðxÞ þ bðxÞ þ xðxÞÞz0ðxÞj

þ d2u0ðmÞdm2

þ ðgbðxÞ � daðxÞÞ du0ðmÞdm

þ ðaðxÞ þ bðxÞ þ xðxÞÞu0ðmÞ��

��; ð3:15Þ

where m = x/e2. Since z0 and u0 are the solutions of the reduced problem

ðaðxÞ þ bðxÞ þ xðxÞÞz0ðxÞ ¼ f ðxÞ
and of the boundary value problem
d2u0ðmÞdm2

þ ðgbð0Þ � dað0ÞÞ du0ðmÞdm

¼ 0; ð3:16Þ

u0ð0Þ ¼ ð/0 � z0ð0ÞÞ; ð3:17aÞu0ð1Þ ¼ 0; ð3:17bÞ

respectively and using Taylor’s series for (gb(x) � da(x)), the above inequality (3.15) reduces to

jLeðzðxÞ � zðxÞÞj 6 e2jz000ðxÞj þ jðgbðxÞ � daðxÞÞz00ðxÞj

þ mðgb0ðnÞ � da0ðnÞÞ du0ðmÞdm

þ ðaðxÞ þ bðxÞ þ xðxÞÞu0ðmÞ��

��; ð3:18Þ


where n 2 (0,1). On solving the boundary value problems (3.16) and (3.17), we get u0(m) = (/0 � y0(0))exp(�m(gb(0) � da(0))). Using this value of u0(m) in the above inequality (3.18), it reduces to

jLeðzðxÞ � zðxÞÞj 6 e2jz000ðxÞj þ jðgbðxÞ � daðxÞÞz00ðxÞj þ j½�ðgb0ðnÞ � da0ðnÞÞm2ðgbð0Þ � dað0ÞÞþ ðaðxÞ þ bðxÞ þ xðxÞÞ�ð/0 � z0ð0ÞÞ � expð�mðgbð0Þ � dað0ÞÞÞj.

Using the fact that texp(�t) 6 exp(�t/2) for all t 2 [0, 1] in the above inequality followed by a simplificationyields

jLeðzðxÞ � zðxÞÞj 6 e2jz000ðxÞj þ jðgbðxÞ � daðxÞÞz00ðxÞj þ jðgb0ðnÞ � da0ðnÞÞð/0 � z0ð0ÞÞj� expðð�xðgbð0Þ � dað0ÞÞÞ=2e2Þ.

Since z000ðxÞ is bounded independently of e for sufficiently smooth (gb(x) � da(x)), (a(x) + b(x) + x(x)) and f(x),there exists a positive constant C1 s.t. jz000ðxÞj 6 C1 for x 2 (0,1) and (gb(x) � da(x)) is of o(e2); so(gb(x) � da(x))/e2 is bounded, i.e., there exists a positive constant such that (gb(x) � da(x))/e2

6 C3. Usingthese facts in the above inequality, we obtain

jLeðzðxÞ � zðxÞÞj 6 e2C2 C0 þ 1

e2expð�xðgbð0Þ � dað0ÞÞ=2e2Þ

� �; ð3:19Þ

where C2 = ja 0(n)(/0 � y0(0))j and C 0 = (C1 + C3)/C2. Now let us introduce the two barrier functions w±(x)defined by

w�ðxÞ ¼ ð1� x=2ÞAe2 þ Be2 expð�Mx=e2Þ � ðzðxÞ � zðxÞÞ;
where A and B are positive constants, We have
Lew�ðxÞ ¼ e2ðw�Þ00ðxÞ þ ðgbðxÞ � daðxÞÞðw�Þ0ðxÞ þ ðaðxÞ þ bðxÞ þ xðxÞÞw�ðxÞ

¼ BM2 expð�Mx=e2Þ � AMe2=2� BMðgbðxÞ � daðxÞÞ � expð�Mx=e2Þ � ðaðxÞ þ bðxÞ þ xðxÞÞ� ½ð1� x=2ÞAe2 þ Be2 expð�Mx=e2Þ� � LeðzðxÞ � zðxÞÞ¼ �AMe2=2þ BM ½M � ðgbðxÞ � daðxÞÞ� expð�Mx=e2Þ þ ðaðxÞ þ bðxÞþ xðxÞÞ½ð1� x=2ÞAe2 þ Be2 expð�Mx=e2Þ� � LeðzðxÞ � zðxÞÞ.

Using the assumption on (gb(x) � da(x)) (i.e., (gb(x) � da(x)) P M > 0) and inequality (3.19) for the boundon LeðzðxÞ � zðxÞÞ, we obtain

Lew�ðxÞ 6 �AMe2=2þ ðaðxÞ þ bðxÞ þ xðxÞÞ½ð1� x=2ÞAe2 þ Be2 � expð�Mx=e2Þ�

� e2C2½C0 þ ðexpð�xðgbð0Þ � dað0ÞÞ=2e2ÞÞ=e2�.

Now since the first and second terms are non-positive, we choose the constants A and B such that the total ofthe moduli of the negative terms dominates the third term. Thus we obtain

Lew�ðxÞ 6 0; x 2 X. ð3:20Þ

We also have

w�ð0Þ ¼ e2ðAþ BÞ � ðzð0Þ � zð0ÞÞ

¼ e2ðAþ BÞ; zð0Þ ¼ zð0Þ

P 0;

w�ð1Þ ¼ e2ðA=2þ B expð�M=e2ÞÞ � ðzð1Þ � zð1ÞÞ

¼ e2ðA=2þ B expð�M=e2ÞÞ; zð1Þ ¼ zð1Þ

P 0.


Thus we have Lew�ðxÞ 6 0; x 2 X and w±(x) P 0 at the both ends of the interval [0, 1]. Then minimum prin-

ciple implies that

w�ðxÞ ¼ ð1� x=2ÞAe2 þ Be2 expð�Mx=e2Þ � ðzðxÞ � zðxÞÞP 0; x 2 X;

which on simplification gives

jzðxÞ � z0ð0Þ � ð/0 � z0ð0ÞÞ expð�xðgbð0Þ � dað0ÞÞ=e2Þj 6 Ce2; ð3:21Þ
for all x 2 [0, 1]. Now since xi = ih, 0 6 i 6 N, so we have
jzðihÞ � z0ð0Þ � ð/0 � z0ð0ÞÞ expð�ihað0Þ=ðe� daðihÞÞÞj 6 Ce2; ð3:22Þ
for all i = 0, . . . ,N. For the particular integer i = n, the above inequality (3.22) gives the required result.
To calculate the fitting parameter, we assume that the solution of (3.10) and (3.11) converges e uniformly tothe solution of the boundary value problems (3.1) and (3.2). This implies that f(xi) � (a(xi) + b(xi) + x(xi))zi isbounded. From (3.10), we have

e2#iðzi�1 � 2zi þ ziþ1Þ=h2 þ ðgbðxiÞ � daðxiÞÞðziþ1 � ziÞ=h ¼ f ðxiÞ � ðaðxiÞ þ bðxiÞ þ xðxiÞÞzi. ð3:23Þ
Multiplying Eq. (3.23) by h for i = n and then taking the limit as h! 0, we obtain
limh!0½ðe2#n=hÞðzn�1 � 2zn þ znþ1Þ þ ðgbðxnÞ � daðxnÞÞðznþ1 � znÞ� ¼ 0.

We use the assumption that the scheme (3.10) and (3.11) is uniformly convergent. So we replace zN�i byz((N � i)h) in the above equation, from which we obtain

limh!0½ðe2#n=hÞðzððn� 1ÞhÞ � 2zðnhÞ þ zððnþ 1ÞhÞÞ þ ðgbðnhÞ � daðnhÞÞðzððnþ 1ÞhÞ � zðnhÞÞ� ¼ 0.

Now using Lemma 3.4 in the above equation, we obtain

limh!0ðh#n=e

2Þð/0 � y0ð0ÞÞ expð�nhðgbð0Þ � dað0ÞÞ=e2Þ½expðhðgbð0Þ � dað0ÞÞ=e2Þ � 2

þ expð�hðgbð0Þ � dað0ÞÞ=e2Þ� þ ðgbð0Þ � dað0ÞÞð/0 � y0ð0ÞÞ� expð�nhðgbð0Þ � dað0ÞÞ=e2Þ½expð�hðgbð0Þ � dað0ÞÞ=e2Þ � 1� ¼ 0;

which implies that

limh!0

#n ¼qð0Þ½1� expð�qð0ÞÞ�

4ðsinhðqð0Þ=2ÞÞ2. � ð3:24Þ

When e = O(1), we have from Eqs. (3.12) and (3.13) that qi! 0 and #i! 1 as N!1 which shows that inthis case the scheme (3.10) and (3.11) is essentially the same as the standard upwind finite difference scheme(3.3) and (3.4) which works nicely provided h < e2. On the other hand when e! 0, since d, g are of small orderof e2 and a(x) and b(x) are sufficiently smooth function, i.e., bounded, therefore (gb(0) � da(0)) is also smallorder of e2. Thus in this case also qi! 0 and #i! 1, i.e., the fitted operator difference scheme (3.10) and (3.11)reduces to the standard upwind finite difference (3.3) and (3.4) and does not reduces to the finite differencescheme for the corresponding reduced problem. Hence the fitted operator scheme (3.10) and (3.11) so con-structed is not e-uniform. Thus for this particular problem one cannot construct an e-uniform convergentscheme using fitted operator approach on a uniform mesh. To sort out this problem, we go for the secondapproach, namely the fitted mesh method which is spelled out in detail in the next section.

3.3. Fitted mesh finite difference method


In this section, we discretize the boundary value problem (3.1) and (3.2) using the fitted mesh finite differ-ence method composed of standard upwind finite difference operator on a piecewise uniform mesh condensingat the boundary points x = 0 and x = 1. The fitted piecewise-uniform mesh XN on the interval [0,1] isconstructed by partitioning the interval into three subintervals (0,s), (s, 1 � s) and (1 � s, 1). On each of these


subintervals, a uniform mesh is placed, i.e., the intervals (0,s) and (1 � s, 1) are divided into N/4 equal meshpoints and the interval (s, 1 � s) is divided into N/2 equal mesh points. The resulting piecewise uniform meshdepends on one parameter which is called the transition parameter and chosen such that s � min[1/4,Ce lnN]with C ¼ 1=

ffiffiffihp

.We assume that N = 2r with r P 3, which guarantees that there is a least one point in the boundary layer.

The fitted mesh finite difference method for the problems (3.1) and (3.2) on the piecewise uniform mesh XN isdefined by

TableThe m

e

2�1

2�2

2�3

2�4

2�5

2�6

2�7

2�8

2�9

2�10

EN

LN3 zi ¼ f ðxiÞ; x 2 X; ð3:25Þ

z0 ¼ /0; ð3:26aÞzN ¼ c1; ð3:26bÞ

where the discrete operator LN3 is defined as

LN3 zi ¼ e2DþD�zi þ ðbðxiÞg� aðxiÞdÞDþzi þ ðaðxiÞ þ bðxiÞ þ xðxiÞÞzi

with

DþD�zi ¼ 2ðDþzi � D�ziÞ=ðhi þ hiþ1Þ; D�zi ¼ ðzi � zi�1Þ=hi;

Dþ ¼ ðziþ1 � ziÞ=hiþ1; XN ¼ fxij1 6 i 6 N � 1g; XN ¼ fxij0 6 i 6 Ng;

xi ¼ihi for 0 6 i 6 N=4;

sþ ði� N=4Þhi for N=4þ 1 6 i 6 3N=4;

1� sþ ði� 3N=4Þhi for 3N=4þ 1 6 i 6 N ;

8><>:

hi ¼4s=N for 0 6 i 6 N=4;

2ð1� 2sÞ=N for N=4þ 1 6 i 6 3N=4;

4s=N for 3N=4 6 i 6 N .

8><>:

3.3.2. Error estimate

Theorem 3.5. The solution ZN ¼ hZiiNi¼0 of the discrete problems (3.25) and (3.26) and the solution z(x) of the

continuous problems (3.1) and (3.2) satisfy the following e-uniform error estimate

kZN � zk 6 CN�1 ln N ;

where C is a constant independent of e.

1aximum absolute error for Example 1, d = 0.5e, g = 0 (Fitted operator finite difference method)

N

64 128 256 512 1024

0.00002070 0.00000518 0.00000129 0.00000032 0.000000080.00014218 0.00003555 0.00000889 0.00000222 0.000000560.00048730 0.00012192 0.00003049 0.00000762 0.000001910.00191038 0.00047866 0.00011973 0.00002994 0.000007480.00757432 0.00191035 0.00047865 0.00011973 0.000029940.02928704 0.00757432 0.00191035 0.00047865 0.00011973

0.10408032 0.02928704 0.00757432 0.00191035 0.000478650.25126281 0.10408032 0.02928704 0.00757432 0.001910350.28328226 0.25126281 0.10408032 0.02928704 0.007574320.17551666 0.28328226 0.25126281 0.10408032 0.02928704

0.28328226 0.2832822 0.25126281 0.10408032 0.02928704

Table 2The maximum absolute error for Example 1, d = 0.5e, g = 0 (Fitted mesh finite difference method)

e N

64 128 256 512 1024

2�1 0.00389693 0.00195231 0.00097704 0.00048875 0.000244432�2 0.01339696 0.00671296 0.00335969 0.00168068 0.000840552�3 0.02483393 0.01248525 0.00626157 0.00313545 0.001568902�4 0.04471265 0.02270532 0.01144523 0.00574649 0.002879312�5 0.08647166 0.04451142 0.02260960 0.01139856 0.005723452�6 0.11856241 0.07441521 0.04407810 0.02260953 0.01139853

2�7 0.11856230 0.07441520 0.04407810 0.02512491 0.014080932�8 0.11856227 0.07441520 0.04407810 0.02512491 0.014080932�9 0.11856226 0.07441520 0.04407810 0.02512491 0.014080932�10 0.11856226 0.07441520 0.04407810 0.02512491 0.01408093

EN 0.11856226 0.07441520 0.04407810 0.02512491 0.01408093

Table 3The maximum absolute error for Example 2, d = 0, g = 0.5e (Fitted operator finite difference method)

e N

64 128 256 512 1024

2�1 0.00001032 0.00000258 0.00000064 0.00000016 0.000000042�2 0.00009934 0.00002484 0.00000621 0.00000155 0.000000392�3 0.00046409 0.00011606 0.00002902 0.00000725 0.000001812�4 0.00173085 0.00043324 0.00010834 0.00002709 0.000006772�5 0.00686780 0.00173027 0.00043309 0.00010831 0.000027082�6 0.02694842 0.00686780 0.00173027 0.00043309 0.00010831

2�7 0.09909360 0.02694842 0.00686780 0.00173027 0.000433092�8 0.32255836 0.09909360 0.02694842 0.00686780 0.001730272�9 0.58984804 0.32255836 0.09909360 0.02694842 0.006867802�10 0.52008235 0.58984804 0.32255836 0.09909360 0.02694842

EN 0.58984804 0.58984804 0.32255836 0.09909360 0.02694842

Table 4The maximum absolute error for Example 2, d = 0, g = 0.5e (Fitted mesh finite difference method)

e N

64 128 256 512 1024

2�1 0.00386431 0.00193903 0.00097117 0.00048600 0.000243112�2 0.01804943 0.00907113 0.00454742 0.00227669 0.001139092�3 0.04540335 0.02290925 0.01150483 0.00576521 0.002885842�4 0.08201893 0.04187433 0.02116545 0.01064349 0.005336432�5 0.15086065 0.07953308 0.04065603 0.02056255 0.010341542�6 0.21678429 0.13360707 0.07873195 0.04063453 0.02055201

2�7 0.21676926 0.13360573 0.07873181 0.04514967 0.025368322�8 0.21676460 0.13360526 0.07873177 0.04514966 0.025368322�9 0.21676325 0.13360510 0.07873175 0.04514966 0.025368322�10 0.21676289 0.13360505 0.07873174 0.04514966 0.02536832

EN 0.21678429 0.13360707 0.07873195 0.04514967 0.02536832


Table 5The maximum absolute error for Example 3, d = 0.5e = g (Fitted operator finite difference method)

e N

64 128 256 512 1024

2�1 0.00004858 0.00001215 0.00000304 0.00000076 0.000000192�2 0.00019912 0.00004980 0.00001245 0.00000311 0.000000782�3 0.00075750 0.00018965 0.00004743 0.00001186 0.000002962�4 0.00300190 0.00075740 0.00018963 0.00004742 0.000011862�5 0.01174547 0.00300190 0.00075740 0.00018963 0.000047422�6 0.04103883 0.01174547 0.00300190 0.00075740 0.00018963

2�7 0.13048121 0.04103883 0.01174547 0.00300190 0.000757402�8 0.20472401 0.13048121 0.04103883 0.01174547 0.003001902�9 0.15822101 0.20472401 0.13048121 0.04103883 0.01174547

EN 0.20472401 0.20472401 0.13048121 0.04103883 0.01174547

Table 6The maximum absolute error for Example 3, d = 0.5e = g (Fitted mesh finite difference method)

e N

64 128 256 512 1024

2�1 0.00712485 0.00357835 0.00179303 0.00089749 0.000448992�2 0.01610028 0.00811222 0.00407261 0.00204045 0.001021252�3 0.02848392 0.01451507 0.00733956 0.00368893 0.001849332�4 0.05289762 0.02789476 0.01424644 0.00720489 0.003622812�5 0.09836276 0.05288973 0.02789043 0.01424457 0.007203942�6 0.13998888 0.08377330 0.05232198 0.02789043 0.014244572�7 0.13998888 0.08377330 0.05232198 0.03091405 0.017533922�8 0.13998888 0.08377330 0.05232198 0.03091405 0.017533922�9 0.13998888 0.08377330 0.05232198 0.03091405 0.01753392

EN 0.13998888 0.08377330 0.05232198 0.03091405 0.01753392


4. Computational results

Some numerical examples are considered and solved using the methods presented here. The exact solutionof the boundary value problems (3.1) and (3.2) for constant coefficients (i.e., a(x) = a, b(x) = b, a(x) = a andx(x) = x are constant), /(x) = 1 = c(x), with f(x) = 1 is

yðxÞ ¼ ðaþ bþ w� 1Þ½ðexpðm2Þ � 1Þ expðm1xÞ � ðexpðm1Þ � 1Þ expðm2xÞ�=½ðaþ bþ wÞðexpðm2Þ � expðm1ÞÞ� þ 1=ðaþ bþ wÞ

and if f(x) = 0 it is

yðxÞ ¼ ½ð1� expðm2ÞÞ expðm1xÞ � ð1� expðm1ÞÞ expðm2xÞ�=ðexpðm1Þ � expðm2ÞÞ;
where
m1 ¼ �ðbg� adÞ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðbg� adÞ2 � 4e2ðaþ bþ xÞ

q� ��2e2;

m2 ¼ �ðbg� adÞ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðbg� adÞ2 � 4e2ðaþ bþ xÞ

q� ��2e2.

Example 1. e2y00(x) � y(x � d) + 0.5y(x) = 0, under the interval and boundary conditions

yðxÞ ¼ 1; �d 6 x 6 0; yð1Þ ¼ 1.


Example 2. e2y00(x) + y(x) � 1.25y(x + g) = 0, under the boundary and interval conditions

yð0Þ ¼ 1; yðxÞ ¼ 1; 1 6 x 6 1þ g.

Example 3. e2y00(x) � y(x � d) + y(x) � 1.25y(x + g) = 1, under the interval conditions

yðxÞ ¼ 1; �d 6 x 6 0; yðxÞ ¼ 1; 1 6 x 6 1þ g.

5. Conclusion

In this paper, boundary value problems for singularly perturbed linear differential difference equation withdelay as well as advance are considered. In Section 3, it is shown that one cannot construct a parameter uni-form numerical scheme based on fitted operator approach while for the same problem a parameter uniformscheme based on fitted mesh approach already developed. This result give a bit of relief to the researchers whoadopt fitted mesh approach rather than the fitted operator approach to deal with the singularly perturbed dif-ferential equations. In Section 4, we present some numerical results in terms of maximum absolute error forthe scheme presented which are tabulated in Tables 1–6. Tables 1, 3 and 5 constructed using fitted operatormethod shows that the convergence of the method depends on value of the parameter e while Tables 2, 4 and 6constructed using fitted mesh method shows that the convergence of the method does not depend on the valueof the parameter e. From Tables 1–6 one draw the conclusions (i) the numerical scheme based on fitted oper-ator approach is not parameter uniform while the numerical scheme based fitted mesh approach is parameteruniform for such type of boundary value problems, (ii) whenever i.e., under the condition that mesh parameterh is smaller than the perturbation parameter e (above the line in Tables 1, 3 and 5) the numerical scheme basedfitted operator approach converges, the rate of convergence is better than the numerical scheme constructedbased fitted mesh approach. Thus the numerical experiments carried out in Section 4 supports the theoreticalfindings in the paper.

References

[1] E.P. Doolan, J.J.H. Miller, W.H.A. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers, BoolePress, Dublin, 1980.

[2] P.A. Farrell, Sufficient conditions for uniform convergence of a class of difference schemes for a singularly perturbed problem, IMAJ. Numer. Anal. 7 (1987) 459–472.

[3] P.A. Frarrell, E. O’Riordan, J.J.H. Miller, Parameter-uniform fitted mesh method for quasilinear differential equations with boundarylayers, Comput. Appl. Math. 1 (2001) 154–172.

[4] M.K. Kadalbajoo, K.K. Sharma, Numerical analysis of boundary-value problems for singularly-perturbed differential-differenceequations with small shifts of mixed type, J. Optim. Theory Appl. 115 (2002) 145–163.

[5] M.K. Kadalbajoo, K.K. Sharma, e uniform fitted mesh method for singularly perturbed differential difference equations with mixedtype of shifts with layer behavior, Int. J. Comput. Math. 81 (2004) 49–62.

[6] J.J.H. Miller, E. Mullarkey, E. O’Riordan, G.I. Shishkin, A simple recipe for uniformly convergent finite difference schemes forsingularly perturbed problems, C. R. Acad. Sci. Paris, 312, Serie I (1991) 643–648.

[7] J.J.H. Miller, E. O’Riordan, G. Shishkin, Fitted Numerical Methods for Singularly Perturbed Problems: Error Estimates in theMaximum Norm for Linear Problems in One and Two Dimension, World Scientific Publication, Singapore, 1996.

[8] H.G. Roos, M. Stynes, L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations: Convection–Diffusion andFlow Problems, Springer-Verlag, New York, 1996.

[9] G.I. Shishkin, A difference scheme for a singularly perturbed equation of parabolic type with a discontinuous initial condition,Sov. Math. Dokl. 37 (1988) 792–796.

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