sextic spline solution of a singularly perturbed boundary-value problems

Applied Mathematics and Computation 181 (2006) 432–439

www.elsevier.com/locate/amc

Sextic spline solution of a singularly perturbedboundary-value problems

Arshad Khan a,*, Islam Khan b, Tariq Aziz b

a Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, Indiab Department of Applied Mathematics, Faculty of Engineering and Technology, Aligarh Muslim University, Aligarh 202 002, India

Abstract

A sixth-order uniform mesh difference scheme using sextic splines for solving a self-adjoint singularly perturbed two-point boundary-value problem arising in the study of chemical reactor theory, of the form

0096-3

doi:10

* CoE-m

� eu00 þ pðxÞu ¼ f ðxÞ; pðxÞ > 0;

uð0Þ ¼ ao; uð1Þ ¼ a1

is derived. Our scheme leads to a pentadiagonal linear system. The convergence analysis is given and the method is shownto have fifth-order convergence. Numerical illustrations are given to confirm the theoretical analysis of our method.� 2006 Elsevier Inc. All rights reserved.

Keywords: Singularly perturbed boundary-value problem; Sextic splines; Monotone matrices; Boundary layers; Uniform convergence

1. Introduction

We consider the following self-adjoint singularly perturbed two-point boundary-value problem:

Lu ¼ �eu00 þ pðxÞu ¼ f ðxÞ; pðxÞP p > 0; 0 < x < 1;

uð0Þ ¼ ao; uð1Þ ¼ a1;ð1Þ

where a0, a1 are given constants and e is a small positive parameter. Further the coefficients f(x) and p(x) aresufficiently smooth functions. We use sextic spline for its solution. The application of exponential splines forthe numerical solution of singularly perturbed two-point boundary-value problems has been described inmany papers [5,8,11–14]. Numerical treatment of singular-perturbation problems has received a great dealof attention in the recent past.

The problem in which a small parameter multiplies to the highest order derivative arises in various fields ofscience and engineering, for instance fluid mechanics, fluid dynamics, elasticity, quantum mechanics, optimal

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

.1016/j.amc.2005.12.059

rresponding author.ail addresses: [email protected], [email protected] (A. Khan), [email protected] (I. Khan).

mailto:[email protected]



A. Khan et al. / Applied Mathematics and Computation 181 (2006) 432–439 433

control, chemical reactor theory, hydrodynamics, convection–diffusion processes etc. A few notable examplesare boundary layer problems, WKB theory, the modelling of steady and unsteady viscous flow problems withlarge Reynolds number and convective-heat transport problems with large Peclet number. General resultsabout the analytical solution of such problems are known and the boundary layer behaviour of solutionu(x, e) has been investigated intensively by many authors. Out of the three principal approaches to solve suchproblems numerically, namely, the finite difference methods, the finite element methods and spline approxima-tion methods, the first two have been used by various authors. Here we use the third one, namely, splineapproach to solve problem of type (1).

In the problems of singularly perturbed nature, it is known that most classical methods fail to yield goodapproximations when e is small relative to mesh size h that is used for discretization of Eq. (1). A variety ofnumerical methods are available in the numerical literature to solve singularly perturbed two-point boundary-value problems for second-order ordinary differential equations. For details, one may refer to survey article byKadalbajoo and Patidar [10].

It is a well known fact that the solution of singularly perturbed boundary-value problem exhibits a multi-scale character. That is, there is a thin layer, where the solution varies rapidly, while away from the layer thesolution behaves regularly and varies slowly. Therefore usual numerical treatment of singular-perturbationproblems gives major computational difficulties, and in recent years, a large number of special purpose meth-ods have been developed to provide accurate numerical solution. Various finite difference methods for singu-larly perturbed self-adjoint problems of the form (1) which are uniformly convergent have been examined inthe literature. Doolan et al. [6] gave sufficient conditions for uniform first-order convergence of a generalthree-point difference scheme. Surla and Stojanovic [13] derived a difference scheme via spline in tensionand obtained the error estimate. It is fact that the non-self adjoint boundary-value problem has a boundarylayer either at x = 0 or at x = 1 depending upon the sign of p(x) (that is, p(x) > 0 or p(x) < 0, for all x). In thecase of a self-adjoint boundary-value problem the boundary layers occur at both end points x = 0 and x = 1.

The occurrence of sharp boundary layers as e, the coefficient of highest derivative, approaches zero createsdifficulty for most standard numerical methods. There are a wide variety of asymptotic expansion methodsavailable for solving the problems of the above type. However, difficulties in applying these methods, suchas finding the approximate asymptotic expansions in inner and outer regions are not easy but requires insightand experimentations. To be more accessible for practicing engineers and applied mathematicians there is aneed for methods, which are easy and ready for computer implementation. The spline technique appears tobe an ideal tool to attain these goals. Because of the presence of boundary layers, difficulties are experiencedin solving problems of above type using numerical methods with uniform mesh. There are two possibilities toobtain small truncation error inside the boundary layer. The first is to choose a fine mesh whereas the secondone is to choose a different formula reflecting the behaviour of solution inside the boundary layer. Presentwork deals with the first approach.

In this problem, we take p(x) = p = constant and 0 < e� 1. Aziz and Khan [3] solve this problem by quin-tic spline method [1,2]. In the present paper, we describe a sixth-order method based on sextic splines. Theadvantage of our method is higher accuracy with the same computational effort. In Section 2, we give briefderivation of sextic spline and some important relations. In Section 3, we present the formulation of ourmethod. In Section 4, we develop fifth-order boundary equations to retain the pentadiagonal structure ofthe system. We establish the convergence of our method in Sections 5 and 6 contains the numerical resultsand discussions.

2. Sextic spline and its relations

Let u(x) be a sufficiently differentiable function defined on [a,b] and S(x) be a sextic spline approximation tou(x). The spline function S(x) is defined by

SðxÞ ¼ QiðxÞ ¼ aiðx� xiÞ6 þ biðx� xiÞ5 þ ciðx� xiÞ4 þ diðx� xiÞ3 þ eiðx� xiÞ2 þ fiðx� xiÞ þ gi

for x 2 ½xi; xiþ1�; i ¼ 0ð1ÞN and SðxÞ 2 C5½a; b�. ð2Þ

The set of sextic polynomials, using a different polynomial in each subinterval [xi,xi+1] defines a smoothapproximation to u(x). We further require that the values of the first-, second-, third-, fourth- and fifth-order

434 A. Khan et al. / Applied Mathematics and Computation 181 (2006) 432–439

derivatives be the same for the pair of sextics that join at each point (xi,ui). We determine the seven coefficientsof (2) in terms of ui, ui+1, mi, Mi, Mi+1, Fi, Fi+1 where we write Q00i ðxjÞ ¼ Mj; Qð4Þi ðxjÞ ¼ F j and Q0iðxjÞ ¼ mj,xj 2 [xi,xi+1]. Using the seven equations namely

ðiÞ QiðxiÞ ¼ ui; ðiiÞ Qiðxiþ1Þ ¼ uiþ1;

ðiiiÞ Q0iðxiÞ ¼ mi; ðivÞ Q00i ðxiÞ ¼ Mi;

ðvÞ Q00i ðxiþ1Þ ¼ Miþ1; ðviÞ Qð4Þi ðxiÞ ¼ F i;

ðviiÞ Qð4Þi ðxiþ1Þ ¼ F iþ1

we obtain via a long but straightforward calculation

ai ¼ðuiþ1 � uiÞ

3h6� mi

3h5� ðMiþ1 þ 2MiÞ

18h4þ ð7F iþ1 þ 8F iÞ

1080h2;

bi ¼�ðuiþ1 � uiÞ

h5þ mi

h4þ ðMiþ1 þ 2MiÞ

6h3� ð4F iþ1 þ 11F iÞ

360h;

ci ¼F i

24;

di ¼5ðuiþ1 � uiÞ

3h3� 5mi

3h2� ð2Miþ1 þ 13MiÞ

18hþ hðF iþ1 � 4F iÞ

216;

ei ¼Mi

2; f i ¼ mi; gi ¼ ui; i ¼ 0ð1ÞN .

ð3Þ

From the continuity of the first, third and fifth derivative at the point x = xi, we derive the relationsrespectively

mi þ mi�1 ¼2ðui � ui�1Þ

hþ hðMi �Mi�1Þ

6� h3ðF i � F i�1Þ

360; ð4Þ

mi þ mi�1 ¼ðuiþ1 � uiÞ

h� hð2Miþ1 þ 21Mi þ 7Mi�1Þ

30þ h3ðF iþ1 � 9F i þ 2F i�1Þ

360; ð5Þ

mi þ mi�1 ¼ðuiþ1 � ui�1Þ

h� hðMiþ1 þ 3Mi þ 2Mi�1Þ

6þ h3ð4F iþ1 þ 21F i þ 5F i�1Þ

360. ð6Þ

From (4)–(6) we obtain on equating the right sides of the equality sign

uiþ1 � 2ui þ ui�1 ¼h2ðMiþ1 þ 13Mi þMi�1Þ

15� h4ðF iþ1 � 8F i þ F i�1Þ

360; ð7Þ

uiþ1 � 2ui þ ui�1 ¼h2ðMiþ1 þ 4Mi þMi�1Þ

6� h4ð2F iþ1 þ 11F i þ 2F i�1Þ

180. ð8Þ

From (7) and (8) we deduce

h2Mi ¼ ðuiþ1 � 2ui þ ui�1Þ �h4ðF iþ1 þ 28F i þ F i�1Þ

360; ð9Þ

h4F i ¼ 20ðuiþ1 � 2ui þ ui�1Þ �2h2ðMiþ1 þ 28Mi þMi�1Þ

3. ð10Þ

We now eliminate Mj, j = i, i ± 1 from (9) using (10) to obtain the recurrence relation

ui�2 � 4ui�1 þ 6ui � 4uiþ1 þ uiþ2 ¼h4ðF i�2 þ 56F i�1 þ 246F i þ 56F iþ1 þ F iþ2Þ

360; i ¼ 2ð1ÞN � 2. ð11Þ

We also obtain the following useful relation:

ui�2 þ 8ui�1 � 18ui þ 8uiþ1 þ uiþ2 ¼h2ðMi�2 þ 56Mi�1 þ 246Mi þ 56Miþ1 þMiþ2Þ

30. ð12Þ


3. Method

At the mesh points xj = a + jh, j = 2(1)N � 2, the differential equation (1) can be discretized by using thespline relations (12) to obtain

ðph2 � 30eÞujþ2 þ 8ð7ph2 � 30eÞujþ1 þ ð246ph2 þ 540eÞuj þ 8ð7ph2 � 30eÞuj�1 þ ðph2 � 30eÞuj�2

¼ h2ðfjþ2 þ 56f jþ1 þ 246f j þ 56f j�1 þ fj�2Þ; j ¼ 2ð1ÞN � 2. ð13Þ

4. Development of the boundary equations

We develop the fifth-order boundary equations at x = 0 as follows:Let T 0ðhÞ ¼ k3u3 þ k2u2 þ k1u1 þ k2u0 þ k3u�1 þ eh2ðk0u000 þ k1u001 þ k2u002 þ k3u003 þ k4u004Þ � h2ðf3 þ 56f 2þ

246f 1 þ 56f 0 þ f�1Þ. We replace the fictitious values u�1 and f�1 by 5u0 � 10u1 + 10u2 � 5u3 + u4 and5f0 � 10f1 + 10f2 � 5f3 + f4 respectively to get

T 0ðhÞ ¼ k3u3 þ k2u2 þ k1u1 þ k2u0 þ k3ð5uo � 10u1 þ 10u2 � 5u3 þ u4Þ þ eh2ðk0u000 þ k1u001 þ k2u002 þ k3u003þ k4u004Þ � h2½f3 þ 56f 2 þ 246f 1 þ 56f 0 þ ð5f 0 � 10f 1 þ 10f 2 � 5f 3 þ f4Þ�;

where k1 = 246ph2 + 540e, k2 = 56ph2 � 240e, k3 = ph2 � 30e, so that

T 0ðhÞ ¼ �30eu4 þ 120eu3 � 540eu2 þ 840eu1 � 390eu0 þ eh2ðk4 þ 1Þu004 þ eh2ðk3 � 4Þu003 þ eh2ðk2 þ 66Þu002þ eh2ðk1 þ 236Þu001 þ eh2ðk0 þ 61Þu000. ð14Þ

By using Taylor series and comparing the coefficients of uð2Þ0 ; uð3Þ0 ; uð4Þ0 ; uð5Þ0 ; uð6Þ0 we get

k0 ¼ �30; k1 ¼ 90; k2 ¼ �90; k3 ¼ 30; k4 ¼ 0.

Putting these values in (14), we get the first fifth-order boundary equation as

ðph2 � 30eÞu4 þ ð26ph2 þ 120eÞu3 � ð24ph2 þ 540eÞu2 þ ð326ph2 þ 840eÞu1

¼ h2ðf4 þ 26f 3 � 24f 2 þ 326f 1 þ 31f 0Þ � a0ð31ph2 � 390eÞ þOðh7Þ. ð15Þ

Similarly, the boundary equation at other end is

ðph2 � 30eÞuN�4 þ ð26ph2 þ 120eÞuN�3 � ð24ph2 þ 540eÞuN�2 þ ð326ph2 þ 840eÞuN�1

¼ h2ðfN�4 þ 26f N�3 � 24f N�2 þ 326f N�1 þ 31f NÞ � a1ð31ph2 � 390eÞ þOðh7Þ. ð16Þ

Using (13) we eliminate u4 from Eq. (15) in order to make the system pentadiagonal, we obtain

ð�30ph2 þ 360eÞu3 � ð270ph2 þ 1080eÞu2 þ ð270ph2 þ 1080eÞu1

¼ h2ð�30f 3 � 270f 2 þ 270f 1 þ 30f 0Þ þ a0ð�30ph2 þ 360eÞ þOðh7Þ ð17Þ

and similarly the other end is

ð�30ph2 þ 360eÞuN�3 � ð270ph2 þ 1080eÞuN�2 þ ð270ph2 þ 1080eÞuN�1

¼ h2ð�30f N�3 � 270f N�2 þ 270f N�1 þ 30f N Þ þ a1ð�30ph2 þ 360eÞ þOðh7Þ. ð18Þ

5. Convergence of the method

Putting the pentadiagonal system (13) in matrix–vector form, we have

AU � h2DF ¼ C; ð19Þ
where A = (aij) is a pentadigonal, diagonally dominant matrix of order N � 1, with ai, i±k = coefficient of ui±k
in (13), k = 0,1,2,


F ¼ ðf1; f2; . . . ; fN�2; fN�1ÞT;U ¼ ðu1; u2; . . . ; uN�2; uN�1ÞT;

D ¼

270 �270 �30

56 246 56 1

1 56 246 56 1

. . . . . . . . . . . . . . . . . . . . .

1 56 246 56

�30 �270 270

2666666664

3777777775

and where

C ¼ ðc1; c2; 0; . . . 0; cN�2; cN�1ÞT;
with
c1 ¼ 30h2f0 � ð30ph2 � 360eÞa0;

c2 ¼ h2f0 � ðph2 � 30eÞa0;

ci ¼ 0; for i ¼ 3ð1ÞN � 3;

cN�2 ¼ h2fN � ðph2 � 30eÞa1;

cN�1 ¼ 30h2fN � ð30ph2 � 360eÞa1.

Also, we have

AU � h2DF ¼ T ðhÞ þ C; ð20Þ
where
U ¼ ðuðx1Þ; uðx2Þ; . . . ; uðxN�1ÞÞT

denotes the exact solution and

T ðhÞ ¼ ½t1ðhÞ; t2ðhÞ; . . . ; tN�1ðhÞ�T;
where tj(h) are defined as follows:
t1 ¼ �ð3=2Þeh7uð7Þðn1Þ; x0 < n1 < x2; ð21aÞti ¼ �ð5=84Þeh8uð8ÞðniÞ; xi�1 < ni < xiþ1; i ¼ 2; 3; . . . ;N � 2; ð21bÞtN�1 ¼ �ð3=2Þeh7uð7ÞðnN�1Þ; xN�2 < nN�1 < xN . ð21cÞ

From (19) and (20), we have

AðU � UÞ ¼ AE ¼ T ðhÞ; ð22Þ
where
E ¼ U � U ¼ ðe1; e2; . . . ; eN�1ÞT.

Clearly, the row sums S1,S2, . . . ,SN�1 of A are

S1 ¼X

j

a1;j ¼ �30ph2 þ 360e;

S2 ¼X

j

a2;j ¼ 359ph2 þ 30e;

Si ¼X

j

ai;j ¼ 360ph2; i ¼ 3ð1ÞN � 3;

SN�2 ¼X

j

aN�2;j ¼ 359ph2 þ 30e;

SN�1 ¼X

j

aN�1;j ¼ �30ph2 þ 360e.


If we choose h to be OðffiffiepÞ, then A becomes irreducible and monotone [7]. It follows that A�1 exists and its

elements are non-negative. Hence, from (22), we have

TableMaxim

e

1/161/321/641/128

E ¼ A�1T ðhÞ. ð23Þ
Also, from the theory of matrices, we have
XN�1

i¼1

a�1k;i Si ¼ 1; k ¼ 1; 2; . . . ;N � 1; ð24Þ

where a�1k;i is the (k, i)th element of the matrix A�1. Therefore,

XN�1

i¼1

a�1k;i 6 1

,min

16i6N�1Si ¼ 1=ðh2Bi0Þ;

where

Bi0 ¼ ð1=h2Þmini

Si > 0

for some i0 between 1 and N � 1.From (21), (23) and (24), we have

ej ¼XN�1

i¼1

a�1j;i T iðhÞ; j ¼ 1; 2; . . . ;N � 1;

and therefore

jejj 6 Kh5=Bio ; j ¼ 1; 2; . . . ;N � 1;

where K is a constant independent of h. It follows that

kEk ¼ Oðh5Þ.
We summarize the above results in the following theorem.
Theorem 5.1. The method given by Eq. (13) for solving the boundary-value problem (1) for sufficiently small h

gives a fifth-order convergent solution.

6. Numerical results and discussion

We have described a numerical method for solving self-adjoint singularly perturbed problem using sexticspline. It is a computationally efficient method and the algorithm can easily be implemented on a computer.The method has been analysed for convergence and proved that the method has fifth-order convergence asconfirmed by the numerical results. Comparison with other existing methods demonstrates the superiorityof the proposed method. To point out that the resulting system is pentadiagonal and for the solution of suchsystems it is highly efficient method exists.

The maximum errors are tabulated in Tables 1a–1g for different values of the parameters e, and N. Intables, we have compared our results with the methods of Refs. [3,4,9,12–14]. To illustrate computationallythe fifth-order convergence of the scheme (13), we consider the following singularly perturbed two-point

1aum absolute errors, Example 6.1: present method

N = 16 N = 32 N = 64 N = 128 N = 256

1.224E�06 6.458E�09 3.407E�11 1.037E�12 8.881E�152.004E�06 1.683E�08 1.367E�10 1.093E�12 1.901E�148.898E�06 1.168E�07 1.203E�09 1.083E�11 9.076E�145.746E�05 9.984E�07 1.182E�08 1.141E�10 9.919E�13

Table 1bMaximum absolute errors, Example 6.1: Aziz and Khan method [3]

e N = 16 N = 32 N = 64 N = 128 N = 256

1/16 1.57E�05 8.79E�07 5.32E�08 3.30E�09 2.05E�101/32 8.27E�06 4.41E�07 2.62E�08 1.62E�09 1.00E�101/64 1.84E�05 8.67E�07 6.65E�08 4.39E�09 2.78E�101/128 1.03E�04 2.61E�06 2.23E�07 1.54E�08 9.44E�10

Table 1cMaximum absolute errors, Example 6.1: Aziz and Khan method [4]

e N = 16 N = 32 N = 64 N = 128 N = 256


Table 1dMaximum absolute errors, Example 6.1: Surla and Stojanovic’s method [13]

e N = 16 N = 32 N = 64 N = 128 N = 256


Table 1eMaximum absolute errors, Example 6.1: Surla et al.’s method [12]

e N = 16 N = 32 N = 64 N = 128 N = 256


Table 1fMaximum absolute errors, Example 6.1: Surla and Vukoslavcevic’s method [14]

e N = 16 N = 32 N = 64 N = 128 N = 256


Table 1gMaximum absolute errors, Example 6.1: Kadalbajoo and Bawa’s method [9]

e N = 16 N = 32 N = 64 N = 128 N = 256




boundary-value problem. The numerical calculations were carried out on ALPHA-VMS 3000/4 AS Computerat the Computer Centre, Aligarh Muslim University, Aligarh, using double precision arithmetic in order toreduce the round-off errors to a minimum.

Example 6.1 (See Doolan et al. [6]). We consider the singularly perturbed two-point boundary-valueproblem

� eu00 þ u ¼ � cos2ðpxÞ � 2ep2 cosð2pxÞ;uð0Þ ¼ uð1Þ ¼ 0.

The exact solution is given by

uðxÞ ¼ ½expð�ð1� xÞ=peÞ þ expð�x=p

eÞ�=½1þ expð�1=p

eÞ� � cos2ðpxÞ.

References

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Optimization Theory and Applications 112 (2002) 517–527.[4] T. Aziz, A. Khan, A spline method for second-order singularly perturbed boundary-value problems, Journal of Computational and

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Applied Mathematics and Computations 130 (2002) 457–510.[11] M.R. Majer, Numerical solution of singularly-perturbed boundary-value problems using a collocation method with tension splines,

Progress in Scientific Computing, vol. 5, Birkhauser, Boston, Massachusetts, 1985, pp. 206–223.[12] K. Surla, D. Herceg, L. Cvetkovic, A Family of Exponential Spline Difference Schemes, Review of Research, Faculty of Science,

Mathematics Series, vol. 19, University of Novi Sad, 1991, pp. 12–23.[13] K. Surla, M. Stojanovic, Solving singularly-perturbed boundary-value problems by splines in tension, Journal of Computational and

Applied Mathematics 24 (1988) 355–363.[14] K. Surla, V. Vukoslavcevic, A Spline Difference Scheme for Boundary Value Problems with a Small Parameter, Review of Research,

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sextic spline solution of a singularly perturbed boundary-value problems

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