Applied Mathematics and Computation 181 (2006) 356–361

www.elsevier.com/locate/amc

A numerical method based on polynomial sextic splinefunctions for the solution of special fifth-order

boundary-value problems

Siraj-ul-Islam a,*, Muhammad Azam Khan b

a University of Engineering and Technology, Peshawar (NWFP), Pakistanb Nescom, Islamabad, Pakistan

Abstract

A second-order accurate numerical scheme is presented for the solution of special fifth-order boundary-value problemswith two-point-boundary conditions. The polynomial sextic spline functions are applied to construct the numerical algo-rithm. Convergence of the method is discussed through standard convergence analysis. A numerical illustration is given toshow the pertinent features of the technique.� 2006 Elsevier Inc. All rights reserved.

Keywords: Fifth-order boundary-value problem; Polynomial sextic spline functions; Numerical method

1. Introduction

A second-order accurate numerical scheme is presented for the solution of fifth-order boundary-value prob-lems with two-point-boundary problems of the form

0096-3

doi:10

* CoE-m

yðvÞ ¼ gðxÞy þ qðxÞ ð1:1Þ

with boundary conditions

yðaÞ ¼ A0; y0ðaÞ ¼ A1; y00ðaÞ ¼ A2; yðbÞ ¼ B0; y0ðbÞ ¼ B1. ð1:2Þ

This type of problems arises in the mathematical modeling of viscoelastic flows (see [1,2]). The literature ofnumerical analysis contains little on the solution of fifth-order boundary-value problems. Theorems, whichlist the condition for the existence and uniqueness of solution of such problems, are thoroughly discussedin a book by Agarwal [3].

In Davis et al. [1] and Karageoghis et al. [2], two numerical algorithms, namely, spectral Galerkin methodsand spectral collocation methods, were applied independently to address the numerical issues related to this

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

.1016/j.amc.2006.01.042

rresponding author.ail addresses: ges0201@giki.edu.pk, sirajuet@yahoo.com (Siraj-ul-Islam), azam_giki@hotmail.com (M. Azam Khan).

Siraj-ul-Islam, M. Azam Khan / Applied Mathematics and Computation 181 (2006) 356–361 357

type of problems. Moreover, Khan [4] investigated the fifth-order boundary-value problem by using finite-difference methods. Wazwaz [5] applied domain decomposition method for solution of such type of bound-ary-value problems. The use of spline functions in the context of fifth-order boundary-value problems wasstudied by Fyfe [6], who used quintic polynomial spline functions to develop consistency relation connectingthe values of solution with fifth-order derivative at the respective nodal points. Following these Calgar et al. [7]have used sixth-degree B-spline functions to develop first-order accurate method for the solution of two-pointspecial fifth-order boundary-value problems.

In this paper, sextic polynomial spline functions are applied to develop a new numerical method for obtain-ing smooth approximations to the solution of fifth-order differential equations. This approach has the advan-tage over the methods of Khan [4] and Calgar et al. [7] that it provides continuous approximations to not onlyfor y(x), but also for y 0, y00 and higher derivatives at every point of the range of integration in addition toimproved accuracy. In Section 2, the new polynomial spline method is developed for solving Eq. (1.1) alongwith boundary conditions (1.2). The convergence analysis of the method is considered in Section 3. Section 4 isdevoted to numerical experiment, discussion and comparison with other known methods.

2. Numerical methods

For simplicity, an interval [a,b] is considered, in order to develop the numerical method for approximatingsolution of a system of differential equations of the type (1.1) and (1.2). For this purpose we define a grid ofn + 1 equally spaced points xi = a + ih, i = 0,1, . . . ,n where h ¼ b�a

nþ1. For each i th segment, the polynomial

Pi(x) has the form

P iðxÞ ¼ aiðx� xiÞ6 þ biðx� xiÞ5 þ ciðx� xiÞ4 þ diðx� xiÞ3 þ eiðx� xiÞ2 þ fiðx� xiÞ þ gi; ð2:1Þ

where ai, bi, ci, di, ei, fi, gi are real finite constants and k is free parameter.

Let yi be an approximation to y(xi), obtained by the segment Pi(x) of the spline function passing throughthe points (xi,yi) and (xi+1,yi+1). To obtain the necessary conditions for the coefficients introduced in Eq. (2.1),we not only require that Pi(x) satisfies Eq. (1.1) at xi and xi+1 and that the boundary conditions (1.2) are ful-filled, but also the continuity of second, third and fourth derivatives at the common nodes (xi,yi).

To determine the coefficients of Eq. (2.1) we first define,

P iðxiÞ ¼ yi; P iðxiþ1Þ ¼ yiþ1; P 0iðxiÞ ¼ Zi;

P 0iðxiþ1Þ ¼ Ziþ1; P 00i ðxiÞ ¼ Mi; P ðvÞi ðxiÞ ¼ Si; P ðvÞi ðxiþ1Þ ¼ Siþ1.

Algebraic manipulation yields the following expressions, whereby i = 0,1,2, . . . ,n:

ai ¼ ð�Si þ Siþ1Þ=720h; bi ¼ Si=120;

ci ¼ �ð3Sih5 þ h5Siþ1 � 120Mih

2 � 480Zih� 240hZiþ1 � 720yi þ 720yiþ1Þ=240h4;

di ¼ ð2Sih5 þ h5Siþ1 � 360Mih

2 � 1080Zih� 360hZiþ1 � 1440yi þ 1440yiþ1Þ=360h3;

ei ¼ Mi=2;

fi ¼ Zi;

gi ¼ yi.

ð2:2Þ

Continuity conditions of the second-, third- and fourth-order derivatives at (xi,yi), that is P ðnÞi�1ðxiÞ ¼ P ðnÞi ðxiÞwhere n = 2,3,4, yield the following equations:

h3Si�1=120þ h3Si=120þMi�1 þ 6Zi�1=hþ 6Zi=hþ 12yi�1=h2 � 12yi=h2 �Mi ¼ 0; ð2:3Þh2Si�1=15þ h2Si=20þ 6Mi�1=hþ 30Zi�1=h2 þ 36Zi=h2 þ 48yi�1=h3 � 24yi=h3

� h2Siþ1=60þ 6Miþ1=hþ 6Ziþ2=h2 � 24yiþ1=h3 ¼ 0; ð2:4Þ2Sih

5 þ 7h5Siþ1 þ 120h2Mi þ 480hZi � 240hZiþ1 þ 720yi � 1440yiþ1 þ h5Siþ2

� 120h2Miþ1 � 240hZiþ2 þ 720yiþ2 ¼ 0. ð2:5Þ

358 Siraj-ul-Islam, M. Azam Khan / Applied Mathematics and Computation 181 (2006) 356–361

In order to get nine additional equations, i is replaced by i � 2, i � 1, i + 1, in each of Eqs. (2.3)–(2.5),respectively.

h3Si�3=120þ h3Si�2=120þMi�3 þ 6Zi�3=hþ 6Zi�2=hþ 12yi�3=h2 � 12yi�2=h2 �Mi�2 ¼ 0; ð2:6Þh3Si�2=120þ h3Si�1=120þMi�2 þ 6Zi�2=hþ 6Zi�1=hþ 12yi�2=h2 � 12yi�1=h2 �Mi�1 ¼ 0; ð2:7Þh3Si=120þ h3Siþ1=120þMi þ 6Zi=hþ 6Ziþ1=hþ 12yi=h2 � 12yiþ1=h2 �Miþ1 ¼ 0; ð2:8Þh2Si�3=15þ h2Si�2=20þ 6Mi�3=hþ 30Zi�3=h2 þ 36Zi�2=h2 þ 48yi�3=h3 � 24yi�2=h3

� h2Si�1=60þ 6Mi�2=hþ 6Zi�1=h2 � 24yi�1=h3 ¼ 0; ð2:9Þh2Si�2=15þ h2Si�1=20þ 6Mi�2=hþ 30Zi�2=h2 þ 36Zi�1=h2 þ 48yi�2=h3

� 24yi�1=h3 � h2Si=60þ 6Mi�1=hþ 6Zi=h2 � 24yi=h3 ¼ 0; ð2:10Þh2Si=15þ h2Siþ1=20þ 6Mi=hþ 30Zi=h2 þ 36Ziþ1=h2 þ 48yi=h3

� 24yiþ1=h3 � Siþ2=60h2 þ 6Miþ1=hþ 6Ziþ2=h2 � 24yiþ2=h3 ¼ 0; ð2:11Þ2h5Si�3 þ 7h5Si�2 þ 120h2Mi�3 þ 480hZi�3 � 240hZi�2 þ 720yi�3 � 1440yi�2 þ h5Si�1

� 120h2Mi�2 � 240hZi�1 þ 720yi�1 ¼ 0; ð2:12Þ2h5Si�2 þ 7h5Si�1 þ 120h2Mi�2 þ 480hZi�2 � 240hZi�1 þ 720yi�2

� 1440yi�1 þ h5Si � 120h2Mi�1 � 240hZi þ 720yi ¼ 0; ð2:13Þ2h5Si þ 7h5Siþ1 þ 120h2Mi þ 480Zih� 240hZiþ1 þ 720yi � 1440yiþ1

þ h5Siþ2 � 120h2Miþ1 � 240hZiþ2 þ 720yiþ2 ¼ 0. ð2:14Þ

Simultaneous solution of Eqs. (2.3)–(2.13) with the help of symbolic tool box Matlab 7 for Mi�3, Mi�2, Mi�1,Mi, Mi+1, Zi�3, Zi�2, Zi�1, Zi, Zi+1 and Zi+2, and elimination M terms and Z terms from Eq. (2.14) gives, afterlengthy calculations, the following recurrence relation:

� yi�3 þ 5yi�2 � 10yi�1 þ 10yi � 5yiþ1 þ yiþ2

¼ h5=720ððSi�3 þ Siþ2Þ þ 57ðSi�2 þ Siþ1Þ þ 302ðSi�1 þ SiÞÞ for i ¼ 3; 5; 6; . . . ; n� 1. ð2:15Þ

In order to have a closed form system, three more equations are needed. These equations are developed bymethod of undetermined coefficients as well as Taylor’s series expansion and, are given by

For n = 1,

10y1 �5

2y2 þ

10

27y3 �

425

54y0 �

55

9hy 00 þ

5

3h2y000 � h5

� � 621889

1088640S0 þ

33127

9072S1 �

636095

72576S2 þ

159169

13608S3 �

90899

10368S4 þ

159011

45360S5 �

271751

217728S6

� �¼ 0.

ð2:16Þ

For n = 2,

26

3y0 �

212

13y1 þ 10y2 �

92

39y3 þ

44

13hy00 �

16

13h2y 000 � h5

� 39349

9072S1 �

36324101

4717440S2 þ

1678981

147420S3 �

5820661

673920S4 þ

10229003

2948400S5 �

303743

524160S6

� �

� h5 13605589

23587200S0 ¼ 0. ð2:17Þ

The nth equation is,

� 5

8yn�3 þ

10

3yn�2 �

15

2yn�1 þ 10yn �

125

24ynþ1 þ

5

2hy0nþ1

� h5 �504101

962280Sn�5 þ

706063

204120Sn�4 �

2628907

653184Sn�3 þ

66587

5832Sn�2 þ

5510009

653184Sn�1 þ

1077641

544320Sn

� �

� 1985653

3265920h5Snþ1 ¼ 0. ð2:18Þ

Siraj-ul-Islam, M. Azam Khan / Applied Mathematics and Computation 181 (2006) 356–361 359

The local truncation errors corresponding to the schemes (2.17) and (2.18) is given by

ti ¼�1

12

� �h7yð7Þi for i ¼ 3; . . . ; n� 1 and ti ¼

7

12h7yðixÞi for i ¼ 1; 2; and n.

3. Convergence analysis

In this section convergence of the method (2.15) along with Eqs. (2.16)–(2.18) is discussed. To do so, thediscretization error en = yn � zn, where zn is numerical approximation to yn obtained by neglecting thetruncation errors in (2.16), is defined. Let Y = (yn), Z = (zn), C = (cn), T = (tn) and E = (en) be n-dimensionalvectors. Define kEk = maxn jenj, where k Æk represents the1-norm of a matrix vectors. Using these notations,Eqs. (2.16)–(2.18) can be rewritten in matrix form as follows:

MY ¼ Cþ T; ð3:1ÞMZ ¼ C; ð3:2ÞME ¼ T; ð3:3Þ

where M is quindiagonal matrix of order n and is given by

M ¼ Aþ h5BD.

Here A = (aij) is a five diagonal matrix of order n and is of the form,

A ¼

10 � 52

1027

�2123

10 �9239

5 �10 10 �5 1

�1 5 �10 10 �5 1

. .. . .

. . .. . .

. . .. . .

.

�1 5 �10 10 �5 1

�1 5 �10 10 �5

� 58

103

� 152

10

266666666666666664

377777777777777775

;

B ¼

� 331279072

63609572576

� 15916913608

9089910368

� 15901145360

127175217728

� 393499072

363241014717440

� 1678981147420

5820661673920

� 102290032948400

303743524160

b c c b

a b c c b a

. .. . .

. . .. . .

. . .. . .

.

a b c c b a

a b c c b504101962280

� 706063204120

2628907653184

� 665875832

5510009653184

� 1077641544320

266666666666666664

377777777777777775

;

where a ¼ �1720; b ¼ �57

720, c ¼ �302

720, and

D ¼ diagðgiÞ; i ¼ 1; 2; . . . ; n.

C ¼

42554

A0 þ 559

hA1 � 53h2A2 þ 621889

1088640h5S0;

� 263

A0 � 4413

hA1 þ 1613

h2A2 þ h5 1360558923587200

S0;

A0 þ aS0; 0; 0; . . . ; 0; B0 þ aSnþ1;12524

B0 � 52hB1 þ 1985653

3265920h5Snþ1

264

375

T

:

The main purpose is to derive bounds on E. For this, the following lemma is needed.

Table 1Maximum errors corresponding to problem 1

H 1/10 1/20 1/40 1/80

Second-order method (18) 2.76(�3) 2.45(�4) 2.01(�5) 1.72(�6)Sextic B-spline collocation [7] 0.1570 0.0747 0.0208 –Fyfe [6] 2.85(�3) 8.75(�5) 1.29(�4) 7.482(�5)Finite difference [4] (fourth-order method) �0.4025(�2) 0.3911(�2) 0.1145(�1) –

360 Siraj-ul-Islam, M. Azam Khan / Applied Mathematics and Computation 181 (2006) 356–361

Lemma 3.1. If W is a matrix of order n and kWk < 1 , then (I + W)�1 and k(I + W)�1k < 1/(I � kWk).Rewriting (3.3) we obtain

E ¼ M�1T ¼ ðAþ h5BDÞ�1T ¼ ð1þ h5A�1BDÞ�1A�1T;

kEk 6 kA�1kkTk1� h5kA�1kkBkkDk

.

Now kBk is a finite number but kA�1k depends numerically on n . However, it follows that the method is second-

order convergent provided h5kA�1kkBkkDk < 1.

4. Numerical results and discussion

In this section the new methods are tested on problems considered by Khan [4] and Calgar et al. [7] whoused finite-difference methods and sixth-degree B-splines functions respectively. The method developed byKhan [4] is shown theoretically fourth-order accurate but experimental results are even worst than the first-order method of Calgar et al. [7]. We have applied Fyfe [6] scheme based on polynomial quintic spline func-tions for solution of problems of the type (1.1) and (1.2), using the boundaries constructed in Eqs. (2.16),(2.17) and (2.18).

Problem 4.1. Consider a linear problem

yðvÞðxÞ ¼ y � 15ex � 10xex; 0 < x < 1;

yð0Þ ¼ 0; y0ð0Þ ¼ 1; y00ð0Þ ¼ 0; yð1Þ ¼ 0; y0ð1Þ ¼ �e

for which the theoretical solution is

yðxÞ ¼ xð1� xÞex.

In Table 1 maximum errors are reported corresponding to a second-order method developed in the previoussections and the methods [4,6,7]. It is verified from Table 1 that on reducing the step size from h to h/2, themaximum error kEk is reduced by a factor 1/2p, where p is the order of the method. Table 1 shows that newmethod performs better than the existing methods of the same order.

5. Conclusion

A new method based on polynomial sextic splines is developed for the solution of special fifth-order bound-ary-value problems. The new method enables us to approximate the solution at every point of the range ofintegration. The method is tested on a problem form literature and the results obtained are very encouragingand the new method performs better than the existing methods.

References

[1] A.R. Davies, A. Karageoghis, T.N. Phillips, Spectral Galerkin methods for the primary two-point boundary-value problem inmodeling viscoelastic flows, Int. J. Numer. Methods Eng. 26 (1988) 647–662.

[2] A. Karageoghis, T.N. Phillips, A.R. Davies, Spectral collocation methods for the primary two-point boundary-value problem inmodeling viscoelastic flows, Int. J. Numer. Methods Eng. 26 (1988) 805–813.

Siraj-ul-Islam, M. Azam Khan / Applied Mathematics and Computation 181 (2006) 356–361 361

[3] R.P. Agarwal, Boundary-Value Problems for Higher Order Differential Equations, World Scientific, Singapore, 1986.[4] M.S. Khan, Finite difference solutions of fifth-order boundary-value problems, PhD Thesis, Brunel University, England, 1994.[5] Abdul Majid Wazwaz, The numerical solution of fifth-order boundary-value problems by domain decomposition method, J. Comput.

Appl. Math. 136 (2001) 259–270.[6] D.J. Fyfe, Linear dependence relations connecting equal interval Nth degree splines and their derivatives, J. Inst. Math. Appl. 7 (1971)

398–406.[7] H.N. Calgar et al., The numerical solution of fifth-order boundary-value problems with sixth-degree B-spline functions, Appl. Math.

Lett. 12 (1999) 25–30.