non polynomial spline approach to the solution of a system of third-order boundary-value problems

Applied Mathematics and Computation 168 (2005) 152–163

www.elsevier.com/locate/amc

Non polynomial spline approach tothe solution of a system of

third-order boundary-value problems

Siraj-ul-Islam a,*, Muhammad Azam Khan a,Ikram A. Tirmizi a, E.H. Twizell b

a GIK Institute of Engineering Sciences and Technology, Topi (NWFP), Pakistanb Department of Mathematical Sciences, Brunel University, Uxbridge, Middlesex UB8 3PH, UK

Abstract

We use a quartic spline equivalent nonpolynomial spline functions to develop a

numerical method for computing approximations to the solution of a system of third-

order boundary-value problems associated with obstacle, unilateral, and contact prob-

lems. We show that the present method gives approximations which are better than

those produced by other collocation, finite-difference, and spline methods. A numerical

example is given to illustrate the applicability and efficiency of the new method.

� 2004 Elsevier Inc. All rights reserved.

Keywords: Nonpolynomial splines; Finite-difference methods; Obstacle problems; Boundary-value

problems

0096-3003/$ - see front matter � 2004 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2004.08.044

* Corresponding author.

E-mail addresses: [email protected] (Siraj-ul-Islam), [email protected] (M.A. Khan),

[email protected] (I.A. Tirmizi), [email protected] (E.H. Twizell).

mailto:[email protected]




Siraj-ul-Islam et al. / Appl. Math. Comput. 168 (2005) 152–163 153

1. Introduction

In this paper, we apply nonpolynomial spline functions to develop a numer-

ical method for obtaining smooth approximations to the solution of a system

of third-order boundary-value problems of the type

y000 ¼f ðxÞ; a 6 x 6 c;

gðxÞyðxÞ þ f ðxÞ þ rðxÞ; c 6 x 6 d;

f ðxÞ; d 6 x 6 b;

8><>: ð1:1Þ

with the boundary conditions

yðaÞ ¼ a1; y0ðaÞ ¼ b1; and y0ðbÞ ¼ b2; ð1:2Þand the continuity conditions of y,y 0 and y00 at c and d. Here, f, g, and r arecontinuous functions on [a,b] and [c,d], respectively. The parameters a1, b1and b2, are real finite constants. Such type of systems arise in the study of

obstacle, unilateral, moving and free boundary-value problems and has impor-

tant applications in other branches of pure and applied sciences, see for exam-

ple ([11,14,16–19] and the references therein). In general it is not possible to

obtain the analytical solution of (1) for arbitrary choices of f(x), g(x) and

r(x). We usually resort to some numerical methods for obtaining an approxi-

mate solution of (1). Al-Said and Noor [1], have used cubic splines functionsto solve a special form of (1), namely,

y000 ¼0; 0 6 x 6 1

4;

yðxÞ � 1; 146 x 6 3

4;

0; 346 x 6 1;

8><>:with boundary conditions

yð0Þ ¼ 0; y0ð0Þ ¼ 0 and y0ð1Þ ¼ 0;

and the continuity conditions of y and y 0 at x ¼ 14and 3

4. Before this, Noor and

Khalifa [2], Al-Said, Noor and Khalifa [3], Al-Said, Noor and Rassias [4], Al-

Said [5], Noor and Al-Said [6], have developed first and second-order methods

for solving problems (1.1). More recently Siraj-ul-Islam and Tirmize [7] has

established and analyzed smooth approximation for third-order nonlinear

boundary-value problems based on nonpolynomial spline which provides bases

for our method.

In the present paper, we apply nonpolynomial spline functions that have a

polynomial and trigonometric part to develop a new numerical method forobtaining smooth approximations to the solution of such system of third-order

differential equations. The new method is of order two for arbitrary a and bsuch that a + b = 1/2. Our method performs better than the other collocation,

finite difference, and spline methods of same order and thus represents an

154 Siraj-ul-Islam et al. / Appl. Math. Comput. 168 (2005) 152–163

improvement over existing methods (see Refs. [1–3]). The spline function we

propose in this paper have the form T n ¼ Spanf1; x; x;2 cos kx; sin kxg where k

is the frequency of the trigonometric part of the splines function which can

be real or pure imaginary and which will be used to raise the accuracy of the

method. Thus in each subinterval xi 6 x 6 xi+1, we have

span 1; x; x2; sin jkjx; cos jkjx� �

;

or span 1; x; x2; sinh jkjx; cosh jkjx� �

;

or span 1; x; x2; x3; x4� �

ðwhen k ¼ 0Þ:

This approach has the advantage over finite difference methods [3] that it pro-

vides continuous approximations to not only for y(x), but also for y 0, y00

and higher derivatives at every point of the range of integration. Also, the

C1-differentiability of the trigonometric part of nonpolynomial splines com-

pensates for the loss of smoothness inherited by polynomial splines. In Section2, we develop the new nonpolynomial spline method for solving (1.1). The con-

vergence analysis of the method is considered in Section 3. Section 4 is devoted

to the application to a system of third-order boundary-value problems. Numer-

ical results and comparison with other known methods are given in Section 5.

2. Numerical method

For simplicity, we take c ¼ 3aþb4

and d ¼ aþ3b4

in order to develop the numer-

ical method for approximating solution of a system of differential equations

(1.1). For this purpose we define a grid of N + 1 equally spaced points

xi = a + ih, i = 0,1, . . .,N, where by h ¼ b�aNþ1

. For each segment, the polynomial

Pi(x) has the form

P iðxÞ ¼ ai cos kðx� xiÞ þ bi sin kðx� xiÞ þ ciðx� xiÞ2 þ diðx� xiÞ þ eii ¼ 0; 1; 2; . . . ;N ;

ð2:1Þwhere ai, bi, ci, di, and ei are constants and k is free parameter. The function

Pi(x), which interpolates y(x) at the mesh points xi depends on k and reduces

to quartic spline in [a,b] as k ! 0.Let yi be an approximation to y(xi), obtained by the segment Pi(x) of the

mixed splines function passing through the points (xi,yi) and (xi+1,yi+1). To ob-

tain the necessary conditions for the coefficients introduced in (2.2), we do not

only require that Pi(x) satisfies (1.1) at xi and xi+1 and that the boundary con-

ditions (1.2) are fulfilled, but also the continuity of first, second, and third

derivatives at the common nodes (xi,yi).

To derive expression for the coefficients of (2.2) in terms of yi, yi+1, Di, Di+1,

Si and Si+1, we first define


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

iðxiþ1Þ ¼ Diþ1;

P 000i ðxiÞ ¼ Si; P 000

i ðxiþ1Þ ¼ Siþ1: ð2:2Þ

From algebraic manipulation we get the following expressions:

ai ¼ h3Siþ1 � Si cosðhÞ

h3 sinðhÞ; bi ¼ �h3

Si

h3;

ci ¼yiþ1 � yi

h2þ hðSiþ1 þ SiÞð1� cosðhÞÞ

h3 sinðhÞ� Di

h� hSi

h2;

di ¼ Di þh2Si

h2; ei ¼ yi � h3

Siþ1 � Si cosðhÞh3 sinðhÞ

; ð2:3Þ

whereby h = kh and i = 0,1, . . .,N � 1.

Using the continuity condition of the first and second derivatives at (xi,yi),

that is P ðnÞi�1ðxiÞ ¼ P ðnÞ

i ðxiÞ where n = 1 and 2, we get the following consistency

relations for i = 1,2, . . .,N:

Di þ Di�1 ¼2

hðyi � yi�1Þ �

h2

h2ðSi�1 þ SiÞ þ

2h2

h3 sinðhÞðSi�1 þ SiÞð1� cosðhÞÞ

ð2:4Þand

Di � Di�1 ¼h2Si cosðhÞh sin hð Þ � h2

2h sin hð Þ Si�1 þ Siþ1ð Þ þ 1

hyi�1 � 2yi þ yiþ1‘

� �þ h2

h3 sinðhÞ1� cos hð Þð Þ Siþ1 � Si�1ð Þ þ h2

h2Si�1 � Sið Þ: ð2:5Þ

Adding Eqs. (2.4) and (2.5), we get

Di ¼1

2hðyiþ1 � yi�1Þ �

h2

h2Si þ

h2

2h3 sin hð Þ1� cos hð Þð Þ Si�1 þ 2Si þ Siþ1ð Þ

þ h2Si cos hð Þ2h sin hð Þ � h2

4h sin hð Þ Si�1 þ Siþ1ð Þ: ð2:6Þ

Similarly

Di�1 ¼1

2hyi � yi�2ð Þ � h2

h2Si�1 þ

h2

2h3 sin hð Þ1� cos hð Þð Þ Si�2 þ 2Si�1 þ Sið Þ

þ h2Si�1 cos hð Þ2h sin hð Þ � h2

4h sin hð Þ Si�2 þ Sið Þ: ð2:7Þ

Di�1 and Di are eliminated from Eq. (2.4) with the help of Eqs. (2.6) and (2.7),

as a result we get the following scheme:


� yi�2 � h3Si�2

cos h� 1

h3 sin hþ 1

2h sin h

� �þ 3yi�1

� h3Si�1

1� cos h

h3 sin h� cos hh sin h

þ 1

2h sin h

� �� 3yi

� h3Si1� cos h


þ 1

2h sin h

� �þ yiþ1

� h3Siþ1

cos h� 1

h3 sin hþ 1

2h sin h

� �¼ 0;

i ¼ 2; . . . ;N � 1; ð2:8Þ

where

Si ¼fi; for 0 6 i 6 n

4and 3n

46 i 6 n;

giyi þ fi þ ri; for n46 i 6 3n

4

�and i = 0,1, . . .,N � 1.

For simplicity we rewrite Eq. (2.8)

�yi�2 � h3Si�2aþ 3yi�1 � h3Si�1b� 3yi � h3Sibþ yiþ1 � h3Siþ1a ¼ 0;

ð2:9Þwhere

a ¼ cos h� 1

h3 sin hþ 1

2h sin h; b ¼ 1� cos h


þ 1

2h sin hand

i ¼ 2; . . . ;N � 1:

The recurrence relation (2.9) gives (n � 2) linear equations in n unknowns yi,

i = 1, . . .,N. We need two more equations at each end of the range of

integration.These two equations are given by [8]

3y0 � 4y1 þ y2 ¼ �2hD0 þh3

12ð3S0 þ 4S1 þ S2Þ; i ¼ 1 ð2:10Þ

and

�3yN�2 þ 8yN�1 � 5yN ¼ �2hDNþ1 þh3

12ð3SN�2 þ 10SN�1 þ 31SN Þ; i ¼ N :

ð2:11ÞThe local truncation errors ti, i = 1, . . .,N, associated with our scheme is

ti ¼

� 110h5yðf1ÞþOðh6Þ;a< f1 < x2; i¼ 1;

h3ð1�2ðaþbÞÞy 000i þh4 � 12þaþb

� �yð4Þi þh5 1

4� 5

2aþ 1

2b

� �� yð5Þi fið Þ

þh6 �112þ 7

6aþ 1

6b

� �yð6Þi ðgiÞþh7 1

40� 17

24aþ 1

24b

� �� yð7Þi ðniÞþOðh8Þ; 26 i6N �1;

� 110h5yðfN ÞþOðh6Þ; xN�2 < fn< b; i¼N ;

8>>><>>>:ð2:12Þ

for any choice of arbitrary a and b whose sum is 12.


3. Convergence analysis

The method is described in matrix form in the following way. Let

A ¼ ðaijÞNi;j¼1 denote the four diagonal matrix. Clearly the system (2.9)–(2.11)

can be expressed in matrix form as

AY ¼ Cþ T; ð3:1Þ

AeY ¼ C; ð3:2Þ

AE ¼ T; ð3:3Þwhere Y = (yi), eY ¼ ðyi

�Þ, C = (ci), T = (ti), E ¼ ðeiÞ ¼ ðyi � yi�Þ be N-dimen-

sional column vectors and A = A0 + Q, Q = h3BG, G = diag(gi), i = 1,2, . . .,Nwith gi 5 0 for ðnþ1Þ

4< i 6 3ðnþ1Þ

4, and the matrices A0 and B are defined by

A0 ¼

�4 1

3 �3 1

�1 3 �3 1

�1 3 �3 1

�1 3 �3 1

�1 3 �3 1

: : : :

: : : :

�1 3 �3 1

�3 8 �5

0BBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCA

; ð3:4Þ

B ¼

�412

�112

�b �b �a

�a �b �b �a

�a �b �b �a

�a �b �b �a

�a �b �b �a

: : : :

: : : :

�a �b �b �a�312

�1012

�3112

0BBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCA

; ð3:5Þ


ci ¼

�3A1 � 2hB1 þ 112h3F 1; i ¼ 1;

A1 þ h3F 2; i ¼ 2;

h3F i; 3 6 i 6 n4� 1 and 3n

4þ 3 6 i 6 n� 1;

h3½F i þ ar�; i ¼ n4

and i ¼ 3n4þ 2

h3½F i þ ðaþ bÞr�; i ¼ n4þ 1 and i ¼ 3n

4þ 1;

h3½F i þ ðaþ 2bÞr�; i ¼ n4þ 2 and i ¼ 3n

4;

h3½F i þ ð2aþ 2bÞr�; n4þ 3 6 i 6 3n

4� 1;

�2hB2 þ 112h3F i; i ¼ N ;

8>>>>>>>>>>>>><>>>>>>>>>>>>>:ð3:6Þ

where

F i ¼3f 0 þ 4f 1 þ f2½ �; i ¼ 1;

afi�2 þ bfi�1 þ bfi þ afiþ1½ �; 2 6 n 6 N � 1;

3f i�2 þ 10f i�1 þ 31f i½ �; i ¼ N :

8><>: ð3:7Þ

Our main purpose is to derive a bound on kEk. From Eq. (3.3), we have

E ¼ A�1T

¼ A0 þ Qð Þ�1T

¼ I þ A�10 Q

� ��1A0T;

kEk 6 I � A�10 Q

� ��1�� : A�1

0

�� : Tk k;

where k Æ k represents the 1-norm in matrix vector. Using the result kIk = 1,

and k(I + A)�1k 6 (I � kAk)�1 we get the following expression:

kEk 6kA�1

0 k:kTk1� A�1

0

�� :kQk : ð3:8Þ

provided that kA�10 k:kQk < 1.

Now from (2.12) we have

kTk ¼ h51

4� 5

2aþ 1

2b

� �� M5; M5 ¼ max yð5ÞðxÞ

: ð3:9Þ

According to Ref. [9], the matrix A0 is nonsingular and its inverse satisfies the

inequality

A�10

6 1

81ðnþ 1Þ 2n2 þ 4nþ 5

� �¼ 2

81ðnþ 1Þ3 1þ 3

2ðnþ 1Þ2

" #

¼ 2

81ðb� aÞ3 1þ 3h2

2ðb� aÞ2

" #h�3 ¼ Oðh�3Þ: ð3:10Þ


Thus, using (3.8)–(3.10) and the fact that kBk = 44, and kGk 6 jg(x)j, we get

Ek k 6A�10

�� M5h2

1� G Bk kgMffi Oðh2Þ; ð3:11Þ

where G ¼ 281ðb� aÞ3 1þ 3h2

2ðb�aÞ

h iand kg(x)k 6 gM provided gM < 81

2kMkðb�aÞ2þ3h2.

This inequality shows that the method developed in this paper is a second-order convergent method for arbitrary values of a and b.

4. Application

To illustrate the application of the numerical method developed in the pre-

vious sections we consider the third-order obstacle boundary value problem of

finding y such that

�y 000 P f on X ¼ ½0; 1�; ð4:1Þ

y P w on X ¼ ½0; 1�; ð4:2Þ

½�y000 � f �½y � w� ¼ 0 on X ¼ ½0; 1�; ð4:3Þ

yð0Þ ¼ 0; y0ð0Þ ¼ 0 and y0ð1Þ ¼ 0; ð4:4Þwhere f(x) is a continuous function and w(x) is the obstacle function. We study

the problem (4.1)–(4.4) in the framework of variational inequality approach.

To do so, we first define the set K as

K ¼ fm : m 2 H 20ðXÞ : m P w on Xg;

which is a closed convex set in H 20ðXÞ, where H 2

0ðXÞ is a Sobolev space, which is

in fact a Hilbert space. For more details, see Refs. [10,12,13]. One can easily

show that the energy functional associated with the problem (4.1)–(4.4) is

I ½m� ¼ �Z 1

0

d3mdx3

� �dmdx

� �dx� 2

Z 1

0

f ðxÞ dmdx

� �dx for all

dmdx

2 K

¼Z 1

0

d2mdx2

� �2

dx� 2

Z 1

0

f ðxÞ dmdx

� �dx

¼ T m; gðmÞi � 2hf ; gðmÞh i;ð4:5Þ

where

hTy; gðmÞi ¼Z 1

0

d2ydx2

� �d2mdx2

� �dx; ð4:6Þ


and g ¼ ddx, is the linear operator. It clear that the operator T defined by (4.6) is

linear, g-symmetric and g-positive. Using the technique of Noor [11,15], one

can easily show that the minimum of the functional I[m] defined by (4.5) asso-

ciated with the problem (4.1)–(4.4) on the closed convex set K can be charac-

terized by the variational inequality

Ty; gðvÞ � gðyÞh i 6 f ; gðvÞ � gðyÞh i for all gðvÞ 2 K: ð4:7Þ

Now using the idea of Lewy and Stampacchia [16], the problem (4.1)–(4.4) may

be written as

�y 000 þ mfy � wgðy � wÞ ¼ f ; 0 < x < 1; ð4:8Þwith boundary conditions y(0) = 0, y 0(0) = 0 and y 0(1) = 0, where

mðtÞ ¼1; for t P 0;

0; for t < 0

�ð4:9Þ

is a discontinuous function and is known as the penalty function, and W is the

given obstacle function defined by

wðxÞ ¼�1; for 0 6 x 6 1

4and 3

46 x 6 1;

1; for 146 x 6 3

4:

(ð4:10Þ

From Eqs. (4.8) and (4.9), we obtain the following system of differential

equation:

y000 ¼f ; for 0 6 x 6 1

4and 3

46 x 6 1;

y þ f � 1; for 146 x 6 3

4;

(ð4:11Þ

with boundary conditions

yð0Þ ¼ 0; y0ð0Þ ¼ 0 and y0ð1Þ ¼ 0; ð4:12Þand the condition of continuity y, y 0 and y00, at x ¼ 1

4and 3

4. Note that the prob-

lem (4.11) is a special form of the system (1) with g(x) = 1 and r = �1.

5. Numerical results and discussion

We consider the system of differential equations (4.11) when f = 0 and

r = �1

y000 ¼0 for 0 6 x 6 1

4and 3

46 x 6 1;

y � 1 for 146 x 6 3

4;

(ð5:1Þ

with the boundary conditions (4.12). The analytical solution for this problem(5.1) is


yðxÞ ¼

12a1x2; 0 6 x 6 1

4;

1þ a2ex þ e�p2 a3 cos

ffiffi3

p

2xþ a4 sin

ffiffi3

p

2x

h i; 1

46 x 6 3

4;

12a5xðx� 2Þ þ a6; 3

46 x 6 1:

8>><>>: ð5:2Þ

To find the constants ai i = 1,2, . . ., 6, we apply the continuity conditions of y,y 0

and y00, at x ¼ 14and 3

4which leads to the following system of linear equations:

132

�S1 �S2CS1 �S2SC1 0 0

14

�S112S2

ffiffiffi3

pSC1þCS1

� �� 1

2S2

ffiffiffi3

pCS1�SC1

� �0 0

1 �S1 � 12S2

ffiffiffi3

pSC1�CS1

� �12S2

ffiffiffi3

pCS1þSC1

� �0 0

0 S3 S4CS2 S4SC21532

�1

0 S3 � 12S4

ffiffiffi3

pSC2þCS2

� �12S4

ffiffiffi3

pCS2�SC2

� �14

0

0 S312S4

ffiffiffi3

pSC2�CS2

� �12S4 �

ffiffiffi3

pCS2�SC2

� ��1 0

266666666664

377777777775

a1a2a3a4a5a6

2666666664

3777777775¼

1

0

0

�1

0

0

2666666664

3777777775;

where

S1 ¼ exp1

4

� �; S2 ¼ exp � 1

8

� �; S3 ¼ exp

3

4

� �; S4 ¼ exp � 3

8

� �;

CS1 ¼ cos

ffiffiffi3

p

8; SC1 ¼ sin

ffiffiffi3

p

8; CS2 ¼ cos

3ffiffiffi3

p

8and SC2 ¼ sin

3ffiffiffi3

p

8:

One can find the exact solution of this system of linear equation by using

Gaussian elimination.

For a variety values of h, the boundary-value problem defined by (5.1) were

solved using the numerical method developed in the previous sections and

some results are given in Table 1. The system of differential equations (5.1)along with the boundary conditions (4.12) was also solved in [1] using cubic

spline functions, [2] using a third-order collocation method with B-spline as

basis function, in [3] using a finite difference method based on the difference

formula

h3y000 ¼ �yi�2 þ 3yi�1 � 3yi þ yiþ1 þOðh4Þ: ð5:3ÞTheir numerical results are also given in Table 1.

Table 1

The observed maximum errors kekh Our method

(2.9) a ¼ � 1350; b ¼ 38

50

Cubic Spline

Ref. [1]

Colloc-Quintic

[2]

Finite Diff Scheme

(5.3) [3]

1/16 7.12 · 10�4 1.23 · 10�3 1.26 · 10�3 6.89 · 10�3

1/32 4.05 · 10�4 5.53 · 10�4 5.60 · 10�4 7.11 · 10�3

1/64 2.24 · 10�4 2.61 · 10�4 3.10 · 10�4 7.27 · 10�3

1/128 1.15 · 10�4 1.27 · 10�4 1.61 · 10�4 7.36 · 10�3


A comparison between our method and previous methods in Table 1 leads to

the conclusion that the present method is better than the other methods [1–3].

6. Conclusion

In this paper, we have developed a new numerical method for solving a system

of third-order boundary-value problems based on non polynomial splines. The

presentmethod enables us to approximate the solution at every point of the range

of integration. The parameter dependency of the present method improves the

accuracy of the schemewhich is evident from the numerical results given in Table

1. A class of obstacle, unilateral, and contact problems can be characterized by

this system of boundary-value problems by using the penalty function method.

The results obtained are very encouraging and our method performs better thanthe existing spline, collocation and finite difference methods.

Acknowledgments

The first author is grateful to Higher Education Commission Pakistan for

granting scholarship for PhD studies and University of Engg & Tech Peshawar

Pakistan for study leave.

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