Applied Mathematics and Computation 179 (2006) 153–160

www.elsevier.com/locate/amc

Quadratic non-polynomial spline approach to the solutionof a system of second-order boundary-value problems

Siraj-ul-Islam a,*, Muhammad Aslam Noor b, Ikram A. Tirmizi c,Muhammad Azam Khan c

a University of Engineering & Technology, Peshawar (NWFP), Pakistanb Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan

c GIK Institute of Engineering Sciences & Technology, Topi (NWFP), Pakistan

Abstract

A quadratic non-polynomial spline functions based method is developed to find approximations solution to a system ofsecond-order boundary-value problems associated with obstacle, unilateral, and contact problems. The present approachhas less computational cost and gives better approximations than those produced by other collocation, finite-difference andspline methods. Convergence analysis of the method is discussed. A numerical example is given to illustrate practical use-fulness of the new method.� 2005 Elsevier Inc. All rights reserved.

Keywords: Non-polynomial splines; Quadratic spline functions; Finite-difference methods; Obstacle problems; Boundary-value problems

1. Introduction

Variational inequality theory has become an effective and powerful tool for studying obstacle and unilateralproblems arising in mathematical and engineering sciences. This theory has developed into an interestingbranch of applicable mathematics, which contains a wealth of new ideas for inspiration and motivation todo research. It has been shown by Kikuchi and Oden [8] that the problem of equilibrium of elastic bodiesin contact with a rigid foundation can be studied in the framework of variation inequality theory. In varia-tional inequality formulation, the location of free boundary (contact problem) becomes an intrinsic part ofthe solution and no special techniques are needed to locate it. Various numerical methods are being developedand applied to find the numerical solutions of the obstacle problems including finite difference techniques andspline based methods. In principle, these methods cannot be applied directly to solve the obstacle problems.However, if the obstacle function is known, one can characterize the obstacle problem by a sequence of

0096-3003/$ - see front matter � 2005 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2005.11.091

* Corresponding author.E-mail addresses: ges0201@giki.edu.pk (Siraj-ul-Islam), noormaslam@hotmail.com (M.A. Noor), tirmizi@giki.edu.pk (I.A. Tirmizi),

azam_giki@hotmail.com (M.A. Khan).

154 Siraj-ul-Islam et al. / Applied Mathematics and Computation 179 (2006) 153–160

boundary-value problems without constraints via the variational inequality and penalty function. The compu-tational advantage of this approach is its simple applicability for solving differential equations. Such types ofpenalty function methods have been used quite effectively by Noor and Tirmzi [10], as a basis for obtainingnumerical solutions for some obstacle problems.

In this paper, non-polynomial spline functions are used to develop a numerical method for obtainingsmooth approximations to the solution of a system of second-order boundary-value problems of the type,

y00 ¼f ðxÞ; a 6 x < c;

gðxÞyðxÞ þ f ðxÞ þ r; c 6 x < d;

f ðxÞ; d 6 x 6 b

8><>: ð1:1Þ

with the boundary conditions,

yðaÞ ¼ a1 and yðbÞ ¼ a2 ð1:2Þ

and the continuity conditions of y and y 0 at c and d. Here, f and g are continuous functions on [a,b] and [c,d]respectively. The parameters a1, a2, and r, are real finite constants. Such type of systems arises in the study ofobstacle, unilateral, moving and free boundary value problems, see, for example [1–6,8] and the referencestherein. In general it is not possible to obtain the analytical solution of (1.1) for arbitrary choices of f(x)and g(x), we usually resort to some numerical methods for obtaining an approximate solution of (1.1).

Noor and Khalifa [9] have solved problem (1.1) using collocation method with cubic splines as basis func-tions. They have shown that this collocation method gives approximation with first order-accuracy. Similarconclusions were pointed out by Noor and Tirmzi [10], where second-order finite difference methods were usedto solve problem (1.1). On the other hand, Al-Said [1,2] has developed and analyzed quadratic and cubicsplines for solving (1.1). He proved that both quadratic and cubic splines methods can be used to producesecond-order smooth approximation for the solution of Eq. (1.1) and its first derivative over the whole rangeof integration. More recently, Siraj-ul-Islam et al. [11–13] have established and analyzed optimal smoothapproximations for systems second, third-order boundary value problems and a class of methods for specialfourth-boundary-value problems based on cubic, quartic and sextic non-polynomial splines and which pro-vides bases for this approach.

In the present paper, quadratic non-polynomial spline functions are applied to develop a new numericalmethod for obtaining smooth approximations to the solution of such system of second-order differential equa-tions. The new method is of order two for arbitrary a and b if 2a + 2b � 1 = 0 and method is of order four ifa ¼ 1

12along with 2a + 2b � 1 = 0. As evident form [1,2] the quadratic and cubic polynomial spline functions

need three and four coefficients evaluation at each subinterval and can produce a numerical scheme of second-order accuracy where as quadratic non-polynomial spline functions need three coefficients evaluation at eachsubinterval [xi,xi+1] and produces a fourth-order scheme with a less computational cost. This improvement isbecause of introduction of parameter k in the trigonometric part of T2 given below. The spline function wepropose in this paper has the form T2 = Span{1,coskx, sinkx} where k is the frequency of the trigonometricpart of the splines function which can be real or pure imaginary and which will be used to raise the accuracy ofthe method. Thus in each subinterval xi 6 x 6 xi+1 , we have

Spanf1; sin x; cos xg;Spanf1; sinh x; cosh xg or

Spanf1; x; x2g; ðwhen k ¼ 0Þ.

This fact is evident when correlation between polynomial and non-polynomial spline bases functions is inves-tigated in the following manner:

T 2 ¼ Spanf1; cos kx; sin kxg ¼ Span 1;1

ksinðkxÞ; 2

k2ð1� cosðkxÞÞ

� �. ð1:3Þ

Form Eq. (1.3), it follows that Ltk!0T2 = {1,x,x2}.The main idea is to use the condition of continuity to get recurrence relation for (1.1). The advantage of

new method is higher accuracy with the less computational effort. In comparison with the finite difference

Siraj-ul-Islam et al. / Applied Mathematics and Computation 179 (2006) 153–160 155

methods, spline solution has its own advantages. For example, once the solution has been computed, the infor-mation required for spline interpolation between mesh points is available. This is particularly important whenthe solution of the boundary-value problem is required at different locations in the interval [a,b]. Thisapproach has the added advantage that it not only provides continuous approximations to y(x), but alsofor y 0, y00 and higher derivatives at every point of the range of integration. Also, the C1-differentiability ofthe trigonometric part of non-polynomial splines compensates for the loss of smoothness inherited by poly-nomial splines. The new method performs better than the other collocation, finite difference, and spline meth-ods of same order and thus represents an improvement over existing methods (see [1,2,7,9,10,12], and thereferences there in). In Section 2, the problem of the type (1.1) is formulated using the penalty function tech-nique of Lewy and Stampacchia [5] to characterize the variational inequality by a sequence of variationalequations. In Section 3 the new method is developed using quadratic non-polynomial spline functions. Con-vergence analysis of the method is considered in Section 4. Numerical validation and comparison with otherknown methods are given in Section 5.

2. Formulation

To illustrate the application of the numerical method developed in the previous sections we consider thesecond-order obstacle boundary-value problem of finding y such that:

�y00 P f ðxÞ; on X ¼ ½0; p�;yðxÞP wðxÞ; on X ¼ ½0; p�;½y00ðxÞ þ f ðxÞ�½yðxÞ � wðxÞ� ¼ 0; on X ¼ ½0; p�;

9>=>;yð0Þ ¼ yðpÞ ¼ 0;

ð2:1Þ

where f(x) is a given force on the string and w(x) is the elastic obstacle function. We study the problem (2.1) inthe framework of variational inequality approach, it can be shown that, (see [2–4,9]), the problem (2.1) isequivalent to the variational inequality problem,

aðy; v� yÞP hf ; v� yi for all v 2 K. ð2:2Þ

This equivalence has been used to study the existence of a unique solution of (2.1) see, for example [3,12].Following the idea and technique of Lewy and Stampacchia [5], the variational inequality (2.2) can bewritten as

y00 � lðy � wÞðy � wÞ ¼ f ðxÞ; 0 < x < p;

yð0Þ ¼ yðpÞ ¼ 0;ð2:3Þ

where l(t) is the discontinuous function defined by

lðtÞ ¼1; for t P 0;

0; for t < 0;

�ð2:4Þ

is known as the penalty function and w is the given obstacle function defined by

wðxÞ ¼�1; for 0 6 x < p

4;

1; for p46 x < 3p

4;

�1; for 3p46 x 6 p.

8><>: ð2:5Þ

Eq. (2.3) describes the equilibrium configuration of an obstacle string pulled at the ends and lying over elas-tic step of constant height 1 and unit rigidity. Since obstacle function w is known, so it is possible to find thesolution of the problem in the interval [0,p].

From Eqs. (2.3)–(2.5), the following system of differential equations is obtained.

y00 ¼f ðxÞ; for 0 6 x < p

4and 3p

46 x 6 p;

y þ f ðxÞ � 1; for p46 x < 3p

4;

(ð2:6Þ

156 Siraj-ul-Islam et al. / Applied Mathematics and Computation 179 (2006) 153–160

with boundary conditions,

yð0Þ ¼ yðpÞ ¼ 0; ð2:7Þ

and the condition of continuity of y and y 0 at x ¼ p

4and 3p

4.

3. Numerical method

For simplicity, the break up points of the interval [a,b] are taken at c ¼ 3aþb4

and d ¼ aþ3b4

to develop thenumerical method for approximating solution of a system of differential equation (1.1). For this purposethe interval [a,b] is divided into n equal subintervals using the grid points xi = a + ih, i = 0,1, . . . ,n, x0 = a,xn = b and h ¼ b�a

n . For each segment, the polynomial Pi(x) has the form

P iðxÞ ¼ ai cos kðx� xiÞ þ bi sin kðx� xiÞ þ ci; i ¼ 0; 1; . . . ; n; ð3:1Þ

where ai, bi and ci are constants and k is free parameter and (3.1) reduces to usual quadratic spline in [a,b]when k! 0 as mentioned in Eq. (1.3).

Let yi be an approximation to y(xi), obtained by the segment Pi(x) of the mixed splines function passingthrough the points (xi,yi) and (xi+1,yi+1). To obtain the necessary conditions for the coefficients introducedin (3.1), it is not only required that Pi(x) satisfies (1.1) at xi and xi+1 and that the boundary conditions(1.2) are fulfilled, but also the continuity of first and second derivatives at the common nodes (xi,yi). To deriveexpression for the coefficients of (3.1) in terms of yi+1/2, Di, and Si+1/2, we first define:

P iðxiþ1=2Þ ¼ yiþ1=2; P 0iðxiÞ ¼ Di and P 00i ðxiþ1=2Þ ¼ Siþ1=2; and

Siþ1=2 ¼fiþ1=2 for 0 6 i < n

4and 3n

46 i < n;

giþ1=2yiþ1=2 þ fiþ1=2 þ riþ1=2 for n46 i < 3n

4.

(ð3:2Þ

From algebraic manipulation of Eq. (3.1) the following expressions follow for the coefficients of the splinefunction introduced in Eq. (3.1):

ai ¼ �Siþ1=2

k2 cosðh=2Þ� tanðh=2Þ

kDi; bi ¼

Di

k; ci ¼ yiþ1=2 þ

Siþ1=2

k2; ð3:3Þ

whereby h = kh and i = 0,1, . . . ,n � 1.The continuity condition of the first and second derivatives at (xi,yi), that is P ðnÞi�1ðxiÞ ¼ P ðnÞi ðxiÞ where n = 0

and 1, yield the following consistency relations for i = 1,2, . . . ,n.

tanðh=2Þk

ðDi þ Di�1Þ ¼ yiþ1=2 � yi�1=2 þ ðSiþ1=2Þ1

k2� 1

k2 cosðh=2Þ

� �þ ðSi�1=2Þ

cosðhÞk2 cosðh=2Þ

� 1

k2

� �; ð3:4Þ

and

Di � Di�1 ¼2Si�1=2 sinðh=2Þ

k. ð3:5Þ

Eqs. (3.4) and (3.5) yield the following scheme:

yiþ1=2 � 2yi�1=2 þ yi�3=2 ¼ Siþ1=2

1

k2 cosðh=2Þ� 1

k2

� �þ Si�1=2

2

k2� cosðhÞ

k2 cosðh=2Þþ 2

k2 cosðh=2Þþ 1

k2 cosðh=2Þ

� �þ h2Si�3=2

1

k2 cosðh=2Þ� 1

k2

� �for i ¼ 2; . . . ; n� 1. ð3:6Þ

The recurrence relation (3.6) gives n � 2, linear equations in n unknowns yi+1/2, i = 2, . . . ,n � 1. Thus, twomore equations one at each end of the range of integration are needed. These two equations are developedby Taylor series and method of undetermined coefficients. For second-order method, these equations are

2y0 � 3y1=2 þ y3=2 ¼h2

24½15S1=2 þ 3S3=2�; at i ¼ 1 ð3:7Þ

Siraj-ul-Islam et al. / Applied Mathematics and Computation 179 (2006) 153–160 157

and

yn�3=2 � 3yn�1=2 þ 2yn ¼h2

24½3Sn�3=2 þ 15Sn�1=2�; at i ¼ n. ð3:8Þ

Employing the same technique, two equation one at each boundary for fourth-order method are given by

57

40y1=2 �

7

20y3=2 �

3

40y5=2 ¼ y0 � h2 �1

240S0 þ

51

160S1=2 þ

13

96S3=2

� �; at i ¼ 1 ð3:9Þ

and

� 3

40yn�5=2 �

7

20yn�3=2 þ

57

40yn�1=2 ¼ �yn � h2 13

96Sn�3=2 þ

51

160Sn�1=2 �

1

240Sn

� �; at i ¼ n. ð3:10Þ

For simplicity Eq. (3.6) is re-written in the form,

�yiþ1=2 þ 2yi�1=2 � yi�3=2 þ h2½aSiþ1=2 þ bSi�1=2 þ aSi�3=2� ¼ 0; ð3:11Þ

where

a ¼ 1

k2 cosðh=2Þ� 1

k2

� �; b ¼ 2

k2� cosðhÞ

k2 cosðh=2Þþ 2

k2 cosðh=2Þþ 1

k2 cosðh=2Þ

� �; i ¼ 2; . . . ; n� 1

and

Siþ1=2 ¼fiþ1=2; for 0 6 i < n

4and 3n

46 i < n;

giþ1=2yiþ1=2 þ fiþ1=2 þ r; for n46 i < 3n

4.

(

The local truncation errors ti, i = 1, . . . ,n, associated with our scheme (3.7), (3.8) and (3.11) is

ti ¼

� 164

h4yð4Þ0 ðf1Þ þOðh5Þ; i ¼ 1;

ð1� 2a� bÞyð2Þi þ h3 12� a� b

2

� �yð3Þi þ h4 �80

384þ 5

4aþ b

8

� �yð4Þi ðfiÞ

þ h5 116� 13a

24� b

48

� �yð5Þi þOðh6Þ; 2 6 i 6 n� 1;

164

h4yð4Þn ðfnÞ þOðh5Þ; i ¼ n.

8>>>>><>>>>>:ð3:12Þ

Thus for any choice of arbitrary a and b satisfying the condition 1 � 2a � b = 0, indicates that method (3.11)is second order convergent. The method is fourth order convergent if 1 � 2a � b = 0 and a ¼ 1

12.

Remarks

(i) When a ¼ 124

and b ¼ 2224

then the new scheme (3.11) reduces to Al-Said [2] second-order method based oncubic polynomial spline functions.

(ii) When a ¼ 18

and b ¼ 68

then the new scheme (3.11) reduces to Al-Said [1] second-order based on quadraticspline functions.

4. Convergence analysis

The method is described in matrix form in the following way. Let A ¼ ðaijÞni;j¼1 denote the tri diagonalmatrix. Clearly the system (3.7), (3.8) and (3.11) can be expressed in matrix form as

AY ¼ Cþ T; ð4:1ÞAeY ¼ C; ð4:2ÞAE ¼ T; ð4:3Þ

158 Siraj-ul-Islam et al. / Applied Mathematics and Computation 179 (2006) 153–160

where Y = (yi+1/2), eY ¼ ð~yiþ1=2Þ, C = (ci), T = (ti), E ¼ ðeiþ1=2Þ ¼ Y� eY be N-dimensional column vectors,ei+1/2 with is the discretization error, A = A0 + Q, Q = h2BG, G = diag(gi�1/2), with gi�1/2 5 0 forn4< i 6 3n

4, i = 1,2, . . . ,n and the matrices A0 and B are defined by

A0 ¼

3 �1

�1 2 �1

�1 2 �1

�1 2 �1

. . .

. . .

. . .

. . .

�1 2 �1

�1 3

26666666666666666664

37777777777777777775

;

B ¼

3 15

a b a

a b a

a b a

. . .

. . .

. . .

. . .

a b a

15 3

26666666666666666664

37777777777777777775

;

ci ¼

2a1 � 124

h2S1; i ¼ 1;

�h2Si; 2 6 i < n4

and 3n4þ 1 < i 6 n� 1;

�h2½Si þ ar�; i ¼ n4

and i ¼ 3n4þ 1;

�h2½Si þ ðaþ bÞr�; i ¼ n4þ 1 and i ¼ 3n

4;

�h2½Si þ ð2aþ bÞr�; n4þ 2 6 i 6 3n

4� 1;

2a2 � 124

h2Sn; i ¼ n;

8>>>>>>>>>><>>>>>>>>>>:

where

Si ¼½3f 1=2 þ 15f 3=2�; i ¼ 1;

½afiþ1=2 þ bfi�1=2 þ afi�3=2�; 2 6 i 6 n� 1;

½15f n�3=2 þ 3f n�1=2�; i ¼ n.

8><>: ð4:4Þ

Here, the main purpose is to derive a bound on kEk. From Eq. (4.3), we have,

E ¼ A�1T ¼ ðA0 þ QÞ�1T ¼ ðI þ A�1

0 Q�1A0T;

kEk 6 kðI � A�10 QÞ�1k � kA�1

0 k � kTk;ð4:5Þ

where kÆk represents the 1-norm in matrix vector. Using the result kIk = 1, and k(I + A)�1k 6 (I � kAk)�1

with kA < 1k, it follows that,

kEk 6 kA�10 k � kTk

1� kA�10 k � kQk

; ð4:6Þ

Siraj-ul-Islam et al. / Applied Mathematics and Computation 179 (2006) 153–160 159

provided that kA�10 k � kQk < 1. It was shown in [2] that,

TableObserv

H

MethoSiraj-uKhanAl-SaiAl-SaiNoor aNumerColloc

kA�10 k 6

ðb� aÞ2 þ h2

8h2

����� ¼ Oðh�2Þ. ð4:7Þ

Now from (3.12) we have

kTk ¼ h4 �5

24þ 5

2aþ b

8

M4; M4 ¼ max jyð4ÞðxÞj. ð4:8Þ

Thus, using Eqs. (4.6)–(4.8) and the fact that kBk = 24, and kGk 6 jg(x)j, it follows that,

kEk 6 kA�10 kM4h2

1� kkBkgMffi Oðh2Þ; ð4:9Þ

provided gM < 1kBkk where k ¼ 1

8½ðb�aÞ2þh2�. The relation (4.9) shows that our method is second order convergent.

5. Example

The system of differential equation (2.6) when f(x) = 0, takes the following form:

y00 ¼0; for 0 6 x < p

4and 3p

46 x 6 p;

y � 1; for p46 x < 3p

4

(ð5:1Þ

with boundary condition (2.7). The analytical solution for this problem is given by

yðxÞ ¼

4xc1; 0 6 x < p

4;

1� 4 coshðp=2�xÞc2

; p46 x < 3p

4;

4ðp�xÞc1

3p46 x 6 p;

8>><>>: ð5:2Þ

where c1 = p + 4 coth(p/4) and c2 = p sinh(p/4) + 4cosh(p/4). The problem (5.1) was solved using the methoddescribed in Sections 2 and 3 with a variety of h values. The observed maximum errors (in absolute value aregiven in Table 1)).

It may be noted from Table 1 that halving step size h reduces the value of the maximum errors associatedwith both yi by a factor of approximately 1/4, which confirms that the new method (3.11) is a second-orderconvergent as predicted in Section 4. The system of differential equation (5.1) along with the boundary con-ditions (2.7) was solved in [1,2] using quadratic and cubic spline methods and the numerical results are alsolisted in Table 1. A comparison between the method (3.11) and the existing methods is given in Table 1. Itis clear from Table1 that the present method gives better results than those given in [1,2,7,9,10,12]. We wouldlike to emphasize that the method (3.11) has the advantage of the ability of approximating the derivative of y

at different values of xi where as other finite difference methods do not have this ability.

1ed maximum errors in absolute values associated with yi

p20

p40

p80

d (3.11) a = 0.1252, b = 1 � 2a 2.390 · 10�4 6.231 · 10�5 1.622 · 10�5

l-Islam et al. [12] 6.43 · 10�4 1.83 · 10�4 4.87 · 10�5

and Aziz [7] 6.43 · 10�4 1.83 · 10�4 4.87 · 10�5

d [2] cubic spline 1.94 · 10�3 4.99 · 10�4 1.27 · 10�4

d [1] quadratic spline 2.2 · 10�3 5.87 · 10�4 1.51 · 10�4

nd Tirmzi [10] 2.50 · 10�2 1.29 · 10�2 6.58 · 10�3

ov [10] 2.32 · 10�2 1.21 · 10�2 6.17 · 10�3

-cubic [9] 1.40 · 10�2 7.71 · 10�3 4.04 · 10�3

160 Siraj-ul-Islam et al. / Applied Mathematics and Computation 179 (2006) 153–160

6. Conclusion

In this paper, a new numerical method is developed for solving a system of second-order boundary-valueproblems based on quadratic non-polynomial splines. The present method enables us to approximate the solu-tion at every point of the range of integration. A class of obstacle, unilateral, and contact problems can becharacterized by this system of boundary-value problems by using the penalty function method. The resultsobtained are very encouraging and new method performs better than other existing methods.

References

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Math. Appl. 42 (2001) 861–869.[3] C. Baiocch, A. Capelo, Variational and Quasi-Variational Inequalities, John Wiley and Sons, New York, 1984.[4] R.W. Cottle, F. Giannessi, J.L. Lions, Variational inequalities and complementarity problems, Theory and Applications, Oxford,

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