non-polynomial splines approach to the solution of sixth-order boundary-value problems

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Applied Mathematics and Computation 195 (2008) 270–284

www.elsevier.com/locate/amc

Non-polynomial splines approach to the solutionof sixth-order boundary-value problems

Siraj-ul-Islam a, Ikram A. Tirmizi b,*, Fazal-i-Haq b, M. Azam Khan c

a University of Engineering & Technology, Peshawar (NWFP), Pakistanb GIK Institute of Engineering Sciences & Technology, Topi (NWFP), Pakistan

c Nescom Islamabad, Pakistan

Abstract

Non-polynomial splines, which are equivalent to seven-degree polynomial splines, are used to develop a class of numer-ical methods for computing approximations to the solution of sixth-order boundary-value problems with two-pointboundary conditions. Second-, fourth- and sixth-order convergence is obtained by using standard procedure. It is shownthat the present methods give approximations, which are better than those produced by other spline and domain decom-position methods. Numerical examples are given to illustrate practical usefulness of the new approach.� 2007 Elsevier Inc. All rights reserved.

Keywords: Sixth-order BVP; Finite-difference methods; Non-polynomial splines

1. Introduction

In this paper, non-polynomial spline functions are applied to develop numerical methods for obtainingsmooth approximations for the following boundary-value problem:

0096-3

doi:10

* CoE-m

Azam_

D6yðxÞ ¼ f ðx; yÞ; a < x < b; D � d=dx; ð1:1Þ
subject to the boundary conditions:
yðaÞ ¼ A0; D2yðaÞ ¼ A2; D4yðaÞ ¼ A4;

yðbÞ ¼ B0; D2yðbÞ ¼ B2; D4yðbÞ ¼ B4;

)ð1:2Þ

where y(x) and f(x,y) are continuous functions defined in the interval x 2 ½a; b�. It is assumed thatf ðx; yÞ 2 C6½a; b� is real and that Ai;Bi; i ¼ 0; 2; 4, are finite real numbers. The literature on the numerical solu-tion of sixth-order boundary-value problems is sparse. Such problems are known to arise in astrophysics; the

003/$ - see front matter � 2007 Elsevier Inc. All rights reserved.

.1016/j.amc.2007.04.093

rresponding author.ail addresses: [email protected] (Siraj-ul-Islam), [email protected] (I.A. Tirmizi), [email protected] (Fazal-i-Haq),[email protected] (M.A. Khan).

mailto:[email protected]




Siraj-ul-Islam et al. / Applied Mathematics and Computation 195 (2008) 270–284 271

narrow convecting layers bounded by stable layers, which are believed to surround A-type stars, may be mod-eled by sixth-order boundary-value problems (Toomre et al. [12]). Also in Glatzmaier [9] it is given that dy-namo action in some stars may be calculated by such equations. Chandrasekhar [6] determined that when aninfinite horizontal layer of fluid is heated from below and is under the action of rotation, instability sets in.When this instability is as ordinary convection, the ordinary differential equation is sixth order.

Theorems, which list the conditions for the existence and uniqueness of solutions of sixth-order boundary-value problems, are thoroughly discussed in the book by Agarwal [1]. Non-numerical techniques for solvingsuch problems are contained in papers by Baldwin [3,4]. Numerical methods of solution are contained implic-itly in the paper by Chawla and Katti [7], although those authors concentrated on numerical methods forfourth-order problems. Twizell [13] developed a second-order method for solving special and general sixth-order problems and in later work Twizell and Boutayeb [14] developed finite-difference methods of ordertwo, four, six and eight for solving such problems. Siddiqi and Twizell [11] used sixth-degree splines, wherespline values at the mid knots of the interpolation interval and the corresponding values of the even orderderivatives were related through consistency relations. M. E Gamel et al. [8] used Sinc-Galerkin method forthe solutions of sixth order boundary-value problems. Wazwaz [15] used decomposition and modified domaindecomposition methods to investigate solution of the sixth-order boundary-value problems.

The spline function proposed in this paper has the form T 7 ¼ Spanf1; x; x2; x3; x4; x5; cos kx; sin kxg where k

is the frequency of the trigonometric part of the splines function. Thus in each subinterval xi 6 x 6 xi+1, wehave:

Spanf1; x; x2; x3; x4; x5; cos kx; sin kxg;Spanf1; x; x2; x3; x4; x5; cosh kx; sinh kxg or

Spanf1; x; x2; x3; x4; x5; x6; x7g; ðwhen k ! 0Þ:

The above correlation can be better explained by the following equation:

T 7 ¼ Spanf1; x; x2; x3; x4; x5; sinðkxÞ; cosðkxÞg ¼ Span1; x; x2; x3; x4; x5; 7!

k7 kx� sinðkxÞ � ðkxÞ36þ ðkxÞ5

120

� �;

8!k8 cosðkxÞ � 1þ ðkxÞ2

2� ðkxÞ4

24þ ðkxÞ6

720

� �:

8><>:

9>=>;ð1:3Þ

In Section 2 of the present paper the special sixth-order, boundary-value problem to be solved is transformedusing non-spline polynomial techniques into linear or non-linear algebraic system. In Section 3 methods ofdifferent orders are categorized. In Section 4, properties of the inverse of coefficient matrix are discussed. InSection 5, convergence of the method is established and numerical results are reported in Section 6.

2. Numerical method

To develop the spline approximation to the problem (1.1) and (1.2), the interval [a,b] is divided into n equalsubintervals, using the grid points xi ¼ aþ ih; i ¼ 0; 1; . . . ; n, where h ¼ b�a

n . For each segment ½xi�1; xi�, thepolynomial Pi�1/2(x) has the form:

P i�1=2ðxÞ ¼ ai�1=2 sin kðx� xi�1=2Þ þ bi�1=2 cos kðx� xi�1=2Þ þ ci�1=2ðx� xi�1=2Þ5 þ di�1=2ðx� xi�1=2Þ4

þ ei�1=2ðx� xi�1=2Þ3 þ fi�1=2ðx� xi�1=2Þ2 þ gi�1=2ðx� xi�1=2Þ þ ri�1=2; ð2:1Þ

where ai�1=2; bi�1=2; ci�1=2; di�1=2; ei�1=2; fi�1=2; gi�1=2 and ri�1/2 are constants and k is free parameter.Let yi�1/2 be an approximation to y(xi�1/2), obtained by the segment Pi�1/2(x) of the mixed splines function

passing through the points ðxi�1=2; yi�1=2Þ and ðxiþ1=2; yiþ1=2Þ. To determine the coefficients in (2.1) at the com-mon nodes ðxi�1=2; yi�1=2Þ, we first define:

P i�1=2ðxi�jÞ ¼ yi�j; P 0i�1=2ðxi�jÞ ¼ Zi�j; P 00i�1=2ðxi�jÞ ¼ Mi�j; P ðviÞi�1=2ðxi�jÞ ¼ Si�j; where j ¼ 1=2:

272 Siraj-ul-Islam et al. / Applied Mathematics and Computation 195 (2008) 270–284

From algebraic manipulation we get the following expressions for coefficients:

ai�1=2 ¼ 1k6 ðSi�1=2 cosðhÞ � Siþ1=2Þ cscðhÞ;

bi�1=2 ¼ �Si�1=2

k6 ;

ci�1=2 ¼ � cscðhÞ2h5k6

h2k2 sinðhÞSiþ1=2 � h2Si�1=2k2 sinðhÞ þ h2Mi�1=2k6 sinðhÞ�h2k6 sinðhÞMiþ1=2 � 6hSi�1=2k þ 6hk cosðhÞSiþ1=2 � 6hkSi�1=2 cosðhÞþ6hkSiþ1=2 þ 6hZi�1=2k6 sinðhÞ þ 6hk6Ziþ1=2 sinðhÞ � 12Siþ1=2 sinðhÞþ12Si�1=2 sinðhÞ þ 12yi�1=2k6 sinðhÞ � 12yiþ1=2k6 sinðhÞ

0BBBBB@

1CCCCCA;

di�1=2 ¼ cscðhÞ2h4k6

2h2k2 sinðhÞSiþ1=2 � 3h2Si�1=2k2 sinðhÞ þ 3h2Mi�1=2k6 sinðhÞ�2h2k6 sinðhÞMiþ1=2 � 14hSi�1=2k þ 14hk cosðhÞSiþ1=2 � 16hkSi�1=2 cosðhÞþ16hkSiþ1=2 þ 16hZi�1=2k6 sinðhÞ þ 14hk6Ziþ1=2 sinðhÞ � 30Siþ1=2 sinðhÞþ30Si�1=2 sinðhÞ þ 30yi�1=2k6 sinðhÞ � 30yiþ1=2k6 sinðhÞ

0BBBBB@

1CCCCCA;

ei�1=2 ¼ � cscðhÞ2h3k6

20yi�1=2k6 sinðhÞ � 20yiþ1=2k6 sinðhÞ � h2k6 sinðhÞMiþ1=2

þh2k2 sinðhÞSiþ1=2 þ 12hkSiþ1=2 þ 8hk cosðhÞSiþ1=2 þ 8hk6Ziþ1=2 sinðhÞ�8hSi�1=2k þ 3h2Mi�1=2k6 sinðhÞ � 3h2Si�1=2k2 sinðhÞ � 12hkSi�1=2 cosðhÞþ12hZi�1=2k6 sinðhÞ � 20Siþ1=2 sinðhÞ þ 20Si�1=2 sinðhÞ

0BBBBB@

1CCCCCA;

fi�1=2 ¼ 12k4 ð�Si�1=2 þMi�1=2k4Þ;

gi�1=2 ¼ cscðhÞk5 ð�Si�1=2 cosðhÞ þ Siþ1=2 þ Zi�1=2k5 sinðhÞÞ;

ri�1=2 ¼Si�1=2þyi�1=2k6

k6 ; whereby h ¼ kh:

9>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>;

ð2:2Þ

Using the continuity condition of the third, fourth and fifth derivatives at ðxi�1=2; yi�1=2Þ, i.e.,P ðnÞi�1=2ðxi�1=2Þ ¼ P ðnÞiþ1=2ðxi�1=2Þ where n ¼ 3; 4; 5, we obtain:

� 1k3 sinðhÞSi�1=2 � 72

h2k5 cotðhÞSiþ1=2 þ 24h2k5 cotðhÞSi�1=2

þ 24h2k5 cotðhÞSiþ3=2 � 1

k3 cosðhÞ cotðhÞSi�1=2 þ 2k3 cotðhÞSiþ1=2

� 18hk4 Siþ1=2 þ 3

hk4 Si�1=2 � 3h Mi�1=2 þ 18

h Miþ1=2 þ 36h2k5 cscðhÞSi�1=2

� 48h2k5 cscðhÞSiþ1=2 � 24

h2 Zi�1=2 þ 120h3k6 Siþ1=2 � 60

h3k6 Si�1=2 � 60h3 yi�1=2 þ 120

h3 yiþ1=2

þ 36h2k5 cscðhÞSþ3=2 � 60

h3 yiþ3=2 � 3h Miþ3=2 þ 3

hk4 Siþ3=2 þ 24h2 Ziþ3=2 � 60

h3k6 Siþ3=2

� 1k3 cscðhÞSiþ3=2

9>>>>>>>>>>>=>>>>>>>>>>>;

¼ 0; ð2:3Þ

� 168h3k5 cotðhÞSiþ3=2 þ 168

h3k5 168 cotðhÞSi�1=2 � 24h2k4 Siþ3=2 þ 24

h2 Miþ3=2

� 192h3k5 cscðhÞSiþ3=2 � 168

h3 Ziþ3=2 þ 360h4k6 Siþ3=2 þ 360

h4 yiþ3=2 þ 24h2k4 Si�1=2 � 24

h2 Mi�1=2

þ 192h3k5 cscðhÞSi�1=2 � 168

h3 Zi�1=2 � 384h3 Ziþ1=2 � 360

h4k6 Si�1=2 � 360h4 yi�1=2

9>>=>>; ¼ 0; ð2:4Þ

1k cscðhÞSi�1=2 þ 1

k cosðhÞ cotðhÞSi�1=2 � 2k cotðhÞSiþ1=2 � 120

h3k4 Siþ1=2

þ 60h3k4 Si�1=2 � 60

h3 Mi�1=2 þ 120h3 Miþ1=2 þ 360

h4k5 cscðhÞSi�1=2 � 720h4k5 cscðhÞSiþ1=2

� 360h4 Zi�1=2 þ 1440

h5k6 Siþ1=2 � 720h5k6 Si�1=2 � 720

k5 yi�1=2 þ 1440h5 yiþ1=2

þ 60h3k4 Siþ3=2 � 60

h3 Miþ3=2 þ 360h4k5 cscðhÞSiþ3=2 þ 360

h4 Ziþ3=2 � 720h5k6 Siþ3=2

� 720h5 Y iþ3=2 � 720

h4k5 cotðhÞSiþ1=2 þ 360h4k5 cotðhÞSi�1=2

þ 360h4k5 cotðhÞSiþ3=2 þ 1

k cscðhÞSiþ3=2

9>>>>>>>>>>>=>>>>>>>>>>>;

¼ 0; ð2:5Þ


Replacing i by iþ 1; i� 1; iþ 2; i� 2, in Eqs. (2.3)–(2.5) we get the following equations:

� 1k3 sinðhÞSiþ1=2 � 72

h2k5 cotðhÞSiþ3=2 þ 24h2k5 cotðhÞSiþ1=2


k3 cosðhÞ cotðhÞSiþ1=2 þ 2k3 cotðhÞSiþ3=2

� 18hk4 Siþ3=2 þ 3

hk4 Siþ1=2 � 3h Miþ1=2 þ 18

h Miþ3=2 þ 36h2k5 cscðhÞSiþ1=2

� 48h2k5 cscðhÞSiþ3=2 � 24

h2 Ziþ1=2 þ 120h3k6 Siþ3=2 � 60

h3k6 Siþ1=2 � 60h3 yiþ1=2 þ 120

h3 yiþ3=2

þ 36h2k5 cscðhÞSiþ5=2 � 60

h3 yiþ5=2 � 3h Miþ5=2 þ 3

hk4 Siþ5=2 þ 24h2 Ziþ5=2 � 60

h3k6 Siþ5=2

� 1k3 cscðhÞSiþ5=2

9>>>>>>>>>>>>=>>>>>>>>>>>>;

¼ 0; ð2:6Þ

� 1k3 sinðhÞSi�3=2 � 72

h2k5 cotðhÞSi�1=2 þ 24h2k5 cotðhÞSi�3=2


k3 cosðhÞ cotðhÞSi�3=2 þ 2k3 cotðhÞSi�1=2

� 18hk4 Si�1=2 þ 3

hk4 Si�3=2 � 3h Mi�3=2 þ 18

h Mi�1=2 þ 36h2k5 cscðhÞSi�3=2 � 48

h2k5 cscðhÞSi�1=2

� 24h2 Zi�3=2 þ 120

h3k6 Si�1=2 � 60h3k6 Si�3=2 � 60

h3 yi�3=2 þ 120h3 yi�1=2 þ 36

h2k5 cscðhÞSiþ1=2

� 60h3 yiþ1=2 � 3

h Miþ1=2 þ 3hk4 Siþ1=2 þ 24

h2 Ziþ1=2 � 60h3k6 Siþ1=2 � 1

k3 cscðhÞSiþ1=2

9>>>>>>>>=>>>>>>>>;¼ 0; ð2:7Þ

� 1k3 sinðhÞSiþ3=2 � 72

h2k5 cotðhÞSiþ5=2 þ 24h2k5 cotðhÞSiþ3=2


k3 cosðhÞ cotðhÞSiþ3=2 þ 2k3 cotðhÞSiþ5=2

� 18hk4 Siþ5=2 þ 3

hk4 Siþ3=2 � 3h Miþ3=2 þ 18

h Miþ5=2 þ 36h2k5 cscðhÞSiþ3=2 � 48


� 24h2 Ziþ3=2 þ 120

h3k6 Siþ5=2 � 60h3k6 Siþ3=2 � 60

h3 yiþ3=2 þ 120h3 yiþ5=2 þ 36


� 60h3 yiþ7=2 � 3

h Miþ7=2 þ 3hk4 Siþ7=2 þ 24

h2 Ziþ7=2 � 60h3k6 Siþ7=2 � 1

k3 cscðhÞSiþ7=2

9>>>>>>>>=>>>>>>>>;¼ 0; ð2:8Þ

� 1k3 sinðhÞSi�5=2 � 72

h2k5 cotðhÞSi�3=2 þ 24h2k5 cotðhÞSi�5=2

þ 24h2k5 cotðhÞSi�1=2 � 1

k3 cosðhÞ cotðhÞSi�5=2 þ 2k3 cotðhÞSi�3=2

� 18hk4 Si�3=2 þ 3

hk4 Si�5=2 � 3h Mi�5=2 þ 18

h Mi�3=2 þ 36h2k5 cscðhÞSi�5=2 � 48


� 24h2 Zi�5=2 þ 120

h3k6 Si�3=2 � 60h3k6 Si�5=2 � 60

h3 yi�5=2 þ 120h3 yi�3=2 þ 36


� 60h3 yi�1=2 � 3

h Mi�1=2 þ 3hk4 Si�1=2 þ 24

h2 Zi�1=2 � 60h3k6 Si�1=2 � 1

k3 cscðhÞSi�1=2

9>>>>>>>>=>>>>>>>>;¼ 0; ð2:9Þ


h3k5 cotðhÞSiþ1=2 � 24h2k4 Siþ5=2 þ 24

h2 Miþ5=2

� 192h3k5 cscðhÞSiþ5=2 � 168

h3 Ziþ5=2 þ 360h4k6 Siþ5=2 þ 360

h4 yiþ5=2 þ 24h2k4 Siþ1=2 � 24

h2 Miþ1=2

þ 192h3k5 cscðhÞSiþ1=2 � 168

h3 Ziþ1=2 � 384h3 Ziþ3=2 � 360

h4k6 Siþ1=2 � 360h4 yiþ1=2

9>>=>>; ¼ 0; ð2:10Þ


h3k5 cotðhÞSi�3=2 � 24h2k4 Siþ1=2 þ 24

h2 Miþ1=2

� 192h3k5 cscðhÞSiþ1=2 � 168

h3 Ziþ1=2 þ 360h4k6 Siþ1=2 þ 360

h4 yiþ1=2 þ 24h2k4 Si�3=2

� 24h2 Mi�3=2 þ 192

h3k5 cscðhÞSi�3=2 � 168h3 Zi�3=2 � 384

h3 Zi�1=2 � 360h4k6 Si�3=2 � 360

h4 yi�3=2

9>>=>>; ¼ 0; ð2:11Þ


h3k5 cotðhÞSiþ3=2 � 24h2k4 Siþ7=2 þ 24

h2 Miþ7=2

� 192h3k5 cscðhÞSiþ7=2 � 168

h3 Ziþ7=2 þ 360h4k6 Siþ7=2 þ 360

h4 yiþ7=2 þ 24h2k4 Siþ3=2

� 24h2 Miþ3=2 þ 192

h3k5 cscðhÞSiþ3=2 � 168h3 Ziþ3=2 � 384

h3 Ziþ5=2 � 360h4k6 Siþ3=2 � 360

h4 yiþ3=2

9>>=>>; ¼ 0; ð2:12Þ

� 168h3k5 cotðhÞSi�1=2 þ 168

h3k5 cotðhÞSi�5=2 � 24h2k4 Si�1=2 þ 24

h2 Mi�1=2

� 192h3k5 cscðhÞSi�1=2 � 168

h3 Zi�1=2 þ 360h4k6 Si�1=2 þ 360

h4 yi�1=2 þ 24h2k4 Si�5=2

� 24h2 Mi�5=2 þ 192

h3k5 cscðhÞSi�5=2 � 168h3 Zi�5=2 � 384

h3 Zi�3=2 � 360h4k6 Si�5=2 � 360

h4 yi�5=2

9>>=>>; ¼ 0; ð2:13Þ


1k sinðhÞSiþ1=2 þ 1

k cosðhÞ cotðhÞSiþ1=2 � 2k cotðhÞSiþ3=2 � 120

h3k4 Siþ3=2

þ 60h3k4 Siþ1=2 � 60

h3 Miþ1=2 þ 120h3 Miþ3=2 þ 360

h4k5 cscðhÞSiþ1=2 � 720h4k5 cscðhÞSiþ3=2

� 360h4 Ziþ1=2 þ 1440

h5k6 Siþ3=2 � 720h5k6 Siþ1=2 � 720

h5 yiþ1=2 þ 1440h5 yiþ3=2 þ 60

h3k4 Siþ5=2

� 60h3 Miþ5=2 þ 360

h4k5 cscðhÞSiþ5=2 þ 360h4 Ziþ5=2 � 720

h5k6 Siþ5=2 � 720h5 yiþ5=2


h4k5 cotðhÞSiþ1=2 þ 360h4k5 cotðhÞSiþ5=2 þ 1

k cscðhÞSiþ5=2

9>>>>>>>>>=>>>>>>>>>;¼ 0; ð2:14Þ

1k sinðhÞSi�3=2 þ 1

k cosðhÞ cotðhÞSi�3=2 � 2k cotðhÞSi�1=2 � 120

h3k4 Si�1=2

þ 60h3k4 Si�3=2 � 60

h3 Mi�3=2 þ 120h3 Mi�1=2 þ 360

h4k5 cscðhÞSi�3=2 � 720h4k5 cscðhÞSi�1=2

� 360h4 Zi�3=2 þ 1440

h5k6 Si�1=2 � 720h5k6 Si�3=2 � 720

h5 yi�3=2 þ 1440h5 yi�1=2 þ 60

h3k4 Siþ1=2

� 60h3 Miþ1=2 þ 360


h5k6 Siþ1=2 � 720h5 yiþ1=2

� 720h4k5 cotðhÞSi�1=2 þ 360

h4k5 cotðhÞSi�3=2 þ 360h4k5 cotðhÞSiþ1=2 þ 1

k cscðhÞSiþ1=2

9>>>>>>>>>=>>>>>>>>>;¼ 0; ð2:15Þ

1k sinðhÞSiþ3=2 þ 1

k cosðhÞ cotðhÞSiþ3=2 � 2k cotðhÞSiþ5=2 � 120

h3k4 Siþ5=2

þ 60h3k4 Siþ3=2 � 60

h3 Miþ3=2 þ 120h3 Miþ5=2 þ 360

h4k5 cscðhÞSiþ3=2 � 720h4k5 cscðhÞSiþ5=2

� 360h4 Ziþ3=2 þ 1440

h5k6 Siþ5=2 � 720h5k6 Siþ3=2 � 720

h5 yiþ3=2 þ 1440h5 yiþ5=2 þ 60

h3k4 Siþ7=2

� 60h3 Miþ7=2 þ 360


h5k6 Siþ7=2 � 720h5 yiþ7=2


h4k5 cotðhÞSiþ3=2 þ 360h4k5 cotðhÞSiþ7=2 þ 1

k cscðhÞSiþ7=2

9>>>>>>>>>=>>>>>>>>>;¼ 0; ð2:16Þ

1k sinðhÞSi�5=2 þ 1

k cosðhÞ cotðhÞSi�5=2 � 2k cotðhÞSi�3=2 � 120

h3k4 Si�3=2

þ 60h3k4 Si�5=2 � 60

h3 Mi�5=2 þ 120h3 Mi�3=2 þ 360

h4k5 cscðhÞSi�5=2 � 720h4k5 cscðhÞSi�3=2

� 360h4 Zi�5=2 þ 1440

h5k6 Si�3=2 � 720h5k6 Si�5=2 � 720

h5 yi�5=2 þ 1440h5 yi�3=2 þ 60

h3k4 Si�1=2

� 60h3 Mi�1=2 þ 360

h4k5 cscðhÞSi�1=2 þ 360h4 Zi�1=2 � 720

h5k6 Si�1=2 � 720=ðh5Þyi�1=2

� 720h5k6 cotðhÞSi�3=2 þ 360

h4k5 cotðhÞSi�5=2 þ 360h4k5 cotðhÞSi�1=2 þ 1

k cscðhÞSi�1=2

9>>>>>>>>>=>>>>>>>>>;¼ 0; ð2:17Þ

For the elimination of Mi�5=2;Mi�3=2;Mi�1=2;Miþ1=2;Miþ3=2;Miþ5=2;Miþ7=2;Zi�5=2;Zi�3=2;Zi�1=2;Ziþ1=2;Ziþ3=2;Ziþ5=2

and Zi+7/2, Eqs. (2.3)–(2.16) are solved simultaneously. The values of Z’s and M’s obtained from these equa-tions are used in Eq. (2.17), which results, after lengthy calculations the following recurrence relation:

yi�7=2 � 6yi�5=2 þ 15yi�3=2 � 20yi�1=2 þ 15yiþ1=2 � 6yiþ3=2 þ yiþ5=2

¼ h6faðSi�7=2 þ Siþ5=2Þ þ bðSi�5=2 þ Siþ3=2Þ þ cðSi�3=2 þ Siþ1=2Þ þ dSi�1=2g; ð2:18Þ

where

a ¼ 1120� 120

h6 � 20h3 sin h

þ 1h sin hþ 120

h5 sin h

� �;

b ¼ 1120� 40

h3 sin h� 2 cos h

h sin h þ 720h6 � 480

h5 sin h� 240 cos h

h5 sin hþ 26

h sin hþ 40 cos hh3 sin h

� �;

c ¼ 1120

100h3 sin h

þ 67h sin hþ 960 cos h

h5 sin hþ 840

h5 sin h� 1800

h6 � 52 cos hh sin h þ 80 cos h

h3 sin h

� �;

d ¼ 1120

2400h6 � 960

h5 sin h� 80

h3 sin h� 132 cos h

h sin h � 240 cos hh3 sin h

þ 52h sin h� 1440 cos h

h5 sin h

� �;

for i ¼ 4; 5; 6; . . . ; n� 4; where n > 5:

9>>>>>>>>>>>>=>>>>>>>>>>>>;

ð2:19Þ

Here limh!0ða; b; c; dÞ ¼ 15040ð1; 120; 1191; 2416Þ, which is the polynomial case as mentioned in Eq. (1.3).


The local truncation error ti; i ¼ 4; 5; 6; . . . ; n� 4, associated with the scheme developed in (2.18) is,

ti ¼C6h6yð6Þi þ C7h7yð7Þi þ C8h8yð8Þi þ C9h9yð9Þi þ C10h10yð10Þ

i þ C11h11yð11Þi þ C12h12yð12Þ

i

þC13h13yð13Þi þ C14h14yð14Þ

i þOðh15Þ;

( )ð2:20Þ

C6 ¼ �ð�1þ 2aþ 2bþ 2cþ dÞ;

C7 ¼ð�1þ 2aþ 2bþ 2cþ dÞ

2;

C8 ¼ �ð�3þ 74aþ 34bþ 10cþ dÞ

8;

C9 ¼ð�7þ 218aþ 98bþ 26cþ dÞ

48;

C10 ¼ð121� 5ð3026aþ 706bþ 82cþ dÞÞ

1920;

C11 ¼ð�77þ 13682aþ 2882bþ 242cþ dÞ

3840;

C12 ¼ð6227� 21ð133274aþ 16354bþ 730cþ dÞÞ

967680;

C13 ¼ð�3353þ 3ð745418aþ 75938bþ 2186cþ dÞÞ

1935360;

C14 ¼ �ð�4681þ 6155426aþ 397186bþ 6562cþ dÞ

10321920:

9>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>;

ð2:21Þ

Thus for different choices of a, b, c, d in scheme (2.18), methods of different order are obtained.The relation (2.18) gives n � 6 linear algebraic equations in n unknowns yiþ1=2; i ¼ 0; 1; . . . ; n� 1. We

require six more equations, three at each end of the range of integration, for the direct computation ofyi+1/2. These equations are developed by Taylor series and method of undetermined coefficients. General formof boundary equations for the main scheme is as follows:

20y0 � 35y1=2 þ 21y3=2 � 7y5=2 þ y7=2 þ sy9=2

¼ a1h2S000 þ b1h4SðivÞ0 þ h6 a1SðviÞ1=2 þ b1SðviÞ

3=2 þ c1SðviÞ5=2 þ d1SðviÞ

7=2 þ e1SðviÞ9=2 þ f1SðviÞ

11=2

� �; ð2:22Þ

� 10y0 þ 21y1=2 � 21y3=2 þ 15y5=2 � 6y7=2 þ y9=2




11=2

� �; ð2:23Þ

2y0 � 7y1=2 þ 15y3=2 � 20y5=2 þ 15y7=2 � 6y9=2 þ y11=2




11=2

� �: ð2:24Þ

The remaining three equations at the other end can be obtained from (2.22)–(2.24) by writing them in reverseorder. The constants ai; bi; ci; di; ei; fi; ai; bi and s are parameters which must be chosen so that the localtruncation errors of (2.22)–(2.24) are identical with (2.18).

3. Numerical methods of different orders

3.1. Second-order methods

a ¼ 146080

; b ¼ 72246080

; c ¼ 1054346080

; d ¼ 2354846080

in the main Eq. (2.18) gives

C6 ¼ C7 ¼ 0; C8 ¼1

24:


The values of the coefficients for these methods are:

a1 ¼7

2; b1 ¼ �

77

96; a1 ¼

13005

cd; b1 ¼

9821

cd; c1 ¼

721

cd; d1 ¼

1

cd; e1 ¼ 0; f 1 ¼ 0;

a2 ¼ �3

4; b2 ¼ �

25

64; a2 ¼ �

9821

cd; b2 ¼

23547

cd; c2 ¼

10543

cd; d2 ¼

722

cd;

e2 ¼1

cd; f 2 ¼ 0; a3 ¼ �

1

4; b3 ¼ �

1

192; a3 ¼

721

cd; b3 ¼

10543

cd; c3 ¼

23548

cd;

d3 ¼10543

cd; e3 ¼

722

cd; f 3 ¼

1

cd; s ¼ 0; cd ¼ 46080: ð3:1Þ

3.2. Fourth-order methods

a ¼ � 720þ 3c

5þ 2d

5; b ¼ 17

20� 8c

5� 9d

10in the main Eq. (2.18) gives C6 ¼ C7 ¼ C8 ¼ C9 ¼ 0;C10 ¼ 151

120� 2c� 3d

2

for arbitrary constants c,d.The values of the coefficients of boundary equations are given by

a1 ¼7

2; b1 ¼ �

77

96; a1 ¼

2685830

cd; b1 ¼

2704842

cd; c1 ¼ �

166502

cd; d1 ¼

50582

cd;

e1 ¼ 0; f 1 ¼ 0; a2 ¼ �3

4; b2 ¼ �

25

64; a2 ¼

12161085

cd; b2 ¼ �

35332965

cd;

c2 ¼64014771

cd; d2 ¼ �

41166795

cd; e2 ¼

10321920

cd; f 2 ¼ 0; a3 ¼ �

1

4; b3 ¼ �

1

192;

a3 ¼51608615

cd; b3 ¼ �

193535007

cd; c3 ¼

273530281

cd; d3 ¼ �

141926257

cd;

e3 ¼10321920

cd; f 3 ¼

10321920

cd; s ¼ 0; cd ¼ 10321920: ð3:2Þ

3.3. Sixth-order methods

a ¼ 11400� d

20; b ¼ � 47

300þ 3d

10and c ¼ 151

240� 3d

4in the main Eq. (2.18) gives C6 ¼ C7 ¼ C8 ¼ C9 ¼ C10 ¼

C11 ¼ 0; C12 ¼ d24� 4153

151200for arbitrary constant d.

The values of the coefficients for these methods are:

a1 ¼7

2; b1 ¼ �

77

96; a1 ¼

119271901

cd; b1 ¼

128562861

cd; c1 ¼ �

18967984

cd;

d1 ¼11537036

cd; e1 ¼ �

3523149

cd; f 1 ¼

483175

cd; a2 ¼ �

3

4; b2 ¼ �

25

64;

a2 ¼80655078

cd; b2 ¼

277312338

cd; c2 ¼

77419113

cd; d2 ¼

19393863

cd; e2 ¼ �

5790447

cd;

f 2 ¼920775

cd; a3 ¼ �

1

4; b3 ¼ �

1

192; a3 ¼

3815581

cd; b3 ¼

100731801

cd; c3 ¼

255355121

cd;

d3 ¼100719251

cd; e3 ¼

3839346

cd; f 3 ¼

5140

cd; s ¼ 1; cd ¼ 464486400: ð3:3Þ

3.4. Eight-order methods

a ¼ 130240

; b ¼ 415040

; c ¼ 218910080

; d ¼ 41537560

in the main Eq. (2.18) gives C6 = C7 = C8 = C9 = C10 = C11 = 0,C12 =0, C13 = 0, C14 5 0. Boundary equations for eight-order method are not constructed, however these equationscan be obtained following the procedure adapted in the previous cases.


4. Properties of the coefficient matrix A0

Consider the following seven-band matrix A0 of order n:

A0 ¼

�35 21 a 0 0

21 �21 b 0 0

aT bT M bTR aTR

0 0 bR �21 21

0 0 aR 21 �35

266666664

377777775; ð4:1Þ

where a ¼ ½�7; 1; 0; . . . ; 0�; b ¼ ½15;�6; 1; 0; . . . ; 0�. Here T denotes the operation of transposition and for arow vector v ¼ ðv1; v2; . . . ; vnÞ; vR ¼ ðvn; vn�1; . . . ; v1Þ. Further, a, b are (n � 4) dimensional row vectors. Thematrix M = (mi,j) is a seven-band matrix of order (n � 4) given by

mi;j ¼

�20; i ¼ j ¼ 1; 2; . . . ; n� 4;

15; ji� jj ¼ 1;

�6; ji� jj ¼ 2;

1; ji� jj ¼ 3;

0; ji� jj > 3:

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

ð4:2Þ

In this section we find matrix A�10 and kA�1

0 k1. The matrix A�10 is likewise of the form

A�10 ¼

e f c g h

j k d l m

cT dT N dTR cTR

m l dR k j

h g cR f e

266666664

377777775; ð4:3Þ

where c = (ci), d = (di) are (n � 4) dimensional row vectors. T and R have the same meaning as defined aboveand e, f, g, h, j, k, l, m are scalars. The matrix N = (ni,j) is of order (n � 4). Using A0A�1

0 ¼ In, we get the fol-lowing equations:

ðiÞ acT � 35eþ 21j ¼ 1;

ðiiÞ acTR � 35hþ 21m ¼ 0;

ðiiiÞ McT þ aTeþ aTRhþ bTjþ bTRm ¼ 0;

ðivÞ bcT þ 21e� 21j ¼ 0;

ðvÞ bcTR þ 21h� 21m ¼ 0;

ðviÞ bdT þ 21f� 21k ¼ 1;

ðviiÞ bdTR þ 21g� 21l ¼ 0;

ðviiiÞ adT � 35fþ 21k ¼ 0;

ðixÞ adTR � 35gþ 21l ¼ 0;

ðxÞ MdT þ aTfþ aTRgþ bTkþ bTRl ¼ 0;

ðxiÞ MN þ aTcþ aTRcR þ bTd þ bTRdR ¼ In�4:

9>>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>>;

ð4:4Þ

Here 0 denotes a (n � 4) dimensional null column vector.


In order to find the unknowns in above equations, we need to use M�1 ¼ ð~mi;jÞ. From [10] it is given by

~mi;j ¼

ðn� 3� iÞðn� 2� iÞðn� 1� iÞjðjþ 1Þðjþ 2Þ240ðnþ 1Þnðn� 1Þðn� 2Þðn� 3Þ

�2iðiþ 2Þðj� 1Þðjþ 3Þðn� 3Þðnþ 1Þ�ðiþ 1Þðiþ 2Þðj� 2Þðj� 1Þnðnþ 1Þ � iðiþ 1Þðjþ 3Þðjþ 4Þðn� 3Þðn� 2Þ

� �; i P j;

ðn� 3� jÞðn� 2� jÞðn� 1� jÞiðiþ 1Þðiþ 2Þ240ðnþ 1Þnðn� 1Þðn� 2Þðn� 3Þ

�2jðjþ 2Þði� 1Þðiþ 3Þðn� 3Þðnþ 1Þ�ðjþ 1Þðjþ 2Þði� 2Þði� 1Þnðnþ 1Þ � jðjþ 1Þðiþ 3Þðiþ 4Þðn� 3Þðn� 2Þ

� �; i 6 j:

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

ð4:5Þ
Algebraic manipulation of Eqs. (4.4((i)–(v)) yields the following expressions
e ¼ � 16n4 þ 40n2 � 11

2880n;

j ¼ � 16n4 � 40n2 þ 9

960n;

h ¼ � 14n4 þ 20n2 þ 11

2880n;

m ¼ � 14n4 � 20n2 � 9

960n;

9>>>>>>>>>>>=>>>>>>>>>>>;

ð4:6Þ

and

ci ¼ �ð2n�2i�3Þ

ð16iþ 24Þn3 þ ð16i2 þ 48iþ 36Þn2

�ð24i3 þ 108i2 þ 112iþ 6Þnþ 6i4 þ 36i3 þ 56i2 þ 6i� 9

� �� 2880n ;

i ¼ 1; 2; . . . ; n� 4:

8>><>>: ð4:7Þ

Eqs. (4.4((vi)–(x)) yield the following results:

f ¼ � 16n4 � 40n2 þ 9

960n;

k ¼ � 16n4 � 120n2 þ 160n� 51

320n;

g ¼ � 14n4 � 20n2 � 9

960n;

l ¼ � 14n4 � 60n2 þ 51

320n;

9>>>>>>>>>>>=>>>>>>>>>>>;

ð4:8Þ

and

di ¼ �ð2n�2i�3Þ

ð16iþ 24Þn3 þ ð16i2 þ 48iþ 36Þn2

�ð24i3 þ 108i2 þ 192iþ 126Þnþ 6i4 þ 36i3 þ 96i2 þ 126iþ 51

� �� 960n ;

i ¼ 1; 2; . . . ; n� 4:

8>><>>: ð4:9Þ

From Eq. (4.4(xi)), we have

N ¼ M�1 þ U ; ð4:10Þ
where
U ¼ �M�1ðaTcþ bTd þ bTRdR þ aTRcRÞ: ð4:11Þ


Hence U = (ui,j) is given by:

ui;j ¼1

2880ðn� 3Þðn� 2Þnðn2 � 1Þ

�12i5 12j5 � 30j4ðn� 3Þ þ 20j3ðn2 � 8nþ 12Þ þ 30j2ð2n2 � 9nþ 9Þ�2jðn4 � 5n3� 15n2 þ 75n� 60Þ � 3ðn4 � 5n3 þ 5n2 þ 5n� 6Þ

�

þ30i4ðn� 3Þ12j5 þ j4ð88� 32nÞ þ 12j3ð2n2� 14nþ 19Þþ8j2ð9n2 � 35nþ 31Þ � 2jð2n4 � 7n3 � 20n2 þ 79n� 54Þ�3ð2n4 � 7n3 þ 4n2 þ 7n� 6Þ

8><>:

9>=>;

þ3ðn4 � 5n3þ 5n2 þ 5n� 6Þ

153þ 12j5 þ j4ð90� 60nÞ�480nþ 360n2 � 48n4

þ180j2ð2n2 � 5nþ 3Þ þ 20j3ð4n2 � 18nþ 15Þ�8jð4n4� 75n2 þ 135n� 60Þ

8>>><>>>:

9>>>=>>>;

þ60i2ðn� 3Þ

�6j5ð2n� 3Þ þ 4j4ð9n2 � 35nþ 31Þþ9ðn� 1Þ2ð2n3 � 5n2� nþ 6Þ � 6j3ð5n3 � 38n2 þ 81n� 51Þþj2ð�6n4� 69n3 þ 420n2 � 679nþ 338Þþjð12n5 � 64n4 þ 41n3þ 238n2 � 413nþ 186Þ

8>>><>>>:

9>>>=>>>;

�20i3

12j5ðn2 � 8nþ 12Þ � 18j4ð2n3 � 20n2 þ 61n� 57Þþ18j2ð5n4 � 53n3 þ 195n2 � 294nþ 153Þþ4j3ð8n4� 103n3 þ 490n2 � 959nþ 645Þþ3ð�4n6 þ 38n5 � 125n4 þ 145n3 þ 39n2 � 183nþ 90Þ�4jð2n6 � 19n5 þ 49n4þ 67n3 � 489n2 þ 711n� 315Þ

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

9>>>>>>=>>>>>>;

þ4i

6j5ðn4 � 5n3 � 15n2 þ 75n� 60Þ�15j4ð2n5 � 13n4 þ n3þ 139n2 � 291nþ 162Þþ20j3ð2n6 � 19n5 þ 49n4þ 67n3 � 489n2 þ 711n� 315Þþ15j2ð12n6� 100n5 þ 233n4 þ 115n3 � 1127n2þ 1425n� 558Þ

�64n8 � 20n7� 55n6 þ 530n5 � 1134n4 þ 600n3

þ825n2 � 1110nþ 360

� �

�2j8n8 � 40n7 � 200n6 þ 1825n5� 4230n4 þ 1650n3

þ6480n2 � 8595nþ 3150

� �:

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

9>>>>>>>>>>>>>=>>>>>>>>>>>>>;

2666666666666666666666666666666666666666666666666666666666664

3777777777777777777777777777777777777777777777777777777777775

ð4:12Þ
We summarize the results of this section in the following lemma.
Lemma 4.1. The matrix A0 is nonsingular with A�10 < 0 and if A�1

0 ¼ ð~ai;jÞ, then ~ai;j are given by ((4.6)–(4.10)).

In order to find kA�10 k1, we first find the sum of elements in the ith row of A�1

0 . The sum of elements in theith row of M�1 is given by,

� 1

720iðiþ 1Þðiþ 2Þðn� i� 1Þðn� i� 2Þðn� i� 3Þ ð4:13Þ

and the sum of the elements of ith row of U is:

� 1

960ðn� 4Þ ð4iþ 6Þn4 þ ð16iþ 24Þn3 � ð8i3 þ 34i2 þ 52iþ 29Þn2 þ ð4i4 � 12i3 � 104i2 � 192i� 116Þn

þ18i4 þ 108i3 þ 258i2 þ 288iþ 120

�: ð4:14Þ

Let Si designate the sum of the elements in the ith row of A�10 in modulus, then Si ¼

Pnj¼1j~ai;jj ¼ �

Pnj¼1~ai;j is

given by

Si ¼1

2880ð12i� 6Þn5 þ ð�20i3 þ 30i2 þ 20i� 15Þn3 þ ð12i5 � 30i4 � 40i3 þ 90i2 þ 16i� 24Þn� 4i6

þ12i5 þ 20i4 � 60i3 � 16i2 þ 48i�: ð4:15Þ


Consider Si as a function of the real variable i. It turns out that Si is maximum for i ¼ 12ðnþ 1Þ. The infinity

norm of A�10 must be restricted to an integral value of i and therefore

kA�10 k1 ¼ max

iSi 6 Sðnþ1Þ=2:

Here equality will hold only if n is odd. We substitute i ¼ 12ðnþ 1Þ into (4.15) to get S(n+1)/2. Hence,

kA�10 k1 6 Sðnþ1Þ=2 ¼

61n6 þ 175n4 þ 259n2 þ 225

46080

¼ 61ðb� aÞ6 þ 175h2ðb� aÞ4 þ 259h4ðb� aÞ2 þ 225h6

46080h6¼ Oðh�6Þ: ð4:16Þ

5. Convergence

Clearly, the family of numerical methods is described by the Eqs. (2.18), boundary equations and the solu-tion vector Y ¼ ½y1=2; y3=2; . . . ; yn�1=2�

T, T denoting transpose, is obtained by solving a non-linear algebraic sys-tem of order n which has the form

A0Y� h6Bf ðx;YÞ � C ¼ 0: ð5:1Þ
Now we investigate the error analysis of the seven-degree non-polynomial spline method described in Section2. To do so we let, y = (y(xi+1/2)), Y = (yi+1/2), C = (ci), T = (ti), and E = (ei+1/2), be n-dimensional columnvectors. Here ei+1/2 = y(xi+1/2) � yi+1/2 is the discretization error for i ¼ 4; . . . ; n� 4. Thus, we can writeour method in the matrix form:
The matrices B and C are given by

B ¼

a1 b1 c1 d1 e1 f1

a2 b2 c2 d2 e2 f2

a3 b3 c3 d3 e3 f3

a b c d b a

. .. . .

. . .. . .

.

. .. . .

. . .. . .

.

. .. . .

. . .. . .

.

a b d c b a

f3 e3 d3 c3 b3 a3

f2 e2 d2 c2 b2 a2

f1 e1 d1 c1 b1 a1

26666666666666666666664

37777777777777777777775

: ð5:2Þ

The column vector C is given by

C ¼

�20A0 þ a1h2A2 þ b1h4A4

10A0 þ a2h2A2 þ b2h4A4

�2A0 þ a3h2A2 þ b3h4A4

0

..

.

..

.

..

.

0

�2B0 þ a3h2B2 þ b3h4B4

10B0 þ a2h2B2 þ b2h4B4

�20B0 þ a1h2B2 þ b1h4B4

26666666666666666666664

37777777777777777777775

: ð5:3Þ


The vector y ¼ ½yðx1=2Þ; yðx3=2Þ; . . . ; yðxn�1=2Þ�T satisfies

TableMaxim

n

10

A0y� h6Bf ðx; yÞ � C� T ¼ 0;

where T ¼ ½t1; t2; . . . ; tn�T is the vector of local truncation errors, and a conventional convergence analysisshows that the norm of the vector

E ¼ y� Y

satisfies:

kEk1 6ðb� aÞ6

46080� ð61ðb� aÞ6 þ 175h2ðb� aÞ4 þ 259h4ðb� aÞ2 þ 225h6ÞB�F �

� C6h6V 6 þ C7h7V 7 þ C8h8V 8 þ C9h9V 9 þ C10h10V 10 þ C11h11V 11 þ C12h12V 12 þ . . . �

; ð5:4Þ

where V i ¼ maxa6x6bj diyðxÞdxi j for i ¼ 1; 2; . . ., B* = kBk, and F � ¼ maxa6x6bj of

oyðxÞ j.The order of convergence of the numerical method is, thus p, if Cp+6 is the first non-vanishing constant on

the right hand side of (2.20) provided

F � <46080

ð61ðb� aÞ6 þ 175h2ðb� aÞ4 þ 259h4ðb� aÞ2 þ 225h6Þ: ð5:5Þ

6. Numerical results

The numerical methods outlined in the previous sections were tested on the following non-linear and linearproblems.

6.1. Non-linear problems

Problem 6.1

D6yðxÞ ¼ e�xy2ðxÞ; 0 < x < 1 ð6:1Þ
subject to the boundary conditions:
yð0Þ ¼ D2ð0Þ ¼ D4ð0Þ ¼ 1;

yð1Þ ¼ D2ð1Þ ¼ D4ð1Þ ¼ e:

The theoretical solution for this problem is

yðxÞ ¼ ex:

The results of maximum absolute errors for this problem are listed in Table 1.

Problem 6.2

D6yðxÞ ¼ 20 exp½�36yðxÞ� � 40ð1þ xÞ�6; 0 < x < 1 ð6:2Þ

with boundary conditions:

yð0Þ ¼ 0; D2yð0Þ ¼ � 1

6; D4yð0Þ ¼ �1; yð1Þ ¼ 1

6ln 2; D2yð1Þ ¼ � 1

24; D4yð1Þ ¼ � 1

16ð6:3Þ

1um absolute errors corresponding to Problem (6.1)

Our sixth-order method (3.3) [15]

3.577 · 10�13 1.299 · 10�4


for which the theoretical solution is

TableMaxim

n

71531

TableMaxim

N

71531

yðxÞ ¼ 1

6lnð1þ xÞ: ð6:4Þ

The results of maximum absolute errors for this problem are listed in Table 2.

6.2. Linear problems

Problem 6.3

D6yðxÞ þ xy ¼ �ð24þ 11xþ x3Þ expðxÞ; 0 6 x 6 1 ð6:5Þ
yð0Þ ¼ 0 ¼ yð1Þ; D2yð0Þ ¼ 0; D2yð1Þ ¼ �4e; D4yð0Þ ¼ �8; D4yð1Þ ¼ �16e: ð6:6Þ
The analytical solution of the above differential equation is
yðxÞ ¼ xð1� xÞ expðxÞ: ð6:7Þ
The maximum errors (in absolute value) for different orders are shown in Table 4.
Problem 6.4

D6yðxÞ þ yðxÞ ¼ 6½2x cosðxÞ þ 5 sinðxÞ�; �1 6 x 6 1 ð6:8Þ
yð�1Þ ¼ 0 ¼ yð1Þ; D2yð�1Þ ¼ �4 cosð�1Þ þ 2 sinð�1Þ;D2yð1Þ ¼ 4 cosð1Þ þ 2 sinð1Þ; D4yð�1Þ ¼ 8 cosð�1Þ � 12 sinð1Þ;D4yð1Þ ¼ �8 cosð1Þ � 12 sinð1Þ: ð6:9Þ

The analytical solution of the above differential equation is

yðxÞ ¼ ðx2 � 1Þ sinðxÞ: ð6:10Þ
The maximum errors (in absolute value) are listed in Table 5.
The interval 0 6 x 6 1 was divided into n equal subintervals each of width h = 2�m with m = 3–5 so thatn = 7, 15, 31 respectively. The value of ky � Yk, where Y is numerical solution, was computed for each valueof n. The results for all second-, fourth-, and sixth-order methods are given in Tables 1–5. In Table 1 the newmethod (2.18) is applied to Problem 6.1 and the results are compared with Wazwaz [15], where as in Table 2the new method is applied to Problem 6.2 and the results are compared with Boutayeb and Twizell [5]. The


Our sixth-order method (3.3) [5]

4.55 · 10�7 2.41 · 10�7

5.96 · 10�9 7.56 · 10�10

2.68 · 10�11 2.25 · 10�11


Second-order method (3.1) Fourth-order method (3.2) Sixth-order method (3.3)

2.3 · 10�3 3.0 · 10�3 4.55 · 10�7

3.4 · 10�4 1.38 · 10�4 5.96 · 10�9

1.54 · 10�4 4.30 · 10�6 2.68 · 10�11

Table 4Maximum absolute errors corresponding to Problem (6.3)

N Second-order method (3.1) Fourth-order method (3.2) Sixth-order method (3.3)

7 2.99 · 10�2 2.39 · 10�4 1.15 · 10�9

15 7.00 · 10�3 3.43 · 10�6 3.95 · 10�12

31 1.80 · 10�3 7.34 · 10�8 4.41 · 10�11

Table 5Maximum absolute errors corresponding to Problem (6.4)

N Second-order method (3.1) Fourth-order method (3.2) Sixth-order method (3.3)

7 1.23 · 10�2 6.97 · 10�4 1.78 · 10�8

15 2.80 · 10�3 3.60 · 10�5 1.37 · 10�10

31 1.6 · 10�3 7.44 · 10�7 9.45 · 10�11

Table 6Maximum absolute errors corresponding to Problem (6.4) and results of [2]

n [2] Sixth-order method (3.3)

8 1.04 · 10�4 9.2 · 10�9

16 6.58 · 10�6 8.45 · 10�11

32 6.86 · 10�7 2.75 · 10�11

64 1.51 · 10�7 1.97 · 10�9


new methods of different orders are applied to Problems 6.2–6.4 and the results are shown in Tables 3–5respectively. Performance of the new method is better than Wazwaz [15] and it is comparable in case ofBoutayeb and Twizell [5]. Akram and Siddiqi [2] solved Problem 6.4 with the same solution but differentboundary conditions. Their results are reported in Table 6. This table shows improved performance of ourmethod.

7. Conclusion

Non-polynomial spline functions are used to develop a class of numerical methods for approximate solu-tion of sixth-order linear and non-linear boundary-value problems, with two-point boundary conditions. Sec-ond-, Fourth- and Sixth- order convergence is obtained. It has been shown that the relative errors in absolutevalue confirm the theoretical convergence.

References

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non-polynomial splines approach to the solution of sixth-order boundary-value problems

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