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

Numerical solution,artificial boundary conditions

Non-polynomial splines approach to the solution

of 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),Pakistan b 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-point boundary conditions.Second-,fourth-and sixth-order convergence is obtained by using standard procedure.It is shown that 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.

Ó2007Elsevier Inc.All rights reserved.

Keywords:Sixth-order BVP;Finite-di?erence methods;Non-polynomial splines

1.Introduction

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

D 6y ðx Þ¼f ðx ;y Þ;a <x <b ;D d =d x ;

ð1:1Þsubject to the boundary conditions:y ða Þ¼A 0;

D 2y ða Þ¼A 2;

D 4

y ða Þ¼A 4;

y ðb Þ¼B 0;D 2y ðb Þ¼B 2;D 4y ðb Þ¼B 4;)

ð1:2Þwhere y (x )and f (x ,y )are continuous functions de?ned in the interval x 2½a ;b .It is assumed that f ðx ;y Þ2C 6½a ;b is real and that A i ;B i ;i ¼0;2;4,are ?nite 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 0096-3003/$-see front matter Ó2007Elsevier Inc.All rights reserved.doi:10.1016/j.amc.2007.04.093*Corresponding author.

E-mail addresses:sirajuet@ (Siraj-ul-Islam),tirmizi_giki@ (I.A.Tirmizi),fhaq2002@ (Fazal-i-Haq),Azam_giki@ (M.A.Khan).

Available online at

Applied Mathematics and Computation 195(2008)

270–284

Numerical solution,artificial boundary conditions

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 an in?nite horizontal layer of?uid is heated from below and is under the action of rotation,instability sets in. When this instability is as ordinary convection,the ordinary di?erential 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 solving such 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 for fourth-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?nite-di?erence methods of order two,four,six and eight for solving such problems.Siddiqi and Twizell[11]used sixth-degree splines,where spline values at the mid knots of the interpolation interval and the corresponding values of the even order derivatives were related through consistency relations.M.E Gamel et al.[8]used Sinc-Galerkin method for the solutions of sixth order boundary-value problems.Wazwaz[15]used decomposition and modi?ed domain decomposition methods to investigate solution of the sixth-order boundary-value problems.

The spline function proposed in this paper has the form T7¼Span f1;x;x2;x3;x4;x5;cos kx;sin kx g where k is the frequency of the trigonometric part of the splines function.Thus in each subinterval x i6x6x i+1,we have:

Span f1;x;x2;x3;x4;x5;cos kx;sin kx g;

Span f1;x;x2;x3;x4;x5;cosh kx;sinh kx g or

Span f1;x;x2;x3;x4;x5;x6;x7g;ðwhen k!0Þ:

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

T7¼Span f1;x;x2;x3;x4;x5;sinðkxÞ;cosðkxÞg¼Span

1;x;x2;x3;x4;x5;7!

k7

kxÀsinðkxÞÀðkxÞ3

6

þðkxÞ5

120

;

8!

k8

cosðkxÞÀ1þðkxÞ2ÀðkxÞ4þðkxÞ6

:

8

><

>:

9

>=

>;

ð1:3Þ

In Section2of the present paper the special sixth-order,boundary-value problem to be solved is transformed using non-spline polynomial techniques into linear or non-linear algebraic system.In Section3methods of di?erent orders are categorized.In Section4,properties of the inverse of coe?cient matrix are discussed.In Section5,convergence of the method is established and numerical results are reported in Section6.

2.Numerical method

To develop the spline approximation to the problem(1.1)and(1.2),the interval[a,b]is divided into n equal subintervals,using the grid points x i¼aþih;i¼0;1;...;n,where h¼bÀa.For each segment½x iÀ1;x i ,the polynomial P iÀ1/2(x)has the form:

P iÀ1=2ðxÞ¼a iÀ1=2sin kðxÀx iÀ1=2Þþb iÀ1=2cos kðxÀx iÀ1=2Þþc iÀ1=2ðxÀx iÀ1=2Þ5þd iÀ1=2ðxÀx iÀ1=2Þ4þe iÀ1=2ðxÀx iÀ1=2Þ3þf iÀ1=2ðxÀx iÀ1=2Þ2þg iÀ1=2ðxÀx iÀ1=2Þþr iÀ1=2;ð2:1Þwhere a iÀ1=2;b iÀ1=2;c iÀ1=2;d iÀ1=2;e iÀ1=2;f iÀ1=2;g iÀ1=2and r iÀ1/2are constants and k is free parameter.

Let y iÀ1/2be an approximation to y(x iÀ1/2),obtained by the segment P iÀ1/2(x)of the mixed splines function

passing through the pointsðx iÀ1=2;y

iÀ1=2Þandðx iþ1=2;y iþ1=2Þ.To determine the coe?cients in(2.1)at the com-

mon nodesðx iÀ1=2;y

iÀ1=2

Þ,we?rst de?ne:

P iÀ1=2ðx iÆjÞ¼y iÆj;P0

iÀ1=2ðx iÆjÞ¼Z iÆj;P00

iÀ1=2

ðx iÆjÞ¼M iÆj;PðviÞ

iÀ1=2

ðx iÆjÞ¼S iÆj;where j¼1=2:

Siraj-ul-Islam et al./Applied Mathematics and Computation195(2008)270–284271

Numerical solution,artificial boundary conditions

From algebraic manipulation we get the following expressions for coe?cients:

a iÀ1=2¼1

k6

ðS iÀ1=2cosðhÞÀS iþ1=2ÞcscðhÞ;

b iÀ1=2¼ÀS iÀ1=2

k6

;

c iÀ1=2¼ÀcscðhÞ

2h5k6

h2k2sinðhÞS iþ1=2Àh2S iÀ1=2k2sinðhÞþh2M iÀ1=2k6sinðhÞ

Àh2k6sinðhÞM iþ1=2À6hS iÀ1=2kþ6hk cosðhÞS iþ1=2À6hkS iÀ1=2cosðhÞ

þ6hkS iþ1=2þ6hZ iÀ1=2k6sinðhÞþ6hk6Z iþ1=2sinðhÞÀ12S iþ1=2sinðhÞ

þ12S iÀ1=2sinðhÞþ12y iÀ1=2k6sinðhÞÀ12y iþ1=2k6sinðhÞ

B B

B B

B@

1

C C

C C

C A;

d iÀ1=2¼cscðhÞ

2h4k6

2h2k2sinðhÞS iþ1=2À3h2S iÀ1=2k2sinðhÞþ3h2M iÀ1=2k6sinðhÞ

À2h2k6sinðhÞM iþ1=2À14hS iÀ1=2kþ14hk cosðhÞS iþ1=2À16hkS iÀ1=2cosðhÞþ16hkS iþ1=2þ16hZ iÀ1=2k6sinðhÞþ14hk6Z iþ1=2sinðhÞÀ30S iþ1=2sinðhÞ

þ30S iÀ1=2sinðhÞþ30y iÀ1=2k6sinðhÞÀ30y iþ1=2k6sinðhÞ

B B

B B

B@

1

C C

C C

C A;

e iÀ1=2¼ÀcscðhÞ

2h3k6

20y

iÀ1=2

k6sinðhÞÀ20y

iþ1=2

k6sinðhÞÀh2k6sinðhÞM iþ1=2

þh2k2sinðhÞS iþ1=2þ12hkS iþ1=2þ8hk cosðhÞS iþ1=2þ8hk6Z iþ1=2sinðhÞ

À8hS iÀ1=2kþ3h2M iÀ1=2k6sinðhÞÀ3h2S iÀ1=2k2sinðhÞÀ12hkS iÀ1=2cosðhÞ

þ12hZ iÀ1=2k6sinðhÞÀ20S iþ1=2sinðhÞþ20S iÀ1=2sinðhÞ

B B

B B

B@

1

C C

C C

C A;

f iÀ1=2¼1

2k

ðÀS iÀ1=2þM iÀ1=2k4Þ;

g iÀ1=2¼cscðhÞ

k5

ðÀS iÀ1=2cosðhÞþS iþ1=2þZ iÀ1=2k5sinðhÞÞ;

r iÀ1=2¼S iÀ1=2þy iÀ1=2k6

k6;whereby h¼kh:

9

>>>

>>>

>>>

>>>

>>>

>>>

>>>

>>>

>>>

>>>

>>>

>>>

>>>

>>>

>>>

>=

>>>

>>>

>>>

>>>

>>>

>>>

>>>

>>>

>>>

>>>

>>>

>>>

>>>

>>>

>>>

>;

ð2:2Þ

Using the continuity condition of the third,fourth and?fth derivatives atðx iÀ1=2;y

iÀ1=2Þ,i.e.,

PðnÞiÀ1=2ðx iÀ1=2Þ¼PðnÞ

iþ1=2

ðx iÀ1=2Þwhere n¼3;4;5,we obtain:

À1

k3

sinðhÞS iÀ1=2À72

h2k5

cotðhÞS iþ1=2þ24

h2k5

cotðhÞS iÀ1=2

þ24

h2k5

cotðhÞS iþ3=2À1

k3

cosðhÞcotðhÞS iÀ1=2þ2

k3

cotðhÞS iþ1=2

À18

hk4

S iþ1=2þ3

hk4

S iÀ1=2À3

h

M iÀ1=2þ18

h

M iþ1=2þ36

h2k5

cscðhÞS iÀ1=2

À48

h2k5

cscðhÞS iþ1=2À24

h2

Z iÀ1=2þ120

h3k6

S iþ1=2À60

h3k6

S iÀ1=2À60

h3

y

iÀ1=2

þ120

h3

y

iþ1=2

þ36

h2k5

cscðhÞSþ3=2À60

h3

y

iþ3=2

À3M iþ3=2þ3

hk4

S iþ3=2þ24

h2

Z iþ3=2À60

h3k6

S iþ3=2

À1

k3

cscðhÞS iþ3=2

9

>>>

>>>

>>>

>>=

>>>

>>>

>>>

>>;

¼0;ð2:3Þ

À168

h3k5

cotðhÞS iþ3=2þ168

h3k5

168cotðhÞS iÀ1=2À24

h2k4

S iþ3=2þ24

h2

M iþ3=2

À192

h3k5

cscðhÞS iþ3=2À168

h3

Z iþ3=2þ360

h4k6

S iþ3=2þ360

h4

y

iþ3=2

þ24

h2k4

S iÀ1=2À24

h2

M iÀ1=2

þ192

h3k5

cscðhÞS iÀ1=2À168

h3

Z iÀ1=2À384

h3

Z iþ1=2À360

h4k6

S iÀ1=2À360

h4

y

iÀ1=2

9

>>=

>>;¼0;ð2:4Þ

1

k

cscðhÞS iÀ1=2þ1

k

cosðhÞcotðhÞS iÀ1=2À2

k

cotðhÞS iþ1=2À120

h3k4

S iþ1=2

þ60

h3k4

S iÀ1=2À60

h3

M iÀ1=2þ120

h3

M iþ1=2þ360

h4k5

cscðhÞS iÀ1=2À720

h4k5

cscðhÞS iþ1=2

À360

h4

Z iÀ1=2þ1440

h5k6

S iþ1=2À720

h5k6

S iÀ1=2À720

k5

y

iÀ1=2

þ1440

h5

y

iþ1=2

þ60

h3k4

S iþ3=2À60

h3

M iþ3=2þ360

h4k5

cscðhÞS iþ3=2þ360

h4

Z iþ3=2À720

h5k6

S iþ3=2

À720

h5

Y iþ3=2À720

h4k5

cotðhÞS iþ1=2þ360

h4k5

cotðhÞS iÀ1=2

þ360

h4k5

cotðhÞS iþ3=2þ1

k

cscðhÞS iþ3=2

9

>>>

>>>

>>>

>>=

>>>

>>>

>>>

>>;

¼0;ð2:5Þ

272Siraj-ul-Islam et al./Applied Mathematics and Computation195(2008)270–284

Numerical solution,artificial boundary conditions

Replacing i by iþ1;iÀ1;iþ2;iÀ2,in Eqs.(2.3)–(2.5)we get the following equations:

À1

k3sinðhÞS iþ1=2À72

h2k5

cotðhÞS iþ3=2þ24

h2k5

cotðhÞS iþ1=2

þ24

h2k5cotðhÞS iþ5=2À1

k3

cosðhÞcotðhÞS iþ1=2þ2

k3

cotðhÞS iþ3=2

À18

hk4S iþ3=2þ3

hk4

S iþ1=2À3M iþ1=2þ18M iþ3=2þ36

h2k5

cscðhÞS iþ1=2

À48

h2k5cscðhÞS iþ3=2À24

h2

Z iþ1=2þ120

h3k6

S iþ3=2À60

h3k6

S iþ1=2À60

h3

y

iþ1=2

þ120

h3

y

iþ3=2

þ36

h2k5cscðhÞS iþ5=2À60

h3

y

iþ5=2

À3M iþ5=2þ3

hk4

S iþ5=2þ24

h2

Z iþ5=2À60

h3k6

S iþ5=2

À1

k3cscðhÞS iþ5=2

9

>>>

>>>

>>>

>>>

=

>>>

>>>

>>>

>>>

;

¼0;ð2:6Þ

À1

k3sinðhÞS iÀ3=2À72

h2k5

cotðhÞS iÀ1=2þ24

h2k5

cotðhÞS iÀ3=2

þ24

h2k5cotðhÞS iþ1=2À1

k3

cosðhÞcotðhÞS iÀ3=2þ2

k3

cotðhÞS iÀ1=2

À18

hk4S iÀ1=2þ3

hk4

S iÀ3=2À3M iÀ3=2þ18M iÀ1=2þ36

h2k5

cscðhÞS iÀ3=2À48

h2k5

cscðhÞS iÀ1=2

À24

h2Z iÀ3=2þ120

h3k6

S iÀ1=2À60

h3k6

S iÀ3=2À60

h3

y

iÀ3=2

þ120

h3

y

iÀ1=2

þ36

h2k5

cscðhÞS iþ1=2

À60

h3y

iþ1=2

À3

h

M iþ1=2þ3

hk4

S iþ1=2þ24

h2

Z iþ1=2À60

h3k6

S iþ1=2À1

k3

cscðhÞS iþ1=2

9

>>>

>>>

>>=

>>>

>>>

>>;

¼0;ð2:7Þ

À1

k3sinðhÞS iþ3=2À72

h2k5

cotðhÞS iþ5=2þ24

h2k5

cotðhÞS iþ3=2

þ24

h2k5cotðhÞS iþ7=2À1

k3

cosðhÞcotðhÞS iþ3=2þ2

k3

cotðhÞS iþ5=2

À18

hk4S iþ5=2þ3

hk4

S iþ3=2À3M iþ3=2þ18M iþ5=2þ36

h2k5

cscðhÞS iþ3=2À48

h2k5

cscðhÞS iþ5=2

À24

h2Z iþ3=2þ120

h3k6

S iþ5=2À60

h3k6

S iþ3=2À60

h3

y

iþ3=2

þ120

h3

y

iþ5=2

þ36

h2k5

cscðhÞS iþ7=2

À60

h3y

iþ7=2

À3M iþ7=2þ3

hk4

S iþ7=2þ24

h2

Z iþ7=2À60

h3k6

S iþ7=2À1

k3

cscðhÞS iþ7=2

9

>>>

>>>

>>=

>>>

>>>

>>;

¼0;ð2:8Þ

À1

k3sinðhÞS iÀ5=2À72

h2k5

cotðhÞS iÀ3=2þ24

h2k5

cotðhÞS iÀ5=2

þ24

h2k5cotðhÞS iÀ1=2À1

k3

cosðhÞcotðhÞS iÀ5=2þ2

k3

cotðhÞS iÀ3=2

À18

hk4S iÀ3=2þ3

hk4

S iÀ5=2À3

h

M iÀ5=2þ18

h

M iÀ3=2þ36

h2k5

cscðhÞS iÀ5=2À48

h2k5

cscðhÞS iÀ3=2

À24

h2Z iÀ5=2þ120

h3k6

S iÀ3=2À60

h3k6

S iÀ5=2À60

h3

y

iÀ5=2

þ120

h3

y

iÀ3=2

þ36

h2k5

cscðhÞS iÀ1=2

À60

h3y

iÀ1=2

À3

h

M iÀ1=2þ3

hk4

S iÀ1=2þ24

h2

Z iÀ1=2À60

h3k6

S iÀ1=2À1

k3

cscðhÞS iÀ1=2

9

>>>

>>>

>>=

>>>

>>>

>>;

¼0;ð2:9Þ

À168

h3k5cotðhÞS iþ5=2þ168

h3k5

cotðhÞS iþ1=2À24

h2k4

S iþ5=2þ24

h2

M iþ5=2

À192

h3k5cscðhÞS iþ5=2À168

h3

Z iþ5=2þ360

h4k6

S iþ5=2þ360

h4

y

iþ5=2

þ24

h2k4

S iþ1=2À24

h2

M iþ1=2

þ192

h3k5cscðhÞS iþ1=2À168

h3

Z iþ1=2À384

h3

Z iþ3=2À360

h4k6

S iþ1=2À360

h4

y

iþ1=2

9

>>=

>>;¼0;ð2:10Þ

À168

h3k5cotðhÞS iþ1=2þ168

h3k5

cotðhÞS iÀ3=2À24

h2k4

S iþ1=2þ24

h2

M iþ1=2

À192

h3k5cscðhÞS iþ1=2À168

h3

Z iþ1=2þ360

h4k6

S iþ1=2þ360

h4

y

iþ1=2

þ24

h2k4

S iÀ3=2

À24

h2M iÀ3=2þ192

h3k5

cscðhÞS iÀ3=2À168

h3

Z iÀ3=2À384

h3

Z iÀ1=2À360

h4k6

S iÀ3=2À360

h4

y

iÀ3=2

9

>>=

>>;¼0;ð2:11Þ

À168

h3k5cotðhÞS iþ7=2þ168

h3k5

cotðhÞS iþ3=2À24

h2k4

S iþ7=2þ24

h2

M iþ7=2

À192

h3k5cscðhÞS iþ7=2À168

h3

Z iþ7=2þ360

h4k6

S iþ7=2þ360

h4

y

iþ7=2

þ24

h2k4

S iþ3=2

À24

h M iþ3=2þ192

h k

cscðhÞS iþ3=2À168

h

Z iþ3=2À384

h

Z iþ5=2À360

h k

S iþ3=2À360

h

y

iþ3=2

9

>>=

>>;¼0;ð2:12Þ

À168

h3k5cotðhÞS iÀ1=2þ168

h3k5

cotðhÞS iÀ5=2À24

h2k4

S iÀ1=2þ24

h2

M iÀ1=2

À192

h3k5cscðhÞS iÀ1=2À168

h3

Z iÀ1=2þ360

h4k6

S iÀ1=2þ360

h4

y

iÀ1=2

þ24

h2k4

S iÀ5=2

À24

h2M iÀ5=2þ192

h3k5

cscðhÞS iÀ5=2À168

h3

Z iÀ5=2À384

h3

Z iÀ3=2À360

h4k6

S iÀ5=2À360

h4

y

iÀ5=2

9

>>=

>>;¼0;ð2:13ÞSiraj-ul-Islam et al./Applied Mathematics and Computation195(2008)270–284273

Numerical solution,artificial boundary conditions

1 k sinðhÞS iþ1=2þ1

k

cosðhÞcotðhÞS iþ1=2À2

k

cotðhÞS iþ3=2À120

h3k4

S iþ3=2

þ60

h3k4

S iþ1=2À60

h3

M iþ1=2þ120

h3

M iþ3=2þ360

h4k5

cscðhÞS iþ1=2À720

h4k5

cscðhÞS iþ3=2

À360

h4

Z iþ1=2þ1440

h5k6

S iþ3=2À720

h5k6

S iþ1=2À720

h5

y

iþ1=2

þ1440

h5

y

iþ3=2

þ60

h3k4

S iþ5=2

À60

h3

M iþ5=2þ360

h4k5

cscðhÞS iþ5=2þ360

h4

Z iþ5=2À720

h5k6

S iþ5=2À720

h5

y

iþ5=2

À720

h4k5

cotðhÞS iþ3=2þ360

h4k5

cotðhÞS iþ1=2þ360

h4k5

cotðhÞS iþ5=2þ1

k

cscðhÞS iþ5=2

9

>>>

>>>

>>>

=

>>>

>>>

>>>

;

¼0;ð2:14Þ

1sinðhÞS iÀ3=2þ1cosðhÞcotðhÞS iÀ3=2À2cotðhÞS iÀ1=2À120

h3k4

S iÀ1=2

þ60

h3k4S iÀ3=2À60

h3

M iÀ3=2þ120

h3

M iÀ1=2þ360

h4k5

cscðhÞS iÀ3=2À720

h4k5

cscðhÞS iÀ1=2

À360

h4Z iÀ3=2þ1440

h5k6

S iÀ1=2À720

h5k6

S iÀ3=2À720

h5

y

iÀ3=2

þ1440

h5

y

iÀ1=2

þ60

h3k4

S iþ1=2

À60

h3M iþ1=2þ360

h4k5

cscðhÞS iþ1=2þ360

h4

Z iþ1=2À720

h5k6

S iþ1=2À720

h5

y

iþ1=2

À720

h4k5cotðhÞS iÀ1=2þ360

h4k5

cotðhÞS iÀ3=2þ360

h4k5

cotðhÞS iþ1=2þ1cscðhÞS iþ1=2

9

>>>

>>>

>>>

=

>>>

>>>

>>>

;

¼0;ð2:15Þ

1sinðhÞS iþ3=2þ1cosðhÞcotðhÞS iþ3=2À2cotðhÞS iþ5=2À120

h3k4

S iþ5=2

þ60

h3k4S iþ3=2À60

h3

M iþ3=2þ120

h3

M iþ5=2þ360

h4k5

cscðhÞS iþ3=2À720

h4k5

cscðhÞS iþ5=2

À360

h4Z iþ3=2þ1440

h5k6

S iþ5=2À720

h5k6

S iþ3=2À720

h5

y

iþ3=2

þ1440

h5

y

iþ5=2

þ60

h3k4

S iþ7=2

À60

h3M iþ7=2þ360

h4k5

cscðhÞS iþ7=2þ360

h4

Z iþ7=2À720

h5k6

S iþ7=2À720

h5

y

iþ7=2

À720

h4k5cotðhÞS iþ5=2þ360

h4k5

cotðhÞS iþ3=2þ360

h4k5

cotðhÞS iþ7=2þ1

k

cscðhÞS iþ7=2

9

>>>

>>>

>>>

=

>>>

>>>

>>>

;

¼0;ð2:16Þ

1 k sinðhÞS iÀ5=2þ1

k

cosðhÞcotðhÞS iÀ5=2À2

k

cotðhÞS iÀ3=2À120

h3k4

S iÀ3=2

þ60

h3k4

S iÀ5=2À60

h3

M iÀ5=2þ120

h3

M iÀ3=2þ360

h4k5

cscðhÞS iÀ5=2À720

h4k5

cscðhÞS iÀ3=2

À360

h4

Z iÀ5=2þ1440

h5k6

S iÀ3=2À720

h5k6

S iÀ5=2À720

h5

y

iÀ5=2

þ1440

h5

y

iÀ3=2

þ60

h3k4

S iÀ1=2

À60

h3

M iÀ1=2þ360

h4k5

cscðhÞS iÀ1=2þ360

h4

Z iÀ1=2À720

h5k6

S iÀ1=2À720=ðh5Þy iÀ1=2

À720

h5k6

cotðhÞS iÀ3=2þ360

h4k5

cotðhÞS iÀ5=2þ360

h4k5

cotðhÞS iÀ1=2þ1cscðhÞS iÀ1=2

9

>>>

>>>

>>>

=

>>>

>>>

>>>

;

¼0;ð2:17Þ

For the elimination of M iÀ5=2;M iÀ3=2;M iÀ1=2;M iþ1=2;M iþ3=2;M iþ5=2;M iþ7=2;Z iÀ5=2;Z iÀ3=2;Z iÀ1=2;Z iþ1=2;Z iþ3=2;Z iþ5=2 and Z i+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: y

iÀ7=2

À6y iÀ5=2þ15y iÀ3=2À20y iÀ1=2þ15y iþ1=2À6y iþ3=2þy iþ5=2

¼h6f aðS iÀ7=2þS iþ5=2ÞþbðS iÀ5=2þS iþ3=2ÞþcðS iÀ3=2þS iþ1=2Þþd S iÀ1=2g;ð2:18Þwhere

a¼1

120

À120

h6

À20

h3sin h

þ1

h sin h

þ120

h5sin h

;

b¼1À40

h3sin h À2cos hþ720

h6

À480

h5sin h

À240cos h

h5sin h

þ26þ40cos h

h3sin h

;

c¼1

120

100

h3sin h

þ67

h sin h

þ960cos h

h5sin h

þ840

h5sin h

À1800

h6

À52cos h

h sin h

þ80cos h

h3sin h

;

d¼12400

h6À960

h5sin h

À80

h3sin h

À132cos hÀ240cos h

h3sin h

þ52À1440cos h

h5sin h

; for i¼4;5;6;...;nÀ4;where n>5:9

>>>

>>>

>>>

>>>

=

>>>

>>>

>>>

>>>

;

ð2:19Þ

Here lim h!0ða;b;c;dÞ¼1

5040ð1;120;1191;2416Þ,which is the polynomial case as mentioned in Eq.(1.3).

274Siraj-ul-Islam et al./Applied Mathematics and Computation195(2008)270–284

Numerical solution,artificial boundary conditions

The local truncation error t i;i¼4;5;6;...;nÀ4,associated with the scheme developed in(2.18)is,

t i¼

C6h6yð6ÞiþC7h7yð7ÞiþC8h8yð8ÞiþC9h9yð9ÞiþC10h10yð10Þ

i

þC11h11yð11Þ

i

þC12h12yð12Þ

i

þC13h13yð13Þ

i

þC14h14yð14Þ

i

þOðh15Þ;

()

ð2:20Þ

C6¼ÀðÀ1þ2aþ2bþ2cþdÞ;

C7¼ðÀ1þ2aþ2bþ2cþdÞ

2

;

C8¼ÀðÀ3þ74aþ34bþ10cþdÞ

8

;

C9¼ðÀ7þ218aþ98bþ26cþdÞ

48

;

C10¼ð121À5ð3026aþ706bþ82cþdÞÞ

1920

;

C11¼ðÀ77þ13682aþ2882bþ242cþdÞ

3840

;

C12¼ð6227À21ð133274aþ16354bþ730cþdÞÞ

967680

;

C13¼ðÀ3353þ3ð745418aþ75938bþ2186cþdÞÞ

1935360

;

C14¼ÀðÀ4681þ6155426aþ397186bþ6562cþdÞ

10321920

:

9

>>>

>>>

>>>

>>>

>>>

>>>

>>>

>>>

>>>

>>>

>>=

>>>

>>>

>>>

>>>

>>>

>>>

>>>

>>>

>>>

>>>

>>;

ð2:21Þ

Thus for di?erent choices of a,b,c,d in scheme(2.18),methods of di?erent order are obtained.

The relation(2.18)gives nÀ6linear algebraic equations in n unknowns y

iþ1=2;i¼0;1;...;nÀ1.We

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

20y

À35y1=2þ21y3=2À7y5=2þy7=2þs y9=2

¼a1h2S00

0þb1h4SðivÞ

þh6a1SðviÞ

1=2

þb1SðviÞ

3=2

þc1SðviÞ

5=2

þd1SðviÞ

7=2

þe1SðviÞ

9=2

þf1SðviÞ

11=2

;ð2:22Þ

À10y0þ21y1=2À21y3=2þ15y5=2À6y7=2þy9=2

¼a2h2S00

0þb2h4SðivÞ

þh6a2SðviÞ

1=2

þb2SðviÞ

3=2

þc2SðviÞ

5=2

þd2SðviÞ

7=2

þe2SðviÞ

9=2

þf2SðviÞ

11=2

;ð2:23Þ

2y

À7y1=2þ15y3=2À20y5=2þ15y7=2À6y9=2þy11=2

¼a3h2S00

0þb3h4SðivÞ

þh6a3SðviÞ

1=2

þb3SðviÞ

3=2

þc3SðviÞ

5=2

þd3SðviÞ

7=2

þe3SðviÞ

9=2

þf3SðviÞ

11=2

:ð2:24Þ

The remaining three equations at the other end can be obtained from(2.22)–(2.24)by writing them in reverse order.The constants a i;b i;c i;d i;e i;f i;a i;b i and s are parameters which must be chosen so that the local truncation errors of(2.22)–(2.24)are identical with(2.18).

3.Numerical methods of di?erent orders

3.1.Second-order methods

a¼1

46080;b¼722

46080

;c¼10543

46080

;d¼23548

46080

in the main Eq.(2.18)gives

C6¼C7¼0;C8¼1 24

:

Siraj-ul-Islam et al./Applied Mathematics and Computation195(2008)270–284275

Numerical solution,artificial boundary conditions

The values of the coe?cients for these methods are:

a1¼7

;b

1

¼À

77

;a1¼

13005

;b1¼

9821

;c1¼

721

;d1¼

1

;e1¼0;f1¼0;

a2¼À3

;b

2

¼À

25

;a2¼À

9821

;b2¼

23547

;c2¼

10543

;d2¼

722

;

e2¼1

cd

;f

2

¼0;a3¼À

1

4

;b

3

¼À

1

192

;a3¼

721

cd

;b3¼

10543

cd

;c3¼

23548

cd

;

d3¼10543

;e3¼

722

cd

;f

3

¼

1

cd

;s¼0;cd¼46080:ð3:1Þ

3.2.Fourth-order methods

a¼À7

20þ3c

5

þ2d

5

;b¼17

20

À8c

5

À9d

10

in the main Eq.(2.18)gives C6¼C7¼C8¼C9¼0;C10¼151

120

À2cÀ3d

2

for arbitrary constants c,d.

The values of the coe?cients of boundary equations are given by

a1¼7

2

;b1¼À

77

96

;a1¼

2685830

cd

;b1¼

2704842

cd

;c1¼À

166502

cd

;d1¼

50582

cd

;

e1¼0;f1¼0;a2¼À3

;b

2

¼À

25

;a2¼

12161085

;b2¼À

35332965

;

c2¼64014771

cd

;d2¼À

41166795

cd

;e2¼

10321920

cd

;f

2

¼0;a3¼À

1

4

;b3¼À

1

192

;

a3¼51608615

cd

;b3¼À

193535007

cd

;c3¼

273530281

cd

;d3¼À

141926257

cd

;

e3¼10321920

cd

;f

3

¼

10321920

cd

;s¼0;cd¼10321920:ð3:2Þ

3.3.Sixth-order methods

a¼11Àd;b¼À47þ3d and c¼151À3d in the main Eq.(2.18)gives C6¼C7¼C8¼C9¼C10¼C11¼0;C12¼dÀ4153for arbitrary constant d.

The values of the coe?cients for these methods are:

a1¼7

2

;b1¼À

77

96

;a1¼

119271901

cd

;b1¼

128562861

cd

;c1¼À

18967984

cd

;

d1¼11537036

cd

;e1¼À

3523149

cd

;f

1

¼

483175

cd

;a2¼À

3

4

;b2¼À

25

64

;

a2¼80655078

cd

;b2¼

277312338

cd

;c2¼

77419113

cd

;d2¼

19393863

cd

;e2¼À

5790447

cd

;

f 2¼

920775

cd

;a3¼À

1

4

;b

3

¼À

1

192

;a3¼

3815581

cd

;b3¼

100731801

cd

;c3¼

255355121

cd

;

d3¼100719251

cd

;e3¼

3839346

cd

;f

3

¼

5140

cd

;s¼1;cd¼464486400:ð3:3Þ

3.4.Eight-order methods

a¼1;b¼41;c¼2189;d¼4153in the main Eq.(2.18)gives C6=C7=C8=C9=C10=C11=0,C12= 0,C13=0,C1450.Boundary equations for eight-order method are not constructed,however these equations can be obtained following the procedure adapted in the previous cases.

276Siraj-ul-Islam et al./Applied Mathematics and Computation195(2008)270–284

Numerical solution,artificial boundary conditions

4.Properties of the coe?cient matrix A0

Consider the following seven-band matrix A0of order n:

A0¼

À3521a00

21À21b00

a T

b T M b TR a TR

00b RÀ2121

00a R21À35

2

66

66

66

64

3

77

77

77

75

;ð4:1Þ

where a¼½À7;1;0;...;0 ;b¼½15;À6;1;0;...;0 .Here T denotes the operation of transposition and for a row vector v¼ðv1;v2;...;v nÞ;v R¼ðv n;v nÀ1;...;v1Þ.Further,a,b are(nÀ4)dimensional row vectors.The matrix M=(m i,j)is a seven-band matrix of order(nÀ4)given by

m i;j¼

À20;i¼j¼1;2;...;nÀ4;

15;j iÀj j¼1;

À6;j iÀj j¼2;

1;j iÀj j¼3;

0;j iÀj j>3:

8

>>>

>>>

><

>>>

>>>

>:

ð4:2Þ

In this section we?nd matrix AÀ1

0and k AÀ1

k

1

.The matrix AÀ1

is likewise of the form

AÀ1

0¼

e f c g h

j k d l m

c T

d T N d TR c TR

m l d R k j

h g c R f e

2

66

66

66

64

3

77

77

77

75

;ð4:3Þ

where c=(c i),d=(d i)are(nÀ4)dimensional row vectors.T and R have the same meaning as de?ned above

and e,f,g,h,j,k,l,m are scalars.The matrix N=(n i,j)is of order(nÀ4).Using A0AÀ1

0¼I n,we get the fol-

lowing equations:

ðiÞac TÀ35eþ21j¼1;

ðiiÞac TRÀ35hþ21m¼0;

ðiiiÞM c Tþa T eþa TR hþb T jþb TR m¼0;

ðivÞbc Tþ21eÀ21j¼0;

ðvÞbc TRþ21hÀ21m¼0;

ðviÞbd Tþ21fÀ21k¼1;

ðviiÞbd TRþ21gÀ21l¼0;

ðviiiÞad TÀ35fþ21k¼0;

ðixÞad TRÀ35gþ21l¼0;

ðxÞM d Tþa T fþa TR gþb T kþb TR l¼0;

ðxiÞMNþa T cþa TR c Rþb T dþb TR d R¼I nÀ4:9

>>>

>>>

>>>

>>>

>>>

>>>

>>>

>>>

>=

>>>

>>>

>>>

>>>

>>>

>>>

>>>

>>>

>;

ð4:4Þ

Here0denotes a(nÀ4)dimensional null column vector.

Siraj-ul-Islam et al./Applied Mathematics and Computation195(2008)270–284277

Numerical solution,artificial boundary conditions

In order to?nd the unknowns in above equations,we need to use MÀ1¼ð~m i;jÞ.From[10]it is given by

~m i;j¼

ðnÀ3ÀiÞðnÀ2ÀiÞðnÀ1ÀiÞjðjþ1Þðjþ2Þ

240ðnþ1ÞnðnÀ1ÞðnÀ2ÞðnÀ3Þ

Â

2iðiþ2ÞðjÀ1Þðjþ3ÞðnÀ3Þðnþ1Þ

Àðiþ1Þðiþ2ÞðjÀ2ÞðjÀ1Þnðnþ1ÞÀiðiþ1Þðjþ3Þðjþ4ÞðnÀ3ÞðnÀ2Þ

!

;i P j;

ðnÀ3ÀjÞðnÀ2ÀjÞðnÀ1ÀjÞiðiþ1Þðiþ2Þ

240ðnþ1ÞnðnÀ1ÞðnÀ2ÞðnÀ3Þ

Â

2jðjþ2ÞðiÀ1Þðiþ3ÞðnÀ3Þðnþ1Þ

Àðjþ1Þðjþ2ÞðiÀ2ÞðiÀ1Þnðnþ1ÞÀjðjþ1Þðiþ3Þðiþ4ÞðnÀ3ÞðnÀ2Þ

!

;i6j:

8

>>>

>>>

>>>

>>>

<

>>>

>>>

>>>

>>>

:

ð4:5Þ

Algebraic manipulation of Eqs.(4.4((i)–(v))yields the following expressions

e¼À16n4þ40n2À11

2880n

;

j¼À16n4À40n2þ9

960n

;

h¼À14n4þ20n2þ11

2880n

;

m¼À14n4À20n2À9

960n

;

9

>>>

>>>

>>>

>>=

>>>

>>>

>>>

>>;

ð4:6Þ

and

c i¼À

ð2nÀ2iÀ3Þ

ð16iþ24Þn3þð16i2þ48iþ36Þn2

Àð24i3þ108i2þ112iþ6Þnþ6i4þ36i3þ56i2þ6iÀ9 &'

; i¼1;2;...;nÀ4:

8

>><

>>:ð4:7ÞEqs.(4.4((vi)–(x))yield the following results:

f¼À16n4À40n2þ9

960n

;

k¼À16n4À120n2þ160nÀ51

320n

;

g¼À14n4À20n2À9

960n

;

l¼À14n4À60n2þ51

320n

;

9

>>>

>>>

>>>

>>=

>>>

>>>

>>>

>>;

ð4:8Þ

and

d i¼À

ð2nÀ2iÀ3Þ

ð16iþ24Þn3þð16i2þ48iþ36Þn2

Àð24i3þ108i2þ192iþ126Þnþ6i4þ36i3þ96i2þ126iþ51 &'

960n

; i¼1;2;...;nÀ4:

8

>><

>>:ð4:9ÞFrom Eq.(4.4(xi)),we have

N¼MÀ1þU;ð4:10Þwhere

U¼ÀMÀ1ða T cþb T dþb TR d Rþa TR c RÞ:ð4:11Þ278Siraj-ul-Islam et al./Applied Mathematics and Computation195(2008)270–284

Numerical solution,artificial boundary conditions

Hence U =(u i ,j )is given by:

u i ;j ¼12880ðn À3Þðn À2Þn ðn 2À1ÞÀ12i 512j 5À30j 4ðn À3Þþ20j 3ðn 2À8n þ12Þþ30j 2ð2n 2À9n þ9ÞÀ2j ðn 4À5n 3À15n 2þ75n À60ÞÀ3ðn 4À5n 3þ5n 2þ5n À6Þ&'þ30i 4ðn À3Þ12j 5þj 4ð88À32n Þþ12j 3ð2n 2À14n þ19Þþ8j 2ð9n 2À35n þ31ÞÀ2j ð2n 4À7n 3À20n 2þ79n À54ÞÀ3ð2n 4À7n 3þ4n 2þ7n À6Þ8><>:9>=>;þ3ðn 4À5n 3þ5n 2þ5n À6Þ153þ12j 5þj 4ð90À60n ÞÀ480n þ360n 2À48n 4þ180j 2ð2n 2À5n þ3Þþ20j 3ð4n 2À18n þ15ÞÀ8j ð4n 4À75n 2þ135n À60Þ8>>><>>>:9>>>=>>>;þ60i 2ðn À3ÞÀ6j 5ð2n À3Þþ4j 4ð9n 2À35n þ31Þþ9ðn À1Þ2ð2n 3À5n 2Àn þ6ÞÀ6j 3ð5n 3À38n 2þ81n À51Þþj 2ðÀ6n 4À69n 3þ420n 2À679n þ338Þþj ð12n 5À64n 4þ41n 3þ238n 2À413n þ186Þ8>>><>>>:9>>>=>>>;À20i 312j 5ðn 2À8n þ12ÞÀ18j 4ð2n 3À20n 2þ61n À57Þþ18j 2ð5n 4À53n 3þ195n 2À294n þ153Þþ4j 3ð8n 4À103n 3þ490n 2À959n þ645Þþ3ðÀ4n 6þ38n 5À125n 4þ145n 3þ39n 2À183n þ90ÞÀ4j ð2n 6À19n 5þ49n 4þ67n 3À489n 2þ711n À315Þ8>>>>>><>>>>>>:9>>>>>>=>>>>>>;þ4i 6j 5ðn 4À5n 3À15n 2þ75n À60ÞÀ15j 4ð2n 5À13n 4þn 3þ139n 2À291n þ162Þþ20j 3ð2n 6À19n 5þ49n 4þ67n 3À489n 2þ711n À315Þþ15j 2ð12n 6À100n 5þ233n 4þ115n 3À1127n 2þ1425n À558ÞÀ64n 8À20n 7À55n 6þ530n 5À1134n 4þ600n 3þ825n 2À1110n þ360 À2j 8n 8À40n 7À200n 6þ1825n 5À4230n 4þ1650n 3þ6480n 2À8595n þ3150 :8>>>>>>>>>>>>><>>>>>>>>>>>>>:9>>>>>>>>>>>>>=>>>>>>>>>>>>>;26666666666666666666666666666666666666666666666666666666666643

777777777777777777777777777777777777777777777777777777777775ð4:12Þ

We summarize the results of this section in the following lemma.

Lemma 4.1.The matrix A 0is nonsingular with A À10<0and if A À10¼ð~

a i ;j Þ,then ~a i ;j are given by ((4.6)–(4.10)).In order to ?nd k A À

10k 1,we ?rst ?nd the sum of elements in the ith row of A À10.The sum of elements in the

ith row of M À1is given by,À1720i ði þ1Þði þ2Þðn Ài À1Þðn Ài À2Þðn Ài À3Þð4:13Þand the sum of the elements of i th row of U is:À1960ðn À4Þð4i þ6Þn 4þð16i þ24Þn 3Àð8i 3þ34i 2þ52i þ29Þn 2þð4i 4À12i 3À104i 2À192i À116Þn Èþ18i 4þ108i 3þ258i 2þ288i þ120É:ð4:14Þ

Let S i designate the sum of the elements in the ith row of A À10in modulus,then S i ¼P n j ¼1j ~a i ;j j ¼ÀP n j ¼1~

a i ;j is given by

S i ¼12880ð12i À6Þn 5þðÀ20i 3þ30i 2þ20i À15Þn 3þð12i 5À30i 4À40i 3þ90i 2þ16i À24Þn À4i 6Èþ12i 5þ20i 4À60i 3À16i 2þ48i É:ð4:15ÞSiraj-ul-Islam et al./Applied Mathematics and Computation 195(2008)270–284279

Numerical solution,artificial boundary conditions

Consider S i as a function of the real variable i.It turns out that S i is maximum for i¼1

2ðnþ1Þ.The in?nity

norm of AÀ1

must be restricted to an integral value of i and therefore

k AÀ1

0k

1

¼max

i

S i6Sðnþ1Þ=2:

Here equality will hold only if n is odd.We substitute i¼1ðnþ1Þinto(4.15)to get S(n+1)/2.Hence,

k AÀ1

0k

1

6S

ðnþ1Þ=2

¼

61n6þ175n4þ259n2þ225

46080

¼

61ðbÀaÞ6þ175h2ðbÀaÞ4þ259h4ðbÀaÞ2þ225h6

46080h6

¼OðhÀ6Þ:ð4:16Þ

5.Convergence

Clearly,the family of numerical methods is described by the Eqs.(2.18),boundary equations and the solu-

tion vector Y¼½y

1=2;y

3=2

;...;y

nÀ1=2

T,T denoting transpose,is obtained by solving a non-linear algebraic sys-

tem of order n which has the form

A0YÀh6B fðx;YÞÀC¼0:ð5:1ÞNow we investigate the error analysis of the seven-degree non-polynomial spline method described in Section 2.To do so we let,y=(y(x i+1/2)),Y=(y i+1/2),C=(c i),T=(t i),and E=(e i+1/2),be n-dimensional column vectors.Here e i+1/2=y(x i+1/2)Ày i+1/2is the discretization error for i¼4;...;nÀ4.Thus,we can write our method in the matrix form:

The matrices B and C are given by

B¼

a1b1c1d1e1f1

a2b2c2d2e2f2

a3b3c3d3e3f3

a b c d b a

..

..

.

..

.

..

.

.

..

..

.

..

.

..

.

.

..

..

.

..

.

..

.

.

a b d c b a

f3e3d3c3b3a3

f2e2d2c2b2a2

f1e1d1c1b1a1

2

66

66

66

66

66

66

66

66

66

66

64

3

77

77

77

77

77

77

77

77

77

77

75

:ð5:2Þ

The column vector C is given by

C¼

À20A0þa1h2A2þb1h4A4

10A0þa2h2A2þb2h4A4

À2A0þa3h2A2þb3h4A4

...

...

...

À2B0þa3h2B2þb3h4B4

10B0þa2h2B2þb2h4B4

À20B0þa1h2B2þb1h4B4

2

66

66

66

66

66

66

66

66

66

66

64

3

77

77

77

77

77

77

77

77

77

77

75

:ð5:3Þ

280Siraj-ul-Islam et al./Applied Mathematics and Computation195(2008)270–284

Numerical solution,artificial boundary conditions

The vector y¼½yðx1=2Þ;yðx3=2Þ;...;yðx nÀ1=2Þ T satis?es

A0yÀh6B fðx;yÞÀCÀT¼0;

where T¼½t1;t2;...;t n T is the vector of local truncation errors,and a conventional convergence analysis shows that the norm of the vector

E¼yÀY

satis?es:

k E k

16

ðbÀaÞ6

46080Àð61ðbÀaÞþ175h2ðbÀaÞþ259h4ðbÀaÞþ225h6ÞBÃFÃ

ÂC6h6V6þC7h7V7þC8h8V8þC9h9V9þC10h10V10þC11h11V11þC12h12V12þ...

ÈÉ

;ð5:4Þ

where V i¼max a6x6b j d i yðxÞ

i

j for i¼1;2;...,B*=k B k,and Fümax a6x6b j o f j.

The order of convergence of the numerical method is,thus p,if C p+6is the?rst non-vanishing constant on the right hand side of(2.20)provided

FÃ<

46080

ð61ðbÀaÞþ175h2ðbÀaÞþ259h4ðbÀaÞþ225h6Þ

:ð5:5Þ

6.Numerical results

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

6.1.Non-linear problems

Problem6.1

D6yðxÞ¼eÀx y2ðxÞ;0<x<1ð6:1Þsubject to the boundary conditions:

yð0Þ¼D2ð0Þ¼D4ð0Þ¼1;

yð1Þ¼D2ð1Þ¼D4ð1Þ¼e:

The theoretical solution for this problem is

yðxÞ¼e x:

The results of maximum absolute errors for this problem are listed in Table1.

Problem6.2

D6yðxÞ¼20exp½À36yðxÞ À40ð1þxÞÀ6;0<x<1ð6:2Þwith boundary conditions:

yð0Þ¼0;D2yð0Þ¼À1

6

;D4yð0Þ¼À1;yð1Þ¼

1

6

ln2;D2yð1Þ¼À

1

24

;D4yð1Þ¼À

1

16

ð6:3Þ

Table1

Maximum absolute errors corresponding to Problem(6.1)

n Our sixth-order method(3.3)[15]

10 3.577·10À13 1.299·10À4

Siraj-ul-Islam et al./Applied Mathematics and Computation195(2008)270–284281

Numerical solution,artificial boundary conditions

for which the theoretical solution is

yðxÞ¼1

6

lnð1þxÞ:ð6:4Þ

The results of maximum absolute errors for this problem are listed in Table2.

6.2.Linear problems

Problem6.3

D6yðxÞþxy¼Àð24þ11xþx3ÞexpðxÞ;06x61ð6:5Þwith boundary conditions:

yð0Þ¼0¼yð1Þ;D2yð0Þ¼0;D2yð1Þ¼À4e;D4yð0Þ¼À8;D4yð1Þ¼À16e:ð6:6ÞThe analytical solution of the above di?erential equation is

yðxÞ¼xð1ÀxÞexpðxÞ:ð6:7ÞThe maximum errors(in absolute value)for di?erent orders are shown in Table4.

Problem6.4

D6yðxÞþyðxÞ¼6½2x cosðxÞþ5sinðxÞ ;À16x61ð6:8Þwith boundary conditions:

yðÀ1Þ¼0¼yð1Þ;D2yðÀ1Þ¼À4cosðÀ1Þþ2sinðÀ1Þ;

D2yð1Þ¼4cosð1Þþ2sinð1Þ;D4yðÀ1Þ¼8cosðÀ1ÞÀ12sinð1Þ;

D4yð1Þ¼À8cosð1ÞÀ12sinð1Þ:ð6:9ÞThe analytical solution of the above di?erential equation is

yðxÞ¼ðx2À1ÞsinðxÞ:ð6:10ÞThe maximum errors(in absolute value)are listed in Table5.

The interval06x61was divided into n equal subintervals each of width h=2Àm with m=3–5so that n=7,15,31respectively.The value of k yÀY k,where Y is numerical solution,was computed for each value of n.The results for all second-,fourth-,and sixth-order methods are given in Tables1–5.In Table1the new method(2.18)is applied to Problem6.1and the results are compared with Wazwaz[15],where as in Table2 the new method is applied to Problem6.2and the results are compared with Boutayeb and Twizell[5].The Table2

Maximum absolute errors corresponding to Problem(6.2)

n Our sixth-order method(3.3)[5]

7 4.55·10À7 2.41·10À7 15 5.96·10À97.56·10À10 31 2.68·10À11 2.25·10À11 Table3

Maximum absolute errors corresponding to Problem(6.2)

N Second-order method(3.1)Fourth-order method(3.2)Sixth-order method(3.3) 7 2.3·10À3 3.0·10À3 4.55·10À7

15 3.4·10À4 1.38·10À4 5.96·10À9

31 1.54·10À4 4.30·10À6 2.68·10À11

282Siraj-ul-Islam et al./Applied Mathematics and Computation195(2008)270–284

Numerical solution,artificial boundary conditions

Siraj-ul-Islam et al./Applied Mathematics and Computation195(2008)270–284283 Table4

Maximum absolute errors corresponding to Problem(6.3)

N Second-order method(3.1)Fourth-order method(3.2)Sixth-order method(3.3) 7 2.99·10À2 2.39·10À4 1.15·10À9

157.00·10À3 3.43·10À6 3.95·10À12

31 1.80·10À37.34·10À8 4.41·10À11

Table5

Maximum absolute errors corresponding to Problem(6.4)

N Second-order method(3.1)Fourth-order method(3.2)Sixth-order method(3.3) 7 1.23·10À2 6.97·10À4 1.78·10À8

15 2.80·10À3 3.60·10À5 1.37·10À10

31 1.6·10À37.44·10À79.45·10À11

Table6

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

n[2]Sixth-order method(3.3) 8 1.04·10À49.2·10À9

16 6.58·10À68.45·10À11

32 6.86·10À7 2.75·10À11

64 1.51·10À7 1.97·10À9

new methods of di?erent orders are applied to Problems6.2–6.4and the results are shown in Tables3–5 respectively.Performance of the new method is better than Wazwaz[15]and it is comparable in case of Boutayeb and Twizell[5].Akram and Siddiqi[2]solved Problem6.4with the same solution but di?erent boundary conditions.Their results are reported in Table6.This table shows improved performance of our method.

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 absolute value con?rm the theoretical convergence.

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