Research ArticleDerivation of Diagonally Implicit Block BackwardDifferentiation Formulas for Solving Stiff Initial Value Problems

Iskandar Shah Mohd Zawawi1 Zarina Bibi Ibrahim12 and Khairil Iskandar Othman3

1Department of Mathematics Faculty of Science Universiti Putra Malaysia 43400 Serdang Selangor Malaysia2Institute for Mathematical Research Universiti Putra Malaysia 43400 Serdang Selangor Malaysia3Department of Mathematics Faculty of Computer and Mathematical Sciences Universiti Teknologi MARA40450 Shah Alam Selangor Malaysia

Correspondence should be addressed to Zarina Bibi Ibrahim zarinabbupmedumy

Received 8 October 2014 Revised 18 February 2015 Accepted 20 February 2015

Academic Editor Gen Qi Xu

Copyright copy 2015 Iskandar Shah Mohd Zawawi et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The diagonally implicit 2-point block backward differentiation formulas (DI2BBDF) of order two order three and order fourare derived for solving stiff initial value problems (IVPs) The stability properties of the derived methods are investigated Theimplementation of the method using Newton iteration is also discussed The performance of the proposed methods in terms ofmaximum error and computational time is compared with the fully implicit block backward differentiation formulas (FIBBDF)and fully implicit block extended backward differentiation formulas (FIBEBDF) The numerical results show that the proposedmethod outperformed both existing methods

1 Introduction

Many scientific and engineering problems which arise inreal-life applications are in the form of ordinary differentialequations (ODEs) where the analytic solution is unknownThe general form of first order ODEs is given in the followingform

1199101015840= 119891 (119909 119910) 119910 (119886) = 119910

0 119886 le 119909 le 119887 (1)

In the early 1950s Curtiss and Hirschfelder [1] realizedthat there is an important class of ODEs which is knownas stiff initial value problems (IVPs) There are variousdefinitions of stiffness given in the literature Generallystiff problems are problems where certain implicit methodsperform better than explicit ones For simplicity we choosethe definition of stiff problem given by Lambert [2]

Definition 1 The system of (1) is said to be stiff if

(1) Re(120582119905) lt 0 119905 = 1 2 119898

(2) max119905|Re(120582

119905)| ≫ min

119905|Re(120582

119905)| where 120582

119905are the

eigenvalues of the Jacobian matrix 119869 = (120597119891120597119910)

Much research has been done by the scientific communityon developing numerical methods which permit an approx-imate solution to (1) The most commonly used numericalmethod is block methodThis classical method is introducedby Milne [3] to compute previous 119896-blocks and calculate thecurrent block where each block contains 119903-pointThe 119896-block119903-point method is given by amatrix finite difference equationof the form

119896

sum

119895=0

119860119895119910119899+119895

= ℎ

119896

sum

119895=0

119861119895119891119899+119895 (2)

where119860119895and 119861

119895are properly chosen 119903times119903matrix coefficients

This method is extended by Shampine and Watts [4] withconvergence and stability properties of one step block implicitmethod Then Fatunla [5] proposed block 119903-point methodfollowed by Majid et al [6 7] with the 2-point blockmethods However the most widely used multistep methodfor solving stiff ODEs is block backward differentiationformulas (BBDF) This method has been claimed by Ibrahim[8] to be one of the suitable numerical methods for solvingstiff IVPs Furthermore Ibrahim et al [9 10] proposed the

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 179231 13 pageshttpdxdoiorg1011552015179231

2 Mathematical Problems in Engineering

fully implicit 119903-point block backward differentiation formulas(FIBBDF) The following equations represent the formulasof fully implicit 2-point block backward differentiation for-mulas of order three (FI2BBDF(3)) and fully implicit 3-point block backward differentiation formula of order three(FI3BBDF(3))

FI2BBDF(3) One has

119910119899+1

= minus1

3119910119899minus1

+ 2119910119899minus2

3119910119899+2

+ 2ℎ119891119899+1

119910119899+2

=2

11119910119899minus1

minus9

11119910119899+18

11119910119899+1

+6

11ℎ119891119899+2

(3)

FI3BBDF(3) One has

119910119899+1

=1

10119910119899minus2

minus3

4119910119899minus1

+ 3119910119899minus3

2119910119899+2

+3

20119910119899+3

+ 3ℎ119891119899+1

119910119899+2

= minus3

65119910119899minus2

+4

13119910119899minus1

minus12

13119910119899+24

13119910119899+1

minus12

65119910119899+3

+12

13ℎ119891119899+2

119910119899+3

=12

137119910119899minus2

minus75

137119910119899minus1

+200

137119910119899minus300

137119910119899+1

+300

65119910119899+2

+60

137ℎ119891119899+3

(4)

In a related study Ibrahim et al [11] plotted the stabilityregion of (3) Since all region in the left half plane is in thestability region the FI2BBDF(3) is A-stable and suitable forsolving stiff problems Nasir et al [12] extended the orderof formula (3) which is called fifth order 2-point BBDFfor solving first order stiff ODEs Recently Musa et al [1314] modified formulas (3) and (4) to compute more thanone solution value per step using extra future point Thismethod is called fully implicit block extended backwarddifferentiation formulas (FIBEBDF) The formulas of fullyimplicit 2-point block extended backward differentiationformula of order three (FI2BEBDF(3)) and fully implicit 3-point block extended backward differentiation formula oforder three (FI3BEBDF(3)) are given in the following forms

FI2BEBDF(3) One has

119910119899+1

=1

9119910119899minus1

minus 119910119899+17

9119910119899+2

minus 2ℎ119891119899+1

minus2

3ℎ119891119899+2

119910119899+2

=17

197119910119899minus1

minus99

197119910119899+279

197119910119899+1

+150

197ℎ119891119899+2

minus18

197ℎ119891119899+3

(5)

FI3BEBDF(3) One has

119910119899+1

= minus1

80119910119899minus2

+1

80119910119899minus1

minus3

4119910119899+25

16119910119899+2

+3

40119910119899+3

minus3

2ℎ119891119899+1

minus3

4ℎ119891119899+2

119910119899+2

= minus3

25119910119899minus2

+ 119910119899minus1

minus 4119910119899+ 12119910

119899+2

minus197

25119910119899+3

+ 12ℎ119891119899+2

+12

5ℎ119891119899+3

119910119899+3

=394

14919119910119899minus2

minus2925

14919119910119899minus1

+9600

14919119910119899minus18700

14919119910119899+1

+26550

14919119910119899+2

+8820

14919ℎ119891119899+3

minus600

14919ℎ119891119899+4

(6)

Numerical results in the literature showed that theFIBEBDF performed better as compared to FIBBDF interms of accuracy Unfortunately the execution time of theFIBEBDF is slower than FIBBDF This is due to the factthat FIBEBDF has an extra future point which requires morecomputation time In order to gain an efficient numericalapproximation in terms of accuracy and computational timethe diagonally implicitmethodmust be consideredThe studyof diagonally implicit for multistep method attracted severalresearchers such as Alexander [15] Ababneh et al [16] andIsmail et al [17] However Lambert [2] stated that there issome confusion over nomenclature to identify the diagonallyimplicit Some authors use the term of diagonally implicit todescribe any semi-implicit method Therefore the definitionof diagonally implicit block method is given by Majid andSuleiman [18] as follows

Definition 2 Method (2) is defined to be diagonally implicitif the coefficients of the upper-diagonal entries are zero

From this motivation we established Definition 2 byintroducing the definition of diagonally implicit 2-pointBBDF method as follows

Definition 3 We consider that 11988611 11988612 11988621 and 119886

22are coeffi-

cients of 119910119899+1

and 119910119899+2

in the matrix form

[1198861111988612

1198862111988622

][119910119899+1

119910119899+2

] (7)

Method (2) is defined to be diagonally implicit if 11988612is zero

whereas 11988611and 11988622

are equal

Therefore the main purpose of this paper is to developa new diagonally implicit multistep method for solving stiffODEs This paper is organized as follows in Section 2 thediagonally implicit two-point block backward differentiationformulas (DI2BBDF) will be derived Next the stabilityproperties of the derived methods are analyzed in Section 3Section 4 discusses the implementation of the methods usingNewton iteration Standard test problems are selected in

Mathematical Problems in Engineering 3

Section 5 whereas the performance of the proposed methodis shown in Section 6 Finally in Section 7 some conclusionsare given

2 Formulation of the Method

In this section we will derive the DI2BBDF of order twoorder three and order four with constant step size to computethe approximated solutions at 119910

119899+1and 119910

119899+2concurrently

Contrary to the fully implicit method that has been proposedby Ibrahim et al [11] the first point of diagonally implicitformula has one less interpolating point The derivationusing polynomial 119875

119896(119909) of degree 119896 in terms of Lagrange

polynomial is defined as follows

119875119896(119909) =

119896

sum

119895=0

119871119896119895(119909) 119891 (119909

119899+1minus119895) (8)

where

119871119896119895(119909) =

119896

prod

119894=0

119894 =119895

(119909 minus 119909119899+1minus119894

)

(119909119899+1minus119895

minus 119909119899+1minus119894

)(9)

for each 119895 = 0 1 119896

21 Derivation of DI2BBDF(2) DI2BBDF(2) will computetwo approximated solutions 119910

119899+1and 119910

119899+2simultaneously in

each block using two back values This formula is derivedusing interpolating points 119909

119899minus1 119909

119899+1to obtain the first

formula 119910119899+1

of DI2BBDF(2)

119875 (119909) =(119909 minus 119909

119899) (119909 minus 119909

119899+1)

(119909119899minus1

minus 119909119899) (119909119899minus1

minus 119909119899+1)119910119899minus1

+(119909 minus 119909

119899minus1) (119909 minus 119909

119899+1)

(119909119899minus 119909119899minus1) (119909119899minus 119909119899+1)119910119899

+(119909 minus 119909

119899minus1) (119909 minus 119909

119899)

(119909119899+1

minus 119909119899minus1) (119909119899+1

minus 119909119899)119910119899+1

(10)

Replacing 119909 = 119904ℎ + 119909119899+1

into (10) yields

119875 (119909119899+1

+ 119904ℎ) =(119904ℎ + ℎ) (119904ℎ)

(minusℎ) (minus2ℎ)119910119899minus1

+(2ℎ + 119904ℎ) (119904ℎ)

(ℎ) (minusℎ)119910119899

+(2ℎ + 119904ℎ) (ℎ + 119904ℎ)

(2ℎ) (ℎ)119910119899+1

(11)

Equation (11) is differentiated once with respect to 119904 at thepoint 119909 = 119909

119899+1 Evaluating 119904 = 0 gives

1198751015840(119909119899+1) =

1

2119910119899minus1

minus 2119910119899+3

2119910119899+1 (12)

The same technique is applied for the second point 119910119899+2

ofDI2BBDF(2) This formula is derived using 119909

119899minus1 119909

119899+2as

the interpolating points and produces

119875 (119909) =(119909 minus 119909

119899) (119909 minus 119909

119899+1) (119909 minus 119909

119899+2)

(119909119899minus1

minus 119909119899) (119909119899minus1

minus 119909119899+1) (119909119899minus1

minus 119909119899+2)119910119899minus1

+(119909 minus 119909

119899minus1) (119909 minus 119909

119899+1) (119909 minus 119909

119899+2)

(119909119899minus 119909119899minus1) (119909119899minus 119909119899+1) (119909119899minus 119909119899+2)119910119899

+(119909 minus 119909

119899minus1) (119909 minus 119909

119899) (119909 minus 119909

119899+2)

(119909119899+1

minus 119909119899minus1) (119909119899+1

minus 119909119899) (119909119899+1

minus 119909119899+2)119910119899+1

+(119909 minus 119909

119899minus1) (119909 minus 119909

119899) (119909 minus 119909

119899+1)

(119909119899+2

minus 119909119899minus1) (119909119899+2

minus 119909119899) (119909119899+2

minus 119909119899+1)119910119899+2

(13)

Substituting 119909 = 119904ℎ + 119909119899+2

into (13) yields

119875 (119909119899+2

+ 119904ℎ)

=(119904ℎ + 2ℎ) (119904ℎ + ℎ) (119904ℎ)

(minusℎ) (minus2ℎ) (minus3ℎ)119910119899minus1

+(119904ℎ + 3ℎ) (119904ℎ + ℎ) (119904ℎ)

(ℎ) (minusℎ) (minus2ℎ)119910119899

+(119904ℎ + 3ℎ) (119904ℎ + 2ℎ) (119904ℎ)

(2ℎ) (ℎ) (minusℎ)119910119899+1

+(119904ℎ + 3ℎ) (119904ℎ + 2ℎ) (119904ℎ + ℎ)

(3ℎ) (2ℎ) (ℎ)119910119899+2

(14)

Differentiating (14) with respect to 119904 at the point 119909 = 119909119899+2

gives

1198751015840(119909119899+2) = minus

1

3119910119899minus1

+3

2119910119899minus 3119910119899+1

+11

6119910119899+2 (15)

Considering ℎ119891119899+1119899+2

= 1198751015840(119909119899+1119899+2

) the corrector formulaof DI2BBDF(2) is given as follows

119910119899+1

= minus1

3119910119899minus1

+4

3119910119899+2

3ℎ119891119899+1

119910119899+2

=2

11119910119899minus1

minus9

11119910119899+18

11119910119899+1

+6

11ℎ119891119899+2

(16)

The order of the method is distinguished by the number ofback values contained in the formulas Adopting a similarapproach as the derivation of DI2BBDF of order two wewill construct the DI2BBDF of orders three and four withdifferent number of interpolating points

4 Mathematical Problems in Engineering

22 Derivation of DI2BBDF(3) The first point 119910119899+1

ofDI2BBDF(3) is derived using interpolating points 119909

119899minus2

119909119899+1

and we have

119875 (119909) =(119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus1)

(119909119899minus2

minus 119909119899+1) (119909119899minus2

minus 119909119899) (119909119899minus2

minus 119909119899minus1)119910119899minus2

+(119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus2)

(119909119899minus1

minus 119909119899+1) (119909119899minus1

minus 119909119899) (119909119899minus1

minus 119909119899minus2)119910119899minus1

+(119909 minus 119909

119899+1) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2)

(119909119899minus 119909119899+1) (119909119899minus 119909119899minus1) (119909119899minus 119909119899minus2)119910119899

+(119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2)

(119909119899+1

minus 119909119899) (119909119899+1

minus 119909119899minus1) (119909119899+1

minus 119909119899minus2)119910119899+1

(17)

Substituting 119909 = 119904ℎ + 119909119899+1

into (17) gives

119875 (119904ℎ + 119909119899+1)

=(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ)

(minus3ℎ) (minus2ℎ) (minusℎ)119910119899minus2

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 3ℎ)

(minus2ℎ) (minusℎ) (ℎ)119910119899minus1

+(119904ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ)

(minusℎ) (ℎ) (2ℎ)119910119899

+(119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ)

(ℎ) (2ℎ) (3ℎ)119910119899+1

(18)

Equation (18) is differentiated once with respect to 119904 at thepoint 119909 = 119909

119899+1 Substituting 119904 = 0 will obtain

1198751015840(119909119899+1) =

11

6119910119899+1

minus1

3119910119899minus2

+3

2119910119899minus1

minus 3119910119899 (19)

The derivation process continues for second point 119910119899+2

ofDI2BBDF(3) using 119909

119899minus3 119909

119899+1as the interpolating points

We have

119875 (119909) =(119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus1)

(119909119899minus2

minus 119909119899+1) (119909119899minus2

minus 119909119899) (119909119899minus2

minus 119909119899minus1)119910119899minus2

+(119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus2)

(119909119899minus1

minus 119909119899+1) (119909119899minus1

minus 119909119899) (119909119899minus1

minus 119909119899minus2)119910119899minus1

+(119909 minus 119909

119899+1) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2)

(119909119899minus 119909119899+1) (119909119899minus 119909119899minus1) (119909119899minus 119909119899minus2)119910119899

+(119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2)

(119909119899+1

minus 119909119899) (119909119899+1

minus 119909119899minus1) (119909119899+1

minus 119909119899minus2)119910119899+1

(20)

Replacing 119909 = 119904ℎ + 119909119899+2

into (20) produces

119875 (119904ℎ + 119909119899+2)

=(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ)

(minus4ℎ) (minus3ℎ) (minus2ℎ) (minusℎ)119910119899minus2

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 3ℎ) (119904ℎ + 4ℎ)

(minus3ℎ) (minus2ℎ) (minusℎ) (ℎ)119910119899minus1

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 4ℎ)

(minus2ℎ) (minusℎ) (ℎ) (2ℎ)119910119899

+(119904ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 4ℎ)

(minusℎ) (ℎ) (2ℎ) (3ℎ)119910119899+1

+(119904ℎ + ℎℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 4ℎ)

(ℎ) (2ℎ) (3ℎ) (4ℎ)119910119899+2

(21)

The resulting polynomial above is differentiated once withrespect to 119904 at the point 119909 = 119909

119899+2 Substituting 119904 = 0 will

give

1198751015840(119909119899+2) =

25

12119910119899+2

+1

4119910119899minus2

minus4

3119910119899minus1

+ 3119910119899minus 4119910119899+1 (22)

Considering ℎ119891119899+1119899+2

= ℎ1198751015840(119909119899+1119899+2

) the corrector formulaof DI2BBDF(3) is given by

119910119899+1

=2

11119910119899minus2

minus9

11119910119899minus1

+18

11119910119899+6

11ℎ119891119899+1

119910119899+2

= minus3

25119910119899minus2

+16

25119910119899minus1

minus36

25119910119899+48

25119910119899+1

+12

25ℎ119891119899+2

(23)

23 Derivation of DI2BBDF(4) The interpolating points119909119899minus3 119909

119899+1are used to obtain the first point 119910

119899+1of

DI2BBDF(4)

119875 (119909)

=(119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2)

(119909119899minus3minus 119909119899+1) (119909119899minus3minus 119909119899) (119909119899minus3minus 119909119899minus1) (119909119899minus3minus 119909119899minus2)119910119899minus3

+(119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus3)

(119909119899minus2minus 119909119899+1) (119909119899minus2minus 119909119899) (119909119899minus2minus 119909119899minus1) (119909119899minus2minus 119909119899minus3)119910119899minus2

+(119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus2) (119909 minus 119909

119899minus3)

(119909119899minus1minus 119909119899+1) (119909119899minus1minus 119909119899) (119909119899minus1minus 119909119899minus2) (119909119899minus1minus 119909119899minus3)119910119899minus1

+(119909 minus 119909

119899+1) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2) (119909 minus 119909

119899minus3)

(119909119899minus 119909119899+1) (119909119899minus 119909119899minus1) (119909119899minus 119909119899minus2) (119909119899minus 119909119899minus3)119910119899

+(119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2) (119909 minus 119909

119899minus3)

(119909119899+1minus 119909119899) (119909119899+1minus 119909119899minus1) (119909119899+1minus 119909119899minus2) (119909119899+1minus 119909119899minus3)119910119899+1

(24)

Mathematical Problems in Engineering 5

We define 119909 = 119904ℎ + 119909119899+1

and produce

119875 (119904ℎ + 119909119899+1)

=(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ)

(minus4ℎ) (minus3ℎ) (minus2ℎ) (minusℎ)119910119899minus3

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 4ℎ)

(minus3ℎ) (minus2ℎ) (minusℎ) (ℎ)119910119899minus2

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 3ℎ) (119904ℎ + 3ℎ)

(minus2ℎ) (minusℎ) (ℎ) (2ℎ)119910119899minus1

+(119904ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 3ℎ)

(minusℎ) (ℎ) (2ℎ) (3ℎ)119910119899

+(119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 3ℎ)

(ℎ) (2ℎ) (3ℎ) (4ℎ)119910119899+1

(25)

Differentiating (25) once with respect to 119904 at the point 119909 =

119909119899+1

and evaluating 119904 = 0 will produce

1198751015840(119909119899+1) =

25

12119910119899+1

+1

4119910119899minus3

minus4

3119910119899minus2

+ 3119910119899minus1

minus 4119910119899 (26)

The derivation continues for the second point 119910119899+2

of formulaby using 119909

119899minus3 119909

119899+2as the interpolating points

119875 (119909) =(119909 minus 119909

119899+2) (119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2)

(119909119899minus3

minus 119909119899+2) (119909119899minus3

minus 119909119899+1) (119909119899minus3

minus 119909119899) (119909119899minus3

minus 119909119899minus1) (119909119899minus3

minus 119909119899minus2)119910119899minus3

+(119909 minus 119909

119899+2) (119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus3)

(119909119899minus2

minus 119909119899+2) (119909119899minus2

minus 119909119899+1) (119909119899minus2

minus 119909119899) (119909119899minus2

minus 119909119899minus1) (119909119899minus2

minus 119909119899minus3)119910119899minus2

+(119909 minus 119909

119899+2) (119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus2) (119909 minus 119909

119899minus3)

(119909119899minus1

minus 119909119899+2) (119909119899minus1

minus 119909119899+1) (119909119899minus1

minus 119909119899) (119909119899minus1

minus 119909119899minus2) (119909119899minus1

minus 119909119899minus3)119910119899minus1

+(119909 minus 119909

119899+2) (119909 minus 119909

119899+1) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2) (119909 minus 119909

119899minus3)

(119909119899minus 119909119899+2) (119909119899minus 119909119899+1) (119909119899minus 119909119899minus1) (119909119899minus 119909119899minus2) (119909119899minus 119909119899minus3)119910119899

+(119909 minus 119909

119899+2) (119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2) (119909 minus 119909

119899minus3)

(119909119899+1

minus 119909119899+2) (119909119899+1

minus 119909119899) (119909119899+1

minus 119909119899minus1) (119909119899+1

minus 119909119899minus2) (119909119899+1

minus 119909119899minus3)119910119899+1

+(119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2) (119909 minus 119909

119899minus3)

(119909119899+2

minus 119909119899+1) (119909119899+2

minus 119909119899) (119909119899+2

minus 119909119899minus1) (119909119899+2

minus 119909119899minus2) (119909119899+2

minus 119909119899minus3)119910119899+2

(27)

Substituting 119909 = ℎ + 119909119899+2

into (27) will produce

119875 (119904ℎ + 119909119899+2)

=(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 4ℎ)

(minus5ℎ) (minus4ℎ) (minus3ℎ) (minus2ℎ) (minusℎ)119910119899minus3

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 5ℎ)

(minus4ℎ) (minus3ℎ) (minus2ℎ) (minusℎ) (ℎ)119910119899minus2

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 4ℎ) (119904ℎ + 5ℎ)

(minus3ℎ) (minus2ℎ) (minusℎ) (ℎ) (2ℎ)119910119899minus1

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 3ℎ) (119904ℎ + 4ℎ) (119904ℎ + 5ℎ)

(minus2ℎ) (minusℎ) (ℎ) (2ℎ) (3ℎ)119910119899

+(119904ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 4ℎ) (119904ℎ + 5ℎ)

(minusℎ) (ℎ) (2ℎ) (3ℎ) (4ℎ)119910119899+1

+(119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 4ℎ) (119904ℎ + 5ℎ)

(ℎ) (2ℎ) (3ℎ) (4ℎ) (5ℎ)119910119899+2

(28)

Equation (28) is differentiated once with respect to 119904 followedby substituting 119904 = 0 we have

1198751015840(119909119899+2) =

137

60119910119899+2

minus1

5119910119899minus3

+5

4119910119899minus2

minus10

3119910119899minus1

+ 5119910119899minus 5119910119899+1

(29)

Therefore the corrector formula of DI2BBDF(4) is obtainedas follows

119910119899+1

= minus3

25119910119899minus3

+16

25119910119899minus2

minus36

25119910119899minus1

+48

25119910119899+12

25ℎ119891119899+1

119910119899+2

=12

137119910119899minus3

minus75

137119910119899minus2

+200

137119910119899minus1

minus300

137119910119899+300

137119910119899+1

+60

137ℎ119891119899+2

(30)

6 Mathematical Problems in Engineering

3 Stability Analysis

In this section we will plot the graph of stability forDI2BBDF(2) DI2BBDF(3) and DI2BBDF(4) using Mathe-matica software Based onDahlquist [19] the linearmultistepmethod (LMM) is able to solve stiff problems if it satisfies thefollowing definition

Definition 4 TheLMMisA-stable if its region of absolute sta-bility contains the whole of the left-hand half-plane Re(ℎ120582) lt0

We consider the simplest test equation

119891 = 1199101015840= 120582119910 (31)

where the eigenvalues 120582119894 119894 = 1 2 119904 satisfy 120582

119894lt 0 to

analyze the stiff stability of the method Substituting (31) into(16) will produce the following matrices form

[

[

1 0

minus18

111

]

]

[119910119899+1

119910119899+2

]

=[[

[

minus1

3

4

32

11minus9

11

]]

]

[119910119899minus1

119910119899

] + ℎ[[

[

2

30

06

11

]]

]

[119891119899+1

119891119899+2

]

(32)

where

119860 =[[

[

1 minus2

3120582ℎ 0

minus18

111 minus

6

11120582ℎ

]]

]

119861 =[[

[

minus1

3

4

32

11minus9

11

]]

]

119862 = [0 0

0 0]

(33)

Let ℎ = ℎ120582 and evaluating the determinant of |119860119905 minus 119861|from (32) the stability polynomial 120588(119905 ℎ) of DI2BBDF(2) isobtained as follows

120588 (119905 ℎ) =1

33minus34119905

33minus8ℎ119905

11+ 1199052minus40ℎ1199052

33+4ℎ21199052

11 (34)

The similar approach is applied to obtain the stability polyno-mial of DI2BBDF(3) and DI2BBDF(4) Substituting (31) into(23) we have

[

[

1 0

minus48

251

]

]

[119910119899+1

119910119899+2

]

=[[

[

02

11

0 minus3

25

]]

]

[119910119899minus3

119910119899minus2

] +[[

[

minus9

11

18

1116

25minus36

25

]]

]

[119910119899minus1

119910119899

]

+ ℎ[[

[

6

110

012

25

]]

]

[119891119899+1

119891119899+2

]

(35)

where

119860 =[[

[

1 minus6

11120582ℎ 0

minus48

251 minus

12

25120582ℎ

]]

]

119861 =[[

[

minus9

11

18

11

16

25minus36

25

]]

]

119862 =[[

[

02

11

03

25

]]

]

(36)

Substituting (31) into (30) will give

[

[

1 0

minus300

1371

]

]

[119910119899+1

119910119899+2

]

=[[

[

minus3

25

16

25

12

137minus75

137

]]

]

[119910119899minus3

119910119899minus2

] +[[

[

minus36

25

48

25

200

137minus300

137

]]

]

[119910119899minus1

119910119899

]

+ ℎ[[

[

12

250

060

137

]]

]

[119891119899+1

119891119899+2

]

(37)

where

119860 =[[

[

1 minus12

25120582ℎ 0

minus300

1371 minus

60

137120582ℎ

]]

]

119861 =[[

[

minus36

25

48

25

200

137minus300

137

]]

]

119862 =[[

[

minus3

25

16

25

12

137minus75

137

]]

]

(38)

We compute the determinant of |1198601199052 minus 119861119905 minus 119862| from(35) and (37) to obtain the stability polynomials 120588(119905 ℎ) ofDI2BBDF(3) and DI2BBDF(4) respectively Consider

120588 (119905 ℎ) =36

275minus86119905

275minus2431199052

275minus324ℎ119905

2

275minus1621199053

275

+36ℎ1199053

275+ 1199054minus282ℎ119905

4

275+72ℎ21199054

275

(39)

120588 (119905 ℎ) =48

137minus176119905

3425minus3871199052

685minus1152ℎ119905

2

685minus2514119905

3

3425

minus216ℎ119905

3

685+ 1199054minus3144ℎ119905

4

3425+144ℎ21199054

685

(40)

Next the boundary of the stability region will be deter-mined by substituting 119905 = 119890

119894120579 into (34) (39) and (40)The graphs of stability region for all formulas are givenin Figures 1 2 and 3

In Figures 1 2 and 3 we observed that the inter-vals of unstable region for DI2BBDF(2) DI2BBDF(3) and

Mathematical Problems in Engineering 7

Unstable

Unstable

Stable

Stable

Stable1 2 3 4 5

Real part

minus3

minus2

minus1

1

2

3Imaginary part

(533333 0)

Figure 1 Graph of stability region for DI2BBDF(2)

Unstable

Unstable

Stable

Stable

Stable2 4 6 8

Real part

minus4

minus2

2

4

Imaginary part

(866667 0)

Figure 2 Graph of stability region for DI2BBDF(3)

DI2BBDF(4) are [0 533333] [0 866667] and [0 1386667]respectively All the graphs of stability regions are combinedas shown in Figure 4 The region outside the green line is thestable region of DI2BBDF(2) the region outside the red lineis the stable region ofDI2BBDF(3) and the region outside theblue line is the stable region of DI2BBDF(4)

Clearly the unstable region becomes larger whenthe order of the method increases From Definition 4DI2BBDF(2) is A-stable while DI2BBDF(3) andDI2BBDF(4) are almost A-stable since the stability regioncovers the entire negative half plane Therefore we canconclude that the proposed method is suitable for solvingstiff problems

Unstable

Unstable

Stable

Stable

2 4 6 8 10 12 14

minus5

5

(1386667 0)

Stable Real part

Imaginary part

Figure 3 Graph of stability region for DI2BBDF(4)

Imaginary part

Real part5 10

minus5

5

10

DI2BBDF(2)

DI2BBDF(3)

DI2BBDF(4)

A-stable region

A-stable region

Figure 4 Graph of stability region for DI2BBDF(2) DI2BBDF(3)and DI2BBDF(4)

4 Implementation of the Method

In this section the proposed methods will be implementedusing Newton iteration We begin by converting (16) (23)and (30) in general form as follows

119910119899+1

= 1205721119910119899+2

+ 1205731ℎ119891119899+1

+ 1205831

119910119899+2

= 1205722119910119899+1

+ 1205732ℎ119891119899+2

+ 1205832

(41)

where 1205831and 120583

2are the back values Equation (41) is trans-

formed into matrix form as follows

[1 0

0 1] [119910119899+1

119910119899+2

]

= [0 1205721

12057220] [119910119899+1

119910119899+2

] + ℎ[12057310

0 1205732

][119891119899+1

119891119899+2

] + [1205831

1205832

]

(42)

8 Mathematical Problems in Engineering

where

119868 = [1 0

0 1] 119884 = [

119910119899+1

119910119899+2

] 119861 = [12057310

0 1205732

]

119865 = [119891119899+1

119891119899+2

] 120583 = [1205831

1205832

]

(43)

Implementing Newtonrsquos method to (42) produces

119865 = (119868 minus 119860)119884 minus ℎ119861119865 + 120583 (44)

Iteration for (44) is given by

119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2= 119865 (119910

(119894)

119899+1119899+2) sdot 1198651015840(119910(119894)

119899+1119899+2)minus1

(45)

where 119910(119894+1)119899+1

denotes the (119894 + 1)th iteration Equation (45) canbe rearranged into the following equation

1198651015840(119910(119894)

119899+1119899+2) (119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2) = 119865 (119910

(119894)

119899+1119899+2) (46)

Replacing (44) into (46) yields

(119868 minus 119860)119884 minus ℎ119861120597119891

120597119910(119910(119894)

119899+1119899+2) (119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2)

= (119868 minus 119860) (119910(119894)

119899+1119899+2) minus ℎ119861 (119910

(119894)

119899+1119899+2) + 120583

(47)

where (120597119891120597119910)(119910(119894)119899+1119899+2

) is the Jacobian matrix of 119891 with re-spect to 119910

41 Newton Iteration of DI2BBDF(2) Formula (16) is writtenin the form of (41)

1198651= 119910119899+1

minus2

3ℎ119891119899+1

+ 1205831

1198652= 119910119899+2

minus18

11119910119899+1

minus6

11ℎ119891119899+2

+ 1205832

(48)

Applying (48) into (47) will yield matrix form as follows

[[[

[

1 minus2

3ℎ120597119891119899+1

120597119910119899+1

0

0 1 minus6

11ℎ120597119891119899+2

120597119910119899+2

]]]

]

[119890(119894+1)

119899+1

119890(119894+1)

119899+2

]

= [

[

minus1 0

18

11minus1

]

]

[119910(119894)

119899+1

119910(119894)

119899+2

] + ℎ[[

[

2

30

06

11

]]

]

[119891(119894)

119899+1

119891(119894)

119899+2

] + [1205831

1205832

]

(49)

with 119890(119894+1)119899+1119899+2

= 119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2being the increment

42 Newton Iteration of DI2BBDF(3) Rearranging formula(23) in the form of (41) we have

1198651= 119910119899+1

minus6

11ℎ119891119899+1

+ 1205831

1198652= 119910119899+2

minus48

25119910119899+1

minus12

25ℎ119891119899+2

+ 1205832

(50)

Replacing (50) into (47) this will yieldmatrix form as follows

[[[

[

1 minus6

11ℎ120597119891119899+1

120597119910119899+1

0

0 1 minus12

25ℎ120597119891119899+2

120597119910119899+2

]]]

]

[119890(119894+1)

119899+1

119890(119894+1)

119899+2

]

= [

[

minus1 0

48

25minus1

]

]

[119910(119894)

119899+1

119910(119894)

119899+2

] + ℎ[[

[

6

110

012

25

]]

]

[119891(119894)

119899+1

119891(119894)

119899+2

] + [1205831

1205832

]

(51)

with 119890(119894+1)119899+1119899+2

= 119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2being the increment

43 Newton Iteration of DI2BBDF(4) We rewrite formula(30) in the form of (41) and obtain

1198651= 119910119899+1

minus12

25ℎ119891119899+1

+ 1205831

1198652= 119910119899+2

minus300

137119910119899+1

minus60

137ℎ119891119899+2

+ 1205832

(52)

Substituting (52) into (47) will produce

[[[

[

1 minus12

25ℎ120597119891119899+1

120597119910119899+1

0

0 1 minus60

137ℎ120597119891119899+2

120597119910119899+2

]]]

]

[119890(119894+1)

119899+1

119890(119894+1)

119899+2

]

= [

[

minus1 0

300

137minus1

]

]

[119910(119894)

119899+1

119910(119894)

119899+2

] + ℎ[[

[

12

250

060

137

]]

]

[119891(119894)

119899+1

119891(119894)

119899+2

] + [1205831

1205832

]

(53)

with 119890(119894+1)119899+1119899+2

= 119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2being the increment

All the formulas derived are implemented in predictor-corrector computation which is symbolized as PECE modeP and C indicate one application of the predictor and thecorrector respectively and E indicates one evaluation of thefunction119891 given119909 and119910The approximation calculations for119910119899+1

and 119910119899+2

in PECE are as follows

(1) P (Predict) 119910(119901)119899+1119899+2

(2) E (Evaluate) 1199101015840119899+1119899+2

= 119891(119909119899+1119899+2

119910(119901)

119899+1119899+2)

(3) C (Correct) 119910(119888)119899+1119899+2

(4) E (Evaluate) 1199101015840119899+1119899+2

= 119891(119909119899+1119899+2

119910(119888)

119899+1119899+2)

To approximate the solution to 119910(119888)

119899+1119899+2 we apply the

two-stage Newton type iteration The iteration process forDI2BBDF(2) is done as follows

Mathematical Problems in Engineering 9

(1) Compute the values for 119890(119894+1)119899+1119899+2

= 119860minus1119861 where

119860 =[[[

[

1 minus2

3ℎ120597119891119899+1

120597119910119899+1

0

0 1 minus6

11ℎ120597119891119899+2

120597119910119899+2

]]]

]

119861 =[[

[

minus119910(119894)

119899+1+2

3ℎ119891(119894)

119899+1+ 1205831

0

18

11119910(119894)

119899+1minus119910(119894)

119899+2+6

11ℎ119891(119894)

119899+2+ 1205832

]]

]

(54)

(2) Calculate the corrected value for 119910(119894+1)119899+1119899+2

with thevalue 119890(119894+1)

119899+1119899+2from Step (1)

(3) Solve 119890(119894+1)119899+1119899+2

= 119860minus1119861 for second stage iteration

(4) The final values for 119910(119894+1)119899+1119899+2

are obtained from thesecond stage iteration of 119890(119894+1)

119899+1119899+2

The similar iteration process is applied for DI2BBDF(3) andDI2BBDF(4)

5 Test Problems

In this section linear and nonlinear stiff problems aretested using C programming to examine the efficiency andreliability of the proposed method The following problemsare commonly found in engineering and physical sciencesparticularly in the studies of vibrations electrical circuits andchemical reaction

Problem 1 (linear) One has

1199101015840= minus10119910 + 10 (55)

Exact solution is

119910 (119909) = 1 + 119890minus10119909

(56)

Initial condition is

119910 (0) = 2

0 le 119909 le 10

(57)

(source Ibrahim [8])

Problem 2 (linear) One has

1199101015840

1= minus5119910

1minus 41199102 119910

1015840

2= minus4119910

1minus 51199102 (58)

Exact solutions are

1199101(119909) =

1

2119890minus9119909

+1

2119890minus119909 119910

2(119909) =

1

2119890minus9119909

minus1

2119890minus119909 (59)

Initial conditions are

1199101(0) = 1 119910

2(0) = 0

0 le 119909 le 20(60)

(source Musa et al [13])

Problem 3 (linear) One has

1199101015840= minus10119909119910 (61)

Exact solution is

119910 (119909) = 119890minus51199092

(62)

Initial condition is119910 (0) = 1

0 le 119909 le 10

(63)

(source Ibrahim et al [10])

Problem 4 (linear) One has

1199101015840

1= 91199101+ 24119910

2+ 5 cos119909 minus 1

3sin119909

1199101015840

2= minus 24119910

1minus 51119910

2minus 9 cos119909 + 1

3sin119909

(64)

Exact solutions are

1199101(119909) = 2119890

minus3119909minus 119890minus39119909

+1

3cos119909

1199102(119909) = minus119890

minus3119909+ 2119890minus39119909

minus1

3cos119909

(65)

Initial conditions are

1199101(0) =

4

3 1199102(0) =

2

3 0 le 119909 le 10 (66)

(source Musa et al [14])

Problem 5 (nonlinear) One has

1199101015840= minus

1199103

2 (67)

Exact solution is

119910 (119909) =1

radic1 + 119909 (68)

Initial condition is119910 (0) = 1

0 le 119909 le 4

(69)

(source Musa et al [14])

Problem 6 (nonlinear) One has

1199101015840=119910 (1 minus 119910)

2119910 minus 1 (70)

Exact solution is

119910 (119909) =1

2+ radic

1

4minus5

36119890minus119909 (71)

Initial condition is

119910 (0) =5

6

0 le 119909 le 1

(72)

(source Musa et al [14])

10 Mathematical Problems in Engineering

Table 1 The accuracy for problem 1

ℎ Methods MAXE TIME

10minus2

FI2BBDF(3) 597499119864 minus 2 136773119864 minus 3

FI2BEBDF(3) 567155119864 minus 2 145525119864 minus 3

DI2BBDF(2) 110568119864 minus 2 472498119864 minus 3

DI2BBDF(3) 166455119864 minus 2 477409119864 minus 3

DI2BBDF(4) 216342119864 minus 2 483298119864 minus 3

10minus4

FI2BBDF(3) 734012119864 minus 4 125797119864 minus 1

FI2BEBDF(3) 323640119864 minus 5 136259119864 minus 1

DI2BBDF(2) 118355119864 minus 6 167260119864 minus 1

DI2BBDF(3) 173430119864 minus 6 169325119864 minus 1

DI2BBDF(4) 225966119864 minus 6 172995119864 minus 1

10minus6

FI2BBDF(3) 735741119864 minus 6 1257001198641

FI2BEBDF(3) 350090119864 minus 7 1362301198641

DI2BBDF(2) 118742119864 minus 10 1491591198641

DI2BBDF(3) 173634119864 minus 10 1544931198641

DI2BBDF(4) 226000119864 minus 10 1546201198641

Table 2 The accuracy for problem 2

ℎ Methods MAXE TIME

10minus2

FI2BBDF(3) 271597119864 minus 2 596467119864 minus 3

FI2BEBDF(3) 149811119864 minus 2 661025119864 minus 3

DI2BBDF(2) 456073119864 minus 3 104480119864 minus 2

DI2BBDF(3) 686126119864 minus 3 108421119864 minus 2

DI2BBDF(4) 893279119864 minus 3 111089119864 minus 2

10minus4

FI2BBDF(3) 340437119864 minus 4 590430119864 minus 1

FI2BEBDF(3) 262649119864 minus 5 654305119864 minus 1

DI2BBDF(2) 485404119864 minus 7 373210119864 minus 1

DI2BBDF(3) 711155119864 minus 7 386556119864 minus 1

DI2BBDF(4) 926476119864 minus 7 387070119864 minus 1

10minus6

FI2BBDF(3) 341146119864 minus 6 5901981198641

FI2BEBDF(3) 244097119864 minus 7 6541061198641

DI2BBDF(2) 486843119864 minus 11 3484511198641

DI2BBDF(3) 711903119864 minus 11 3604901198641

DI2BBDF(4) 926598119864 minus 11 3606491198641

6 Numerical Results

The performance of the derived methods is compared withthe existing methods in terms of maximum error and exe-cution time We consider 10minus2 10minus4 and 10minus6 as the stepsize ℎ Tables 1 and 2 present the performance compar-ison of DI2BBDF(2) DI2BBDF(3) and DI2BBDF(4) withFI2BBDF(3) and FI2BEBDF(3) whereas Tables 3ndash6 exhibitthe comparison of proposed methods with FI3BBDF(3) andFI3BEBDF(3) In addition the graphs of log(MAXE) againstlog(ℎ) are illustrated as shown in Figures 5ndash10 The followingnotations are used in Tables 1ndash6

ℎ step sizeMAXE maximum errorTIME time execution using high performance com-puter (HPC)

Table 3 The accuracy for problem 3

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 356692119864 minus 2 141642119864 minus 3

FI3BEBDF(3) 124084119864 minus 2 302167119864 minus 3

DI2BBDF(2) 117256119864 minus 3 448489119864 minus 3

DI2BBDF(3) 172278119864 minus 3 481200119864 minus 3

DI2BBDF(4) 225003119864 minus 3 485301119864 minus 3

10minus4

FI3BBDF(3) 435640119864 minus 4 130103119864 minus 1

FI3BEBDF(3) 706516119864 minus 5 208115119864 minus 1

DI2BBDF(2) 118749119864 minus 7 160304119864 minus 1

DI2BBDF(3) 173636119864 minus 7 177445119864 minus 1

DI2BBDF(4) 226000119864 minus 7 178064119864 minus 1

10minus6

FI3BBDF(3) 434625119864 minus 6 1297751198641

FI3BEBDF(3) 703257119864 minus 7 2076091198640

DI2BBDF(2) 124989119864 minus 11 1507391198641

DI2BBDF(3) 173636119864 minus 11 1550731198641

DI2BBDF(4) 225999119864 minus 11 1602161198641

Table 4 The accuracy for problem 4

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 66269411986499 701775119864 minus 3

FI3BEBDF(3) 168449119864 minus 1 137847119864 minus 2

DI2BBDF(2) 280830119864 minus 1 106871119864 minus 2

DI2BBDF(3) 400389119864 minus 1 124969119864 minus 2

DI2BBDF(4) 477342119864 minus 1 126502119864 minus 2

10minus4

FI3BBDF(3) 845376119864 minus 3 676737119864 minus 1

FI3BEBDF(3) 695725119864 minus 3 135671119864 + 0

DI2BBDF(2) 356255119864 minus 5 432359119864 minus 1

DI2BBDF(3) 524241119864 minus 5 442599119864 minus 1

DI2BBDF(4) 685105119864 minus 5 456207119864 minus 1

10minus6

FI3BBDF(3) 854545119864 minus 5 6768191198641

FI3BEBDF(3) 718859119864 minus 5 1358621198642

DI2BBDF(2) 360130119864 minus 9 4000101198641

DI2BBDF(3) 526673119864 minus 9 4001921198641

DI2BBDF(4) 685529119864 minus 9 4103221198641

Table 5 The accuracy for problem 5

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 525481119864 minus 3 578083119864 minus 4

FI3BEBDF(3) 388475119864 minus 3 916583119864 minus 4

DI2BBDF(2) 856253119864 minus 5 229836119864 minus 4

DI2BBDF(3) 121629119864 minus 4 189066119864 minus 4

DI2BBDF(4) 165094119864 minus 4 230074119864 minus 4

10minus4

FI3BBDF(3) 551558119864 minus 5 483860119864 minus 2

FI3BEBDF(3) 427506119864 minus 5 774877119864 minus 2

DI2BBDF(2) 889803119864 minus 9 117438119864 minus 2

DI2BBDF(3) 130124119864 minus 8 131578119864 minus 2

DI2BBDF(4) 169456119864 minus 8 126071119864 minus 2

10minus6

FI3BBDF(3) 551803119864 minus 7 547768119864 + 0

FI3BEBDF(3) 427924119864 minus 7 777643119864 + 0

DI2BBDF(2) 126115119864 minus 11 119318119864 + 0

DI2BBDF(3) 627106119864 minus 11 133892119864 + 0

DI2BBDF(4) 104222119864 minus 11 126146119864 + 0

Mathematical Problems in Engineering 11

Table 6 The accuracy for problem 6

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 218555119864 minus 3 283084119864 minus 4

FI3BEBDF(3) 152719119864 minus 3 428500119864 minus 4

DI2BBDF(2) 387041119864 minus 5 538826119864 minus 5

DI2BBDF(3) 547958119864 minus 5 531673119864 minus 5

DI2BBDF(4) 747696119864 minus 5 618877119864 minus 5

10minus4

FI3BBDF(3) 229812119864 minus 5 148627119864 minus 2

FI3BEBDF(3) 178708119864 minus 5 232184119864 minus 2

DI2BBDF(2) 401638119864 minus 9 349712119864 minus 3

DI2BBDF(3) 587347119864 minus 9 375295119864 minus 3

DI2BBDF(4) 764927119864 minus 9 364804119864 minus 3

10minus6

FI3BBDF(3) 229936119864 minus 7 146660119864 + 0

FI3BEBDF(3) 178973119864 minus 7 228775119864 + 0

DI2BBDF(2) 990807119864 minus 12 338157119864 minus 1

DI2BBDF(3) 377716119864 minus 11 369837119864 minus 1

DI2BBDF(4) 462796119864 minus 12 356802119864 minus 1

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

log(h)minus6 minus5 minus4 minus3 minus2

FI2BBDF(3)FI2BEBDF(3)

DI2BBDF(2)DI2BBDF(3)DI2BBDF(4)

Figure 5 Graph of log(MAXE) versus log(ℎ) for problem 1

DI2BBDF(2) diagonally implicit 2-point block back-ward differentiation formulas of order 2DI2BBDF(3) diagonally implicit 2-point block back-ward differentiation formulas of order 3DI2BBDF(4) diagonally implicit 2-point block back-ward differentiation formulas of order 4FI2BBDF(3) fully implicit 2-point block backwarddifferentiation formulas of order 3FI2BEBDF(3) fully implicit 2-point block extendedbackward differentiation formulas of order 3FI3BBDF(3) fully implicit 3-point block backwarddifferentiation formulas of order 3

log(h)minus6 minus5 minus4 minus3 minus2

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

FI2BBDF(3)FI2BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

Figure 6 Graph of log(MAXE) versus log(ℎ) for problem 2

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus11minus12

minus10minus9minus8minus7minus6minus5minus4minus3minus2minus10

log(

MA

XE)

Figure 7 Graph of log(MAXE) versus log(ℎ) for problem 3

FI3BEBDF(3) fully implicit 3-point block extendedbackward differentiation formulas of order 3

7 Discussion

This section is divided into discussion of maximum error andcomputational time

71 Maximum Error From the numerical results in Tables1ndash6 we observe that DI2BBDF(2) DI2BBDF(3) andDI2BBDF(4) outperformed the existing methods in termsof maximum error This is expected because the diagonally

12 Mathematical Problems in Engineering

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

Figure 8 Graph of log(MAXE) versus log(ℎ) for problem 4

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus11

minus12

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

Figure 9 Graph of log(MAXE) versus log(ℎ) for problem 5

implicit method has less differentiation coefficients in orderto prevent the cumulative error The graphs in Figures 5ndash10depict the scales of maximum error versus step size h for theproposed methods as compared to existing methods Thereis a slight drop in accuracy for the proposed methods as theorder increases This is due to the increase in cumulativeerrors during the computation as more interpolating pointsare used Among the methods of order three DI2BBDF(3)is more accurate compared with FI2BBDF(3) FI3BBDF(3)FI2BEBDF(3) and FI3BEBDF(3) For all test problems we

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus11minus12

minus10minus9minus8minus7minus6minus5minus4minus3minus2minus10

log(

MA

XE)

Figure 10 Graph of log(MAXE) versus log(ℎ) for problem 6

conclude that the accuracy of the proposedmethod increasesas the step size becomes smaller

72 Computational Time In Tables 1-2 it can be seen thatthe execution time taken by DI2BBDF(2) DI2BBDF(3) andDI2BBDF(4) is comparable with that of FI2BBDF(3) andFI2BEBDF(3) The computational time increases as the stepsize becomes smaller However Tables 3ndash6 show that theproposed methods compute faster than FI3BBDF(3) andFI3BEBDF(3) This could be justified by the fact that the3-point method has to perform extra computation on theJacobian matrix since it involves a 3 by 3 matrix while the 2-point method has to compute Jacobian matrix of dimension2 by 2 only Among the proposed methods DI2BBDF(2)computes faster than DI2BBDF(3) and DI2BBDF(4) In factthe number of back values involved in every formula willaffect the computational time Since DI2BBDF(4) has moreback values than DI2BBDF(3) and DI2BBDF(2) it requiresextra time to compute the approximated solutions

8 Conclusion

Research conducted in this paper shows the capability ofconstructing the diagonally implicit BBDF method with A-stable properties From the results obtained via the numericalexperiment we can conclude that the proposed methodserves the purpose of significant alternative numericalmethod for solving linear and nonlinear stiff IVPs occurringin the fields of engineering and physical sciences

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 13

Acknowledgment

This researchwas supported by the Institute forMathematicalResearch (INSPEM) Department of Mathematics UniversitiPutra Malaysia (UPM) under Fundamental Grant Scheme(Project code 02-02-13-1380FR)

References

[1] C F Curtiss and J O Hirschfelder ldquoIntegration of stiff equa-tionsrdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 38 no 3 pp 235ndash243 1952

[2] J D Lambert Numercial Methods for Ordinary DifferentialSystems The Initial Value Problem John Wiley amp Sons NewYork NY USA 1991

[3] W E Milne Numerical Solution of Differential Equations JohnWiley amp Sons New York NY USA 1953

[4] L F Shampine and H A Watts ldquoBlock implicit one-stepmethodsrdquo Mathematics of Computation vol 23 pp 731ndash7401969

[5] S O Fatunla ldquoBlock methods for second order ODEsrdquo Inter-national Journal of Computer Mathematics vol 41 pp 55ndash631990

[6] Z A Majid M B Suleiman F I Ismail and K I Othmanldquo2-Point 1 block diagonally and 2-point 1 block fully implicitmethod for solving first order ordinary differential equationsrdquoin Proceedings of the 12th National Symposium on MathematicalScience vol 12 2004

[7] Z Abdul Majid N Z Mokhtar and M Suleiman ldquoDirecttwo-point block one-step method for solving general second-order ordinary differential equationsrdquo Mathematical Problemsin Engineering vol 2012 Article ID 184253 16 pages 2012

[8] Z B Ibrahim Block method for multistep formulas for solvingordinary differential equations [PhD thesis] Universiti PutraMalaysia Seri Kembangan Malaysia 2006

[9] Z B Ibrahim M Suleiman and K I Othman ldquoFixed coeffi-cients block backward differentiation formulas for the numer-ical solution of stiff ordinary differential equationsrdquo EuropeanJournal of Scientific Research vol 21 no 3 pp 508ndash520 2008

[10] Z B Ibrahim M B Suleiman and K I Othman ldquoImplicit r-point block backward differentiation formula for solving first-order stiff ODEsrdquo Applied Mathematics and Computation vol186 no 1 pp 558ndash565 2007

[11] Z B Ibrahim R Johari and F Ismail ldquoOn the stability of blockbackward differentiation formulaerdquo Matematika vol 19 no 2pp 83ndash89 2003

[12] N A Nasir Z B Ibrahim M B Suleiman and K I OthmanldquoFifth order two-point block backward differentiation formulasfor solving ordinary differential equationsrdquo Applied Mathemat-ical Sciences vol 5 no 69ndash72 pp 3505ndash3518 2011

[13] H Musa M B Suleiman and N Senu ldquoA-stable 2-Point blockextended backward differentiation formulas for stiff ordinarydifferential equationsrdquo in Proceedings of the 5th InternationalConference on Research and education in Mathematics vol 1450of AIP Conference Proceedings pp 254ndash258 2012

[14] H Musa M B Suleiman and N Senu ldquoFully implicit 3-pointblock extended backward differentiation formula for stiff initialvalue problemsrdquo Applied Mathematical Sciences vol 6 no 85ndash88 pp 4211ndash4227 2012

[15] R Alexander ldquoDiagonally implicit Runge-kutta for stiff ordi-nary differential equationsrdquo SIAM Journal on Numerical Analy-sis vol 14 no 6 pp 1006ndash1021 1977

[16] O Y Ababneh R Ahmad and E S Ismail ldquoDesign of newdiagonally implicit Runge-Kutta methods for stiff problemsrdquoApplied Mathematical Sciences vol 3 no 45 pp 2241ndash22532009

[17] F Ismail A L I Raed Al-Khasawneh M Suleiman and A BU Malik Hassan ldquoEmbedded pair of diagonally implicit runge-kutta method for solving ordinary differential equationsrdquo SainsMalaysiana vol 39 no 6 pp 1049ndash1054 2010

[18] Z A Majid and M B Suleiman ldquoPerformance of four pointdiagonally implicit block method for solving first order stiffordinary differential equationsrdquo Department of MathematicsUTM vol 22 no 2 pp 137ndash146 2006

[19] G G Dahlquist ldquoA special stability problem for linearmultistepmethodsrdquo BIT Numerical Mathematics vol 3 no 1 pp 27ndash431963

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Mathematical Problems in Engineering

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

fully implicit 119903-point block backward differentiation formulas(FIBBDF) The following equations represent the formulasof fully implicit 2-point block backward differentiation for-mulas of order three (FI2BBDF(3)) and fully implicit 3-point block backward differentiation formula of order three(FI3BBDF(3))

FI2BBDF(3) One has

119910119899+1

= minus1

3119910119899minus1

+ 2119910119899minus2

3119910119899+2

+ 2ℎ119891119899+1

119910119899+2

=2

11119910119899minus1

minus9

11119910119899+18

11119910119899+1

+6

11ℎ119891119899+2

(3)

FI3BBDF(3) One has

119910119899+1

=1

10119910119899minus2

minus3

4119910119899minus1

+ 3119910119899minus3

2119910119899+2

+3

20119910119899+3

+ 3ℎ119891119899+1

119910119899+2

= minus3

65119910119899minus2

+4

13119910119899minus1

minus12

13119910119899+24

13119910119899+1

minus12

65119910119899+3

+12

13ℎ119891119899+2

119910119899+3

=12

137119910119899minus2

minus75

137119910119899minus1

+200

137119910119899minus300

137119910119899+1

+300

65119910119899+2

+60

137ℎ119891119899+3

(4)

In a related study Ibrahim et al [11] plotted the stabilityregion of (3) Since all region in the left half plane is in thestability region the FI2BBDF(3) is A-stable and suitable forsolving stiff problems Nasir et al [12] extended the orderof formula (3) which is called fifth order 2-point BBDFfor solving first order stiff ODEs Recently Musa et al [1314] modified formulas (3) and (4) to compute more thanone solution value per step using extra future point Thismethod is called fully implicit block extended backwarddifferentiation formulas (FIBEBDF) The formulas of fullyimplicit 2-point block extended backward differentiationformula of order three (FI2BEBDF(3)) and fully implicit 3-point block extended backward differentiation formula oforder three (FI3BEBDF(3)) are given in the following forms

FI2BEBDF(3) One has

119910119899+1

=1

9119910119899minus1

minus 119910119899+17

9119910119899+2

minus 2ℎ119891119899+1

minus2

3ℎ119891119899+2

119910119899+2

=17

197119910119899minus1

minus99

197119910119899+279

197119910119899+1

+150

197ℎ119891119899+2

minus18

197ℎ119891119899+3

(5)

FI3BEBDF(3) One has

119910119899+1

= minus1

80119910119899minus2

+1

80119910119899minus1

minus3

4119910119899+25

16119910119899+2

+3

40119910119899+3

minus3

2ℎ119891119899+1

minus3

4ℎ119891119899+2

119910119899+2

= minus3

25119910119899minus2

+ 119910119899minus1

minus 4119910119899+ 12119910

119899+2

minus197

25119910119899+3

+ 12ℎ119891119899+2

+12

5ℎ119891119899+3

119910119899+3

=394

14919119910119899minus2

minus2925

14919119910119899minus1

+9600

14919119910119899minus18700

14919119910119899+1

+26550

14919119910119899+2

+8820

14919ℎ119891119899+3

minus600

14919ℎ119891119899+4

(6)

Numerical results in the literature showed that theFIBEBDF performed better as compared to FIBBDF interms of accuracy Unfortunately the execution time of theFIBEBDF is slower than FIBBDF This is due to the factthat FIBEBDF has an extra future point which requires morecomputation time In order to gain an efficient numericalapproximation in terms of accuracy and computational timethe diagonally implicitmethodmust be consideredThe studyof diagonally implicit for multistep method attracted severalresearchers such as Alexander [15] Ababneh et al [16] andIsmail et al [17] However Lambert [2] stated that there issome confusion over nomenclature to identify the diagonallyimplicit Some authors use the term of diagonally implicit todescribe any semi-implicit method Therefore the definitionof diagonally implicit block method is given by Majid andSuleiman [18] as follows

Definition 2 Method (2) is defined to be diagonally implicitif the coefficients of the upper-diagonal entries are zero

From this motivation we established Definition 2 byintroducing the definition of diagonally implicit 2-pointBBDF method as follows

Definition 3 We consider that 11988611 11988612 11988621 and 119886

22are coeffi-

cients of 119910119899+1

and 119910119899+2

in the matrix form

[1198861111988612

1198862111988622

][119910119899+1

119910119899+2

] (7)

Method (2) is defined to be diagonally implicit if 11988612is zero

whereas 11988611and 11988622

are equal

Therefore the main purpose of this paper is to developa new diagonally implicit multistep method for solving stiffODEs This paper is organized as follows in Section 2 thediagonally implicit two-point block backward differentiationformulas (DI2BBDF) will be derived Next the stabilityproperties of the derived methods are analyzed in Section 3Section 4 discusses the implementation of the methods usingNewton iteration Standard test problems are selected in

Mathematical Problems in Engineering 3

Section 5 whereas the performance of the proposed methodis shown in Section 6 Finally in Section 7 some conclusionsare given

2 Formulation of the Method

In this section we will derive the DI2BBDF of order twoorder three and order four with constant step size to computethe approximated solutions at 119910

119899+1and 119910

119899+2concurrently

Contrary to the fully implicit method that has been proposedby Ibrahim et al [11] the first point of diagonally implicitformula has one less interpolating point The derivationusing polynomial 119875

119896(119909) of degree 119896 in terms of Lagrange

polynomial is defined as follows

119875119896(119909) =

119896

sum

119895=0

119871119896119895(119909) 119891 (119909

119899+1minus119895) (8)

where

119871119896119895(119909) =

119896

prod

119894=0

119894 =119895

(119909 minus 119909119899+1minus119894

)

(119909119899+1minus119895

minus 119909119899+1minus119894

)(9)

for each 119895 = 0 1 119896

21 Derivation of DI2BBDF(2) DI2BBDF(2) will computetwo approximated solutions 119910

119899+1and 119910

119899+2simultaneously in

each block using two back values This formula is derivedusing interpolating points 119909

119899minus1 119909

119899+1to obtain the first

formula 119910119899+1

of DI2BBDF(2)

119875 (119909) =(119909 minus 119909

119899) (119909 minus 119909

119899+1)

(119909119899minus1

minus 119909119899) (119909119899minus1

minus 119909119899+1)119910119899minus1

+(119909 minus 119909

119899minus1) (119909 minus 119909

119899+1)

(119909119899minus 119909119899minus1) (119909119899minus 119909119899+1)119910119899

+(119909 minus 119909

119899minus1) (119909 minus 119909

119899)

(119909119899+1

minus 119909119899minus1) (119909119899+1

minus 119909119899)119910119899+1

(10)

Replacing 119909 = 119904ℎ + 119909119899+1

into (10) yields

119875 (119909119899+1

+ 119904ℎ) =(119904ℎ + ℎ) (119904ℎ)

(minusℎ) (minus2ℎ)119910119899minus1

+(2ℎ + 119904ℎ) (119904ℎ)

(ℎ) (minusℎ)119910119899

+(2ℎ + 119904ℎ) (ℎ + 119904ℎ)

(2ℎ) (ℎ)119910119899+1

(11)

Equation (11) is differentiated once with respect to 119904 at thepoint 119909 = 119909

119899+1 Evaluating 119904 = 0 gives

1198751015840(119909119899+1) =

1

2119910119899minus1

minus 2119910119899+3

2119910119899+1 (12)

The same technique is applied for the second point 119910119899+2

ofDI2BBDF(2) This formula is derived using 119909

119899minus1 119909

119899+2as

the interpolating points and produces

119875 (119909) =(119909 minus 119909

119899) (119909 minus 119909

119899+1) (119909 minus 119909

119899+2)

(119909119899minus1

minus 119909119899) (119909119899minus1

minus 119909119899+1) (119909119899minus1

minus 119909119899+2)119910119899minus1

+(119909 minus 119909

119899minus1) (119909 minus 119909

119899+1) (119909 minus 119909

119899+2)

(119909119899minus 119909119899minus1) (119909119899minus 119909119899+1) (119909119899minus 119909119899+2)119910119899

+(119909 minus 119909

119899minus1) (119909 minus 119909

119899) (119909 minus 119909

119899+2)

(119909119899+1

minus 119909119899minus1) (119909119899+1

minus 119909119899) (119909119899+1

minus 119909119899+2)119910119899+1

+(119909 minus 119909

119899minus1) (119909 minus 119909

119899) (119909 minus 119909

119899+1)

(119909119899+2

minus 119909119899minus1) (119909119899+2

minus 119909119899) (119909119899+2

minus 119909119899+1)119910119899+2

(13)

Substituting 119909 = 119904ℎ + 119909119899+2

into (13) yields

119875 (119909119899+2

+ 119904ℎ)

=(119904ℎ + 2ℎ) (119904ℎ + ℎ) (119904ℎ)

(minusℎ) (minus2ℎ) (minus3ℎ)119910119899minus1

+(119904ℎ + 3ℎ) (119904ℎ + ℎ) (119904ℎ)

(ℎ) (minusℎ) (minus2ℎ)119910119899

+(119904ℎ + 3ℎ) (119904ℎ + 2ℎ) (119904ℎ)

(2ℎ) (ℎ) (minusℎ)119910119899+1

+(119904ℎ + 3ℎ) (119904ℎ + 2ℎ) (119904ℎ + ℎ)

(3ℎ) (2ℎ) (ℎ)119910119899+2

(14)

Differentiating (14) with respect to 119904 at the point 119909 = 119909119899+2

gives

1198751015840(119909119899+2) = minus

1

3119910119899minus1

+3

2119910119899minus 3119910119899+1

+11

6119910119899+2 (15)

Considering ℎ119891119899+1119899+2

= 1198751015840(119909119899+1119899+2

) the corrector formulaof DI2BBDF(2) is given as follows

119910119899+1

= minus1

3119910119899minus1

+4

3119910119899+2

3ℎ119891119899+1

119910119899+2

=2

11119910119899minus1

minus9

11119910119899+18

11119910119899+1

+6

11ℎ119891119899+2

(16)

The order of the method is distinguished by the number ofback values contained in the formulas Adopting a similarapproach as the derivation of DI2BBDF of order two wewill construct the DI2BBDF of orders three and four withdifferent number of interpolating points

4 Mathematical Problems in Engineering

22 Derivation of DI2BBDF(3) The first point 119910119899+1

ofDI2BBDF(3) is derived using interpolating points 119909

119899minus2

119909119899+1

and we have

119875 (119909) =(119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus1)

(119909119899minus2

minus 119909119899+1) (119909119899minus2

minus 119909119899) (119909119899minus2

minus 119909119899minus1)119910119899minus2

+(119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus2)

(119909119899minus1

minus 119909119899+1) (119909119899minus1

minus 119909119899) (119909119899minus1

minus 119909119899minus2)119910119899minus1

+(119909 minus 119909

119899+1) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2)

(119909119899minus 119909119899+1) (119909119899minus 119909119899minus1) (119909119899minus 119909119899minus2)119910119899

+(119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2)

(119909119899+1

minus 119909119899) (119909119899+1

minus 119909119899minus1) (119909119899+1

minus 119909119899minus2)119910119899+1

(17)

Substituting 119909 = 119904ℎ + 119909119899+1

into (17) gives

119875 (119904ℎ + 119909119899+1)

=(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ)

(minus3ℎ) (minus2ℎ) (minusℎ)119910119899minus2

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 3ℎ)

(minus2ℎ) (minusℎ) (ℎ)119910119899minus1

+(119904ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ)

(minusℎ) (ℎ) (2ℎ)119910119899

+(119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ)

(ℎ) (2ℎ) (3ℎ)119910119899+1

(18)

Equation (18) is differentiated once with respect to 119904 at thepoint 119909 = 119909

119899+1 Substituting 119904 = 0 will obtain

1198751015840(119909119899+1) =

11

6119910119899+1

minus1

3119910119899minus2

+3

2119910119899minus1

minus 3119910119899 (19)

The derivation process continues for second point 119910119899+2

ofDI2BBDF(3) using 119909

119899minus3 119909

119899+1as the interpolating points

We have

119875 (119909) =(119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus1)

(119909119899minus2

minus 119909119899+1) (119909119899minus2

minus 119909119899) (119909119899minus2

minus 119909119899minus1)119910119899minus2

+(119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus2)

(119909119899minus1

minus 119909119899+1) (119909119899minus1

minus 119909119899) (119909119899minus1

minus 119909119899minus2)119910119899minus1

+(119909 minus 119909

119899+1) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2)

(119909119899minus 119909119899+1) (119909119899minus 119909119899minus1) (119909119899minus 119909119899minus2)119910119899

+(119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2)

(119909119899+1

minus 119909119899) (119909119899+1

minus 119909119899minus1) (119909119899+1

minus 119909119899minus2)119910119899+1

(20)

Replacing 119909 = 119904ℎ + 119909119899+2

into (20) produces

119875 (119904ℎ + 119909119899+2)

=(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ)

(minus4ℎ) (minus3ℎ) (minus2ℎ) (minusℎ)119910119899minus2

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 3ℎ) (119904ℎ + 4ℎ)

(minus3ℎ) (minus2ℎ) (minusℎ) (ℎ)119910119899minus1

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 4ℎ)

(minus2ℎ) (minusℎ) (ℎ) (2ℎ)119910119899

+(119904ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 4ℎ)

(minusℎ) (ℎ) (2ℎ) (3ℎ)119910119899+1

+(119904ℎ + ℎℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 4ℎ)

(ℎ) (2ℎ) (3ℎ) (4ℎ)119910119899+2

(21)

The resulting polynomial above is differentiated once withrespect to 119904 at the point 119909 = 119909

119899+2 Substituting 119904 = 0 will

give

1198751015840(119909119899+2) =

25

12119910119899+2

+1

4119910119899minus2

minus4

3119910119899minus1

+ 3119910119899minus 4119910119899+1 (22)

Considering ℎ119891119899+1119899+2

= ℎ1198751015840(119909119899+1119899+2

) the corrector formulaof DI2BBDF(3) is given by

119910119899+1

=2

11119910119899minus2

minus9

11119910119899minus1

+18

11119910119899+6

11ℎ119891119899+1

119910119899+2

= minus3

25119910119899minus2

+16

25119910119899minus1

minus36

25119910119899+48

25119910119899+1

+12

25ℎ119891119899+2

(23)

23 Derivation of DI2BBDF(4) The interpolating points119909119899minus3 119909

119899+1are used to obtain the first point 119910

119899+1of

DI2BBDF(4)

119875 (119909)

=(119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2)

(119909119899minus3minus 119909119899+1) (119909119899minus3minus 119909119899) (119909119899minus3minus 119909119899minus1) (119909119899minus3minus 119909119899minus2)119910119899minus3

+(119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus3)

(119909119899minus2minus 119909119899+1) (119909119899minus2minus 119909119899) (119909119899minus2minus 119909119899minus1) (119909119899minus2minus 119909119899minus3)119910119899minus2

+(119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus2) (119909 minus 119909

119899minus3)

(119909119899minus1minus 119909119899+1) (119909119899minus1minus 119909119899) (119909119899minus1minus 119909119899minus2) (119909119899minus1minus 119909119899minus3)119910119899minus1

+(119909 minus 119909

119899+1) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2) (119909 minus 119909

119899minus3)

(119909119899minus 119909119899+1) (119909119899minus 119909119899minus1) (119909119899minus 119909119899minus2) (119909119899minus 119909119899minus3)119910119899

+(119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2) (119909 minus 119909

119899minus3)

(119909119899+1minus 119909119899) (119909119899+1minus 119909119899minus1) (119909119899+1minus 119909119899minus2) (119909119899+1minus 119909119899minus3)119910119899+1

(24)

Mathematical Problems in Engineering 5

We define 119909 = 119904ℎ + 119909119899+1

and produce

119875 (119904ℎ + 119909119899+1)

=(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ)

(minus4ℎ) (minus3ℎ) (minus2ℎ) (minusℎ)119910119899minus3

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 4ℎ)

(minus3ℎ) (minus2ℎ) (minusℎ) (ℎ)119910119899minus2

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 3ℎ) (119904ℎ + 3ℎ)

(minus2ℎ) (minusℎ) (ℎ) (2ℎ)119910119899minus1

+(119904ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 3ℎ)

(minusℎ) (ℎ) (2ℎ) (3ℎ)119910119899

+(119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 3ℎ)

(ℎ) (2ℎ) (3ℎ) (4ℎ)119910119899+1

(25)

Differentiating (25) once with respect to 119904 at the point 119909 =

119909119899+1

and evaluating 119904 = 0 will produce

1198751015840(119909119899+1) =

25

12119910119899+1

+1

4119910119899minus3

minus4

3119910119899minus2

+ 3119910119899minus1

minus 4119910119899 (26)

The derivation continues for the second point 119910119899+2

of formulaby using 119909

119899minus3 119909

119899+2as the interpolating points

119875 (119909) =(119909 minus 119909

119899+2) (119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2)

(119909119899minus3

minus 119909119899+2) (119909119899minus3

minus 119909119899+1) (119909119899minus3

minus 119909119899) (119909119899minus3

minus 119909119899minus1) (119909119899minus3

minus 119909119899minus2)119910119899minus3

+(119909 minus 119909

119899+2) (119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus3)

(119909119899minus2

minus 119909119899+2) (119909119899minus2

minus 119909119899+1) (119909119899minus2

minus 119909119899) (119909119899minus2

minus 119909119899minus1) (119909119899minus2

minus 119909119899minus3)119910119899minus2

+(119909 minus 119909

119899+2) (119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus2) (119909 minus 119909

119899minus3)

(119909119899minus1

minus 119909119899+2) (119909119899minus1

minus 119909119899+1) (119909119899minus1

minus 119909119899) (119909119899minus1

minus 119909119899minus2) (119909119899minus1

minus 119909119899minus3)119910119899minus1

+(119909 minus 119909

119899+2) (119909 minus 119909

119899+1) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2) (119909 minus 119909

119899minus3)

(119909119899minus 119909119899+2) (119909119899minus 119909119899+1) (119909119899minus 119909119899minus1) (119909119899minus 119909119899minus2) (119909119899minus 119909119899minus3)119910119899

+(119909 minus 119909

119899+2) (119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2) (119909 minus 119909

119899minus3)

(119909119899+1

minus 119909119899+2) (119909119899+1

minus 119909119899) (119909119899+1

minus 119909119899minus1) (119909119899+1

minus 119909119899minus2) (119909119899+1

minus 119909119899minus3)119910119899+1

+(119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2) (119909 minus 119909

119899minus3)

(119909119899+2

minus 119909119899+1) (119909119899+2

minus 119909119899) (119909119899+2

minus 119909119899minus1) (119909119899+2

minus 119909119899minus2) (119909119899+2

minus 119909119899minus3)119910119899+2

(27)

Substituting 119909 = ℎ + 119909119899+2

into (27) will produce

119875 (119904ℎ + 119909119899+2)

=(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 4ℎ)

(minus5ℎ) (minus4ℎ) (minus3ℎ) (minus2ℎ) (minusℎ)119910119899minus3

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 5ℎ)

(minus4ℎ) (minus3ℎ) (minus2ℎ) (minusℎ) (ℎ)119910119899minus2

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 4ℎ) (119904ℎ + 5ℎ)

(minus3ℎ) (minus2ℎ) (minusℎ) (ℎ) (2ℎ)119910119899minus1

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 3ℎ) (119904ℎ + 4ℎ) (119904ℎ + 5ℎ)

(minus2ℎ) (minusℎ) (ℎ) (2ℎ) (3ℎ)119910119899

+(119904ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 4ℎ) (119904ℎ + 5ℎ)

(minusℎ) (ℎ) (2ℎ) (3ℎ) (4ℎ)119910119899+1

+(119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 4ℎ) (119904ℎ + 5ℎ)

(ℎ) (2ℎ) (3ℎ) (4ℎ) (5ℎ)119910119899+2

(28)

Equation (28) is differentiated once with respect to 119904 followedby substituting 119904 = 0 we have

1198751015840(119909119899+2) =

137

60119910119899+2

minus1

5119910119899minus3

+5

4119910119899minus2

minus10

3119910119899minus1

+ 5119910119899minus 5119910119899+1

(29)

Therefore the corrector formula of DI2BBDF(4) is obtainedas follows

119910119899+1

= minus3

25119910119899minus3

+16

25119910119899minus2

minus36

25119910119899minus1

+48

25119910119899+12

25ℎ119891119899+1

119910119899+2

=12

137119910119899minus3

minus75

137119910119899minus2

+200

137119910119899minus1

minus300

137119910119899+300

137119910119899+1

+60

137ℎ119891119899+2

(30)

6 Mathematical Problems in Engineering

3 Stability Analysis

In this section we will plot the graph of stability forDI2BBDF(2) DI2BBDF(3) and DI2BBDF(4) using Mathe-matica software Based onDahlquist [19] the linearmultistepmethod (LMM) is able to solve stiff problems if it satisfies thefollowing definition

Definition 4 TheLMMisA-stable if its region of absolute sta-bility contains the whole of the left-hand half-plane Re(ℎ120582) lt0

We consider the simplest test equation

119891 = 1199101015840= 120582119910 (31)

where the eigenvalues 120582119894 119894 = 1 2 119904 satisfy 120582

119894lt 0 to

analyze the stiff stability of the method Substituting (31) into(16) will produce the following matrices form

[

[

1 0

minus18

111

]

]

[119910119899+1

119910119899+2

]

=[[

[

minus1

3

4

32

11minus9

11

]]

]

[119910119899minus1

119910119899

] + ℎ[[

[

2

30

06

11

]]

]

[119891119899+1

119891119899+2

]

(32)

where

119860 =[[

[

1 minus2

3120582ℎ 0

minus18

111 minus

6

11120582ℎ

]]

]

119861 =[[

[

minus1

3

4

32

11minus9

11

]]

]

119862 = [0 0

0 0]

(33)

Let ℎ = ℎ120582 and evaluating the determinant of |119860119905 minus 119861|from (32) the stability polynomial 120588(119905 ℎ) of DI2BBDF(2) isobtained as follows

120588 (119905 ℎ) =1

33minus34119905

33minus8ℎ119905

11+ 1199052minus40ℎ1199052

33+4ℎ21199052

11 (34)

The similar approach is applied to obtain the stability polyno-mial of DI2BBDF(3) and DI2BBDF(4) Substituting (31) into(23) we have

[

[

1 0

minus48

251

]

]

[119910119899+1

119910119899+2

]

=[[

[

02

11

0 minus3

25

]]

]

[119910119899minus3

119910119899minus2

] +[[

[

minus9

11

18

1116

25minus36

25

]]

]

[119910119899minus1

119910119899

]

+ ℎ[[

[

6

110

012

25

]]

]

[119891119899+1

119891119899+2

]

(35)

where

119860 =[[

[

1 minus6

11120582ℎ 0

minus48

251 minus

12

25120582ℎ

]]

]

119861 =[[

[

minus9

11

18

11

16

25minus36

25

]]

]

119862 =[[

[

02

11

03

25

]]

]

(36)

Substituting (31) into (30) will give

[

[

1 0

minus300

1371

]

]

[119910119899+1

119910119899+2

]

=[[

[

minus3

25

16

25

12

137minus75

137

]]

]

[119910119899minus3

119910119899minus2

] +[[

[

minus36

25

48

25

200

137minus300

137

]]

]

[119910119899minus1

119910119899

]

+ ℎ[[

[

12

250

060

137

]]

]

[119891119899+1

119891119899+2

]

(37)

where

119860 =[[

[

1 minus12

25120582ℎ 0

minus300

1371 minus

60

137120582ℎ

]]

]

119861 =[[

[

minus36

25

48

25

200

137minus300

137

]]

]

119862 =[[

[

minus3

25

16

25

12

137minus75

137

]]

]

(38)

We compute the determinant of |1198601199052 minus 119861119905 minus 119862| from(35) and (37) to obtain the stability polynomials 120588(119905 ℎ) ofDI2BBDF(3) and DI2BBDF(4) respectively Consider

120588 (119905 ℎ) =36

275minus86119905

275minus2431199052

275minus324ℎ119905

2

275minus1621199053

275

+36ℎ1199053

275+ 1199054minus282ℎ119905

4

275+72ℎ21199054

275

(39)

120588 (119905 ℎ) =48

137minus176119905

3425minus3871199052

685minus1152ℎ119905

2

685minus2514119905

3

3425

minus216ℎ119905

3

685+ 1199054minus3144ℎ119905

4

3425+144ℎ21199054

685

(40)

Next the boundary of the stability region will be deter-mined by substituting 119905 = 119890

119894120579 into (34) (39) and (40)The graphs of stability region for all formulas are givenin Figures 1 2 and 3

In Figures 1 2 and 3 we observed that the inter-vals of unstable region for DI2BBDF(2) DI2BBDF(3) and

Mathematical Problems in Engineering 7

Unstable

Unstable

Stable

Stable

Stable1 2 3 4 5

Real part

minus3

minus2

minus1

1

2

3Imaginary part

(533333 0)

Figure 1 Graph of stability region for DI2BBDF(2)

Unstable

Unstable

Stable

Stable

Stable2 4 6 8

Real part

minus4

minus2

2

4

Imaginary part

(866667 0)

Figure 2 Graph of stability region for DI2BBDF(3)

DI2BBDF(4) are [0 533333] [0 866667] and [0 1386667]respectively All the graphs of stability regions are combinedas shown in Figure 4 The region outside the green line is thestable region of DI2BBDF(2) the region outside the red lineis the stable region ofDI2BBDF(3) and the region outside theblue line is the stable region of DI2BBDF(4)

Clearly the unstable region becomes larger whenthe order of the method increases From Definition 4DI2BBDF(2) is A-stable while DI2BBDF(3) andDI2BBDF(4) are almost A-stable since the stability regioncovers the entire negative half plane Therefore we canconclude that the proposed method is suitable for solvingstiff problems

Unstable

Unstable

Stable

Stable

2 4 6 8 10 12 14

minus5

5

(1386667 0)

Stable Real part

Imaginary part

Figure 3 Graph of stability region for DI2BBDF(4)

Imaginary part

Real part5 10

minus5

5

10

DI2BBDF(2)

DI2BBDF(3)

DI2BBDF(4)

A-stable region

A-stable region

Figure 4 Graph of stability region for DI2BBDF(2) DI2BBDF(3)and DI2BBDF(4)

4 Implementation of the Method

In this section the proposed methods will be implementedusing Newton iteration We begin by converting (16) (23)and (30) in general form as follows

119910119899+1

= 1205721119910119899+2

+ 1205731ℎ119891119899+1

+ 1205831

119910119899+2

= 1205722119910119899+1

+ 1205732ℎ119891119899+2

+ 1205832

(41)

where 1205831and 120583

2are the back values Equation (41) is trans-

formed into matrix form as follows

[1 0

0 1] [119910119899+1

119910119899+2

]

= [0 1205721

12057220] [119910119899+1

119910119899+2

] + ℎ[12057310

0 1205732

][119891119899+1

119891119899+2

] + [1205831

1205832

]

(42)

8 Mathematical Problems in Engineering

where

119868 = [1 0

0 1] 119884 = [

119910119899+1

119910119899+2

] 119861 = [12057310

0 1205732

]

119865 = [119891119899+1

119891119899+2

] 120583 = [1205831

1205832

]

(43)

Implementing Newtonrsquos method to (42) produces

119865 = (119868 minus 119860)119884 minus ℎ119861119865 + 120583 (44)

Iteration for (44) is given by

119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2= 119865 (119910

(119894)

119899+1119899+2) sdot 1198651015840(119910(119894)

119899+1119899+2)minus1

(45)

where 119910(119894+1)119899+1

denotes the (119894 + 1)th iteration Equation (45) canbe rearranged into the following equation

1198651015840(119910(119894)

119899+1119899+2) (119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2) = 119865 (119910

(119894)

119899+1119899+2) (46)

Replacing (44) into (46) yields

(119868 minus 119860)119884 minus ℎ119861120597119891

120597119910(119910(119894)

119899+1119899+2) (119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2)

= (119868 minus 119860) (119910(119894)

119899+1119899+2) minus ℎ119861 (119910

(119894)

119899+1119899+2) + 120583

(47)

where (120597119891120597119910)(119910(119894)119899+1119899+2

) is the Jacobian matrix of 119891 with re-spect to 119910

41 Newton Iteration of DI2BBDF(2) Formula (16) is writtenin the form of (41)

1198651= 119910119899+1

minus2

3ℎ119891119899+1

+ 1205831

1198652= 119910119899+2

minus18

11119910119899+1

minus6

11ℎ119891119899+2

+ 1205832

(48)

Applying (48) into (47) will yield matrix form as follows

[[[

[

1 minus2

3ℎ120597119891119899+1

120597119910119899+1

0

0 1 minus6

11ℎ120597119891119899+2

120597119910119899+2

]]]

]

[119890(119894+1)

119899+1

119890(119894+1)

119899+2

]

= [

[

minus1 0

18

11minus1

]

]

[119910(119894)

119899+1

119910(119894)

119899+2

] + ℎ[[

[

2

30

06

11

]]

]

[119891(119894)

119899+1

119891(119894)

119899+2

] + [1205831

1205832

]

(49)

with 119890(119894+1)119899+1119899+2

= 119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2being the increment

42 Newton Iteration of DI2BBDF(3) Rearranging formula(23) in the form of (41) we have

1198651= 119910119899+1

minus6

11ℎ119891119899+1

+ 1205831

1198652= 119910119899+2

minus48

25119910119899+1

minus12

25ℎ119891119899+2

+ 1205832

(50)

Replacing (50) into (47) this will yieldmatrix form as follows

[[[

[

1 minus6

11ℎ120597119891119899+1

120597119910119899+1

0

0 1 minus12

25ℎ120597119891119899+2

120597119910119899+2

]]]

]

[119890(119894+1)

119899+1

119890(119894+1)

119899+2

]

= [

[

minus1 0

48

25minus1

]

]

[119910(119894)

119899+1

119910(119894)

119899+2

] + ℎ[[

[

6

110

012

25

]]

]

[119891(119894)

119899+1

119891(119894)

119899+2

] + [1205831

1205832

]

(51)

with 119890(119894+1)119899+1119899+2

= 119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2being the increment

43 Newton Iteration of DI2BBDF(4) We rewrite formula(30) in the form of (41) and obtain

1198651= 119910119899+1

minus12

25ℎ119891119899+1

+ 1205831

1198652= 119910119899+2

minus300

137119910119899+1

minus60

137ℎ119891119899+2

+ 1205832

(52)

Substituting (52) into (47) will produce

[[[

[

1 minus12

25ℎ120597119891119899+1

120597119910119899+1

0

0 1 minus60

137ℎ120597119891119899+2

120597119910119899+2

]]]

]

[119890(119894+1)

119899+1

119890(119894+1)

119899+2

]

= [

[

minus1 0

300

137minus1

]

]

[119910(119894)

119899+1

119910(119894)

119899+2

] + ℎ[[

[

12

250

060

137

]]

]

[119891(119894)

119899+1

119891(119894)

119899+2

] + [1205831

1205832

]

(53)

with 119890(119894+1)119899+1119899+2

= 119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2being the increment

All the formulas derived are implemented in predictor-corrector computation which is symbolized as PECE modeP and C indicate one application of the predictor and thecorrector respectively and E indicates one evaluation of thefunction119891 given119909 and119910The approximation calculations for119910119899+1

and 119910119899+2

in PECE are as follows

(1) P (Predict) 119910(119901)119899+1119899+2

(2) E (Evaluate) 1199101015840119899+1119899+2

= 119891(119909119899+1119899+2

119910(119901)

119899+1119899+2)

(3) C (Correct) 119910(119888)119899+1119899+2

(4) E (Evaluate) 1199101015840119899+1119899+2

= 119891(119909119899+1119899+2

119910(119888)

119899+1119899+2)

To approximate the solution to 119910(119888)

119899+1119899+2 we apply the

two-stage Newton type iteration The iteration process forDI2BBDF(2) is done as follows

Mathematical Problems in Engineering 9

(1) Compute the values for 119890(119894+1)119899+1119899+2

= 119860minus1119861 where

119860 =[[[

[

1 minus2

3ℎ120597119891119899+1

120597119910119899+1

0

0 1 minus6

11ℎ120597119891119899+2

120597119910119899+2

]]]

]

119861 =[[

[

minus119910(119894)

119899+1+2

3ℎ119891(119894)

119899+1+ 1205831

0

18

11119910(119894)

119899+1minus119910(119894)

119899+2+6

11ℎ119891(119894)

119899+2+ 1205832

]]

]

(54)

(2) Calculate the corrected value for 119910(119894+1)119899+1119899+2

with thevalue 119890(119894+1)

119899+1119899+2from Step (1)

(3) Solve 119890(119894+1)119899+1119899+2

= 119860minus1119861 for second stage iteration

(4) The final values for 119910(119894+1)119899+1119899+2

are obtained from thesecond stage iteration of 119890(119894+1)

119899+1119899+2

The similar iteration process is applied for DI2BBDF(3) andDI2BBDF(4)

5 Test Problems

In this section linear and nonlinear stiff problems aretested using C programming to examine the efficiency andreliability of the proposed method The following problemsare commonly found in engineering and physical sciencesparticularly in the studies of vibrations electrical circuits andchemical reaction

Problem 1 (linear) One has

1199101015840= minus10119910 + 10 (55)

Exact solution is

119910 (119909) = 1 + 119890minus10119909

(56)

Initial condition is

119910 (0) = 2

0 le 119909 le 10

(57)

(source Ibrahim [8])

Problem 2 (linear) One has

1199101015840

1= minus5119910

1minus 41199102 119910

1015840

2= minus4119910

1minus 51199102 (58)

Exact solutions are

1199101(119909) =

1

2119890minus9119909

+1

2119890minus119909 119910

2(119909) =

1

2119890minus9119909

minus1

2119890minus119909 (59)

Initial conditions are

1199101(0) = 1 119910

2(0) = 0

0 le 119909 le 20(60)

(source Musa et al [13])

Problem 3 (linear) One has

1199101015840= minus10119909119910 (61)

Exact solution is

119910 (119909) = 119890minus51199092

(62)

Initial condition is119910 (0) = 1

0 le 119909 le 10

(63)

(source Ibrahim et al [10])

Problem 4 (linear) One has

1199101015840

1= 91199101+ 24119910

2+ 5 cos119909 minus 1

3sin119909

1199101015840

2= minus 24119910

1minus 51119910

2minus 9 cos119909 + 1

3sin119909

(64)

Exact solutions are

1199101(119909) = 2119890

minus3119909minus 119890minus39119909

+1

3cos119909

1199102(119909) = minus119890

minus3119909+ 2119890minus39119909

minus1

3cos119909

(65)

Initial conditions are

1199101(0) =

4

3 1199102(0) =

2

3 0 le 119909 le 10 (66)

(source Musa et al [14])

Problem 5 (nonlinear) One has

1199101015840= minus

1199103

2 (67)

Exact solution is

119910 (119909) =1

radic1 + 119909 (68)

Initial condition is119910 (0) = 1

0 le 119909 le 4

(69)

(source Musa et al [14])

Problem 6 (nonlinear) One has

1199101015840=119910 (1 minus 119910)

2119910 minus 1 (70)

Exact solution is

119910 (119909) =1

2+ radic

1

4minus5

36119890minus119909 (71)

Initial condition is

119910 (0) =5

6

0 le 119909 le 1

(72)

(source Musa et al [14])

10 Mathematical Problems in Engineering

Table 1 The accuracy for problem 1

ℎ Methods MAXE TIME

10minus2

FI2BBDF(3) 597499119864 minus 2 136773119864 minus 3

FI2BEBDF(3) 567155119864 minus 2 145525119864 minus 3

DI2BBDF(2) 110568119864 minus 2 472498119864 minus 3

DI2BBDF(3) 166455119864 minus 2 477409119864 minus 3

DI2BBDF(4) 216342119864 minus 2 483298119864 minus 3

10minus4

FI2BBDF(3) 734012119864 minus 4 125797119864 minus 1

FI2BEBDF(3) 323640119864 minus 5 136259119864 minus 1

DI2BBDF(2) 118355119864 minus 6 167260119864 minus 1

DI2BBDF(3) 173430119864 minus 6 169325119864 minus 1

DI2BBDF(4) 225966119864 minus 6 172995119864 minus 1

10minus6

FI2BBDF(3) 735741119864 minus 6 1257001198641

FI2BEBDF(3) 350090119864 minus 7 1362301198641

DI2BBDF(2) 118742119864 minus 10 1491591198641

DI2BBDF(3) 173634119864 minus 10 1544931198641

DI2BBDF(4) 226000119864 minus 10 1546201198641

Table 2 The accuracy for problem 2

ℎ Methods MAXE TIME

10minus2

FI2BBDF(3) 271597119864 minus 2 596467119864 minus 3

FI2BEBDF(3) 149811119864 minus 2 661025119864 minus 3

DI2BBDF(2) 456073119864 minus 3 104480119864 minus 2

DI2BBDF(3) 686126119864 minus 3 108421119864 minus 2

DI2BBDF(4) 893279119864 minus 3 111089119864 minus 2

10minus4

FI2BBDF(3) 340437119864 minus 4 590430119864 minus 1

FI2BEBDF(3) 262649119864 minus 5 654305119864 minus 1

DI2BBDF(2) 485404119864 minus 7 373210119864 minus 1

DI2BBDF(3) 711155119864 minus 7 386556119864 minus 1

DI2BBDF(4) 926476119864 minus 7 387070119864 minus 1

10minus6

FI2BBDF(3) 341146119864 minus 6 5901981198641

FI2BEBDF(3) 244097119864 minus 7 6541061198641

DI2BBDF(2) 486843119864 minus 11 3484511198641

DI2BBDF(3) 711903119864 minus 11 3604901198641

DI2BBDF(4) 926598119864 minus 11 3606491198641

6 Numerical Results

The performance of the derived methods is compared withthe existing methods in terms of maximum error and exe-cution time We consider 10minus2 10minus4 and 10minus6 as the stepsize ℎ Tables 1 and 2 present the performance compar-ison of DI2BBDF(2) DI2BBDF(3) and DI2BBDF(4) withFI2BBDF(3) and FI2BEBDF(3) whereas Tables 3ndash6 exhibitthe comparison of proposed methods with FI3BBDF(3) andFI3BEBDF(3) In addition the graphs of log(MAXE) againstlog(ℎ) are illustrated as shown in Figures 5ndash10 The followingnotations are used in Tables 1ndash6

ℎ step sizeMAXE maximum errorTIME time execution using high performance com-puter (HPC)

Table 3 The accuracy for problem 3

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 356692119864 minus 2 141642119864 minus 3

FI3BEBDF(3) 124084119864 minus 2 302167119864 minus 3

DI2BBDF(2) 117256119864 minus 3 448489119864 minus 3

DI2BBDF(3) 172278119864 minus 3 481200119864 minus 3

DI2BBDF(4) 225003119864 minus 3 485301119864 minus 3

10minus4

FI3BBDF(3) 435640119864 minus 4 130103119864 minus 1

FI3BEBDF(3) 706516119864 minus 5 208115119864 minus 1

DI2BBDF(2) 118749119864 minus 7 160304119864 minus 1

DI2BBDF(3) 173636119864 minus 7 177445119864 minus 1

DI2BBDF(4) 226000119864 minus 7 178064119864 minus 1

10minus6

FI3BBDF(3) 434625119864 minus 6 1297751198641

FI3BEBDF(3) 703257119864 minus 7 2076091198640

DI2BBDF(2) 124989119864 minus 11 1507391198641

DI2BBDF(3) 173636119864 minus 11 1550731198641

DI2BBDF(4) 225999119864 minus 11 1602161198641

Table 4 The accuracy for problem 4

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 66269411986499 701775119864 minus 3

FI3BEBDF(3) 168449119864 minus 1 137847119864 minus 2

DI2BBDF(2) 280830119864 minus 1 106871119864 minus 2

DI2BBDF(3) 400389119864 minus 1 124969119864 minus 2

DI2BBDF(4) 477342119864 minus 1 126502119864 minus 2

10minus4

FI3BBDF(3) 845376119864 minus 3 676737119864 minus 1

FI3BEBDF(3) 695725119864 minus 3 135671119864 + 0

DI2BBDF(2) 356255119864 minus 5 432359119864 minus 1

DI2BBDF(3) 524241119864 minus 5 442599119864 minus 1

DI2BBDF(4) 685105119864 minus 5 456207119864 minus 1

10minus6

FI3BBDF(3) 854545119864 minus 5 6768191198641

FI3BEBDF(3) 718859119864 minus 5 1358621198642

DI2BBDF(2) 360130119864 minus 9 4000101198641

DI2BBDF(3) 526673119864 minus 9 4001921198641

DI2BBDF(4) 685529119864 minus 9 4103221198641

Table 5 The accuracy for problem 5

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 525481119864 minus 3 578083119864 minus 4

FI3BEBDF(3) 388475119864 minus 3 916583119864 minus 4

DI2BBDF(2) 856253119864 minus 5 229836119864 minus 4

DI2BBDF(3) 121629119864 minus 4 189066119864 minus 4

DI2BBDF(4) 165094119864 minus 4 230074119864 minus 4

10minus4

FI3BBDF(3) 551558119864 minus 5 483860119864 minus 2

FI3BEBDF(3) 427506119864 minus 5 774877119864 minus 2

DI2BBDF(2) 889803119864 minus 9 117438119864 minus 2

DI2BBDF(3) 130124119864 minus 8 131578119864 minus 2

DI2BBDF(4) 169456119864 minus 8 126071119864 minus 2

10minus6

FI3BBDF(3) 551803119864 minus 7 547768119864 + 0

FI3BEBDF(3) 427924119864 minus 7 777643119864 + 0

DI2BBDF(2) 126115119864 minus 11 119318119864 + 0

DI2BBDF(3) 627106119864 minus 11 133892119864 + 0

DI2BBDF(4) 104222119864 minus 11 126146119864 + 0

Mathematical Problems in Engineering 11

Table 6 The accuracy for problem 6

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 218555119864 minus 3 283084119864 minus 4

FI3BEBDF(3) 152719119864 minus 3 428500119864 minus 4

DI2BBDF(2) 387041119864 minus 5 538826119864 minus 5

DI2BBDF(3) 547958119864 minus 5 531673119864 minus 5

DI2BBDF(4) 747696119864 minus 5 618877119864 minus 5

10minus4

FI3BBDF(3) 229812119864 minus 5 148627119864 minus 2

FI3BEBDF(3) 178708119864 minus 5 232184119864 minus 2

DI2BBDF(2) 401638119864 minus 9 349712119864 minus 3

DI2BBDF(3) 587347119864 minus 9 375295119864 minus 3

DI2BBDF(4) 764927119864 minus 9 364804119864 minus 3

10minus6

FI3BBDF(3) 229936119864 minus 7 146660119864 + 0

FI3BEBDF(3) 178973119864 minus 7 228775119864 + 0

DI2BBDF(2) 990807119864 minus 12 338157119864 minus 1

DI2BBDF(3) 377716119864 minus 11 369837119864 minus 1

DI2BBDF(4) 462796119864 minus 12 356802119864 minus 1

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

log(h)minus6 minus5 minus4 minus3 minus2

FI2BBDF(3)FI2BEBDF(3)

DI2BBDF(2)DI2BBDF(3)DI2BBDF(4)

Figure 5 Graph of log(MAXE) versus log(ℎ) for problem 1

DI2BBDF(2) diagonally implicit 2-point block back-ward differentiation formulas of order 2DI2BBDF(3) diagonally implicit 2-point block back-ward differentiation formulas of order 3DI2BBDF(4) diagonally implicit 2-point block back-ward differentiation formulas of order 4FI2BBDF(3) fully implicit 2-point block backwarddifferentiation formulas of order 3FI2BEBDF(3) fully implicit 2-point block extendedbackward differentiation formulas of order 3FI3BBDF(3) fully implicit 3-point block backwarddifferentiation formulas of order 3

log(h)minus6 minus5 minus4 minus3 minus2

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

FI2BBDF(3)FI2BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

Figure 6 Graph of log(MAXE) versus log(ℎ) for problem 2

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus11minus12

minus10minus9minus8minus7minus6minus5minus4minus3minus2minus10

log(

MA

XE)

Figure 7 Graph of log(MAXE) versus log(ℎ) for problem 3

FI3BEBDF(3) fully implicit 3-point block extendedbackward differentiation formulas of order 3

7 Discussion

This section is divided into discussion of maximum error andcomputational time

71 Maximum Error From the numerical results in Tables1ndash6 we observe that DI2BBDF(2) DI2BBDF(3) andDI2BBDF(4) outperformed the existing methods in termsof maximum error This is expected because the diagonally

12 Mathematical Problems in Engineering

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

Figure 8 Graph of log(MAXE) versus log(ℎ) for problem 4

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus11

minus12

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

Figure 9 Graph of log(MAXE) versus log(ℎ) for problem 5

implicit method has less differentiation coefficients in orderto prevent the cumulative error The graphs in Figures 5ndash10depict the scales of maximum error versus step size h for theproposed methods as compared to existing methods Thereis a slight drop in accuracy for the proposed methods as theorder increases This is due to the increase in cumulativeerrors during the computation as more interpolating pointsare used Among the methods of order three DI2BBDF(3)is more accurate compared with FI2BBDF(3) FI3BBDF(3)FI2BEBDF(3) and FI3BEBDF(3) For all test problems we

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus11minus12

minus10minus9minus8minus7minus6minus5minus4minus3minus2minus10

log(

MA

XE)

Figure 10 Graph of log(MAXE) versus log(ℎ) for problem 6

conclude that the accuracy of the proposedmethod increasesas the step size becomes smaller

72 Computational Time In Tables 1-2 it can be seen thatthe execution time taken by DI2BBDF(2) DI2BBDF(3) andDI2BBDF(4) is comparable with that of FI2BBDF(3) andFI2BEBDF(3) The computational time increases as the stepsize becomes smaller However Tables 3ndash6 show that theproposed methods compute faster than FI3BBDF(3) andFI3BEBDF(3) This could be justified by the fact that the3-point method has to perform extra computation on theJacobian matrix since it involves a 3 by 3 matrix while the 2-point method has to compute Jacobian matrix of dimension2 by 2 only Among the proposed methods DI2BBDF(2)computes faster than DI2BBDF(3) and DI2BBDF(4) In factthe number of back values involved in every formula willaffect the computational time Since DI2BBDF(4) has moreback values than DI2BBDF(3) and DI2BBDF(2) it requiresextra time to compute the approximated solutions

8 Conclusion

Research conducted in this paper shows the capability ofconstructing the diagonally implicit BBDF method with A-stable properties From the results obtained via the numericalexperiment we can conclude that the proposed methodserves the purpose of significant alternative numericalmethod for solving linear and nonlinear stiff IVPs occurringin the fields of engineering and physical sciences

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 13

Acknowledgment

This researchwas supported by the Institute forMathematicalResearch (INSPEM) Department of Mathematics UniversitiPutra Malaysia (UPM) under Fundamental Grant Scheme(Project code 02-02-13-1380FR)

References

[1] C F Curtiss and J O Hirschfelder ldquoIntegration of stiff equa-tionsrdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 38 no 3 pp 235ndash243 1952

[2] J D Lambert Numercial Methods for Ordinary DifferentialSystems The Initial Value Problem John Wiley amp Sons NewYork NY USA 1991

[3] W E Milne Numerical Solution of Differential Equations JohnWiley amp Sons New York NY USA 1953

[4] L F Shampine and H A Watts ldquoBlock implicit one-stepmethodsrdquo Mathematics of Computation vol 23 pp 731ndash7401969

[5] S O Fatunla ldquoBlock methods for second order ODEsrdquo Inter-national Journal of Computer Mathematics vol 41 pp 55ndash631990

[6] Z A Majid M B Suleiman F I Ismail and K I Othmanldquo2-Point 1 block diagonally and 2-point 1 block fully implicitmethod for solving first order ordinary differential equationsrdquoin Proceedings of the 12th National Symposium on MathematicalScience vol 12 2004

[7] Z Abdul Majid N Z Mokhtar and M Suleiman ldquoDirecttwo-point block one-step method for solving general second-order ordinary differential equationsrdquo Mathematical Problemsin Engineering vol 2012 Article ID 184253 16 pages 2012

[8] Z B Ibrahim Block method for multistep formulas for solvingordinary differential equations [PhD thesis] Universiti PutraMalaysia Seri Kembangan Malaysia 2006

[9] Z B Ibrahim M Suleiman and K I Othman ldquoFixed coeffi-cients block backward differentiation formulas for the numer-ical solution of stiff ordinary differential equationsrdquo EuropeanJournal of Scientific Research vol 21 no 3 pp 508ndash520 2008

[10] Z B Ibrahim M B Suleiman and K I Othman ldquoImplicit r-point block backward differentiation formula for solving first-order stiff ODEsrdquo Applied Mathematics and Computation vol186 no 1 pp 558ndash565 2007

[11] Z B Ibrahim R Johari and F Ismail ldquoOn the stability of blockbackward differentiation formulaerdquo Matematika vol 19 no 2pp 83ndash89 2003

[12] N A Nasir Z B Ibrahim M B Suleiman and K I OthmanldquoFifth order two-point block backward differentiation formulasfor solving ordinary differential equationsrdquo Applied Mathemat-ical Sciences vol 5 no 69ndash72 pp 3505ndash3518 2011

[13] H Musa M B Suleiman and N Senu ldquoA-stable 2-Point blockextended backward differentiation formulas for stiff ordinarydifferential equationsrdquo in Proceedings of the 5th InternationalConference on Research and education in Mathematics vol 1450of AIP Conference Proceedings pp 254ndash258 2012

[14] H Musa M B Suleiman and N Senu ldquoFully implicit 3-pointblock extended backward differentiation formula for stiff initialvalue problemsrdquo Applied Mathematical Sciences vol 6 no 85ndash88 pp 4211ndash4227 2012

[15] R Alexander ldquoDiagonally implicit Runge-kutta for stiff ordi-nary differential equationsrdquo SIAM Journal on Numerical Analy-sis vol 14 no 6 pp 1006ndash1021 1977

[16] O Y Ababneh R Ahmad and E S Ismail ldquoDesign of newdiagonally implicit Runge-Kutta methods for stiff problemsrdquoApplied Mathematical Sciences vol 3 no 45 pp 2241ndash22532009

[17] F Ismail A L I Raed Al-Khasawneh M Suleiman and A BU Malik Hassan ldquoEmbedded pair of diagonally implicit runge-kutta method for solving ordinary differential equationsrdquo SainsMalaysiana vol 39 no 6 pp 1049ndash1054 2010

[18] Z A Majid and M B Suleiman ldquoPerformance of four pointdiagonally implicit block method for solving first order stiffordinary differential equationsrdquo Department of MathematicsUTM vol 22 no 2 pp 137ndash146 2006

[19] G G Dahlquist ldquoA special stability problem for linearmultistepmethodsrdquo BIT Numerical Mathematics vol 3 no 1 pp 27ndash431963

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Section 5 whereas the performance of the proposed methodis shown in Section 6 Finally in Section 7 some conclusionsare given

2 Formulation of the Method

In this section we will derive the DI2BBDF of order twoorder three and order four with constant step size to computethe approximated solutions at 119910

119899+1and 119910

119899+2concurrently

Contrary to the fully implicit method that has been proposedby Ibrahim et al [11] the first point of diagonally implicitformula has one less interpolating point The derivationusing polynomial 119875

119896(119909) of degree 119896 in terms of Lagrange

polynomial is defined as follows

119875119896(119909) =

119896

sum

119895=0

119871119896119895(119909) 119891 (119909

119899+1minus119895) (8)

where

119871119896119895(119909) =

119896

prod

119894=0

119894 =119895

(119909 minus 119909119899+1minus119894

)

(119909119899+1minus119895

minus 119909119899+1minus119894

)(9)

for each 119895 = 0 1 119896

21 Derivation of DI2BBDF(2) DI2BBDF(2) will computetwo approximated solutions 119910

119899+1and 119910

119899+2simultaneously in

each block using two back values This formula is derivedusing interpolating points 119909

119899minus1 119909

119899+1to obtain the first

formula 119910119899+1

of DI2BBDF(2)

119875 (119909) =(119909 minus 119909

119899) (119909 minus 119909

119899+1)

(119909119899minus1

minus 119909119899) (119909119899minus1

minus 119909119899+1)119910119899minus1

+(119909 minus 119909

119899minus1) (119909 minus 119909

119899+1)

(119909119899minus 119909119899minus1) (119909119899minus 119909119899+1)119910119899

+(119909 minus 119909

119899minus1) (119909 minus 119909

119899)

(119909119899+1

minus 119909119899minus1) (119909119899+1

minus 119909119899)119910119899+1

(10)

Replacing 119909 = 119904ℎ + 119909119899+1

into (10) yields

119875 (119909119899+1

+ 119904ℎ) =(119904ℎ + ℎ) (119904ℎ)

(minusℎ) (minus2ℎ)119910119899minus1

+(2ℎ + 119904ℎ) (119904ℎ)

(ℎ) (minusℎ)119910119899

+(2ℎ + 119904ℎ) (ℎ + 119904ℎ)

(2ℎ) (ℎ)119910119899+1

(11)

Equation (11) is differentiated once with respect to 119904 at thepoint 119909 = 119909

119899+1 Evaluating 119904 = 0 gives

1198751015840(119909119899+1) =

1

2119910119899minus1

minus 2119910119899+3

2119910119899+1 (12)

The same technique is applied for the second point 119910119899+2

ofDI2BBDF(2) This formula is derived using 119909

119899minus1 119909

119899+2as

the interpolating points and produces

119875 (119909) =(119909 minus 119909

119899) (119909 minus 119909

119899+1) (119909 minus 119909

119899+2)

(119909119899minus1

minus 119909119899) (119909119899minus1

minus 119909119899+1) (119909119899minus1

minus 119909119899+2)119910119899minus1

+(119909 minus 119909

119899minus1) (119909 minus 119909

119899+1) (119909 minus 119909

119899+2)

(119909119899minus 119909119899minus1) (119909119899minus 119909119899+1) (119909119899minus 119909119899+2)119910119899

+(119909 minus 119909

119899minus1) (119909 minus 119909

119899) (119909 minus 119909

119899+2)

(119909119899+1

minus 119909119899minus1) (119909119899+1

minus 119909119899) (119909119899+1

minus 119909119899+2)119910119899+1

+(119909 minus 119909

119899minus1) (119909 minus 119909

119899) (119909 minus 119909

119899+1)

(119909119899+2

minus 119909119899minus1) (119909119899+2

minus 119909119899) (119909119899+2

minus 119909119899+1)119910119899+2

(13)

Substituting 119909 = 119904ℎ + 119909119899+2

into (13) yields

119875 (119909119899+2

+ 119904ℎ)

=(119904ℎ + 2ℎ) (119904ℎ + ℎ) (119904ℎ)

(minusℎ) (minus2ℎ) (minus3ℎ)119910119899minus1

+(119904ℎ + 3ℎ) (119904ℎ + ℎ) (119904ℎ)

(ℎ) (minusℎ) (minus2ℎ)119910119899

+(119904ℎ + 3ℎ) (119904ℎ + 2ℎ) (119904ℎ)

(2ℎ) (ℎ) (minusℎ)119910119899+1

+(119904ℎ + 3ℎ) (119904ℎ + 2ℎ) (119904ℎ + ℎ)

(3ℎ) (2ℎ) (ℎ)119910119899+2

(14)

Differentiating (14) with respect to 119904 at the point 119909 = 119909119899+2

gives

1198751015840(119909119899+2) = minus

1

3119910119899minus1

+3

2119910119899minus 3119910119899+1

+11

6119910119899+2 (15)

Considering ℎ119891119899+1119899+2

= 1198751015840(119909119899+1119899+2

) the corrector formulaof DI2BBDF(2) is given as follows

119910119899+1

= minus1

3119910119899minus1

+4

3119910119899+2

3ℎ119891119899+1

119910119899+2

=2

11119910119899minus1

minus9

11119910119899+18

11119910119899+1

+6

11ℎ119891119899+2

(16)

The order of the method is distinguished by the number ofback values contained in the formulas Adopting a similarapproach as the derivation of DI2BBDF of order two wewill construct the DI2BBDF of orders three and four withdifferent number of interpolating points

4 Mathematical Problems in Engineering

22 Derivation of DI2BBDF(3) The first point 119910119899+1

ofDI2BBDF(3) is derived using interpolating points 119909

119899minus2

119909119899+1

and we have

119875 (119909) =(119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus1)

(119909119899minus2

minus 119909119899+1) (119909119899minus2

minus 119909119899) (119909119899minus2

minus 119909119899minus1)119910119899minus2

+(119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus2)

(119909119899minus1

minus 119909119899+1) (119909119899minus1

minus 119909119899) (119909119899minus1

minus 119909119899minus2)119910119899minus1

+(119909 minus 119909

119899+1) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2)

(119909119899minus 119909119899+1) (119909119899minus 119909119899minus1) (119909119899minus 119909119899minus2)119910119899

+(119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2)

(119909119899+1

minus 119909119899) (119909119899+1

minus 119909119899minus1) (119909119899+1

minus 119909119899minus2)119910119899+1

(17)

Substituting 119909 = 119904ℎ + 119909119899+1

into (17) gives

119875 (119904ℎ + 119909119899+1)

=(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ)

(minus3ℎ) (minus2ℎ) (minusℎ)119910119899minus2

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 3ℎ)

(minus2ℎ) (minusℎ) (ℎ)119910119899minus1

+(119904ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ)

(minusℎ) (ℎ) (2ℎ)119910119899

+(119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ)

(ℎ) (2ℎ) (3ℎ)119910119899+1

(18)

Equation (18) is differentiated once with respect to 119904 at thepoint 119909 = 119909

119899+1 Substituting 119904 = 0 will obtain

1198751015840(119909119899+1) =

11

6119910119899+1

minus1

3119910119899minus2

+3

2119910119899minus1

minus 3119910119899 (19)

The derivation process continues for second point 119910119899+2

ofDI2BBDF(3) using 119909

119899minus3 119909

119899+1as the interpolating points

We have

119875 (119909) =(119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus1)

(119909119899minus2

minus 119909119899+1) (119909119899minus2

minus 119909119899) (119909119899minus2

minus 119909119899minus1)119910119899minus2

+(119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus2)

(119909119899minus1

minus 119909119899+1) (119909119899minus1

minus 119909119899) (119909119899minus1

minus 119909119899minus2)119910119899minus1

+(119909 minus 119909

119899+1) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2)

(119909119899minus 119909119899+1) (119909119899minus 119909119899minus1) (119909119899minus 119909119899minus2)119910119899

+(119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2)

(119909119899+1

minus 119909119899) (119909119899+1

minus 119909119899minus1) (119909119899+1

minus 119909119899minus2)119910119899+1

(20)

Replacing 119909 = 119904ℎ + 119909119899+2

into (20) produces

119875 (119904ℎ + 119909119899+2)

=(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ)

(minus4ℎ) (minus3ℎ) (minus2ℎ) (minusℎ)119910119899minus2

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 3ℎ) (119904ℎ + 4ℎ)

(minus3ℎ) (minus2ℎ) (minusℎ) (ℎ)119910119899minus1

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 4ℎ)

(minus2ℎ) (minusℎ) (ℎ) (2ℎ)119910119899

+(119904ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 4ℎ)

(minusℎ) (ℎ) (2ℎ) (3ℎ)119910119899+1

+(119904ℎ + ℎℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 4ℎ)

(ℎ) (2ℎ) (3ℎ) (4ℎ)119910119899+2

(21)

The resulting polynomial above is differentiated once withrespect to 119904 at the point 119909 = 119909

119899+2 Substituting 119904 = 0 will

give

1198751015840(119909119899+2) =

25

12119910119899+2

+1

4119910119899minus2

minus4

3119910119899minus1

+ 3119910119899minus 4119910119899+1 (22)

Considering ℎ119891119899+1119899+2

= ℎ1198751015840(119909119899+1119899+2

) the corrector formulaof DI2BBDF(3) is given by

119910119899+1

=2

11119910119899minus2

minus9

11119910119899minus1

+18

11119910119899+6

11ℎ119891119899+1

119910119899+2

= minus3

25119910119899minus2

+16

25119910119899minus1

minus36

25119910119899+48

25119910119899+1

+12

25ℎ119891119899+2

(23)

23 Derivation of DI2BBDF(4) The interpolating points119909119899minus3 119909

119899+1are used to obtain the first point 119910

119899+1of

DI2BBDF(4)

119875 (119909)

=(119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2)

(119909119899minus3minus 119909119899+1) (119909119899minus3minus 119909119899) (119909119899minus3minus 119909119899minus1) (119909119899minus3minus 119909119899minus2)119910119899minus3

+(119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus3)

(119909119899minus2minus 119909119899+1) (119909119899minus2minus 119909119899) (119909119899minus2minus 119909119899minus1) (119909119899minus2minus 119909119899minus3)119910119899minus2

+(119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus2) (119909 minus 119909

119899minus3)

(119909119899minus1minus 119909119899+1) (119909119899minus1minus 119909119899) (119909119899minus1minus 119909119899minus2) (119909119899minus1minus 119909119899minus3)119910119899minus1

+(119909 minus 119909

119899+1) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2) (119909 minus 119909

119899minus3)

(119909119899minus 119909119899+1) (119909119899minus 119909119899minus1) (119909119899minus 119909119899minus2) (119909119899minus 119909119899minus3)119910119899

+(119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2) (119909 minus 119909

119899minus3)

(119909119899+1minus 119909119899) (119909119899+1minus 119909119899minus1) (119909119899+1minus 119909119899minus2) (119909119899+1minus 119909119899minus3)119910119899+1

(24)

Mathematical Problems in Engineering 5

We define 119909 = 119904ℎ + 119909119899+1

and produce

119875 (119904ℎ + 119909119899+1)

=(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ)

(minus4ℎ) (minus3ℎ) (minus2ℎ) (minusℎ)119910119899minus3

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 4ℎ)

(minus3ℎ) (minus2ℎ) (minusℎ) (ℎ)119910119899minus2

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 3ℎ) (119904ℎ + 3ℎ)

(minus2ℎ) (minusℎ) (ℎ) (2ℎ)119910119899minus1

+(119904ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 3ℎ)

(minusℎ) (ℎ) (2ℎ) (3ℎ)119910119899

+(119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 3ℎ)

(ℎ) (2ℎ) (3ℎ) (4ℎ)119910119899+1

(25)

Differentiating (25) once with respect to 119904 at the point 119909 =

119909119899+1

and evaluating 119904 = 0 will produce

1198751015840(119909119899+1) =

25

12119910119899+1

+1

4119910119899minus3

minus4

3119910119899minus2

+ 3119910119899minus1

minus 4119910119899 (26)

The derivation continues for the second point 119910119899+2

of formulaby using 119909

119899minus3 119909

119899+2as the interpolating points

119875 (119909) =(119909 minus 119909

119899+2) (119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2)

(119909119899minus3

minus 119909119899+2) (119909119899minus3

minus 119909119899+1) (119909119899minus3

minus 119909119899) (119909119899minus3

minus 119909119899minus1) (119909119899minus3

minus 119909119899minus2)119910119899minus3

+(119909 minus 119909

119899+2) (119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus3)

(119909119899minus2

minus 119909119899+2) (119909119899minus2

minus 119909119899+1) (119909119899minus2

minus 119909119899) (119909119899minus2

minus 119909119899minus1) (119909119899minus2

minus 119909119899minus3)119910119899minus2

+(119909 minus 119909

119899+2) (119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus2) (119909 minus 119909

119899minus3)

(119909119899minus1

minus 119909119899+2) (119909119899minus1

minus 119909119899+1) (119909119899minus1

minus 119909119899) (119909119899minus1

minus 119909119899minus2) (119909119899minus1

minus 119909119899minus3)119910119899minus1

+(119909 minus 119909

119899+2) (119909 minus 119909

119899+1) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2) (119909 minus 119909

119899minus3)

(119909119899minus 119909119899+2) (119909119899minus 119909119899+1) (119909119899minus 119909119899minus1) (119909119899minus 119909119899minus2) (119909119899minus 119909119899minus3)119910119899

+(119909 minus 119909

119899+2) (119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2) (119909 minus 119909

119899minus3)

(119909119899+1

minus 119909119899+2) (119909119899+1

minus 119909119899) (119909119899+1

minus 119909119899minus1) (119909119899+1

minus 119909119899minus2) (119909119899+1

minus 119909119899minus3)119910119899+1

+(119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2) (119909 minus 119909

119899minus3)

(119909119899+2

minus 119909119899+1) (119909119899+2

minus 119909119899) (119909119899+2

minus 119909119899minus1) (119909119899+2

minus 119909119899minus2) (119909119899+2

minus 119909119899minus3)119910119899+2

(27)

Substituting 119909 = ℎ + 119909119899+2

into (27) will produce

119875 (119904ℎ + 119909119899+2)

=(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 4ℎ)

(minus5ℎ) (minus4ℎ) (minus3ℎ) (minus2ℎ) (minusℎ)119910119899minus3

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 5ℎ)

(minus4ℎ) (minus3ℎ) (minus2ℎ) (minusℎ) (ℎ)119910119899minus2

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 4ℎ) (119904ℎ + 5ℎ)

(minus3ℎ) (minus2ℎ) (minusℎ) (ℎ) (2ℎ)119910119899minus1

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 3ℎ) (119904ℎ + 4ℎ) (119904ℎ + 5ℎ)

(minus2ℎ) (minusℎ) (ℎ) (2ℎ) (3ℎ)119910119899

+(119904ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 4ℎ) (119904ℎ + 5ℎ)

(minusℎ) (ℎ) (2ℎ) (3ℎ) (4ℎ)119910119899+1

+(119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 4ℎ) (119904ℎ + 5ℎ)

(ℎ) (2ℎ) (3ℎ) (4ℎ) (5ℎ)119910119899+2

(28)

Equation (28) is differentiated once with respect to 119904 followedby substituting 119904 = 0 we have

1198751015840(119909119899+2) =

137

60119910119899+2

minus1

5119910119899minus3

+5

4119910119899minus2

minus10

3119910119899minus1

+ 5119910119899minus 5119910119899+1

(29)

Therefore the corrector formula of DI2BBDF(4) is obtainedas follows

119910119899+1

= minus3

25119910119899minus3

+16

25119910119899minus2

minus36

25119910119899minus1

+48

25119910119899+12

25ℎ119891119899+1

119910119899+2

=12

137119910119899minus3

minus75

137119910119899minus2

+200

137119910119899minus1

minus300

137119910119899+300

137119910119899+1

+60

137ℎ119891119899+2

(30)

6 Mathematical Problems in Engineering

3 Stability Analysis

In this section we will plot the graph of stability forDI2BBDF(2) DI2BBDF(3) and DI2BBDF(4) using Mathe-matica software Based onDahlquist [19] the linearmultistepmethod (LMM) is able to solve stiff problems if it satisfies thefollowing definition

Definition 4 TheLMMisA-stable if its region of absolute sta-bility contains the whole of the left-hand half-plane Re(ℎ120582) lt0

We consider the simplest test equation

119891 = 1199101015840= 120582119910 (31)

where the eigenvalues 120582119894 119894 = 1 2 119904 satisfy 120582

119894lt 0 to

analyze the stiff stability of the method Substituting (31) into(16) will produce the following matrices form

[

[

1 0

minus18

111

]

]

[119910119899+1

119910119899+2

]

=[[

[

minus1

3

4

32

11minus9

11

]]

]

[119910119899minus1

119910119899

] + ℎ[[

[

2

30

06

11

]]

]

[119891119899+1

119891119899+2

]

(32)

where

119860 =[[

[

1 minus2

3120582ℎ 0

minus18

111 minus

6

11120582ℎ

]]

]

119861 =[[

[

minus1

3

4

32

11minus9

11

]]

]

119862 = [0 0

0 0]

(33)

Let ℎ = ℎ120582 and evaluating the determinant of |119860119905 minus 119861|from (32) the stability polynomial 120588(119905 ℎ) of DI2BBDF(2) isobtained as follows

120588 (119905 ℎ) =1

33minus34119905

33minus8ℎ119905

11+ 1199052minus40ℎ1199052

33+4ℎ21199052

11 (34)

The similar approach is applied to obtain the stability polyno-mial of DI2BBDF(3) and DI2BBDF(4) Substituting (31) into(23) we have

[

[

1 0

minus48

251

]

]

[119910119899+1

119910119899+2

]

=[[

[

02

11

0 minus3

25

]]

]

[119910119899minus3

119910119899minus2

] +[[

[

minus9

11

18

1116

25minus36

25

]]

]

[119910119899minus1

119910119899

]

+ ℎ[[

[

6

110

012

25

]]

]

[119891119899+1

119891119899+2

]

(35)

where

119860 =[[

[

1 minus6

11120582ℎ 0

minus48

251 minus

12

25120582ℎ

]]

]

119861 =[[

[

minus9

11

18

11

16

25minus36

25

]]

]

119862 =[[

[

02

11

03

25

]]

]

(36)

Substituting (31) into (30) will give

[

[

1 0

minus300

1371

]

]

[119910119899+1

119910119899+2

]

=[[

[

minus3

25

16

25

12

137minus75

137

]]

]

[119910119899minus3

119910119899minus2

] +[[

[

minus36

25

48

25

200

137minus300

137

]]

]

[119910119899minus1

119910119899

]

+ ℎ[[

[

12

250

060

137

]]

]

[119891119899+1

119891119899+2

]

(37)

where

119860 =[[

[

1 minus12

25120582ℎ 0

minus300

1371 minus

60

137120582ℎ

]]

]

119861 =[[

[

minus36

25

48

25

200

137minus300

137

]]

]

119862 =[[

[

minus3

25

16

25

12

137minus75

137

]]

]

(38)

We compute the determinant of |1198601199052 minus 119861119905 minus 119862| from(35) and (37) to obtain the stability polynomials 120588(119905 ℎ) ofDI2BBDF(3) and DI2BBDF(4) respectively Consider

120588 (119905 ℎ) =36

275minus86119905

275minus2431199052

275minus324ℎ119905

2

275minus1621199053

275

+36ℎ1199053

275+ 1199054minus282ℎ119905

4

275+72ℎ21199054

275

(39)

120588 (119905 ℎ) =48

137minus176119905

3425minus3871199052

685minus1152ℎ119905

2

685minus2514119905

3

3425

minus216ℎ119905

3

685+ 1199054minus3144ℎ119905

4

3425+144ℎ21199054

685

(40)

Next the boundary of the stability region will be deter-mined by substituting 119905 = 119890

119894120579 into (34) (39) and (40)The graphs of stability region for all formulas are givenin Figures 1 2 and 3

In Figures 1 2 and 3 we observed that the inter-vals of unstable region for DI2BBDF(2) DI2BBDF(3) and

Mathematical Problems in Engineering 7

Unstable

Unstable

Stable

Stable

Stable1 2 3 4 5

Real part

minus3

minus2

minus1

1

2

3Imaginary part

(533333 0)

Figure 1 Graph of stability region for DI2BBDF(2)

Unstable

Unstable

Stable

Stable

Stable2 4 6 8

Real part

minus4

minus2

2

4

Imaginary part

(866667 0)

Figure 2 Graph of stability region for DI2BBDF(3)

DI2BBDF(4) are [0 533333] [0 866667] and [0 1386667]respectively All the graphs of stability regions are combinedas shown in Figure 4 The region outside the green line is thestable region of DI2BBDF(2) the region outside the red lineis the stable region ofDI2BBDF(3) and the region outside theblue line is the stable region of DI2BBDF(4)

Clearly the unstable region becomes larger whenthe order of the method increases From Definition 4DI2BBDF(2) is A-stable while DI2BBDF(3) andDI2BBDF(4) are almost A-stable since the stability regioncovers the entire negative half plane Therefore we canconclude that the proposed method is suitable for solvingstiff problems

Unstable

Unstable

Stable

Stable

2 4 6 8 10 12 14

minus5

5

(1386667 0)

Stable Real part

Imaginary part

Figure 3 Graph of stability region for DI2BBDF(4)

Imaginary part

Real part5 10

minus5

5

10

DI2BBDF(2)

DI2BBDF(3)

DI2BBDF(4)

A-stable region

A-stable region

Figure 4 Graph of stability region for DI2BBDF(2) DI2BBDF(3)and DI2BBDF(4)

4 Implementation of the Method

In this section the proposed methods will be implementedusing Newton iteration We begin by converting (16) (23)and (30) in general form as follows

119910119899+1

= 1205721119910119899+2

+ 1205731ℎ119891119899+1

+ 1205831

119910119899+2

= 1205722119910119899+1

+ 1205732ℎ119891119899+2

+ 1205832

(41)

where 1205831and 120583

2are the back values Equation (41) is trans-

formed into matrix form as follows

[1 0

0 1] [119910119899+1

119910119899+2

]

= [0 1205721

12057220] [119910119899+1

119910119899+2

] + ℎ[12057310

0 1205732

][119891119899+1

119891119899+2

] + [1205831

1205832

]

(42)

8 Mathematical Problems in Engineering

where

119868 = [1 0

0 1] 119884 = [

119910119899+1

119910119899+2

] 119861 = [12057310

0 1205732

]

119865 = [119891119899+1

119891119899+2

] 120583 = [1205831

1205832

]

(43)

Implementing Newtonrsquos method to (42) produces

119865 = (119868 minus 119860)119884 minus ℎ119861119865 + 120583 (44)

Iteration for (44) is given by

119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2= 119865 (119910

(119894)

119899+1119899+2) sdot 1198651015840(119910(119894)

119899+1119899+2)minus1

(45)

where 119910(119894+1)119899+1

denotes the (119894 + 1)th iteration Equation (45) canbe rearranged into the following equation

1198651015840(119910(119894)

119899+1119899+2) (119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2) = 119865 (119910

(119894)

119899+1119899+2) (46)

Replacing (44) into (46) yields

(119868 minus 119860)119884 minus ℎ119861120597119891

120597119910(119910(119894)

119899+1119899+2) (119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2)

= (119868 minus 119860) (119910(119894)

119899+1119899+2) minus ℎ119861 (119910

(119894)

119899+1119899+2) + 120583

(47)

where (120597119891120597119910)(119910(119894)119899+1119899+2

) is the Jacobian matrix of 119891 with re-spect to 119910

41 Newton Iteration of DI2BBDF(2) Formula (16) is writtenin the form of (41)

1198651= 119910119899+1

minus2

3ℎ119891119899+1

+ 1205831

1198652= 119910119899+2

minus18

11119910119899+1

minus6

11ℎ119891119899+2

+ 1205832

(48)

Applying (48) into (47) will yield matrix form as follows

[[[

[

1 minus2

3ℎ120597119891119899+1

120597119910119899+1

0

0 1 minus6

11ℎ120597119891119899+2

120597119910119899+2

]]]

]

[119890(119894+1)

119899+1

119890(119894+1)

119899+2

]

= [

[

minus1 0

18

11minus1

]

]

[119910(119894)

119899+1

119910(119894)

119899+2

] + ℎ[[

[

2

30

06

11

]]

]

[119891(119894)

119899+1

119891(119894)

119899+2

] + [1205831

1205832

]

(49)

with 119890(119894+1)119899+1119899+2

= 119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2being the increment

42 Newton Iteration of DI2BBDF(3) Rearranging formula(23) in the form of (41) we have

1198651= 119910119899+1

minus6

11ℎ119891119899+1

+ 1205831

1198652= 119910119899+2

minus48

25119910119899+1

minus12

25ℎ119891119899+2

+ 1205832

(50)

Replacing (50) into (47) this will yieldmatrix form as follows

[[[

[

1 minus6

11ℎ120597119891119899+1

120597119910119899+1

0

0 1 minus12

25ℎ120597119891119899+2

120597119910119899+2

]]]

]

[119890(119894+1)

119899+1

119890(119894+1)

119899+2

]

= [

[

minus1 0

48

25minus1

]

]

[119910(119894)

119899+1

119910(119894)

119899+2

] + ℎ[[

[

6

110

012

25

]]

]

[119891(119894)

119899+1

119891(119894)

119899+2

] + [1205831

1205832

]

(51)

with 119890(119894+1)119899+1119899+2

= 119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2being the increment

43 Newton Iteration of DI2BBDF(4) We rewrite formula(30) in the form of (41) and obtain

1198651= 119910119899+1

minus12

25ℎ119891119899+1

+ 1205831

1198652= 119910119899+2

minus300

137119910119899+1

minus60

137ℎ119891119899+2

+ 1205832

(52)

Substituting (52) into (47) will produce

[[[

[

1 minus12

25ℎ120597119891119899+1

120597119910119899+1

0

0 1 minus60

137ℎ120597119891119899+2

120597119910119899+2

]]]

]

[119890(119894+1)

119899+1

119890(119894+1)

119899+2

]

= [

[

minus1 0

300

137minus1

]

]

[119910(119894)

119899+1

119910(119894)

119899+2

] + ℎ[[

[

12

250

060

137

]]

]

[119891(119894)

119899+1

119891(119894)

119899+2

] + [1205831

1205832

]

(53)

with 119890(119894+1)119899+1119899+2

= 119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2being the increment

All the formulas derived are implemented in predictor-corrector computation which is symbolized as PECE modeP and C indicate one application of the predictor and thecorrector respectively and E indicates one evaluation of thefunction119891 given119909 and119910The approximation calculations for119910119899+1

and 119910119899+2

in PECE are as follows

(1) P (Predict) 119910(119901)119899+1119899+2

(2) E (Evaluate) 1199101015840119899+1119899+2

= 119891(119909119899+1119899+2

119910(119901)

119899+1119899+2)

(3) C (Correct) 119910(119888)119899+1119899+2

(4) E (Evaluate) 1199101015840119899+1119899+2

= 119891(119909119899+1119899+2

119910(119888)

119899+1119899+2)

To approximate the solution to 119910(119888)

119899+1119899+2 we apply the

two-stage Newton type iteration The iteration process forDI2BBDF(2) is done as follows

Mathematical Problems in Engineering 9

(1) Compute the values for 119890(119894+1)119899+1119899+2

= 119860minus1119861 where

119860 =[[[

[

1 minus2

3ℎ120597119891119899+1

120597119910119899+1

0

0 1 minus6

11ℎ120597119891119899+2

120597119910119899+2

]]]

]

119861 =[[

[

minus119910(119894)

119899+1+2

3ℎ119891(119894)

119899+1+ 1205831

0

18

11119910(119894)

119899+1minus119910(119894)

119899+2+6

11ℎ119891(119894)

119899+2+ 1205832

]]

]

(54)

(2) Calculate the corrected value for 119910(119894+1)119899+1119899+2

with thevalue 119890(119894+1)

119899+1119899+2from Step (1)

(3) Solve 119890(119894+1)119899+1119899+2

= 119860minus1119861 for second stage iteration

(4) The final values for 119910(119894+1)119899+1119899+2

are obtained from thesecond stage iteration of 119890(119894+1)

119899+1119899+2

The similar iteration process is applied for DI2BBDF(3) andDI2BBDF(4)

5 Test Problems

In this section linear and nonlinear stiff problems aretested using C programming to examine the efficiency andreliability of the proposed method The following problemsare commonly found in engineering and physical sciencesparticularly in the studies of vibrations electrical circuits andchemical reaction

Problem 1 (linear) One has

1199101015840= minus10119910 + 10 (55)

Exact solution is

119910 (119909) = 1 + 119890minus10119909

(56)

Initial condition is

119910 (0) = 2

0 le 119909 le 10

(57)

(source Ibrahim [8])

Problem 2 (linear) One has

1199101015840

1= minus5119910

1minus 41199102 119910

1015840

2= minus4119910

1minus 51199102 (58)

Exact solutions are

1199101(119909) =

1

2119890minus9119909

+1

2119890minus119909 119910

2(119909) =

1

2119890minus9119909

minus1

2119890minus119909 (59)

Initial conditions are

1199101(0) = 1 119910

2(0) = 0

0 le 119909 le 20(60)

(source Musa et al [13])

Problem 3 (linear) One has

1199101015840= minus10119909119910 (61)

Exact solution is

119910 (119909) = 119890minus51199092

(62)

Initial condition is119910 (0) = 1

0 le 119909 le 10

(63)

(source Ibrahim et al [10])

Problem 4 (linear) One has

1199101015840

1= 91199101+ 24119910

2+ 5 cos119909 minus 1

3sin119909

1199101015840

2= minus 24119910

1minus 51119910

2minus 9 cos119909 + 1

3sin119909

(64)

Exact solutions are

1199101(119909) = 2119890

minus3119909minus 119890minus39119909

+1

3cos119909

1199102(119909) = minus119890

minus3119909+ 2119890minus39119909

minus1

3cos119909

(65)

Initial conditions are

1199101(0) =

4

3 1199102(0) =

2

3 0 le 119909 le 10 (66)

(source Musa et al [14])

Problem 5 (nonlinear) One has

1199101015840= minus

1199103

2 (67)

Exact solution is

119910 (119909) =1

radic1 + 119909 (68)

Initial condition is119910 (0) = 1

0 le 119909 le 4

(69)

(source Musa et al [14])

Problem 6 (nonlinear) One has

1199101015840=119910 (1 minus 119910)

2119910 minus 1 (70)

Exact solution is

119910 (119909) =1

2+ radic

1

4minus5

36119890minus119909 (71)

Initial condition is

119910 (0) =5

6

0 le 119909 le 1

(72)

(source Musa et al [14])

10 Mathematical Problems in Engineering

Table 1 The accuracy for problem 1

ℎ Methods MAXE TIME

10minus2

FI2BBDF(3) 597499119864 minus 2 136773119864 minus 3

FI2BEBDF(3) 567155119864 minus 2 145525119864 minus 3

DI2BBDF(2) 110568119864 minus 2 472498119864 minus 3

DI2BBDF(3) 166455119864 minus 2 477409119864 minus 3

DI2BBDF(4) 216342119864 minus 2 483298119864 minus 3

10minus4

FI2BBDF(3) 734012119864 minus 4 125797119864 minus 1

FI2BEBDF(3) 323640119864 minus 5 136259119864 minus 1

DI2BBDF(2) 118355119864 minus 6 167260119864 minus 1

DI2BBDF(3) 173430119864 minus 6 169325119864 minus 1

DI2BBDF(4) 225966119864 minus 6 172995119864 minus 1

10minus6

FI2BBDF(3) 735741119864 minus 6 1257001198641

FI2BEBDF(3) 350090119864 minus 7 1362301198641

DI2BBDF(2) 118742119864 minus 10 1491591198641

DI2BBDF(3) 173634119864 minus 10 1544931198641

DI2BBDF(4) 226000119864 minus 10 1546201198641

Table 2 The accuracy for problem 2

ℎ Methods MAXE TIME

10minus2

FI2BBDF(3) 271597119864 minus 2 596467119864 minus 3

FI2BEBDF(3) 149811119864 minus 2 661025119864 minus 3

DI2BBDF(2) 456073119864 minus 3 104480119864 minus 2

DI2BBDF(3) 686126119864 minus 3 108421119864 minus 2

DI2BBDF(4) 893279119864 minus 3 111089119864 minus 2

10minus4

FI2BBDF(3) 340437119864 minus 4 590430119864 minus 1

FI2BEBDF(3) 262649119864 minus 5 654305119864 minus 1

DI2BBDF(2) 485404119864 minus 7 373210119864 minus 1

DI2BBDF(3) 711155119864 minus 7 386556119864 minus 1

DI2BBDF(4) 926476119864 minus 7 387070119864 minus 1

10minus6

FI2BBDF(3) 341146119864 minus 6 5901981198641

FI2BEBDF(3) 244097119864 minus 7 6541061198641

DI2BBDF(2) 486843119864 minus 11 3484511198641

DI2BBDF(3) 711903119864 minus 11 3604901198641

DI2BBDF(4) 926598119864 minus 11 3606491198641

6 Numerical Results

The performance of the derived methods is compared withthe existing methods in terms of maximum error and exe-cution time We consider 10minus2 10minus4 and 10minus6 as the stepsize ℎ Tables 1 and 2 present the performance compar-ison of DI2BBDF(2) DI2BBDF(3) and DI2BBDF(4) withFI2BBDF(3) and FI2BEBDF(3) whereas Tables 3ndash6 exhibitthe comparison of proposed methods with FI3BBDF(3) andFI3BEBDF(3) In addition the graphs of log(MAXE) againstlog(ℎ) are illustrated as shown in Figures 5ndash10 The followingnotations are used in Tables 1ndash6

ℎ step sizeMAXE maximum errorTIME time execution using high performance com-puter (HPC)

Table 3 The accuracy for problem 3

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 356692119864 minus 2 141642119864 minus 3

FI3BEBDF(3) 124084119864 minus 2 302167119864 minus 3

DI2BBDF(2) 117256119864 minus 3 448489119864 minus 3

DI2BBDF(3) 172278119864 minus 3 481200119864 minus 3

DI2BBDF(4) 225003119864 minus 3 485301119864 minus 3

10minus4

FI3BBDF(3) 435640119864 minus 4 130103119864 minus 1

FI3BEBDF(3) 706516119864 minus 5 208115119864 minus 1

DI2BBDF(2) 118749119864 minus 7 160304119864 minus 1

DI2BBDF(3) 173636119864 minus 7 177445119864 minus 1

DI2BBDF(4) 226000119864 minus 7 178064119864 minus 1

10minus6

FI3BBDF(3) 434625119864 minus 6 1297751198641

FI3BEBDF(3) 703257119864 minus 7 2076091198640

DI2BBDF(2) 124989119864 minus 11 1507391198641

DI2BBDF(3) 173636119864 minus 11 1550731198641

DI2BBDF(4) 225999119864 minus 11 1602161198641

Table 4 The accuracy for problem 4

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 66269411986499 701775119864 minus 3

FI3BEBDF(3) 168449119864 minus 1 137847119864 minus 2

DI2BBDF(2) 280830119864 minus 1 106871119864 minus 2

DI2BBDF(3) 400389119864 minus 1 124969119864 minus 2

DI2BBDF(4) 477342119864 minus 1 126502119864 minus 2

10minus4

FI3BBDF(3) 845376119864 minus 3 676737119864 minus 1

FI3BEBDF(3) 695725119864 minus 3 135671119864 + 0

DI2BBDF(2) 356255119864 minus 5 432359119864 minus 1

DI2BBDF(3) 524241119864 minus 5 442599119864 minus 1

DI2BBDF(4) 685105119864 minus 5 456207119864 minus 1

10minus6

FI3BBDF(3) 854545119864 minus 5 6768191198641

FI3BEBDF(3) 718859119864 minus 5 1358621198642

DI2BBDF(2) 360130119864 minus 9 4000101198641

DI2BBDF(3) 526673119864 minus 9 4001921198641

DI2BBDF(4) 685529119864 minus 9 4103221198641

Table 5 The accuracy for problem 5

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 525481119864 minus 3 578083119864 minus 4

FI3BEBDF(3) 388475119864 minus 3 916583119864 minus 4

DI2BBDF(2) 856253119864 minus 5 229836119864 minus 4

DI2BBDF(3) 121629119864 minus 4 189066119864 minus 4

DI2BBDF(4) 165094119864 minus 4 230074119864 minus 4

10minus4

FI3BBDF(3) 551558119864 minus 5 483860119864 minus 2

FI3BEBDF(3) 427506119864 minus 5 774877119864 minus 2

DI2BBDF(2) 889803119864 minus 9 117438119864 minus 2

DI2BBDF(3) 130124119864 minus 8 131578119864 minus 2

DI2BBDF(4) 169456119864 minus 8 126071119864 minus 2

10minus6

FI3BBDF(3) 551803119864 minus 7 547768119864 + 0

FI3BEBDF(3) 427924119864 minus 7 777643119864 + 0

DI2BBDF(2) 126115119864 minus 11 119318119864 + 0

DI2BBDF(3) 627106119864 minus 11 133892119864 + 0

DI2BBDF(4) 104222119864 minus 11 126146119864 + 0

Mathematical Problems in Engineering 11

Table 6 The accuracy for problem 6

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 218555119864 minus 3 283084119864 minus 4

FI3BEBDF(3) 152719119864 minus 3 428500119864 minus 4

DI2BBDF(2) 387041119864 minus 5 538826119864 minus 5

DI2BBDF(3) 547958119864 minus 5 531673119864 minus 5

DI2BBDF(4) 747696119864 minus 5 618877119864 minus 5

10minus4

FI3BBDF(3) 229812119864 minus 5 148627119864 minus 2

FI3BEBDF(3) 178708119864 minus 5 232184119864 minus 2

DI2BBDF(2) 401638119864 minus 9 349712119864 minus 3

DI2BBDF(3) 587347119864 minus 9 375295119864 minus 3

DI2BBDF(4) 764927119864 minus 9 364804119864 minus 3

10minus6

FI3BBDF(3) 229936119864 minus 7 146660119864 + 0

FI3BEBDF(3) 178973119864 minus 7 228775119864 + 0

DI2BBDF(2) 990807119864 minus 12 338157119864 minus 1

DI2BBDF(3) 377716119864 minus 11 369837119864 minus 1

DI2BBDF(4) 462796119864 minus 12 356802119864 minus 1

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

log(h)minus6 minus5 minus4 minus3 minus2

FI2BBDF(3)FI2BEBDF(3)

DI2BBDF(2)DI2BBDF(3)DI2BBDF(4)

Figure 5 Graph of log(MAXE) versus log(ℎ) for problem 1

DI2BBDF(2) diagonally implicit 2-point block back-ward differentiation formulas of order 2DI2BBDF(3) diagonally implicit 2-point block back-ward differentiation formulas of order 3DI2BBDF(4) diagonally implicit 2-point block back-ward differentiation formulas of order 4FI2BBDF(3) fully implicit 2-point block backwarddifferentiation formulas of order 3FI2BEBDF(3) fully implicit 2-point block extendedbackward differentiation formulas of order 3FI3BBDF(3) fully implicit 3-point block backwarddifferentiation formulas of order 3

log(h)minus6 minus5 minus4 minus3 minus2

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

FI2BBDF(3)FI2BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

Figure 6 Graph of log(MAXE) versus log(ℎ) for problem 2

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus11minus12

minus10minus9minus8minus7minus6minus5minus4minus3minus2minus10

log(

MA

XE)

Figure 7 Graph of log(MAXE) versus log(ℎ) for problem 3

FI3BEBDF(3) fully implicit 3-point block extendedbackward differentiation formulas of order 3

7 Discussion

This section is divided into discussion of maximum error andcomputational time

71 Maximum Error From the numerical results in Tables1ndash6 we observe that DI2BBDF(2) DI2BBDF(3) andDI2BBDF(4) outperformed the existing methods in termsof maximum error This is expected because the diagonally

12 Mathematical Problems in Engineering

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

Figure 8 Graph of log(MAXE) versus log(ℎ) for problem 4

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus11

minus12

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

Figure 9 Graph of log(MAXE) versus log(ℎ) for problem 5

implicit method has less differentiation coefficients in orderto prevent the cumulative error The graphs in Figures 5ndash10depict the scales of maximum error versus step size h for theproposed methods as compared to existing methods Thereis a slight drop in accuracy for the proposed methods as theorder increases This is due to the increase in cumulativeerrors during the computation as more interpolating pointsare used Among the methods of order three DI2BBDF(3)is more accurate compared with FI2BBDF(3) FI3BBDF(3)FI2BEBDF(3) and FI3BEBDF(3) For all test problems we

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus11minus12

minus10minus9minus8minus7minus6minus5minus4minus3minus2minus10

log(

MA

XE)

Figure 10 Graph of log(MAXE) versus log(ℎ) for problem 6

conclude that the accuracy of the proposedmethod increasesas the step size becomes smaller

72 Computational Time In Tables 1-2 it can be seen thatthe execution time taken by DI2BBDF(2) DI2BBDF(3) andDI2BBDF(4) is comparable with that of FI2BBDF(3) andFI2BEBDF(3) The computational time increases as the stepsize becomes smaller However Tables 3ndash6 show that theproposed methods compute faster than FI3BBDF(3) andFI3BEBDF(3) This could be justified by the fact that the3-point method has to perform extra computation on theJacobian matrix since it involves a 3 by 3 matrix while the 2-point method has to compute Jacobian matrix of dimension2 by 2 only Among the proposed methods DI2BBDF(2)computes faster than DI2BBDF(3) and DI2BBDF(4) In factthe number of back values involved in every formula willaffect the computational time Since DI2BBDF(4) has moreback values than DI2BBDF(3) and DI2BBDF(2) it requiresextra time to compute the approximated solutions

8 Conclusion

Research conducted in this paper shows the capability ofconstructing the diagonally implicit BBDF method with A-stable properties From the results obtained via the numericalexperiment we can conclude that the proposed methodserves the purpose of significant alternative numericalmethod for solving linear and nonlinear stiff IVPs occurringin the fields of engineering and physical sciences

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 13

Acknowledgment

This researchwas supported by the Institute forMathematicalResearch (INSPEM) Department of Mathematics UniversitiPutra Malaysia (UPM) under Fundamental Grant Scheme(Project code 02-02-13-1380FR)

References

[1] C F Curtiss and J O Hirschfelder ldquoIntegration of stiff equa-tionsrdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 38 no 3 pp 235ndash243 1952

[2] J D Lambert Numercial Methods for Ordinary DifferentialSystems The Initial Value Problem John Wiley amp Sons NewYork NY USA 1991

[3] W E Milne Numerical Solution of Differential Equations JohnWiley amp Sons New York NY USA 1953

[4] L F Shampine and H A Watts ldquoBlock implicit one-stepmethodsrdquo Mathematics of Computation vol 23 pp 731ndash7401969

[5] S O Fatunla ldquoBlock methods for second order ODEsrdquo Inter-national Journal of Computer Mathematics vol 41 pp 55ndash631990

[6] Z A Majid M B Suleiman F I Ismail and K I Othmanldquo2-Point 1 block diagonally and 2-point 1 block fully implicitmethod for solving first order ordinary differential equationsrdquoin Proceedings of the 12th National Symposium on MathematicalScience vol 12 2004

[7] Z Abdul Majid N Z Mokhtar and M Suleiman ldquoDirecttwo-point block one-step method for solving general second-order ordinary differential equationsrdquo Mathematical Problemsin Engineering vol 2012 Article ID 184253 16 pages 2012

[8] Z B Ibrahim Block method for multistep formulas for solvingordinary differential equations [PhD thesis] Universiti PutraMalaysia Seri Kembangan Malaysia 2006

[9] Z B Ibrahim M Suleiman and K I Othman ldquoFixed coeffi-cients block backward differentiation formulas for the numer-ical solution of stiff ordinary differential equationsrdquo EuropeanJournal of Scientific Research vol 21 no 3 pp 508ndash520 2008

[10] Z B Ibrahim M B Suleiman and K I Othman ldquoImplicit r-point block backward differentiation formula for solving first-order stiff ODEsrdquo Applied Mathematics and Computation vol186 no 1 pp 558ndash565 2007

[11] Z B Ibrahim R Johari and F Ismail ldquoOn the stability of blockbackward differentiation formulaerdquo Matematika vol 19 no 2pp 83ndash89 2003

[12] N A Nasir Z B Ibrahim M B Suleiman and K I OthmanldquoFifth order two-point block backward differentiation formulasfor solving ordinary differential equationsrdquo Applied Mathemat-ical Sciences vol 5 no 69ndash72 pp 3505ndash3518 2011

[13] H Musa M B Suleiman and N Senu ldquoA-stable 2-Point blockextended backward differentiation formulas for stiff ordinarydifferential equationsrdquo in Proceedings of the 5th InternationalConference on Research and education in Mathematics vol 1450of AIP Conference Proceedings pp 254ndash258 2012

[14] H Musa M B Suleiman and N Senu ldquoFully implicit 3-pointblock extended backward differentiation formula for stiff initialvalue problemsrdquo Applied Mathematical Sciences vol 6 no 85ndash88 pp 4211ndash4227 2012

[15] R Alexander ldquoDiagonally implicit Runge-kutta for stiff ordi-nary differential equationsrdquo SIAM Journal on Numerical Analy-sis vol 14 no 6 pp 1006ndash1021 1977

[16] O Y Ababneh R Ahmad and E S Ismail ldquoDesign of newdiagonally implicit Runge-Kutta methods for stiff problemsrdquoApplied Mathematical Sciences vol 3 no 45 pp 2241ndash22532009

[17] F Ismail A L I Raed Al-Khasawneh M Suleiman and A BU Malik Hassan ldquoEmbedded pair of diagonally implicit runge-kutta method for solving ordinary differential equationsrdquo SainsMalaysiana vol 39 no 6 pp 1049ndash1054 2010

[18] Z A Majid and M B Suleiman ldquoPerformance of four pointdiagonally implicit block method for solving first order stiffordinary differential equationsrdquo Department of MathematicsUTM vol 22 no 2 pp 137ndash146 2006

[19] G G Dahlquist ldquoA special stability problem for linearmultistepmethodsrdquo BIT Numerical Mathematics vol 3 no 1 pp 27ndash431963

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

22 Derivation of DI2BBDF(3) The first point 119910119899+1

ofDI2BBDF(3) is derived using interpolating points 119909

119899minus2

119909119899+1

and we have

119875 (119909) =(119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus1)

(119909119899minus2

minus 119909119899+1) (119909119899minus2

minus 119909119899) (119909119899minus2

minus 119909119899minus1)119910119899minus2

+(119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus2)

(119909119899minus1

minus 119909119899+1) (119909119899minus1

minus 119909119899) (119909119899minus1

minus 119909119899minus2)119910119899minus1

+(119909 minus 119909

119899+1) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2)

(119909119899minus 119909119899+1) (119909119899minus 119909119899minus1) (119909119899minus 119909119899minus2)119910119899

+(119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2)

(119909119899+1

minus 119909119899) (119909119899+1

minus 119909119899minus1) (119909119899+1

minus 119909119899minus2)119910119899+1

(17)

Substituting 119909 = 119904ℎ + 119909119899+1

into (17) gives

119875 (119904ℎ + 119909119899+1)

=(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ)

(minus3ℎ) (minus2ℎ) (minusℎ)119910119899minus2

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 3ℎ)

(minus2ℎ) (minusℎ) (ℎ)119910119899minus1

+(119904ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ)

(minusℎ) (ℎ) (2ℎ)119910119899

+(119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ)

(ℎ) (2ℎ) (3ℎ)119910119899+1

(18)

Equation (18) is differentiated once with respect to 119904 at thepoint 119909 = 119909

119899+1 Substituting 119904 = 0 will obtain

1198751015840(119909119899+1) =

11

6119910119899+1

minus1

3119910119899minus2

+3

2119910119899minus1

minus 3119910119899 (19)

The derivation process continues for second point 119910119899+2

ofDI2BBDF(3) using 119909

119899minus3 119909

119899+1as the interpolating points

We have

119875 (119909) =(119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus1)

(119909119899minus2

minus 119909119899+1) (119909119899minus2

minus 119909119899) (119909119899minus2

minus 119909119899minus1)119910119899minus2

+(119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus2)

(119909119899minus1

minus 119909119899+1) (119909119899minus1

minus 119909119899) (119909119899minus1

minus 119909119899minus2)119910119899minus1

+(119909 minus 119909

119899+1) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2)

(119909119899minus 119909119899+1) (119909119899minus 119909119899minus1) (119909119899minus 119909119899minus2)119910119899

+(119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2)

(119909119899+1

minus 119909119899) (119909119899+1

minus 119909119899minus1) (119909119899+1

minus 119909119899minus2)119910119899+1

(20)

Replacing 119909 = 119904ℎ + 119909119899+2

into (20) produces

119875 (119904ℎ + 119909119899+2)

=(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ)

(minus4ℎ) (minus3ℎ) (minus2ℎ) (minusℎ)119910119899minus2

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 3ℎ) (119904ℎ + 4ℎ)

(minus3ℎ) (minus2ℎ) (minusℎ) (ℎ)119910119899minus1

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 4ℎ)

(minus2ℎ) (minusℎ) (ℎ) (2ℎ)119910119899

+(119904ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 4ℎ)

(minusℎ) (ℎ) (2ℎ) (3ℎ)119910119899+1

+(119904ℎ + ℎℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 4ℎ)

(ℎ) (2ℎ) (3ℎ) (4ℎ)119910119899+2

(21)

The resulting polynomial above is differentiated once withrespect to 119904 at the point 119909 = 119909

119899+2 Substituting 119904 = 0 will

give

1198751015840(119909119899+2) =

25

12119910119899+2

+1

4119910119899minus2

minus4

3119910119899minus1

+ 3119910119899minus 4119910119899+1 (22)

Considering ℎ119891119899+1119899+2

= ℎ1198751015840(119909119899+1119899+2

) the corrector formulaof DI2BBDF(3) is given by

119910119899+1

=2

11119910119899minus2

minus9

11119910119899minus1

+18

11119910119899+6

11ℎ119891119899+1

119910119899+2

= minus3

25119910119899minus2

+16

25119910119899minus1

minus36

25119910119899+48

25119910119899+1

+12

25ℎ119891119899+2

(23)

23 Derivation of DI2BBDF(4) The interpolating points119909119899minus3 119909

119899+1are used to obtain the first point 119910

119899+1of

DI2BBDF(4)

119875 (119909)

=(119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2)

(119909119899minus3minus 119909119899+1) (119909119899minus3minus 119909119899) (119909119899minus3minus 119909119899minus1) (119909119899minus3minus 119909119899minus2)119910119899minus3

+(119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus3)

(119909119899minus2minus 119909119899+1) (119909119899minus2minus 119909119899) (119909119899minus2minus 119909119899minus1) (119909119899minus2minus 119909119899minus3)119910119899minus2

+(119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus2) (119909 minus 119909

119899minus3)

(119909119899minus1minus 119909119899+1) (119909119899minus1minus 119909119899) (119909119899minus1minus 119909119899minus2) (119909119899minus1minus 119909119899minus3)119910119899minus1

+(119909 minus 119909

119899+1) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2) (119909 minus 119909

119899minus3)

(119909119899minus 119909119899+1) (119909119899minus 119909119899minus1) (119909119899minus 119909119899minus2) (119909119899minus 119909119899minus3)119910119899

+(119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2) (119909 minus 119909

119899minus3)

(119909119899+1minus 119909119899) (119909119899+1minus 119909119899minus1) (119909119899+1minus 119909119899minus2) (119909119899+1minus 119909119899minus3)119910119899+1

(24)

Mathematical Problems in Engineering 5

We define 119909 = 119904ℎ + 119909119899+1

and produce

119875 (119904ℎ + 119909119899+1)

=(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ)

(minus4ℎ) (minus3ℎ) (minus2ℎ) (minusℎ)119910119899minus3

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 4ℎ)

(minus3ℎ) (minus2ℎ) (minusℎ) (ℎ)119910119899minus2

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 3ℎ) (119904ℎ + 3ℎ)

(minus2ℎ) (minusℎ) (ℎ) (2ℎ)119910119899minus1

+(119904ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 3ℎ)

(minusℎ) (ℎ) (2ℎ) (3ℎ)119910119899

+(119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 3ℎ)

(ℎ) (2ℎ) (3ℎ) (4ℎ)119910119899+1

(25)

Differentiating (25) once with respect to 119904 at the point 119909 =

119909119899+1

and evaluating 119904 = 0 will produce

1198751015840(119909119899+1) =

25

12119910119899+1

+1

4119910119899minus3

minus4

3119910119899minus2

+ 3119910119899minus1

minus 4119910119899 (26)

The derivation continues for the second point 119910119899+2

of formulaby using 119909

119899minus3 119909

119899+2as the interpolating points

119875 (119909) =(119909 minus 119909

119899+2) (119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2)

(119909119899minus3

minus 119909119899+2) (119909119899minus3

minus 119909119899+1) (119909119899minus3

minus 119909119899) (119909119899minus3

minus 119909119899minus1) (119909119899minus3

minus 119909119899minus2)119910119899minus3

+(119909 minus 119909

119899+2) (119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus3)

(119909119899minus2

minus 119909119899+2) (119909119899minus2

minus 119909119899+1) (119909119899minus2

minus 119909119899) (119909119899minus2

minus 119909119899minus1) (119909119899minus2

minus 119909119899minus3)119910119899minus2

+(119909 minus 119909

119899+2) (119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus2) (119909 minus 119909

119899minus3)

(119909119899minus1

minus 119909119899+2) (119909119899minus1

minus 119909119899+1) (119909119899minus1

minus 119909119899) (119909119899minus1

minus 119909119899minus2) (119909119899minus1

minus 119909119899minus3)119910119899minus1

+(119909 minus 119909

119899+2) (119909 minus 119909

119899+1) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2) (119909 minus 119909

119899minus3)

(119909119899minus 119909119899+2) (119909119899minus 119909119899+1) (119909119899minus 119909119899minus1) (119909119899minus 119909119899minus2) (119909119899minus 119909119899minus3)119910119899

+(119909 minus 119909

119899+2) (119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2) (119909 minus 119909

119899minus3)

(119909119899+1

minus 119909119899+2) (119909119899+1

minus 119909119899) (119909119899+1

minus 119909119899minus1) (119909119899+1

minus 119909119899minus2) (119909119899+1

minus 119909119899minus3)119910119899+1

+(119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2) (119909 minus 119909

119899minus3)

(119909119899+2

minus 119909119899+1) (119909119899+2

minus 119909119899) (119909119899+2

minus 119909119899minus1) (119909119899+2

minus 119909119899minus2) (119909119899+2

minus 119909119899minus3)119910119899+2

(27)

Substituting 119909 = ℎ + 119909119899+2

into (27) will produce

119875 (119904ℎ + 119909119899+2)

=(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 4ℎ)

(minus5ℎ) (minus4ℎ) (minus3ℎ) (minus2ℎ) (minusℎ)119910119899minus3

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 5ℎ)

(minus4ℎ) (minus3ℎ) (minus2ℎ) (minusℎ) (ℎ)119910119899minus2

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 4ℎ) (119904ℎ + 5ℎ)

(minus3ℎ) (minus2ℎ) (minusℎ) (ℎ) (2ℎ)119910119899minus1

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 3ℎ) (119904ℎ + 4ℎ) (119904ℎ + 5ℎ)

(minus2ℎ) (minusℎ) (ℎ) (2ℎ) (3ℎ)119910119899

+(119904ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 4ℎ) (119904ℎ + 5ℎ)

(minusℎ) (ℎ) (2ℎ) (3ℎ) (4ℎ)119910119899+1

+(119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 4ℎ) (119904ℎ + 5ℎ)

(ℎ) (2ℎ) (3ℎ) (4ℎ) (5ℎ)119910119899+2

(28)

Equation (28) is differentiated once with respect to 119904 followedby substituting 119904 = 0 we have

1198751015840(119909119899+2) =

137

60119910119899+2

minus1

5119910119899minus3

+5

4119910119899minus2

minus10

3119910119899minus1

+ 5119910119899minus 5119910119899+1

(29)

Therefore the corrector formula of DI2BBDF(4) is obtainedas follows

119910119899+1

= minus3

25119910119899minus3

+16

25119910119899minus2

minus36

25119910119899minus1

+48

25119910119899+12

25ℎ119891119899+1

119910119899+2

=12

137119910119899minus3

minus75

137119910119899minus2

+200

137119910119899minus1

minus300

137119910119899+300

137119910119899+1

+60

137ℎ119891119899+2

(30)

6 Mathematical Problems in Engineering

3 Stability Analysis

In this section we will plot the graph of stability forDI2BBDF(2) DI2BBDF(3) and DI2BBDF(4) using Mathe-matica software Based onDahlquist [19] the linearmultistepmethod (LMM) is able to solve stiff problems if it satisfies thefollowing definition

Definition 4 TheLMMisA-stable if its region of absolute sta-bility contains the whole of the left-hand half-plane Re(ℎ120582) lt0

We consider the simplest test equation

119891 = 1199101015840= 120582119910 (31)

where the eigenvalues 120582119894 119894 = 1 2 119904 satisfy 120582

119894lt 0 to

analyze the stiff stability of the method Substituting (31) into(16) will produce the following matrices form

[

[

1 0

minus18

111

]

]

[119910119899+1

119910119899+2

]

=[[

[

minus1

3

4

32

11minus9

11

]]

]

[119910119899minus1

119910119899

] + ℎ[[

[

2

30

06

11

]]

]

[119891119899+1

119891119899+2

]

(32)

where

119860 =[[

[

1 minus2

3120582ℎ 0

minus18

111 minus

6

11120582ℎ

]]

]

119861 =[[

[

minus1

3

4

32

11minus9

11

]]

]

119862 = [0 0

0 0]

(33)

Let ℎ = ℎ120582 and evaluating the determinant of |119860119905 minus 119861|from (32) the stability polynomial 120588(119905 ℎ) of DI2BBDF(2) isobtained as follows

120588 (119905 ℎ) =1

33minus34119905

33minus8ℎ119905

11+ 1199052minus40ℎ1199052

33+4ℎ21199052

11 (34)

The similar approach is applied to obtain the stability polyno-mial of DI2BBDF(3) and DI2BBDF(4) Substituting (31) into(23) we have

[

[

1 0

minus48

251

]

]

[119910119899+1

119910119899+2

]

=[[

[

02

11

0 minus3

25

]]

]

[119910119899minus3

119910119899minus2

] +[[

[

minus9

11

18

1116

25minus36

25

]]

]

[119910119899minus1

119910119899

]

+ ℎ[[

[

6

110

012

25

]]

]

[119891119899+1

119891119899+2

]

(35)

where

119860 =[[

[

1 minus6

11120582ℎ 0

minus48

251 minus

12

25120582ℎ

]]

]

119861 =[[

[

minus9

11

18

11

16

25minus36

25

]]

]

119862 =[[

[

02

11

03

25

]]

]

(36)

Substituting (31) into (30) will give

[

[

1 0

minus300

1371

]

]

[119910119899+1

119910119899+2

]

=[[

[

minus3

25

16

25

12

137minus75

137

]]

]

[119910119899minus3

119910119899minus2

] +[[

[

minus36

25

48

25

200

137minus300

137

]]

]

[119910119899minus1

119910119899

]

+ ℎ[[

[

12

250

060

137

]]

]

[119891119899+1

119891119899+2

]

(37)

where

119860 =[[

[

1 minus12

25120582ℎ 0

minus300

1371 minus

60

137120582ℎ

]]

]

119861 =[[

[

minus36

25

48

25

200

137minus300

137

]]

]

119862 =[[

[

minus3

25

16

25

12

137minus75

137

]]

]

(38)

We compute the determinant of |1198601199052 minus 119861119905 minus 119862| from(35) and (37) to obtain the stability polynomials 120588(119905 ℎ) ofDI2BBDF(3) and DI2BBDF(4) respectively Consider

120588 (119905 ℎ) =36

275minus86119905

275minus2431199052

275minus324ℎ119905

2

275minus1621199053

275

+36ℎ1199053

275+ 1199054minus282ℎ119905

4

275+72ℎ21199054

275

(39)

120588 (119905 ℎ) =48

137minus176119905

3425minus3871199052

685minus1152ℎ119905

2

685minus2514119905

3

3425

minus216ℎ119905

3

685+ 1199054minus3144ℎ119905

4

3425+144ℎ21199054

685

(40)

Next the boundary of the stability region will be deter-mined by substituting 119905 = 119890

119894120579 into (34) (39) and (40)The graphs of stability region for all formulas are givenin Figures 1 2 and 3

In Figures 1 2 and 3 we observed that the inter-vals of unstable region for DI2BBDF(2) DI2BBDF(3) and

Mathematical Problems in Engineering 7

Unstable

Unstable

Stable

Stable

Stable1 2 3 4 5

Real part

minus3

minus2

minus1

1

2

3Imaginary part

(533333 0)

Figure 1 Graph of stability region for DI2BBDF(2)

Unstable

Unstable

Stable

Stable

Stable2 4 6 8

Real part

minus4

minus2

2

4

Imaginary part

(866667 0)

Figure 2 Graph of stability region for DI2BBDF(3)

DI2BBDF(4) are [0 533333] [0 866667] and [0 1386667]respectively All the graphs of stability regions are combinedas shown in Figure 4 The region outside the green line is thestable region of DI2BBDF(2) the region outside the red lineis the stable region ofDI2BBDF(3) and the region outside theblue line is the stable region of DI2BBDF(4)

Clearly the unstable region becomes larger whenthe order of the method increases From Definition 4DI2BBDF(2) is A-stable while DI2BBDF(3) andDI2BBDF(4) are almost A-stable since the stability regioncovers the entire negative half plane Therefore we canconclude that the proposed method is suitable for solvingstiff problems

Unstable

Unstable

Stable

Stable

2 4 6 8 10 12 14

minus5

5

(1386667 0)

Stable Real part

Imaginary part

Figure 3 Graph of stability region for DI2BBDF(4)

Imaginary part

Real part5 10

minus5

5

10

DI2BBDF(2)

DI2BBDF(3)

DI2BBDF(4)

A-stable region

A-stable region

Figure 4 Graph of stability region for DI2BBDF(2) DI2BBDF(3)and DI2BBDF(4)

4 Implementation of the Method

In this section the proposed methods will be implementedusing Newton iteration We begin by converting (16) (23)and (30) in general form as follows

119910119899+1

= 1205721119910119899+2

+ 1205731ℎ119891119899+1

+ 1205831

119910119899+2

= 1205722119910119899+1

+ 1205732ℎ119891119899+2

+ 1205832

(41)

where 1205831and 120583

2are the back values Equation (41) is trans-

formed into matrix form as follows

[1 0

0 1] [119910119899+1

119910119899+2

]

= [0 1205721

12057220] [119910119899+1

119910119899+2

] + ℎ[12057310

0 1205732

][119891119899+1

119891119899+2

] + [1205831

1205832

]

(42)

8 Mathematical Problems in Engineering

where

119868 = [1 0

0 1] 119884 = [

119910119899+1

119910119899+2

] 119861 = [12057310

0 1205732

]

119865 = [119891119899+1

119891119899+2

] 120583 = [1205831

1205832

]

(43)

Implementing Newtonrsquos method to (42) produces

119865 = (119868 minus 119860)119884 minus ℎ119861119865 + 120583 (44)

Iteration for (44) is given by

119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2= 119865 (119910

(119894)

119899+1119899+2) sdot 1198651015840(119910(119894)

119899+1119899+2)minus1

(45)

where 119910(119894+1)119899+1

denotes the (119894 + 1)th iteration Equation (45) canbe rearranged into the following equation

1198651015840(119910(119894)

119899+1119899+2) (119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2) = 119865 (119910

(119894)

119899+1119899+2) (46)

Replacing (44) into (46) yields

(119868 minus 119860)119884 minus ℎ119861120597119891

120597119910(119910(119894)

119899+1119899+2) (119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2)

= (119868 minus 119860) (119910(119894)

119899+1119899+2) minus ℎ119861 (119910

(119894)

119899+1119899+2) + 120583

(47)

where (120597119891120597119910)(119910(119894)119899+1119899+2

) is the Jacobian matrix of 119891 with re-spect to 119910

41 Newton Iteration of DI2BBDF(2) Formula (16) is writtenin the form of (41)

1198651= 119910119899+1

minus2

3ℎ119891119899+1

+ 1205831

1198652= 119910119899+2

minus18

11119910119899+1

minus6

11ℎ119891119899+2

+ 1205832

(48)

Applying (48) into (47) will yield matrix form as follows

[[[

[

1 minus2

3ℎ120597119891119899+1

120597119910119899+1

0

0 1 minus6

11ℎ120597119891119899+2

120597119910119899+2

]]]

]

[119890(119894+1)

119899+1

119890(119894+1)

119899+2

]

= [

[

minus1 0

18

11minus1

]

]

[119910(119894)

119899+1

119910(119894)

119899+2

] + ℎ[[

[

2

30

06

11

]]

]

[119891(119894)

119899+1

119891(119894)

119899+2

] + [1205831

1205832

]

(49)

with 119890(119894+1)119899+1119899+2

= 119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2being the increment

42 Newton Iteration of DI2BBDF(3) Rearranging formula(23) in the form of (41) we have

1198651= 119910119899+1

minus6

11ℎ119891119899+1

+ 1205831

1198652= 119910119899+2

minus48

25119910119899+1

minus12

25ℎ119891119899+2

+ 1205832

(50)

Replacing (50) into (47) this will yieldmatrix form as follows

[[[

[

1 minus6

11ℎ120597119891119899+1

120597119910119899+1

0

0 1 minus12

25ℎ120597119891119899+2

120597119910119899+2

]]]

]

[119890(119894+1)

119899+1

119890(119894+1)

119899+2

]

= [

[

minus1 0

48

25minus1

]

]

[119910(119894)

119899+1

119910(119894)

119899+2

] + ℎ[[

[

6

110

012

25

]]

]

[119891(119894)

119899+1

119891(119894)

119899+2

] + [1205831

1205832

]

(51)

with 119890(119894+1)119899+1119899+2

= 119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2being the increment

43 Newton Iteration of DI2BBDF(4) We rewrite formula(30) in the form of (41) and obtain

1198651= 119910119899+1

minus12

25ℎ119891119899+1

+ 1205831

1198652= 119910119899+2

minus300

137119910119899+1

minus60

137ℎ119891119899+2

+ 1205832

(52)

Substituting (52) into (47) will produce

[[[

[

1 minus12

25ℎ120597119891119899+1

120597119910119899+1

0

0 1 minus60

137ℎ120597119891119899+2

120597119910119899+2

]]]

]

[119890(119894+1)

119899+1

119890(119894+1)

119899+2

]

= [

[

minus1 0

300

137minus1

]

]

[119910(119894)

119899+1

119910(119894)

119899+2

] + ℎ[[

[

12

250

060

137

]]

]

[119891(119894)

119899+1

119891(119894)

119899+2

] + [1205831

1205832

]

(53)

with 119890(119894+1)119899+1119899+2

= 119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2being the increment

All the formulas derived are implemented in predictor-corrector computation which is symbolized as PECE modeP and C indicate one application of the predictor and thecorrector respectively and E indicates one evaluation of thefunction119891 given119909 and119910The approximation calculations for119910119899+1

and 119910119899+2

in PECE are as follows

(1) P (Predict) 119910(119901)119899+1119899+2

(2) E (Evaluate) 1199101015840119899+1119899+2

= 119891(119909119899+1119899+2

119910(119901)

119899+1119899+2)

(3) C (Correct) 119910(119888)119899+1119899+2

(4) E (Evaluate) 1199101015840119899+1119899+2

= 119891(119909119899+1119899+2

119910(119888)

119899+1119899+2)

To approximate the solution to 119910(119888)

119899+1119899+2 we apply the

two-stage Newton type iteration The iteration process forDI2BBDF(2) is done as follows

Mathematical Problems in Engineering 9

(1) Compute the values for 119890(119894+1)119899+1119899+2

= 119860minus1119861 where

119860 =[[[

[

1 minus2

3ℎ120597119891119899+1

120597119910119899+1

0

0 1 minus6

11ℎ120597119891119899+2

120597119910119899+2

]]]

]

119861 =[[

[

minus119910(119894)

119899+1+2

3ℎ119891(119894)

119899+1+ 1205831

0

18

11119910(119894)

119899+1minus119910(119894)

119899+2+6

11ℎ119891(119894)

119899+2+ 1205832

]]

]

(54)

(2) Calculate the corrected value for 119910(119894+1)119899+1119899+2

with thevalue 119890(119894+1)

119899+1119899+2from Step (1)

(3) Solve 119890(119894+1)119899+1119899+2

= 119860minus1119861 for second stage iteration

(4) The final values for 119910(119894+1)119899+1119899+2

are obtained from thesecond stage iteration of 119890(119894+1)

119899+1119899+2

The similar iteration process is applied for DI2BBDF(3) andDI2BBDF(4)

5 Test Problems

In this section linear and nonlinear stiff problems aretested using C programming to examine the efficiency andreliability of the proposed method The following problemsare commonly found in engineering and physical sciencesparticularly in the studies of vibrations electrical circuits andchemical reaction

Problem 1 (linear) One has

1199101015840= minus10119910 + 10 (55)

Exact solution is

119910 (119909) = 1 + 119890minus10119909

(56)

Initial condition is

119910 (0) = 2

0 le 119909 le 10

(57)

(source Ibrahim [8])

Problem 2 (linear) One has

1199101015840

1= minus5119910

1minus 41199102 119910

1015840

2= minus4119910

1minus 51199102 (58)

Exact solutions are

1199101(119909) =

1

2119890minus9119909

+1

2119890minus119909 119910

2(119909) =

1

2119890minus9119909

minus1

2119890minus119909 (59)

Initial conditions are

1199101(0) = 1 119910

2(0) = 0

0 le 119909 le 20(60)

(source Musa et al [13])

Problem 3 (linear) One has

1199101015840= minus10119909119910 (61)

Exact solution is

119910 (119909) = 119890minus51199092

(62)

Initial condition is119910 (0) = 1

0 le 119909 le 10

(63)

(source Ibrahim et al [10])

Problem 4 (linear) One has

1199101015840

1= 91199101+ 24119910

2+ 5 cos119909 minus 1

3sin119909

1199101015840

2= minus 24119910

1minus 51119910

2minus 9 cos119909 + 1

3sin119909

(64)

Exact solutions are

1199101(119909) = 2119890

minus3119909minus 119890minus39119909

+1

3cos119909

1199102(119909) = minus119890

minus3119909+ 2119890minus39119909

minus1

3cos119909

(65)

Initial conditions are

1199101(0) =

4

3 1199102(0) =

2

3 0 le 119909 le 10 (66)

(source Musa et al [14])

Problem 5 (nonlinear) One has

1199101015840= minus

1199103

2 (67)

Exact solution is

119910 (119909) =1

radic1 + 119909 (68)

Initial condition is119910 (0) = 1

0 le 119909 le 4

(69)

(source Musa et al [14])

Problem 6 (nonlinear) One has

1199101015840=119910 (1 minus 119910)

2119910 minus 1 (70)

Exact solution is

119910 (119909) =1

2+ radic

1

4minus5

36119890minus119909 (71)

Initial condition is

119910 (0) =5

6

0 le 119909 le 1

(72)

(source Musa et al [14])

10 Mathematical Problems in Engineering

Table 1 The accuracy for problem 1

ℎ Methods MAXE TIME

10minus2

FI2BBDF(3) 597499119864 minus 2 136773119864 minus 3

FI2BEBDF(3) 567155119864 minus 2 145525119864 minus 3

DI2BBDF(2) 110568119864 minus 2 472498119864 minus 3

DI2BBDF(3) 166455119864 minus 2 477409119864 minus 3

DI2BBDF(4) 216342119864 minus 2 483298119864 minus 3

10minus4

FI2BBDF(3) 734012119864 minus 4 125797119864 minus 1

FI2BEBDF(3) 323640119864 minus 5 136259119864 minus 1

DI2BBDF(2) 118355119864 minus 6 167260119864 minus 1

DI2BBDF(3) 173430119864 minus 6 169325119864 minus 1

DI2BBDF(4) 225966119864 minus 6 172995119864 minus 1

10minus6

FI2BBDF(3) 735741119864 minus 6 1257001198641

FI2BEBDF(3) 350090119864 minus 7 1362301198641

DI2BBDF(2) 118742119864 minus 10 1491591198641

DI2BBDF(3) 173634119864 minus 10 1544931198641

DI2BBDF(4) 226000119864 minus 10 1546201198641

Table 2 The accuracy for problem 2

ℎ Methods MAXE TIME

10minus2

FI2BBDF(3) 271597119864 minus 2 596467119864 minus 3

FI2BEBDF(3) 149811119864 minus 2 661025119864 minus 3

DI2BBDF(2) 456073119864 minus 3 104480119864 minus 2

DI2BBDF(3) 686126119864 minus 3 108421119864 minus 2

DI2BBDF(4) 893279119864 minus 3 111089119864 minus 2

10minus4

FI2BBDF(3) 340437119864 minus 4 590430119864 minus 1

FI2BEBDF(3) 262649119864 minus 5 654305119864 minus 1

DI2BBDF(2) 485404119864 minus 7 373210119864 minus 1

DI2BBDF(3) 711155119864 minus 7 386556119864 minus 1

DI2BBDF(4) 926476119864 minus 7 387070119864 minus 1

10minus6

FI2BBDF(3) 341146119864 minus 6 5901981198641

FI2BEBDF(3) 244097119864 minus 7 6541061198641

DI2BBDF(2) 486843119864 minus 11 3484511198641

DI2BBDF(3) 711903119864 minus 11 3604901198641

DI2BBDF(4) 926598119864 minus 11 3606491198641

6 Numerical Results

The performance of the derived methods is compared withthe existing methods in terms of maximum error and exe-cution time We consider 10minus2 10minus4 and 10minus6 as the stepsize ℎ Tables 1 and 2 present the performance compar-ison of DI2BBDF(2) DI2BBDF(3) and DI2BBDF(4) withFI2BBDF(3) and FI2BEBDF(3) whereas Tables 3ndash6 exhibitthe comparison of proposed methods with FI3BBDF(3) andFI3BEBDF(3) In addition the graphs of log(MAXE) againstlog(ℎ) are illustrated as shown in Figures 5ndash10 The followingnotations are used in Tables 1ndash6

ℎ step sizeMAXE maximum errorTIME time execution using high performance com-puter (HPC)

Table 3 The accuracy for problem 3

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 356692119864 minus 2 141642119864 minus 3

FI3BEBDF(3) 124084119864 minus 2 302167119864 minus 3

DI2BBDF(2) 117256119864 minus 3 448489119864 minus 3

DI2BBDF(3) 172278119864 minus 3 481200119864 minus 3

DI2BBDF(4) 225003119864 minus 3 485301119864 minus 3

10minus4

FI3BBDF(3) 435640119864 minus 4 130103119864 minus 1

FI3BEBDF(3) 706516119864 minus 5 208115119864 minus 1

DI2BBDF(2) 118749119864 minus 7 160304119864 minus 1

DI2BBDF(3) 173636119864 minus 7 177445119864 minus 1

DI2BBDF(4) 226000119864 minus 7 178064119864 minus 1

10minus6

FI3BBDF(3) 434625119864 minus 6 1297751198641

FI3BEBDF(3) 703257119864 minus 7 2076091198640

DI2BBDF(2) 124989119864 minus 11 1507391198641

DI2BBDF(3) 173636119864 minus 11 1550731198641

DI2BBDF(4) 225999119864 minus 11 1602161198641

Table 4 The accuracy for problem 4

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 66269411986499 701775119864 minus 3

FI3BEBDF(3) 168449119864 minus 1 137847119864 minus 2

DI2BBDF(2) 280830119864 minus 1 106871119864 minus 2

DI2BBDF(3) 400389119864 minus 1 124969119864 minus 2

DI2BBDF(4) 477342119864 minus 1 126502119864 minus 2

10minus4

FI3BBDF(3) 845376119864 minus 3 676737119864 minus 1

FI3BEBDF(3) 695725119864 minus 3 135671119864 + 0

DI2BBDF(2) 356255119864 minus 5 432359119864 minus 1

DI2BBDF(3) 524241119864 minus 5 442599119864 minus 1

DI2BBDF(4) 685105119864 minus 5 456207119864 minus 1

10minus6

FI3BBDF(3) 854545119864 minus 5 6768191198641

FI3BEBDF(3) 718859119864 minus 5 1358621198642

DI2BBDF(2) 360130119864 minus 9 4000101198641

DI2BBDF(3) 526673119864 minus 9 4001921198641

DI2BBDF(4) 685529119864 minus 9 4103221198641

Table 5 The accuracy for problem 5

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 525481119864 minus 3 578083119864 minus 4

FI3BEBDF(3) 388475119864 minus 3 916583119864 minus 4

DI2BBDF(2) 856253119864 minus 5 229836119864 minus 4

DI2BBDF(3) 121629119864 minus 4 189066119864 minus 4

DI2BBDF(4) 165094119864 minus 4 230074119864 minus 4

10minus4

FI3BBDF(3) 551558119864 minus 5 483860119864 minus 2

FI3BEBDF(3) 427506119864 minus 5 774877119864 minus 2

DI2BBDF(2) 889803119864 minus 9 117438119864 minus 2

DI2BBDF(3) 130124119864 minus 8 131578119864 minus 2

DI2BBDF(4) 169456119864 minus 8 126071119864 minus 2

10minus6

FI3BBDF(3) 551803119864 minus 7 547768119864 + 0

FI3BEBDF(3) 427924119864 minus 7 777643119864 + 0

DI2BBDF(2) 126115119864 minus 11 119318119864 + 0

DI2BBDF(3) 627106119864 minus 11 133892119864 + 0

DI2BBDF(4) 104222119864 minus 11 126146119864 + 0

Mathematical Problems in Engineering 11

Table 6 The accuracy for problem 6

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 218555119864 minus 3 283084119864 minus 4

FI3BEBDF(3) 152719119864 minus 3 428500119864 minus 4

DI2BBDF(2) 387041119864 minus 5 538826119864 minus 5

DI2BBDF(3) 547958119864 minus 5 531673119864 minus 5

DI2BBDF(4) 747696119864 minus 5 618877119864 minus 5

10minus4

FI3BBDF(3) 229812119864 minus 5 148627119864 minus 2

FI3BEBDF(3) 178708119864 minus 5 232184119864 minus 2

DI2BBDF(2) 401638119864 minus 9 349712119864 minus 3

DI2BBDF(3) 587347119864 minus 9 375295119864 minus 3

DI2BBDF(4) 764927119864 minus 9 364804119864 minus 3

10minus6

FI3BBDF(3) 229936119864 minus 7 146660119864 + 0

FI3BEBDF(3) 178973119864 minus 7 228775119864 + 0

DI2BBDF(2) 990807119864 minus 12 338157119864 minus 1

DI2BBDF(3) 377716119864 minus 11 369837119864 minus 1

DI2BBDF(4) 462796119864 minus 12 356802119864 minus 1

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

log(h)minus6 minus5 minus4 minus3 minus2

FI2BBDF(3)FI2BEBDF(3)

DI2BBDF(2)DI2BBDF(3)DI2BBDF(4)

Figure 5 Graph of log(MAXE) versus log(ℎ) for problem 1

DI2BBDF(2) diagonally implicit 2-point block back-ward differentiation formulas of order 2DI2BBDF(3) diagonally implicit 2-point block back-ward differentiation formulas of order 3DI2BBDF(4) diagonally implicit 2-point block back-ward differentiation formulas of order 4FI2BBDF(3) fully implicit 2-point block backwarddifferentiation formulas of order 3FI2BEBDF(3) fully implicit 2-point block extendedbackward differentiation formulas of order 3FI3BBDF(3) fully implicit 3-point block backwarddifferentiation formulas of order 3

log(h)minus6 minus5 minus4 minus3 minus2

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

FI2BBDF(3)FI2BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

Figure 6 Graph of log(MAXE) versus log(ℎ) for problem 2

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus11minus12

minus10minus9minus8minus7minus6minus5minus4minus3minus2minus10

log(

MA

XE)

Figure 7 Graph of log(MAXE) versus log(ℎ) for problem 3

FI3BEBDF(3) fully implicit 3-point block extendedbackward differentiation formulas of order 3

7 Discussion

This section is divided into discussion of maximum error andcomputational time

71 Maximum Error From the numerical results in Tables1ndash6 we observe that DI2BBDF(2) DI2BBDF(3) andDI2BBDF(4) outperformed the existing methods in termsof maximum error This is expected because the diagonally

12 Mathematical Problems in Engineering

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

Figure 8 Graph of log(MAXE) versus log(ℎ) for problem 4

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus11

minus12

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

Figure 9 Graph of log(MAXE) versus log(ℎ) for problem 5

implicit method has less differentiation coefficients in orderto prevent the cumulative error The graphs in Figures 5ndash10depict the scales of maximum error versus step size h for theproposed methods as compared to existing methods Thereis a slight drop in accuracy for the proposed methods as theorder increases This is due to the increase in cumulativeerrors during the computation as more interpolating pointsare used Among the methods of order three DI2BBDF(3)is more accurate compared with FI2BBDF(3) FI3BBDF(3)FI2BEBDF(3) and FI3BEBDF(3) For all test problems we

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus11minus12

minus10minus9minus8minus7minus6minus5minus4minus3minus2minus10

log(

MA

XE)

Figure 10 Graph of log(MAXE) versus log(ℎ) for problem 6

conclude that the accuracy of the proposedmethod increasesas the step size becomes smaller

72 Computational Time In Tables 1-2 it can be seen thatthe execution time taken by DI2BBDF(2) DI2BBDF(3) andDI2BBDF(4) is comparable with that of FI2BBDF(3) andFI2BEBDF(3) The computational time increases as the stepsize becomes smaller However Tables 3ndash6 show that theproposed methods compute faster than FI3BBDF(3) andFI3BEBDF(3) This could be justified by the fact that the3-point method has to perform extra computation on theJacobian matrix since it involves a 3 by 3 matrix while the 2-point method has to compute Jacobian matrix of dimension2 by 2 only Among the proposed methods DI2BBDF(2)computes faster than DI2BBDF(3) and DI2BBDF(4) In factthe number of back values involved in every formula willaffect the computational time Since DI2BBDF(4) has moreback values than DI2BBDF(3) and DI2BBDF(2) it requiresextra time to compute the approximated solutions

8 Conclusion

Research conducted in this paper shows the capability ofconstructing the diagonally implicit BBDF method with A-stable properties From the results obtained via the numericalexperiment we can conclude that the proposed methodserves the purpose of significant alternative numericalmethod for solving linear and nonlinear stiff IVPs occurringin the fields of engineering and physical sciences

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 13

Acknowledgment

This researchwas supported by the Institute forMathematicalResearch (INSPEM) Department of Mathematics UniversitiPutra Malaysia (UPM) under Fundamental Grant Scheme(Project code 02-02-13-1380FR)

References

[1] C F Curtiss and J O Hirschfelder ldquoIntegration of stiff equa-tionsrdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 38 no 3 pp 235ndash243 1952

[2] J D Lambert Numercial Methods for Ordinary DifferentialSystems The Initial Value Problem John Wiley amp Sons NewYork NY USA 1991

[3] W E Milne Numerical Solution of Differential Equations JohnWiley amp Sons New York NY USA 1953

[4] L F Shampine and H A Watts ldquoBlock implicit one-stepmethodsrdquo Mathematics of Computation vol 23 pp 731ndash7401969

[5] S O Fatunla ldquoBlock methods for second order ODEsrdquo Inter-national Journal of Computer Mathematics vol 41 pp 55ndash631990

[6] Z A Majid M B Suleiman F I Ismail and K I Othmanldquo2-Point 1 block diagonally and 2-point 1 block fully implicitmethod for solving first order ordinary differential equationsrdquoin Proceedings of the 12th National Symposium on MathematicalScience vol 12 2004

[7] Z Abdul Majid N Z Mokhtar and M Suleiman ldquoDirecttwo-point block one-step method for solving general second-order ordinary differential equationsrdquo Mathematical Problemsin Engineering vol 2012 Article ID 184253 16 pages 2012

[8] Z B Ibrahim Block method for multistep formulas for solvingordinary differential equations [PhD thesis] Universiti PutraMalaysia Seri Kembangan Malaysia 2006

[9] Z B Ibrahim M Suleiman and K I Othman ldquoFixed coeffi-cients block backward differentiation formulas for the numer-ical solution of stiff ordinary differential equationsrdquo EuropeanJournal of Scientific Research vol 21 no 3 pp 508ndash520 2008

[10] Z B Ibrahim M B Suleiman and K I Othman ldquoImplicit r-point block backward differentiation formula for solving first-order stiff ODEsrdquo Applied Mathematics and Computation vol186 no 1 pp 558ndash565 2007

[11] Z B Ibrahim R Johari and F Ismail ldquoOn the stability of blockbackward differentiation formulaerdquo Matematika vol 19 no 2pp 83ndash89 2003

[12] N A Nasir Z B Ibrahim M B Suleiman and K I OthmanldquoFifth order two-point block backward differentiation formulasfor solving ordinary differential equationsrdquo Applied Mathemat-ical Sciences vol 5 no 69ndash72 pp 3505ndash3518 2011

[13] H Musa M B Suleiman and N Senu ldquoA-stable 2-Point blockextended backward differentiation formulas for stiff ordinarydifferential equationsrdquo in Proceedings of the 5th InternationalConference on Research and education in Mathematics vol 1450of AIP Conference Proceedings pp 254ndash258 2012

[14] H Musa M B Suleiman and N Senu ldquoFully implicit 3-pointblock extended backward differentiation formula for stiff initialvalue problemsrdquo Applied Mathematical Sciences vol 6 no 85ndash88 pp 4211ndash4227 2012

[15] R Alexander ldquoDiagonally implicit Runge-kutta for stiff ordi-nary differential equationsrdquo SIAM Journal on Numerical Analy-sis vol 14 no 6 pp 1006ndash1021 1977

[16] O Y Ababneh R Ahmad and E S Ismail ldquoDesign of newdiagonally implicit Runge-Kutta methods for stiff problemsrdquoApplied Mathematical Sciences vol 3 no 45 pp 2241ndash22532009

[17] F Ismail A L I Raed Al-Khasawneh M Suleiman and A BU Malik Hassan ldquoEmbedded pair of diagonally implicit runge-kutta method for solving ordinary differential equationsrdquo SainsMalaysiana vol 39 no 6 pp 1049ndash1054 2010

[18] Z A Majid and M B Suleiman ldquoPerformance of four pointdiagonally implicit block method for solving first order stiffordinary differential equationsrdquo Department of MathematicsUTM vol 22 no 2 pp 137ndash146 2006

[19] G G Dahlquist ldquoA special stability problem for linearmultistepmethodsrdquo BIT Numerical Mathematics vol 3 no 1 pp 27ndash431963

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

We define 119909 = 119904ℎ + 119909119899+1

and produce

119875 (119904ℎ + 119909119899+1)

=(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ)

(minus4ℎ) (minus3ℎ) (minus2ℎ) (minusℎ)119910119899minus3

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 4ℎ)

(minus3ℎ) (minus2ℎ) (minusℎ) (ℎ)119910119899minus2

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 3ℎ) (119904ℎ + 3ℎ)

(minus2ℎ) (minusℎ) (ℎ) (2ℎ)119910119899minus1

+(119904ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 3ℎ)

(minusℎ) (ℎ) (2ℎ) (3ℎ)119910119899

+(119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 3ℎ)

(ℎ) (2ℎ) (3ℎ) (4ℎ)119910119899+1

(25)

Differentiating (25) once with respect to 119904 at the point 119909 =

119909119899+1

and evaluating 119904 = 0 will produce

1198751015840(119909119899+1) =

25

12119910119899+1

+1

4119910119899minus3

minus4

3119910119899minus2

+ 3119910119899minus1

minus 4119910119899 (26)

The derivation continues for the second point 119910119899+2

of formulaby using 119909

119899minus3 119909

119899+2as the interpolating points

119875 (119909) =(119909 minus 119909

119899+2) (119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2)

(119909119899minus3

minus 119909119899+2) (119909119899minus3

minus 119909119899+1) (119909119899minus3

minus 119909119899) (119909119899minus3

minus 119909119899minus1) (119909119899minus3

minus 119909119899minus2)119910119899minus3

+(119909 minus 119909

119899+2) (119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus3)

(119909119899minus2

minus 119909119899+2) (119909119899minus2

minus 119909119899+1) (119909119899minus2

minus 119909119899) (119909119899minus2

minus 119909119899minus1) (119909119899minus2

minus 119909119899minus3)119910119899minus2

+(119909 minus 119909

119899+2) (119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus2) (119909 minus 119909

119899minus3)

(119909119899minus1

minus 119909119899+2) (119909119899minus1

minus 119909119899+1) (119909119899minus1

minus 119909119899) (119909119899minus1

minus 119909119899minus2) (119909119899minus1

minus 119909119899minus3)119910119899minus1

+(119909 minus 119909

119899+2) (119909 minus 119909

119899+1) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2) (119909 minus 119909

119899minus3)

(119909119899minus 119909119899+2) (119909119899minus 119909119899+1) (119909119899minus 119909119899minus1) (119909119899minus 119909119899minus2) (119909119899minus 119909119899minus3)119910119899

+(119909 minus 119909

119899+2) (119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2) (119909 minus 119909

119899minus3)

(119909119899+1

minus 119909119899+2) (119909119899+1

minus 119909119899) (119909119899+1

minus 119909119899minus1) (119909119899+1

minus 119909119899minus2) (119909119899+1

minus 119909119899minus3)119910119899+1

+(119909 minus 119909

119899+1) (119909 minus 119909

119899) (119909 minus 119909

119899minus1) (119909 minus 119909

119899minus2) (119909 minus 119909

119899minus3)

(119909119899+2

minus 119909119899+1) (119909119899+2

minus 119909119899) (119909119899+2

minus 119909119899minus1) (119909119899+2

minus 119909119899minus2) (119909119899+2

minus 119909119899minus3)119910119899+2

(27)

Substituting 119909 = ℎ + 119909119899+2

into (27) will produce

119875 (119904ℎ + 119909119899+2)

=(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 4ℎ)

(minus5ℎ) (minus4ℎ) (minus3ℎ) (minus2ℎ) (minusℎ)119910119899minus3

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 5ℎ)

(minus4ℎ) (minus3ℎ) (minus2ℎ) (minusℎ) (ℎ)119910119899minus2

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 4ℎ) (119904ℎ + 5ℎ)

(minus3ℎ) (minus2ℎ) (minusℎ) (ℎ) (2ℎ)119910119899minus1

+(119904ℎ) (119904ℎ + ℎ) (119904ℎ + 3ℎ) (119904ℎ + 4ℎ) (119904ℎ + 5ℎ)

(minus2ℎ) (minusℎ) (ℎ) (2ℎ) (3ℎ)119910119899

+(119904ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 4ℎ) (119904ℎ + 5ℎ)

(minusℎ) (ℎ) (2ℎ) (3ℎ) (4ℎ)119910119899+1

+(119904ℎ + ℎ) (119904ℎ + 2ℎ) (119904ℎ + 3ℎ) (119904ℎ + 4ℎ) (119904ℎ + 5ℎ)

(ℎ) (2ℎ) (3ℎ) (4ℎ) (5ℎ)119910119899+2

(28)

Equation (28) is differentiated once with respect to 119904 followedby substituting 119904 = 0 we have

1198751015840(119909119899+2) =

137

60119910119899+2

minus1

5119910119899minus3

+5

4119910119899minus2

minus10

3119910119899minus1

+ 5119910119899minus 5119910119899+1

(29)

Therefore the corrector formula of DI2BBDF(4) is obtainedas follows

119910119899+1

= minus3

25119910119899minus3

+16

25119910119899minus2

minus36

25119910119899minus1

+48

25119910119899+12

25ℎ119891119899+1

119910119899+2

=12

137119910119899minus3

minus75

137119910119899minus2

+200

137119910119899minus1

minus300

137119910119899+300

137119910119899+1

+60

137ℎ119891119899+2

(30)

6 Mathematical Problems in Engineering

3 Stability Analysis

In this section we will plot the graph of stability forDI2BBDF(2) DI2BBDF(3) and DI2BBDF(4) using Mathe-matica software Based onDahlquist [19] the linearmultistepmethod (LMM) is able to solve stiff problems if it satisfies thefollowing definition

Definition 4 TheLMMisA-stable if its region of absolute sta-bility contains the whole of the left-hand half-plane Re(ℎ120582) lt0

We consider the simplest test equation

119891 = 1199101015840= 120582119910 (31)

where the eigenvalues 120582119894 119894 = 1 2 119904 satisfy 120582

119894lt 0 to

analyze the stiff stability of the method Substituting (31) into(16) will produce the following matrices form

[

[

1 0

minus18

111

]

]

[119910119899+1

119910119899+2

]

=[[

[

minus1

3

4

32

11minus9

11

]]

]

[119910119899minus1

119910119899

] + ℎ[[

[

2

30

06

11

]]

]

[119891119899+1

119891119899+2

]

(32)

where

119860 =[[

[

1 minus2

3120582ℎ 0

minus18

111 minus

6

11120582ℎ

]]

]

119861 =[[

[

minus1

3

4

32

11minus9

11

]]

]

119862 = [0 0

0 0]

(33)

Let ℎ = ℎ120582 and evaluating the determinant of |119860119905 minus 119861|from (32) the stability polynomial 120588(119905 ℎ) of DI2BBDF(2) isobtained as follows

120588 (119905 ℎ) =1

33minus34119905

33minus8ℎ119905

11+ 1199052minus40ℎ1199052

33+4ℎ21199052

11 (34)

The similar approach is applied to obtain the stability polyno-mial of DI2BBDF(3) and DI2BBDF(4) Substituting (31) into(23) we have

[

[

1 0

minus48

251

]

]

[119910119899+1

119910119899+2

]

=[[

[

02

11

0 minus3

25

]]

]

[119910119899minus3

119910119899minus2

] +[[

[

minus9

11

18

1116

25minus36

25

]]

]

[119910119899minus1

119910119899

]

+ ℎ[[

[

6

110

012

25

]]

]

[119891119899+1

119891119899+2

]

(35)

where

119860 =[[

[

1 minus6

11120582ℎ 0

minus48

251 minus

12

25120582ℎ

]]

]

119861 =[[

[

minus9

11

18

11

16

25minus36

25

]]

]

119862 =[[

[

02

11

03

25

]]

]

(36)

Substituting (31) into (30) will give

[

[

1 0

minus300

1371

]

]

[119910119899+1

119910119899+2

]

=[[

[

minus3

25

16

25

12

137minus75

137

]]

]

[119910119899minus3

119910119899minus2

] +[[

[

minus36

25

48

25

200

137minus300

137

]]

]

[119910119899minus1

119910119899

]

+ ℎ[[

[

12

250

060

137

]]

]

[119891119899+1

119891119899+2

]

(37)

where

119860 =[[

[

1 minus12

25120582ℎ 0

minus300

1371 minus

60

137120582ℎ

]]

]

119861 =[[

[

minus36

25

48

25

200

137minus300

137

]]

]

119862 =[[

[

minus3

25

16

25

12

137minus75

137

]]

]

(38)

We compute the determinant of |1198601199052 minus 119861119905 minus 119862| from(35) and (37) to obtain the stability polynomials 120588(119905 ℎ) ofDI2BBDF(3) and DI2BBDF(4) respectively Consider

120588 (119905 ℎ) =36

275minus86119905

275minus2431199052

275minus324ℎ119905

2

275minus1621199053

275

+36ℎ1199053

275+ 1199054minus282ℎ119905

4

275+72ℎ21199054

275

(39)

120588 (119905 ℎ) =48

137minus176119905

3425minus3871199052

685minus1152ℎ119905

2

685minus2514119905

3

3425

minus216ℎ119905

3

685+ 1199054minus3144ℎ119905

4

3425+144ℎ21199054

685

(40)

Next the boundary of the stability region will be deter-mined by substituting 119905 = 119890

119894120579 into (34) (39) and (40)The graphs of stability region for all formulas are givenin Figures 1 2 and 3

In Figures 1 2 and 3 we observed that the inter-vals of unstable region for DI2BBDF(2) DI2BBDF(3) and

Mathematical Problems in Engineering 7

Unstable

Unstable

Stable

Stable

Stable1 2 3 4 5

Real part

minus3

minus2

minus1

1

2

3Imaginary part

(533333 0)

Figure 1 Graph of stability region for DI2BBDF(2)

Unstable

Unstable

Stable

Stable

Stable2 4 6 8

Real part

minus4

minus2

2

4

Imaginary part

(866667 0)

Figure 2 Graph of stability region for DI2BBDF(3)

DI2BBDF(4) are [0 533333] [0 866667] and [0 1386667]respectively All the graphs of stability regions are combinedas shown in Figure 4 The region outside the green line is thestable region of DI2BBDF(2) the region outside the red lineis the stable region ofDI2BBDF(3) and the region outside theblue line is the stable region of DI2BBDF(4)

Clearly the unstable region becomes larger whenthe order of the method increases From Definition 4DI2BBDF(2) is A-stable while DI2BBDF(3) andDI2BBDF(4) are almost A-stable since the stability regioncovers the entire negative half plane Therefore we canconclude that the proposed method is suitable for solvingstiff problems

Unstable

Unstable

Stable

Stable

2 4 6 8 10 12 14

minus5

5

(1386667 0)

Stable Real part

Imaginary part

Figure 3 Graph of stability region for DI2BBDF(4)

Imaginary part

Real part5 10

minus5

5

10

DI2BBDF(2)

DI2BBDF(3)

DI2BBDF(4)

A-stable region

A-stable region

Figure 4 Graph of stability region for DI2BBDF(2) DI2BBDF(3)and DI2BBDF(4)

4 Implementation of the Method

In this section the proposed methods will be implementedusing Newton iteration We begin by converting (16) (23)and (30) in general form as follows

119910119899+1

= 1205721119910119899+2

+ 1205731ℎ119891119899+1

+ 1205831

119910119899+2

= 1205722119910119899+1

+ 1205732ℎ119891119899+2

+ 1205832

(41)

where 1205831and 120583

2are the back values Equation (41) is trans-

formed into matrix form as follows

[1 0

0 1] [119910119899+1

119910119899+2

]

= [0 1205721

12057220] [119910119899+1

119910119899+2

] + ℎ[12057310

0 1205732

][119891119899+1

119891119899+2

] + [1205831

1205832

]

(42)

8 Mathematical Problems in Engineering

where

119868 = [1 0

0 1] 119884 = [

119910119899+1

119910119899+2

] 119861 = [12057310

0 1205732

]

119865 = [119891119899+1

119891119899+2

] 120583 = [1205831

1205832

]

(43)

Implementing Newtonrsquos method to (42) produces

119865 = (119868 minus 119860)119884 minus ℎ119861119865 + 120583 (44)

Iteration for (44) is given by

119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2= 119865 (119910

(119894)

119899+1119899+2) sdot 1198651015840(119910(119894)

119899+1119899+2)minus1

(45)

where 119910(119894+1)119899+1

denotes the (119894 + 1)th iteration Equation (45) canbe rearranged into the following equation

1198651015840(119910(119894)

119899+1119899+2) (119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2) = 119865 (119910

(119894)

119899+1119899+2) (46)

Replacing (44) into (46) yields

(119868 minus 119860)119884 minus ℎ119861120597119891

120597119910(119910(119894)

119899+1119899+2) (119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2)

= (119868 minus 119860) (119910(119894)

119899+1119899+2) minus ℎ119861 (119910

(119894)

119899+1119899+2) + 120583

(47)

where (120597119891120597119910)(119910(119894)119899+1119899+2

) is the Jacobian matrix of 119891 with re-spect to 119910

41 Newton Iteration of DI2BBDF(2) Formula (16) is writtenin the form of (41)

1198651= 119910119899+1

minus2

3ℎ119891119899+1

+ 1205831

1198652= 119910119899+2

minus18

11119910119899+1

minus6

11ℎ119891119899+2

+ 1205832

(48)

Applying (48) into (47) will yield matrix form as follows

[[[

[

1 minus2

3ℎ120597119891119899+1

120597119910119899+1

0

0 1 minus6

11ℎ120597119891119899+2

120597119910119899+2

]]]

]

[119890(119894+1)

119899+1

119890(119894+1)

119899+2

]

= [

[

minus1 0

18

11minus1

]

]

[119910(119894)

119899+1

119910(119894)

119899+2

] + ℎ[[

[

2

30

06

11

]]

]

[119891(119894)

119899+1

119891(119894)

119899+2

] + [1205831

1205832

]

(49)

with 119890(119894+1)119899+1119899+2

= 119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2being the increment

42 Newton Iteration of DI2BBDF(3) Rearranging formula(23) in the form of (41) we have

1198651= 119910119899+1

minus6

11ℎ119891119899+1

+ 1205831

1198652= 119910119899+2

minus48

25119910119899+1

minus12

25ℎ119891119899+2

+ 1205832

(50)

Replacing (50) into (47) this will yieldmatrix form as follows

[[[

[

1 minus6

11ℎ120597119891119899+1

120597119910119899+1

0

0 1 minus12

25ℎ120597119891119899+2

120597119910119899+2

]]]

]

[119890(119894+1)

119899+1

119890(119894+1)

119899+2

]

= [

[

minus1 0

48

25minus1

]

]

[119910(119894)

119899+1

119910(119894)

119899+2

] + ℎ[[

[

6

110

012

25

]]

]

[119891(119894)

119899+1

119891(119894)

119899+2

] + [1205831

1205832

]

(51)

with 119890(119894+1)119899+1119899+2

= 119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2being the increment

43 Newton Iteration of DI2BBDF(4) We rewrite formula(30) in the form of (41) and obtain

1198651= 119910119899+1

minus12

25ℎ119891119899+1

+ 1205831

1198652= 119910119899+2

minus300

137119910119899+1

minus60

137ℎ119891119899+2

+ 1205832

(52)

Substituting (52) into (47) will produce

[[[

[

1 minus12

25ℎ120597119891119899+1

120597119910119899+1

0

0 1 minus60

137ℎ120597119891119899+2

120597119910119899+2

]]]

]

[119890(119894+1)

119899+1

119890(119894+1)

119899+2

]

= [

[

minus1 0

300

137minus1

]

]

[119910(119894)

119899+1

119910(119894)

119899+2

] + ℎ[[

[

12

250

060

137

]]

]

[119891(119894)

119899+1

119891(119894)

119899+2

] + [1205831

1205832

]

(53)

with 119890(119894+1)119899+1119899+2

= 119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2being the increment

All the formulas derived are implemented in predictor-corrector computation which is symbolized as PECE modeP and C indicate one application of the predictor and thecorrector respectively and E indicates one evaluation of thefunction119891 given119909 and119910The approximation calculations for119910119899+1

and 119910119899+2

in PECE are as follows

(1) P (Predict) 119910(119901)119899+1119899+2

(2) E (Evaluate) 1199101015840119899+1119899+2

= 119891(119909119899+1119899+2

119910(119901)

119899+1119899+2)

(3) C (Correct) 119910(119888)119899+1119899+2

(4) E (Evaluate) 1199101015840119899+1119899+2

= 119891(119909119899+1119899+2

119910(119888)

119899+1119899+2)

To approximate the solution to 119910(119888)

119899+1119899+2 we apply the

two-stage Newton type iteration The iteration process forDI2BBDF(2) is done as follows

Mathematical Problems in Engineering 9

(1) Compute the values for 119890(119894+1)119899+1119899+2

= 119860minus1119861 where

119860 =[[[

[

1 minus2

3ℎ120597119891119899+1

120597119910119899+1

0

0 1 minus6

11ℎ120597119891119899+2

120597119910119899+2

]]]

]

119861 =[[

[

minus119910(119894)

119899+1+2

3ℎ119891(119894)

119899+1+ 1205831

0

18

11119910(119894)

119899+1minus119910(119894)

119899+2+6

11ℎ119891(119894)

119899+2+ 1205832

]]

]

(54)

(2) Calculate the corrected value for 119910(119894+1)119899+1119899+2

with thevalue 119890(119894+1)

119899+1119899+2from Step (1)

(3) Solve 119890(119894+1)119899+1119899+2

= 119860minus1119861 for second stage iteration

(4) The final values for 119910(119894+1)119899+1119899+2

are obtained from thesecond stage iteration of 119890(119894+1)

119899+1119899+2

The similar iteration process is applied for DI2BBDF(3) andDI2BBDF(4)

5 Test Problems

In this section linear and nonlinear stiff problems aretested using C programming to examine the efficiency andreliability of the proposed method The following problemsare commonly found in engineering and physical sciencesparticularly in the studies of vibrations electrical circuits andchemical reaction

Problem 1 (linear) One has

1199101015840= minus10119910 + 10 (55)

Exact solution is

119910 (119909) = 1 + 119890minus10119909

(56)

Initial condition is

119910 (0) = 2

0 le 119909 le 10

(57)

(source Ibrahim [8])

Problem 2 (linear) One has

1199101015840

1= minus5119910

1minus 41199102 119910

1015840

2= minus4119910

1minus 51199102 (58)

Exact solutions are

1199101(119909) =

1

2119890minus9119909

+1

2119890minus119909 119910

2(119909) =

1

2119890minus9119909

minus1

2119890minus119909 (59)

Initial conditions are

1199101(0) = 1 119910

2(0) = 0

0 le 119909 le 20(60)

(source Musa et al [13])

Problem 3 (linear) One has

1199101015840= minus10119909119910 (61)

Exact solution is

119910 (119909) = 119890minus51199092

(62)

Initial condition is119910 (0) = 1

0 le 119909 le 10

(63)

(source Ibrahim et al [10])

Problem 4 (linear) One has

1199101015840

1= 91199101+ 24119910

2+ 5 cos119909 minus 1

3sin119909

1199101015840

2= minus 24119910

1minus 51119910

2minus 9 cos119909 + 1

3sin119909

(64)

Exact solutions are

1199101(119909) = 2119890

minus3119909minus 119890minus39119909

+1

3cos119909

1199102(119909) = minus119890

minus3119909+ 2119890minus39119909

minus1

3cos119909

(65)

Initial conditions are

1199101(0) =

4

3 1199102(0) =

2

3 0 le 119909 le 10 (66)

(source Musa et al [14])

Problem 5 (nonlinear) One has

1199101015840= minus

1199103

2 (67)

Exact solution is

119910 (119909) =1

radic1 + 119909 (68)

Initial condition is119910 (0) = 1

0 le 119909 le 4

(69)

(source Musa et al [14])

Problem 6 (nonlinear) One has

1199101015840=119910 (1 minus 119910)

2119910 minus 1 (70)

Exact solution is

119910 (119909) =1

2+ radic

1

4minus5

36119890minus119909 (71)

Initial condition is

119910 (0) =5

6

0 le 119909 le 1

(72)

(source Musa et al [14])

10 Mathematical Problems in Engineering

Table 1 The accuracy for problem 1

ℎ Methods MAXE TIME

10minus2

FI2BBDF(3) 597499119864 minus 2 136773119864 minus 3

FI2BEBDF(3) 567155119864 minus 2 145525119864 minus 3

DI2BBDF(2) 110568119864 minus 2 472498119864 minus 3

DI2BBDF(3) 166455119864 minus 2 477409119864 minus 3

DI2BBDF(4) 216342119864 minus 2 483298119864 minus 3

10minus4

FI2BBDF(3) 734012119864 minus 4 125797119864 minus 1

FI2BEBDF(3) 323640119864 minus 5 136259119864 minus 1

DI2BBDF(2) 118355119864 minus 6 167260119864 minus 1

DI2BBDF(3) 173430119864 minus 6 169325119864 minus 1

DI2BBDF(4) 225966119864 minus 6 172995119864 minus 1

10minus6

FI2BBDF(3) 735741119864 minus 6 1257001198641

FI2BEBDF(3) 350090119864 minus 7 1362301198641

DI2BBDF(2) 118742119864 minus 10 1491591198641

DI2BBDF(3) 173634119864 minus 10 1544931198641

DI2BBDF(4) 226000119864 minus 10 1546201198641

Table 2 The accuracy for problem 2

ℎ Methods MAXE TIME

10minus2

FI2BBDF(3) 271597119864 minus 2 596467119864 minus 3

FI2BEBDF(3) 149811119864 minus 2 661025119864 minus 3

DI2BBDF(2) 456073119864 minus 3 104480119864 minus 2

DI2BBDF(3) 686126119864 minus 3 108421119864 minus 2

DI2BBDF(4) 893279119864 minus 3 111089119864 minus 2

10minus4

FI2BBDF(3) 340437119864 minus 4 590430119864 minus 1

FI2BEBDF(3) 262649119864 minus 5 654305119864 minus 1

DI2BBDF(2) 485404119864 minus 7 373210119864 minus 1

DI2BBDF(3) 711155119864 minus 7 386556119864 minus 1

DI2BBDF(4) 926476119864 minus 7 387070119864 minus 1

10minus6

FI2BBDF(3) 341146119864 minus 6 5901981198641

FI2BEBDF(3) 244097119864 minus 7 6541061198641

DI2BBDF(2) 486843119864 minus 11 3484511198641

DI2BBDF(3) 711903119864 minus 11 3604901198641

DI2BBDF(4) 926598119864 minus 11 3606491198641

6 Numerical Results

The performance of the derived methods is compared withthe existing methods in terms of maximum error and exe-cution time We consider 10minus2 10minus4 and 10minus6 as the stepsize ℎ Tables 1 and 2 present the performance compar-ison of DI2BBDF(2) DI2BBDF(3) and DI2BBDF(4) withFI2BBDF(3) and FI2BEBDF(3) whereas Tables 3ndash6 exhibitthe comparison of proposed methods with FI3BBDF(3) andFI3BEBDF(3) In addition the graphs of log(MAXE) againstlog(ℎ) are illustrated as shown in Figures 5ndash10 The followingnotations are used in Tables 1ndash6

ℎ step sizeMAXE maximum errorTIME time execution using high performance com-puter (HPC)

Table 3 The accuracy for problem 3

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 356692119864 minus 2 141642119864 minus 3

FI3BEBDF(3) 124084119864 minus 2 302167119864 minus 3

DI2BBDF(2) 117256119864 minus 3 448489119864 minus 3

DI2BBDF(3) 172278119864 minus 3 481200119864 minus 3

DI2BBDF(4) 225003119864 minus 3 485301119864 minus 3

10minus4

FI3BBDF(3) 435640119864 minus 4 130103119864 minus 1

FI3BEBDF(3) 706516119864 minus 5 208115119864 minus 1

DI2BBDF(2) 118749119864 minus 7 160304119864 minus 1

DI2BBDF(3) 173636119864 minus 7 177445119864 minus 1

DI2BBDF(4) 226000119864 minus 7 178064119864 minus 1

10minus6

FI3BBDF(3) 434625119864 minus 6 1297751198641

FI3BEBDF(3) 703257119864 minus 7 2076091198640

DI2BBDF(2) 124989119864 minus 11 1507391198641

DI2BBDF(3) 173636119864 minus 11 1550731198641

DI2BBDF(4) 225999119864 minus 11 1602161198641

Table 4 The accuracy for problem 4

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 66269411986499 701775119864 minus 3

FI3BEBDF(3) 168449119864 minus 1 137847119864 minus 2

DI2BBDF(2) 280830119864 minus 1 106871119864 minus 2

DI2BBDF(3) 400389119864 minus 1 124969119864 minus 2

DI2BBDF(4) 477342119864 minus 1 126502119864 minus 2

10minus4

FI3BBDF(3) 845376119864 minus 3 676737119864 minus 1

FI3BEBDF(3) 695725119864 minus 3 135671119864 + 0

DI2BBDF(2) 356255119864 minus 5 432359119864 minus 1

DI2BBDF(3) 524241119864 minus 5 442599119864 minus 1

DI2BBDF(4) 685105119864 minus 5 456207119864 minus 1

10minus6

FI3BBDF(3) 854545119864 minus 5 6768191198641

FI3BEBDF(3) 718859119864 minus 5 1358621198642

DI2BBDF(2) 360130119864 minus 9 4000101198641

DI2BBDF(3) 526673119864 minus 9 4001921198641

DI2BBDF(4) 685529119864 minus 9 4103221198641

Table 5 The accuracy for problem 5

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 525481119864 minus 3 578083119864 minus 4

FI3BEBDF(3) 388475119864 minus 3 916583119864 minus 4

DI2BBDF(2) 856253119864 minus 5 229836119864 minus 4

DI2BBDF(3) 121629119864 minus 4 189066119864 minus 4

DI2BBDF(4) 165094119864 minus 4 230074119864 minus 4

10minus4

FI3BBDF(3) 551558119864 minus 5 483860119864 minus 2

FI3BEBDF(3) 427506119864 minus 5 774877119864 minus 2

DI2BBDF(2) 889803119864 minus 9 117438119864 minus 2

DI2BBDF(3) 130124119864 minus 8 131578119864 minus 2

DI2BBDF(4) 169456119864 minus 8 126071119864 minus 2

10minus6

FI3BBDF(3) 551803119864 minus 7 547768119864 + 0

FI3BEBDF(3) 427924119864 minus 7 777643119864 + 0

DI2BBDF(2) 126115119864 minus 11 119318119864 + 0

DI2BBDF(3) 627106119864 minus 11 133892119864 + 0

DI2BBDF(4) 104222119864 minus 11 126146119864 + 0

Mathematical Problems in Engineering 11

Table 6 The accuracy for problem 6

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 218555119864 minus 3 283084119864 minus 4

FI3BEBDF(3) 152719119864 minus 3 428500119864 minus 4

DI2BBDF(2) 387041119864 minus 5 538826119864 minus 5

DI2BBDF(3) 547958119864 minus 5 531673119864 minus 5

DI2BBDF(4) 747696119864 minus 5 618877119864 minus 5

10minus4

FI3BBDF(3) 229812119864 minus 5 148627119864 minus 2

FI3BEBDF(3) 178708119864 minus 5 232184119864 minus 2

DI2BBDF(2) 401638119864 minus 9 349712119864 minus 3

DI2BBDF(3) 587347119864 minus 9 375295119864 minus 3

DI2BBDF(4) 764927119864 minus 9 364804119864 minus 3

10minus6

FI3BBDF(3) 229936119864 minus 7 146660119864 + 0

FI3BEBDF(3) 178973119864 minus 7 228775119864 + 0

DI2BBDF(2) 990807119864 minus 12 338157119864 minus 1

DI2BBDF(3) 377716119864 minus 11 369837119864 minus 1

DI2BBDF(4) 462796119864 minus 12 356802119864 minus 1

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

log(h)minus6 minus5 minus4 minus3 minus2

FI2BBDF(3)FI2BEBDF(3)

DI2BBDF(2)DI2BBDF(3)DI2BBDF(4)

Figure 5 Graph of log(MAXE) versus log(ℎ) for problem 1

DI2BBDF(2) diagonally implicit 2-point block back-ward differentiation formulas of order 2DI2BBDF(3) diagonally implicit 2-point block back-ward differentiation formulas of order 3DI2BBDF(4) diagonally implicit 2-point block back-ward differentiation formulas of order 4FI2BBDF(3) fully implicit 2-point block backwarddifferentiation formulas of order 3FI2BEBDF(3) fully implicit 2-point block extendedbackward differentiation formulas of order 3FI3BBDF(3) fully implicit 3-point block backwarddifferentiation formulas of order 3

log(h)minus6 minus5 minus4 minus3 minus2

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

FI2BBDF(3)FI2BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

Figure 6 Graph of log(MAXE) versus log(ℎ) for problem 2

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus11minus12

minus10minus9minus8minus7minus6minus5minus4minus3minus2minus10

log(

MA

XE)

Figure 7 Graph of log(MAXE) versus log(ℎ) for problem 3

FI3BEBDF(3) fully implicit 3-point block extendedbackward differentiation formulas of order 3

7 Discussion

This section is divided into discussion of maximum error andcomputational time

71 Maximum Error From the numerical results in Tables1ndash6 we observe that DI2BBDF(2) DI2BBDF(3) andDI2BBDF(4) outperformed the existing methods in termsof maximum error This is expected because the diagonally

12 Mathematical Problems in Engineering

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

Figure 8 Graph of log(MAXE) versus log(ℎ) for problem 4

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus11

minus12

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

Figure 9 Graph of log(MAXE) versus log(ℎ) for problem 5

implicit method has less differentiation coefficients in orderto prevent the cumulative error The graphs in Figures 5ndash10depict the scales of maximum error versus step size h for theproposed methods as compared to existing methods Thereis a slight drop in accuracy for the proposed methods as theorder increases This is due to the increase in cumulativeerrors during the computation as more interpolating pointsare used Among the methods of order three DI2BBDF(3)is more accurate compared with FI2BBDF(3) FI3BBDF(3)FI2BEBDF(3) and FI3BEBDF(3) For all test problems we

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus11minus12

minus10minus9minus8minus7minus6minus5minus4minus3minus2minus10

log(

MA

XE)

Figure 10 Graph of log(MAXE) versus log(ℎ) for problem 6

conclude that the accuracy of the proposedmethod increasesas the step size becomes smaller

72 Computational Time In Tables 1-2 it can be seen thatthe execution time taken by DI2BBDF(2) DI2BBDF(3) andDI2BBDF(4) is comparable with that of FI2BBDF(3) andFI2BEBDF(3) The computational time increases as the stepsize becomes smaller However Tables 3ndash6 show that theproposed methods compute faster than FI3BBDF(3) andFI3BEBDF(3) This could be justified by the fact that the3-point method has to perform extra computation on theJacobian matrix since it involves a 3 by 3 matrix while the 2-point method has to compute Jacobian matrix of dimension2 by 2 only Among the proposed methods DI2BBDF(2)computes faster than DI2BBDF(3) and DI2BBDF(4) In factthe number of back values involved in every formula willaffect the computational time Since DI2BBDF(4) has moreback values than DI2BBDF(3) and DI2BBDF(2) it requiresextra time to compute the approximated solutions

8 Conclusion

Research conducted in this paper shows the capability ofconstructing the diagonally implicit BBDF method with A-stable properties From the results obtained via the numericalexperiment we can conclude that the proposed methodserves the purpose of significant alternative numericalmethod for solving linear and nonlinear stiff IVPs occurringin the fields of engineering and physical sciences

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 13

Acknowledgment

This researchwas supported by the Institute forMathematicalResearch (INSPEM) Department of Mathematics UniversitiPutra Malaysia (UPM) under Fundamental Grant Scheme(Project code 02-02-13-1380FR)

References

[1] C F Curtiss and J O Hirschfelder ldquoIntegration of stiff equa-tionsrdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 38 no 3 pp 235ndash243 1952

[2] J D Lambert Numercial Methods for Ordinary DifferentialSystems The Initial Value Problem John Wiley amp Sons NewYork NY USA 1991

[3] W E Milne Numerical Solution of Differential Equations JohnWiley amp Sons New York NY USA 1953

[4] L F Shampine and H A Watts ldquoBlock implicit one-stepmethodsrdquo Mathematics of Computation vol 23 pp 731ndash7401969

[5] S O Fatunla ldquoBlock methods for second order ODEsrdquo Inter-national Journal of Computer Mathematics vol 41 pp 55ndash631990

[6] Z A Majid M B Suleiman F I Ismail and K I Othmanldquo2-Point 1 block diagonally and 2-point 1 block fully implicitmethod for solving first order ordinary differential equationsrdquoin Proceedings of the 12th National Symposium on MathematicalScience vol 12 2004

[7] Z Abdul Majid N Z Mokhtar and M Suleiman ldquoDirecttwo-point block one-step method for solving general second-order ordinary differential equationsrdquo Mathematical Problemsin Engineering vol 2012 Article ID 184253 16 pages 2012

[8] Z B Ibrahim Block method for multistep formulas for solvingordinary differential equations [PhD thesis] Universiti PutraMalaysia Seri Kembangan Malaysia 2006

[9] Z B Ibrahim M Suleiman and K I Othman ldquoFixed coeffi-cients block backward differentiation formulas for the numer-ical solution of stiff ordinary differential equationsrdquo EuropeanJournal of Scientific Research vol 21 no 3 pp 508ndash520 2008

[10] Z B Ibrahim M B Suleiman and K I Othman ldquoImplicit r-point block backward differentiation formula for solving first-order stiff ODEsrdquo Applied Mathematics and Computation vol186 no 1 pp 558ndash565 2007

[11] Z B Ibrahim R Johari and F Ismail ldquoOn the stability of blockbackward differentiation formulaerdquo Matematika vol 19 no 2pp 83ndash89 2003

[12] N A Nasir Z B Ibrahim M B Suleiman and K I OthmanldquoFifth order two-point block backward differentiation formulasfor solving ordinary differential equationsrdquo Applied Mathemat-ical Sciences vol 5 no 69ndash72 pp 3505ndash3518 2011

[13] H Musa M B Suleiman and N Senu ldquoA-stable 2-Point blockextended backward differentiation formulas for stiff ordinarydifferential equationsrdquo in Proceedings of the 5th InternationalConference on Research and education in Mathematics vol 1450of AIP Conference Proceedings pp 254ndash258 2012

[14] H Musa M B Suleiman and N Senu ldquoFully implicit 3-pointblock extended backward differentiation formula for stiff initialvalue problemsrdquo Applied Mathematical Sciences vol 6 no 85ndash88 pp 4211ndash4227 2012

[15] R Alexander ldquoDiagonally implicit Runge-kutta for stiff ordi-nary differential equationsrdquo SIAM Journal on Numerical Analy-sis vol 14 no 6 pp 1006ndash1021 1977

[16] O Y Ababneh R Ahmad and E S Ismail ldquoDesign of newdiagonally implicit Runge-Kutta methods for stiff problemsrdquoApplied Mathematical Sciences vol 3 no 45 pp 2241ndash22532009

[17] F Ismail A L I Raed Al-Khasawneh M Suleiman and A BU Malik Hassan ldquoEmbedded pair of diagonally implicit runge-kutta method for solving ordinary differential equationsrdquo SainsMalaysiana vol 39 no 6 pp 1049ndash1054 2010

[18] Z A Majid and M B Suleiman ldquoPerformance of four pointdiagonally implicit block method for solving first order stiffordinary differential equationsrdquo Department of MathematicsUTM vol 22 no 2 pp 137ndash146 2006

[19] G G Dahlquist ldquoA special stability problem for linearmultistepmethodsrdquo BIT Numerical Mathematics vol 3 no 1 pp 27ndash431963

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

3 Stability Analysis

In this section we will plot the graph of stability forDI2BBDF(2) DI2BBDF(3) and DI2BBDF(4) using Mathe-matica software Based onDahlquist [19] the linearmultistepmethod (LMM) is able to solve stiff problems if it satisfies thefollowing definition

Definition 4 TheLMMisA-stable if its region of absolute sta-bility contains the whole of the left-hand half-plane Re(ℎ120582) lt0

We consider the simplest test equation

119891 = 1199101015840= 120582119910 (31)

where the eigenvalues 120582119894 119894 = 1 2 119904 satisfy 120582

119894lt 0 to

analyze the stiff stability of the method Substituting (31) into(16) will produce the following matrices form

[

[

1 0

minus18

111

]

]

[119910119899+1

119910119899+2

]

=[[

[

minus1

3

4

32

11minus9

11

]]

]

[119910119899minus1

119910119899

] + ℎ[[

[

2

30

06

11

]]

]

[119891119899+1

119891119899+2

]

(32)

where

119860 =[[

[

1 minus2

3120582ℎ 0

minus18

111 minus

6

11120582ℎ

]]

]

119861 =[[

[

minus1

3

4

32

11minus9

11

]]

]

119862 = [0 0

0 0]

(33)

Let ℎ = ℎ120582 and evaluating the determinant of |119860119905 minus 119861|from (32) the stability polynomial 120588(119905 ℎ) of DI2BBDF(2) isobtained as follows

120588 (119905 ℎ) =1

33minus34119905

33minus8ℎ119905

11+ 1199052minus40ℎ1199052

33+4ℎ21199052

11 (34)

The similar approach is applied to obtain the stability polyno-mial of DI2BBDF(3) and DI2BBDF(4) Substituting (31) into(23) we have

[

[

1 0

minus48

251

]

]

[119910119899+1

119910119899+2

]

=[[

[

02

11

0 minus3

25

]]

]

[119910119899minus3

119910119899minus2

] +[[

[

minus9

11

18

1116

25minus36

25

]]

]

[119910119899minus1

119910119899

]

+ ℎ[[

[

6

110

012

25

]]

]

[119891119899+1

119891119899+2

]

(35)

where

119860 =[[

[

1 minus6

11120582ℎ 0

minus48

251 minus

12

25120582ℎ

]]

]

119861 =[[

[

minus9

11

18

11

16

25minus36

25

]]

]

119862 =[[

[

02

11

03

25

]]

]

(36)

Substituting (31) into (30) will give

[

[

1 0

minus300

1371

]

]

[119910119899+1

119910119899+2

]

=[[

[

minus3

25

16

25

12

137minus75

137

]]

]

[119910119899minus3

119910119899minus2

] +[[

[

minus36

25

48

25

200

137minus300

137

]]

]

[119910119899minus1

119910119899

]

+ ℎ[[

[

12

250

060

137

]]

]

[119891119899+1

119891119899+2

]

(37)

where

119860 =[[

[

1 minus12

25120582ℎ 0

minus300

1371 minus

60

137120582ℎ

]]

]

119861 =[[

[

minus36

25

48

25

200

137minus300

137

]]

]

119862 =[[

[

minus3

25

16

25

12

137minus75

137

]]

]

(38)

We compute the determinant of |1198601199052 minus 119861119905 minus 119862| from(35) and (37) to obtain the stability polynomials 120588(119905 ℎ) ofDI2BBDF(3) and DI2BBDF(4) respectively Consider

120588 (119905 ℎ) =36

275minus86119905

275minus2431199052

275minus324ℎ119905

2

275minus1621199053

275

+36ℎ1199053

275+ 1199054minus282ℎ119905

4

275+72ℎ21199054

275

(39)

120588 (119905 ℎ) =48

137minus176119905

3425minus3871199052

685minus1152ℎ119905

2

685minus2514119905

3

3425

minus216ℎ119905

3

685+ 1199054minus3144ℎ119905

4

3425+144ℎ21199054

685

(40)

Next the boundary of the stability region will be deter-mined by substituting 119905 = 119890

119894120579 into (34) (39) and (40)The graphs of stability region for all formulas are givenin Figures 1 2 and 3

In Figures 1 2 and 3 we observed that the inter-vals of unstable region for DI2BBDF(2) DI2BBDF(3) and

Mathematical Problems in Engineering 7

Unstable

Unstable

Stable

Stable

Stable1 2 3 4 5

Real part

minus3

minus2

minus1

1

2

3Imaginary part

(533333 0)

Figure 1 Graph of stability region for DI2BBDF(2)

Unstable

Unstable

Stable

Stable

Stable2 4 6 8

Real part

minus4

minus2

2

4

Imaginary part

(866667 0)

Figure 2 Graph of stability region for DI2BBDF(3)

DI2BBDF(4) are [0 533333] [0 866667] and [0 1386667]respectively All the graphs of stability regions are combinedas shown in Figure 4 The region outside the green line is thestable region of DI2BBDF(2) the region outside the red lineis the stable region ofDI2BBDF(3) and the region outside theblue line is the stable region of DI2BBDF(4)

Clearly the unstable region becomes larger whenthe order of the method increases From Definition 4DI2BBDF(2) is A-stable while DI2BBDF(3) andDI2BBDF(4) are almost A-stable since the stability regioncovers the entire negative half plane Therefore we canconclude that the proposed method is suitable for solvingstiff problems

Unstable

Unstable

Stable

Stable

2 4 6 8 10 12 14

minus5

5

(1386667 0)

Stable Real part

Imaginary part

Figure 3 Graph of stability region for DI2BBDF(4)

Imaginary part

Real part5 10

minus5

5

10

DI2BBDF(2)

DI2BBDF(3)

DI2BBDF(4)

A-stable region

A-stable region

Figure 4 Graph of stability region for DI2BBDF(2) DI2BBDF(3)and DI2BBDF(4)

4 Implementation of the Method

In this section the proposed methods will be implementedusing Newton iteration We begin by converting (16) (23)and (30) in general form as follows

119910119899+1

= 1205721119910119899+2

+ 1205731ℎ119891119899+1

+ 1205831

119910119899+2

= 1205722119910119899+1

+ 1205732ℎ119891119899+2

+ 1205832

(41)

where 1205831and 120583

2are the back values Equation (41) is trans-

formed into matrix form as follows

[1 0

0 1] [119910119899+1

119910119899+2

]

= [0 1205721

12057220] [119910119899+1

119910119899+2

] + ℎ[12057310

0 1205732

][119891119899+1

119891119899+2

] + [1205831

1205832

]

(42)

8 Mathematical Problems in Engineering

where

119868 = [1 0

0 1] 119884 = [

119910119899+1

119910119899+2

] 119861 = [12057310

0 1205732

]

119865 = [119891119899+1

119891119899+2

] 120583 = [1205831

1205832

]

(43)

Implementing Newtonrsquos method to (42) produces

119865 = (119868 minus 119860)119884 minus ℎ119861119865 + 120583 (44)

Iteration for (44) is given by

119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2= 119865 (119910

(119894)

119899+1119899+2) sdot 1198651015840(119910(119894)

119899+1119899+2)minus1

(45)

where 119910(119894+1)119899+1

denotes the (119894 + 1)th iteration Equation (45) canbe rearranged into the following equation

1198651015840(119910(119894)

119899+1119899+2) (119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2) = 119865 (119910

(119894)

119899+1119899+2) (46)

Replacing (44) into (46) yields

(119868 minus 119860)119884 minus ℎ119861120597119891

120597119910(119910(119894)

119899+1119899+2) (119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2)

= (119868 minus 119860) (119910(119894)

119899+1119899+2) minus ℎ119861 (119910

(119894)

119899+1119899+2) + 120583

(47)

where (120597119891120597119910)(119910(119894)119899+1119899+2

) is the Jacobian matrix of 119891 with re-spect to 119910

41 Newton Iteration of DI2BBDF(2) Formula (16) is writtenin the form of (41)

1198651= 119910119899+1

minus2

3ℎ119891119899+1

+ 1205831

1198652= 119910119899+2

minus18

11119910119899+1

minus6

11ℎ119891119899+2

+ 1205832

(48)

Applying (48) into (47) will yield matrix form as follows

[[[

[

1 minus2

3ℎ120597119891119899+1

120597119910119899+1

0

0 1 minus6

11ℎ120597119891119899+2

120597119910119899+2

]]]

]

[119890(119894+1)

119899+1

119890(119894+1)

119899+2

]

= [

[

minus1 0

18

11minus1

]

]

[119910(119894)

119899+1

119910(119894)

119899+2

] + ℎ[[

[

2

30

06

11

]]

]

[119891(119894)

119899+1

119891(119894)

119899+2

] + [1205831

1205832

]

(49)

with 119890(119894+1)119899+1119899+2

= 119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2being the increment

42 Newton Iteration of DI2BBDF(3) Rearranging formula(23) in the form of (41) we have

1198651= 119910119899+1

minus6

11ℎ119891119899+1

+ 1205831

1198652= 119910119899+2

minus48

25119910119899+1

minus12

25ℎ119891119899+2

+ 1205832

(50)

Replacing (50) into (47) this will yieldmatrix form as follows

[[[

[

1 minus6

11ℎ120597119891119899+1

120597119910119899+1

0

0 1 minus12

25ℎ120597119891119899+2

120597119910119899+2

]]]

]

[119890(119894+1)

119899+1

119890(119894+1)

119899+2

]

= [

[

minus1 0

48

25minus1

]

]

[119910(119894)

119899+1

119910(119894)

119899+2

] + ℎ[[

[

6

110

012

25

]]

]

[119891(119894)

119899+1

119891(119894)

119899+2

] + [1205831

1205832

]

(51)

with 119890(119894+1)119899+1119899+2

= 119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2being the increment

43 Newton Iteration of DI2BBDF(4) We rewrite formula(30) in the form of (41) and obtain

1198651= 119910119899+1

minus12

25ℎ119891119899+1

+ 1205831

1198652= 119910119899+2

minus300

137119910119899+1

minus60

137ℎ119891119899+2

+ 1205832

(52)

Substituting (52) into (47) will produce

[[[

[

1 minus12

25ℎ120597119891119899+1

120597119910119899+1

0

0 1 minus60

137ℎ120597119891119899+2

120597119910119899+2

]]]

]

[119890(119894+1)

119899+1

119890(119894+1)

119899+2

]

= [

[

minus1 0

300

137minus1

]

]

[119910(119894)

119899+1

119910(119894)

119899+2

] + ℎ[[

[

12

250

060

137

]]

]

[119891(119894)

119899+1

119891(119894)

119899+2

] + [1205831

1205832

]

(53)

with 119890(119894+1)119899+1119899+2

= 119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2being the increment

All the formulas derived are implemented in predictor-corrector computation which is symbolized as PECE modeP and C indicate one application of the predictor and thecorrector respectively and E indicates one evaluation of thefunction119891 given119909 and119910The approximation calculations for119910119899+1

and 119910119899+2

in PECE are as follows

(1) P (Predict) 119910(119901)119899+1119899+2

(2) E (Evaluate) 1199101015840119899+1119899+2

= 119891(119909119899+1119899+2

119910(119901)

119899+1119899+2)

(3) C (Correct) 119910(119888)119899+1119899+2

(4) E (Evaluate) 1199101015840119899+1119899+2

= 119891(119909119899+1119899+2

119910(119888)

119899+1119899+2)

To approximate the solution to 119910(119888)

119899+1119899+2 we apply the

two-stage Newton type iteration The iteration process forDI2BBDF(2) is done as follows

Mathematical Problems in Engineering 9

(1) Compute the values for 119890(119894+1)119899+1119899+2

= 119860minus1119861 where

119860 =[[[

[

1 minus2

3ℎ120597119891119899+1

120597119910119899+1

0

0 1 minus6

11ℎ120597119891119899+2

120597119910119899+2

]]]

]

119861 =[[

[

minus119910(119894)

119899+1+2

3ℎ119891(119894)

119899+1+ 1205831

0

18

11119910(119894)

119899+1minus119910(119894)

119899+2+6

11ℎ119891(119894)

119899+2+ 1205832

]]

]

(54)

(2) Calculate the corrected value for 119910(119894+1)119899+1119899+2

with thevalue 119890(119894+1)

119899+1119899+2from Step (1)

(3) Solve 119890(119894+1)119899+1119899+2

= 119860minus1119861 for second stage iteration

(4) The final values for 119910(119894+1)119899+1119899+2

are obtained from thesecond stage iteration of 119890(119894+1)

119899+1119899+2

The similar iteration process is applied for DI2BBDF(3) andDI2BBDF(4)

5 Test Problems

In this section linear and nonlinear stiff problems aretested using C programming to examine the efficiency andreliability of the proposed method The following problemsare commonly found in engineering and physical sciencesparticularly in the studies of vibrations electrical circuits andchemical reaction

Problem 1 (linear) One has

1199101015840= minus10119910 + 10 (55)

Exact solution is

119910 (119909) = 1 + 119890minus10119909

(56)

Initial condition is

119910 (0) = 2

0 le 119909 le 10

(57)

(source Ibrahim [8])

Problem 2 (linear) One has

1199101015840

1= minus5119910

1minus 41199102 119910

1015840

2= minus4119910

1minus 51199102 (58)

Exact solutions are

1199101(119909) =

1

2119890minus9119909

+1

2119890minus119909 119910

2(119909) =

1

2119890minus9119909

minus1

2119890minus119909 (59)

Initial conditions are

1199101(0) = 1 119910

2(0) = 0

0 le 119909 le 20(60)

(source Musa et al [13])

Problem 3 (linear) One has

1199101015840= minus10119909119910 (61)

Exact solution is

119910 (119909) = 119890minus51199092

(62)

Initial condition is119910 (0) = 1

0 le 119909 le 10

(63)

(source Ibrahim et al [10])

Problem 4 (linear) One has

1199101015840

1= 91199101+ 24119910

2+ 5 cos119909 minus 1

3sin119909

1199101015840

2= minus 24119910

1minus 51119910

2minus 9 cos119909 + 1

3sin119909

(64)

Exact solutions are

1199101(119909) = 2119890

minus3119909minus 119890minus39119909

+1

3cos119909

1199102(119909) = minus119890

minus3119909+ 2119890minus39119909

minus1

3cos119909

(65)

Initial conditions are

1199101(0) =

4

3 1199102(0) =

2

3 0 le 119909 le 10 (66)

(source Musa et al [14])

Problem 5 (nonlinear) One has

1199101015840= minus

1199103

2 (67)

Exact solution is

119910 (119909) =1

radic1 + 119909 (68)

Initial condition is119910 (0) = 1

0 le 119909 le 4

(69)

(source Musa et al [14])

Problem 6 (nonlinear) One has

1199101015840=119910 (1 minus 119910)

2119910 minus 1 (70)

Exact solution is

119910 (119909) =1

2+ radic

1

4minus5

36119890minus119909 (71)

Initial condition is

119910 (0) =5

6

0 le 119909 le 1

(72)

(source Musa et al [14])

10 Mathematical Problems in Engineering

Table 1 The accuracy for problem 1

ℎ Methods MAXE TIME

10minus2

FI2BBDF(3) 597499119864 minus 2 136773119864 minus 3

FI2BEBDF(3) 567155119864 minus 2 145525119864 minus 3

DI2BBDF(2) 110568119864 minus 2 472498119864 minus 3

DI2BBDF(3) 166455119864 minus 2 477409119864 minus 3

DI2BBDF(4) 216342119864 minus 2 483298119864 minus 3

10minus4

FI2BBDF(3) 734012119864 minus 4 125797119864 minus 1

FI2BEBDF(3) 323640119864 minus 5 136259119864 minus 1

DI2BBDF(2) 118355119864 minus 6 167260119864 minus 1

DI2BBDF(3) 173430119864 minus 6 169325119864 minus 1

DI2BBDF(4) 225966119864 minus 6 172995119864 minus 1

10minus6

FI2BBDF(3) 735741119864 minus 6 1257001198641

FI2BEBDF(3) 350090119864 minus 7 1362301198641

DI2BBDF(2) 118742119864 minus 10 1491591198641

DI2BBDF(3) 173634119864 minus 10 1544931198641

DI2BBDF(4) 226000119864 minus 10 1546201198641

Table 2 The accuracy for problem 2

ℎ Methods MAXE TIME

10minus2

FI2BBDF(3) 271597119864 minus 2 596467119864 minus 3

FI2BEBDF(3) 149811119864 minus 2 661025119864 minus 3

DI2BBDF(2) 456073119864 minus 3 104480119864 minus 2

DI2BBDF(3) 686126119864 minus 3 108421119864 minus 2

DI2BBDF(4) 893279119864 minus 3 111089119864 minus 2

10minus4

FI2BBDF(3) 340437119864 minus 4 590430119864 minus 1

FI2BEBDF(3) 262649119864 minus 5 654305119864 minus 1

DI2BBDF(2) 485404119864 minus 7 373210119864 minus 1

DI2BBDF(3) 711155119864 minus 7 386556119864 minus 1

DI2BBDF(4) 926476119864 minus 7 387070119864 minus 1

10minus6

FI2BBDF(3) 341146119864 minus 6 5901981198641

FI2BEBDF(3) 244097119864 minus 7 6541061198641

DI2BBDF(2) 486843119864 minus 11 3484511198641

DI2BBDF(3) 711903119864 minus 11 3604901198641

DI2BBDF(4) 926598119864 minus 11 3606491198641

6 Numerical Results

The performance of the derived methods is compared withthe existing methods in terms of maximum error and exe-cution time We consider 10minus2 10minus4 and 10minus6 as the stepsize ℎ Tables 1 and 2 present the performance compar-ison of DI2BBDF(2) DI2BBDF(3) and DI2BBDF(4) withFI2BBDF(3) and FI2BEBDF(3) whereas Tables 3ndash6 exhibitthe comparison of proposed methods with FI3BBDF(3) andFI3BEBDF(3) In addition the graphs of log(MAXE) againstlog(ℎ) are illustrated as shown in Figures 5ndash10 The followingnotations are used in Tables 1ndash6

ℎ step sizeMAXE maximum errorTIME time execution using high performance com-puter (HPC)

Table 3 The accuracy for problem 3

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 356692119864 minus 2 141642119864 minus 3

FI3BEBDF(3) 124084119864 minus 2 302167119864 minus 3

DI2BBDF(2) 117256119864 minus 3 448489119864 minus 3

DI2BBDF(3) 172278119864 minus 3 481200119864 minus 3

DI2BBDF(4) 225003119864 minus 3 485301119864 minus 3

10minus4

FI3BBDF(3) 435640119864 minus 4 130103119864 minus 1

FI3BEBDF(3) 706516119864 minus 5 208115119864 minus 1

DI2BBDF(2) 118749119864 minus 7 160304119864 minus 1

DI2BBDF(3) 173636119864 minus 7 177445119864 minus 1

DI2BBDF(4) 226000119864 minus 7 178064119864 minus 1

10minus6

FI3BBDF(3) 434625119864 minus 6 1297751198641

FI3BEBDF(3) 703257119864 minus 7 2076091198640

DI2BBDF(2) 124989119864 minus 11 1507391198641

DI2BBDF(3) 173636119864 minus 11 1550731198641

DI2BBDF(4) 225999119864 minus 11 1602161198641

Table 4 The accuracy for problem 4

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 66269411986499 701775119864 minus 3

FI3BEBDF(3) 168449119864 minus 1 137847119864 minus 2

DI2BBDF(2) 280830119864 minus 1 106871119864 minus 2

DI2BBDF(3) 400389119864 minus 1 124969119864 minus 2

DI2BBDF(4) 477342119864 minus 1 126502119864 minus 2

10minus4

FI3BBDF(3) 845376119864 minus 3 676737119864 minus 1

FI3BEBDF(3) 695725119864 minus 3 135671119864 + 0

DI2BBDF(2) 356255119864 minus 5 432359119864 minus 1

DI2BBDF(3) 524241119864 minus 5 442599119864 minus 1

DI2BBDF(4) 685105119864 minus 5 456207119864 minus 1

10minus6

FI3BBDF(3) 854545119864 minus 5 6768191198641

FI3BEBDF(3) 718859119864 minus 5 1358621198642

DI2BBDF(2) 360130119864 minus 9 4000101198641

DI2BBDF(3) 526673119864 minus 9 4001921198641

DI2BBDF(4) 685529119864 minus 9 4103221198641

Table 5 The accuracy for problem 5

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 525481119864 minus 3 578083119864 minus 4

FI3BEBDF(3) 388475119864 minus 3 916583119864 minus 4

DI2BBDF(2) 856253119864 minus 5 229836119864 minus 4

DI2BBDF(3) 121629119864 minus 4 189066119864 minus 4

DI2BBDF(4) 165094119864 minus 4 230074119864 minus 4

10minus4

FI3BBDF(3) 551558119864 minus 5 483860119864 minus 2

FI3BEBDF(3) 427506119864 minus 5 774877119864 minus 2

DI2BBDF(2) 889803119864 minus 9 117438119864 minus 2

DI2BBDF(3) 130124119864 minus 8 131578119864 minus 2

DI2BBDF(4) 169456119864 minus 8 126071119864 minus 2

10minus6

FI3BBDF(3) 551803119864 minus 7 547768119864 + 0

FI3BEBDF(3) 427924119864 minus 7 777643119864 + 0

DI2BBDF(2) 126115119864 minus 11 119318119864 + 0

DI2BBDF(3) 627106119864 minus 11 133892119864 + 0

DI2BBDF(4) 104222119864 minus 11 126146119864 + 0

Mathematical Problems in Engineering 11

Table 6 The accuracy for problem 6

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 218555119864 minus 3 283084119864 minus 4

FI3BEBDF(3) 152719119864 minus 3 428500119864 minus 4

DI2BBDF(2) 387041119864 minus 5 538826119864 minus 5

DI2BBDF(3) 547958119864 minus 5 531673119864 minus 5

DI2BBDF(4) 747696119864 minus 5 618877119864 minus 5

10minus4

FI3BBDF(3) 229812119864 minus 5 148627119864 minus 2

FI3BEBDF(3) 178708119864 minus 5 232184119864 minus 2

DI2BBDF(2) 401638119864 minus 9 349712119864 minus 3

DI2BBDF(3) 587347119864 minus 9 375295119864 minus 3

DI2BBDF(4) 764927119864 minus 9 364804119864 minus 3

10minus6

FI3BBDF(3) 229936119864 minus 7 146660119864 + 0

FI3BEBDF(3) 178973119864 minus 7 228775119864 + 0

DI2BBDF(2) 990807119864 minus 12 338157119864 minus 1

DI2BBDF(3) 377716119864 minus 11 369837119864 minus 1

DI2BBDF(4) 462796119864 minus 12 356802119864 minus 1

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

log(h)minus6 minus5 minus4 minus3 minus2

FI2BBDF(3)FI2BEBDF(3)

DI2BBDF(2)DI2BBDF(3)DI2BBDF(4)

Figure 5 Graph of log(MAXE) versus log(ℎ) for problem 1

DI2BBDF(2) diagonally implicit 2-point block back-ward differentiation formulas of order 2DI2BBDF(3) diagonally implicit 2-point block back-ward differentiation formulas of order 3DI2BBDF(4) diagonally implicit 2-point block back-ward differentiation formulas of order 4FI2BBDF(3) fully implicit 2-point block backwarddifferentiation formulas of order 3FI2BEBDF(3) fully implicit 2-point block extendedbackward differentiation formulas of order 3FI3BBDF(3) fully implicit 3-point block backwarddifferentiation formulas of order 3

log(h)minus6 minus5 minus4 minus3 minus2

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

FI2BBDF(3)FI2BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

Figure 6 Graph of log(MAXE) versus log(ℎ) for problem 2

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus11minus12

minus10minus9minus8minus7minus6minus5minus4minus3minus2minus10

log(

MA

XE)

Figure 7 Graph of log(MAXE) versus log(ℎ) for problem 3

FI3BEBDF(3) fully implicit 3-point block extendedbackward differentiation formulas of order 3

7 Discussion

This section is divided into discussion of maximum error andcomputational time

71 Maximum Error From the numerical results in Tables1ndash6 we observe that DI2BBDF(2) DI2BBDF(3) andDI2BBDF(4) outperformed the existing methods in termsof maximum error This is expected because the diagonally

12 Mathematical Problems in Engineering

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

Figure 8 Graph of log(MAXE) versus log(ℎ) for problem 4

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus11

minus12

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

Figure 9 Graph of log(MAXE) versus log(ℎ) for problem 5

implicit method has less differentiation coefficients in orderto prevent the cumulative error The graphs in Figures 5ndash10depict the scales of maximum error versus step size h for theproposed methods as compared to existing methods Thereis a slight drop in accuracy for the proposed methods as theorder increases This is due to the increase in cumulativeerrors during the computation as more interpolating pointsare used Among the methods of order three DI2BBDF(3)is more accurate compared with FI2BBDF(3) FI3BBDF(3)FI2BEBDF(3) and FI3BEBDF(3) For all test problems we

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus11minus12

minus10minus9minus8minus7minus6minus5minus4minus3minus2minus10

log(

MA

XE)

Figure 10 Graph of log(MAXE) versus log(ℎ) for problem 6

conclude that the accuracy of the proposedmethod increasesas the step size becomes smaller

72 Computational Time In Tables 1-2 it can be seen thatthe execution time taken by DI2BBDF(2) DI2BBDF(3) andDI2BBDF(4) is comparable with that of FI2BBDF(3) andFI2BEBDF(3) The computational time increases as the stepsize becomes smaller However Tables 3ndash6 show that theproposed methods compute faster than FI3BBDF(3) andFI3BEBDF(3) This could be justified by the fact that the3-point method has to perform extra computation on theJacobian matrix since it involves a 3 by 3 matrix while the 2-point method has to compute Jacobian matrix of dimension2 by 2 only Among the proposed methods DI2BBDF(2)computes faster than DI2BBDF(3) and DI2BBDF(4) In factthe number of back values involved in every formula willaffect the computational time Since DI2BBDF(4) has moreback values than DI2BBDF(3) and DI2BBDF(2) it requiresextra time to compute the approximated solutions

8 Conclusion

Research conducted in this paper shows the capability ofconstructing the diagonally implicit BBDF method with A-stable properties From the results obtained via the numericalexperiment we can conclude that the proposed methodserves the purpose of significant alternative numericalmethod for solving linear and nonlinear stiff IVPs occurringin the fields of engineering and physical sciences

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 13

Acknowledgment

This researchwas supported by the Institute forMathematicalResearch (INSPEM) Department of Mathematics UniversitiPutra Malaysia (UPM) under Fundamental Grant Scheme(Project code 02-02-13-1380FR)

References

[1] C F Curtiss and J O Hirschfelder ldquoIntegration of stiff equa-tionsrdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 38 no 3 pp 235ndash243 1952

[2] J D Lambert Numercial Methods for Ordinary DifferentialSystems The Initial Value Problem John Wiley amp Sons NewYork NY USA 1991

[3] W E Milne Numerical Solution of Differential Equations JohnWiley amp Sons New York NY USA 1953

[4] L F Shampine and H A Watts ldquoBlock implicit one-stepmethodsrdquo Mathematics of Computation vol 23 pp 731ndash7401969

[5] S O Fatunla ldquoBlock methods for second order ODEsrdquo Inter-national Journal of Computer Mathematics vol 41 pp 55ndash631990

[6] Z A Majid M B Suleiman F I Ismail and K I Othmanldquo2-Point 1 block diagonally and 2-point 1 block fully implicitmethod for solving first order ordinary differential equationsrdquoin Proceedings of the 12th National Symposium on MathematicalScience vol 12 2004

[7] Z Abdul Majid N Z Mokhtar and M Suleiman ldquoDirecttwo-point block one-step method for solving general second-order ordinary differential equationsrdquo Mathematical Problemsin Engineering vol 2012 Article ID 184253 16 pages 2012

[8] Z B Ibrahim Block method for multistep formulas for solvingordinary differential equations [PhD thesis] Universiti PutraMalaysia Seri Kembangan Malaysia 2006

[9] Z B Ibrahim M Suleiman and K I Othman ldquoFixed coeffi-cients block backward differentiation formulas for the numer-ical solution of stiff ordinary differential equationsrdquo EuropeanJournal of Scientific Research vol 21 no 3 pp 508ndash520 2008

[10] Z B Ibrahim M B Suleiman and K I Othman ldquoImplicit r-point block backward differentiation formula for solving first-order stiff ODEsrdquo Applied Mathematics and Computation vol186 no 1 pp 558ndash565 2007

[11] Z B Ibrahim R Johari and F Ismail ldquoOn the stability of blockbackward differentiation formulaerdquo Matematika vol 19 no 2pp 83ndash89 2003

[12] N A Nasir Z B Ibrahim M B Suleiman and K I OthmanldquoFifth order two-point block backward differentiation formulasfor solving ordinary differential equationsrdquo Applied Mathemat-ical Sciences vol 5 no 69ndash72 pp 3505ndash3518 2011

[13] H Musa M B Suleiman and N Senu ldquoA-stable 2-Point blockextended backward differentiation formulas for stiff ordinarydifferential equationsrdquo in Proceedings of the 5th InternationalConference on Research and education in Mathematics vol 1450of AIP Conference Proceedings pp 254ndash258 2012

[14] H Musa M B Suleiman and N Senu ldquoFully implicit 3-pointblock extended backward differentiation formula for stiff initialvalue problemsrdquo Applied Mathematical Sciences vol 6 no 85ndash88 pp 4211ndash4227 2012

[15] R Alexander ldquoDiagonally implicit Runge-kutta for stiff ordi-nary differential equationsrdquo SIAM Journal on Numerical Analy-sis vol 14 no 6 pp 1006ndash1021 1977

[16] O Y Ababneh R Ahmad and E S Ismail ldquoDesign of newdiagonally implicit Runge-Kutta methods for stiff problemsrdquoApplied Mathematical Sciences vol 3 no 45 pp 2241ndash22532009

[17] F Ismail A L I Raed Al-Khasawneh M Suleiman and A BU Malik Hassan ldquoEmbedded pair of diagonally implicit runge-kutta method for solving ordinary differential equationsrdquo SainsMalaysiana vol 39 no 6 pp 1049ndash1054 2010

[18] Z A Majid and M B Suleiman ldquoPerformance of four pointdiagonally implicit block method for solving first order stiffordinary differential equationsrdquo Department of MathematicsUTM vol 22 no 2 pp 137ndash146 2006

[19] G G Dahlquist ldquoA special stability problem for linearmultistepmethodsrdquo BIT Numerical Mathematics vol 3 no 1 pp 27ndash431963

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Unstable

Unstable

Stable

Stable

Stable1 2 3 4 5

Real part

minus3

minus2

minus1

1

2

3Imaginary part

(533333 0)

Figure 1 Graph of stability region for DI2BBDF(2)

Unstable

Unstable

Stable

Stable

Stable2 4 6 8

Real part

minus4

minus2

2

4

Imaginary part

(866667 0)

Figure 2 Graph of stability region for DI2BBDF(3)

DI2BBDF(4) are [0 533333] [0 866667] and [0 1386667]respectively All the graphs of stability regions are combinedas shown in Figure 4 The region outside the green line is thestable region of DI2BBDF(2) the region outside the red lineis the stable region ofDI2BBDF(3) and the region outside theblue line is the stable region of DI2BBDF(4)

Clearly the unstable region becomes larger whenthe order of the method increases From Definition 4DI2BBDF(2) is A-stable while DI2BBDF(3) andDI2BBDF(4) are almost A-stable since the stability regioncovers the entire negative half plane Therefore we canconclude that the proposed method is suitable for solvingstiff problems

Unstable

Unstable

Stable

Stable

2 4 6 8 10 12 14

minus5

5

(1386667 0)

Stable Real part

Imaginary part

Figure 3 Graph of stability region for DI2BBDF(4)

Imaginary part

Real part5 10

minus5

5

10

DI2BBDF(2)

DI2BBDF(3)

DI2BBDF(4)

A-stable region

A-stable region

Figure 4 Graph of stability region for DI2BBDF(2) DI2BBDF(3)and DI2BBDF(4)

4 Implementation of the Method

In this section the proposed methods will be implementedusing Newton iteration We begin by converting (16) (23)and (30) in general form as follows

119910119899+1

= 1205721119910119899+2

+ 1205731ℎ119891119899+1

+ 1205831

119910119899+2

= 1205722119910119899+1

+ 1205732ℎ119891119899+2

+ 1205832

(41)

where 1205831and 120583

2are the back values Equation (41) is trans-

formed into matrix form as follows

[1 0

0 1] [119910119899+1

119910119899+2

]

= [0 1205721

12057220] [119910119899+1

119910119899+2

] + ℎ[12057310

0 1205732

][119891119899+1

119891119899+2

] + [1205831

1205832

]

(42)

8 Mathematical Problems in Engineering

where

119868 = [1 0

0 1] 119884 = [

119910119899+1

119910119899+2

] 119861 = [12057310

0 1205732

]

119865 = [119891119899+1

119891119899+2

] 120583 = [1205831

1205832

]

(43)

Implementing Newtonrsquos method to (42) produces

119865 = (119868 minus 119860)119884 minus ℎ119861119865 + 120583 (44)

Iteration for (44) is given by

119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2= 119865 (119910

(119894)

119899+1119899+2) sdot 1198651015840(119910(119894)

119899+1119899+2)minus1

(45)

where 119910(119894+1)119899+1

denotes the (119894 + 1)th iteration Equation (45) canbe rearranged into the following equation

1198651015840(119910(119894)

119899+1119899+2) (119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2) = 119865 (119910

(119894)

119899+1119899+2) (46)

Replacing (44) into (46) yields

(119868 minus 119860)119884 minus ℎ119861120597119891

120597119910(119910(119894)

119899+1119899+2) (119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2)

= (119868 minus 119860) (119910(119894)

119899+1119899+2) minus ℎ119861 (119910

(119894)

119899+1119899+2) + 120583

(47)

where (120597119891120597119910)(119910(119894)119899+1119899+2

) is the Jacobian matrix of 119891 with re-spect to 119910

41 Newton Iteration of DI2BBDF(2) Formula (16) is writtenin the form of (41)

1198651= 119910119899+1

minus2

3ℎ119891119899+1

+ 1205831

1198652= 119910119899+2

minus18

11119910119899+1

minus6

11ℎ119891119899+2

+ 1205832

(48)

Applying (48) into (47) will yield matrix form as follows

[[[

[

1 minus2

3ℎ120597119891119899+1

120597119910119899+1

0

0 1 minus6

11ℎ120597119891119899+2

120597119910119899+2

]]]

]

[119890(119894+1)

119899+1

119890(119894+1)

119899+2

]

= [

[

minus1 0

18

11minus1

]

]

[119910(119894)

119899+1

119910(119894)

119899+2

] + ℎ[[

[

2

30

06

11

]]

]

[119891(119894)

119899+1

119891(119894)

119899+2

] + [1205831

1205832

]

(49)

with 119890(119894+1)119899+1119899+2

= 119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2being the increment

42 Newton Iteration of DI2BBDF(3) Rearranging formula(23) in the form of (41) we have

1198651= 119910119899+1

minus6

11ℎ119891119899+1

+ 1205831

1198652= 119910119899+2

minus48

25119910119899+1

minus12

25ℎ119891119899+2

+ 1205832

(50)

Replacing (50) into (47) this will yieldmatrix form as follows

[[[

[

1 minus6

11ℎ120597119891119899+1

120597119910119899+1

0

0 1 minus12

25ℎ120597119891119899+2

120597119910119899+2

]]]

]

[119890(119894+1)

119899+1

119890(119894+1)

119899+2

]

= [

[

minus1 0

48

25minus1

]

]

[119910(119894)

119899+1

119910(119894)

119899+2

] + ℎ[[

[

6

110

012

25

]]

]

[119891(119894)

119899+1

119891(119894)

119899+2

] + [1205831

1205832

]

(51)

with 119890(119894+1)119899+1119899+2

= 119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2being the increment

43 Newton Iteration of DI2BBDF(4) We rewrite formula(30) in the form of (41) and obtain

1198651= 119910119899+1

minus12

25ℎ119891119899+1

+ 1205831

1198652= 119910119899+2

minus300

137119910119899+1

minus60

137ℎ119891119899+2

+ 1205832

(52)

Substituting (52) into (47) will produce

[[[

[

1 minus12

25ℎ120597119891119899+1

120597119910119899+1

0

0 1 minus60

137ℎ120597119891119899+2

120597119910119899+2

]]]

]

[119890(119894+1)

119899+1

119890(119894+1)

119899+2

]

= [

[

minus1 0

300

137minus1

]

]

[119910(119894)

119899+1

119910(119894)

119899+2

] + ℎ[[

[

12

250

060

137

]]

]

[119891(119894)

119899+1

119891(119894)

119899+2

] + [1205831

1205832

]

(53)

with 119890(119894+1)119899+1119899+2

= 119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2being the increment

All the formulas derived are implemented in predictor-corrector computation which is symbolized as PECE modeP and C indicate one application of the predictor and thecorrector respectively and E indicates one evaluation of thefunction119891 given119909 and119910The approximation calculations for119910119899+1

and 119910119899+2

in PECE are as follows

(1) P (Predict) 119910(119901)119899+1119899+2

(2) E (Evaluate) 1199101015840119899+1119899+2

= 119891(119909119899+1119899+2

119910(119901)

119899+1119899+2)

(3) C (Correct) 119910(119888)119899+1119899+2

(4) E (Evaluate) 1199101015840119899+1119899+2

= 119891(119909119899+1119899+2

119910(119888)

119899+1119899+2)

To approximate the solution to 119910(119888)

119899+1119899+2 we apply the

two-stage Newton type iteration The iteration process forDI2BBDF(2) is done as follows

Mathematical Problems in Engineering 9

(1) Compute the values for 119890(119894+1)119899+1119899+2

= 119860minus1119861 where

119860 =[[[

[

1 minus2

3ℎ120597119891119899+1

120597119910119899+1

0

0 1 minus6

11ℎ120597119891119899+2

120597119910119899+2

]]]

]

119861 =[[

[

minus119910(119894)

119899+1+2

3ℎ119891(119894)

119899+1+ 1205831

0

18

11119910(119894)

119899+1minus119910(119894)

119899+2+6

11ℎ119891(119894)

119899+2+ 1205832

]]

]

(54)

(2) Calculate the corrected value for 119910(119894+1)119899+1119899+2

with thevalue 119890(119894+1)

119899+1119899+2from Step (1)

(3) Solve 119890(119894+1)119899+1119899+2

= 119860minus1119861 for second stage iteration

(4) The final values for 119910(119894+1)119899+1119899+2

are obtained from thesecond stage iteration of 119890(119894+1)

119899+1119899+2

The similar iteration process is applied for DI2BBDF(3) andDI2BBDF(4)

5 Test Problems

In this section linear and nonlinear stiff problems aretested using C programming to examine the efficiency andreliability of the proposed method The following problemsare commonly found in engineering and physical sciencesparticularly in the studies of vibrations electrical circuits andchemical reaction

Problem 1 (linear) One has

1199101015840= minus10119910 + 10 (55)

Exact solution is

119910 (119909) = 1 + 119890minus10119909

(56)

Initial condition is

119910 (0) = 2

0 le 119909 le 10

(57)

(source Ibrahim [8])

Problem 2 (linear) One has

1199101015840

1= minus5119910

1minus 41199102 119910

1015840

2= minus4119910

1minus 51199102 (58)

Exact solutions are

1199101(119909) =

1

2119890minus9119909

+1

2119890minus119909 119910

2(119909) =

1

2119890minus9119909

minus1

2119890minus119909 (59)

Initial conditions are

1199101(0) = 1 119910

2(0) = 0

0 le 119909 le 20(60)

(source Musa et al [13])

Problem 3 (linear) One has

1199101015840= minus10119909119910 (61)

Exact solution is

119910 (119909) = 119890minus51199092

(62)

Initial condition is119910 (0) = 1

0 le 119909 le 10

(63)

(source Ibrahim et al [10])

Problem 4 (linear) One has

1199101015840

1= 91199101+ 24119910

2+ 5 cos119909 minus 1

3sin119909

1199101015840

2= minus 24119910

1minus 51119910

2minus 9 cos119909 + 1

3sin119909

(64)

Exact solutions are

1199101(119909) = 2119890

minus3119909minus 119890minus39119909

+1

3cos119909

1199102(119909) = minus119890

minus3119909+ 2119890minus39119909

minus1

3cos119909

(65)

Initial conditions are

1199101(0) =

4

3 1199102(0) =

2

3 0 le 119909 le 10 (66)

(source Musa et al [14])

Problem 5 (nonlinear) One has

1199101015840= minus

1199103

2 (67)

Exact solution is

119910 (119909) =1

radic1 + 119909 (68)

Initial condition is119910 (0) = 1

0 le 119909 le 4

(69)

(source Musa et al [14])

Problem 6 (nonlinear) One has

1199101015840=119910 (1 minus 119910)

2119910 minus 1 (70)

Exact solution is

119910 (119909) =1

2+ radic

1

4minus5

36119890minus119909 (71)

Initial condition is

119910 (0) =5

6

0 le 119909 le 1

(72)

(source Musa et al [14])

10 Mathematical Problems in Engineering

Table 1 The accuracy for problem 1

ℎ Methods MAXE TIME

10minus2

FI2BBDF(3) 597499119864 minus 2 136773119864 minus 3

FI2BEBDF(3) 567155119864 minus 2 145525119864 minus 3

DI2BBDF(2) 110568119864 minus 2 472498119864 minus 3

DI2BBDF(3) 166455119864 minus 2 477409119864 minus 3

DI2BBDF(4) 216342119864 minus 2 483298119864 minus 3

10minus4

FI2BBDF(3) 734012119864 minus 4 125797119864 minus 1

FI2BEBDF(3) 323640119864 minus 5 136259119864 minus 1

DI2BBDF(2) 118355119864 minus 6 167260119864 minus 1

DI2BBDF(3) 173430119864 minus 6 169325119864 minus 1

DI2BBDF(4) 225966119864 minus 6 172995119864 minus 1

10minus6

FI2BBDF(3) 735741119864 minus 6 1257001198641

FI2BEBDF(3) 350090119864 minus 7 1362301198641

DI2BBDF(2) 118742119864 minus 10 1491591198641

DI2BBDF(3) 173634119864 minus 10 1544931198641

DI2BBDF(4) 226000119864 minus 10 1546201198641

Table 2 The accuracy for problem 2

ℎ Methods MAXE TIME

10minus2

FI2BBDF(3) 271597119864 minus 2 596467119864 minus 3

FI2BEBDF(3) 149811119864 minus 2 661025119864 minus 3

DI2BBDF(2) 456073119864 minus 3 104480119864 minus 2

DI2BBDF(3) 686126119864 minus 3 108421119864 minus 2

DI2BBDF(4) 893279119864 minus 3 111089119864 minus 2

10minus4

FI2BBDF(3) 340437119864 minus 4 590430119864 minus 1

FI2BEBDF(3) 262649119864 minus 5 654305119864 minus 1

DI2BBDF(2) 485404119864 minus 7 373210119864 minus 1

DI2BBDF(3) 711155119864 minus 7 386556119864 minus 1

DI2BBDF(4) 926476119864 minus 7 387070119864 minus 1

10minus6

FI2BBDF(3) 341146119864 minus 6 5901981198641

FI2BEBDF(3) 244097119864 minus 7 6541061198641

DI2BBDF(2) 486843119864 minus 11 3484511198641

DI2BBDF(3) 711903119864 minus 11 3604901198641

DI2BBDF(4) 926598119864 minus 11 3606491198641

6 Numerical Results

The performance of the derived methods is compared withthe existing methods in terms of maximum error and exe-cution time We consider 10minus2 10minus4 and 10minus6 as the stepsize ℎ Tables 1 and 2 present the performance compar-ison of DI2BBDF(2) DI2BBDF(3) and DI2BBDF(4) withFI2BBDF(3) and FI2BEBDF(3) whereas Tables 3ndash6 exhibitthe comparison of proposed methods with FI3BBDF(3) andFI3BEBDF(3) In addition the graphs of log(MAXE) againstlog(ℎ) are illustrated as shown in Figures 5ndash10 The followingnotations are used in Tables 1ndash6

ℎ step sizeMAXE maximum errorTIME time execution using high performance com-puter (HPC)

Table 3 The accuracy for problem 3

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 356692119864 minus 2 141642119864 minus 3

FI3BEBDF(3) 124084119864 minus 2 302167119864 minus 3

DI2BBDF(2) 117256119864 minus 3 448489119864 minus 3

DI2BBDF(3) 172278119864 minus 3 481200119864 minus 3

DI2BBDF(4) 225003119864 minus 3 485301119864 minus 3

10minus4

FI3BBDF(3) 435640119864 minus 4 130103119864 minus 1

FI3BEBDF(3) 706516119864 minus 5 208115119864 minus 1

DI2BBDF(2) 118749119864 minus 7 160304119864 minus 1

DI2BBDF(3) 173636119864 minus 7 177445119864 minus 1

DI2BBDF(4) 226000119864 minus 7 178064119864 minus 1

10minus6

FI3BBDF(3) 434625119864 minus 6 1297751198641

FI3BEBDF(3) 703257119864 minus 7 2076091198640

DI2BBDF(2) 124989119864 minus 11 1507391198641

DI2BBDF(3) 173636119864 minus 11 1550731198641

DI2BBDF(4) 225999119864 minus 11 1602161198641

Table 4 The accuracy for problem 4

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 66269411986499 701775119864 minus 3

FI3BEBDF(3) 168449119864 minus 1 137847119864 minus 2

DI2BBDF(2) 280830119864 minus 1 106871119864 minus 2

DI2BBDF(3) 400389119864 minus 1 124969119864 minus 2

DI2BBDF(4) 477342119864 minus 1 126502119864 minus 2

10minus4

FI3BBDF(3) 845376119864 minus 3 676737119864 minus 1

FI3BEBDF(3) 695725119864 minus 3 135671119864 + 0

DI2BBDF(2) 356255119864 minus 5 432359119864 minus 1

DI2BBDF(3) 524241119864 minus 5 442599119864 minus 1

DI2BBDF(4) 685105119864 minus 5 456207119864 minus 1

10minus6

FI3BBDF(3) 854545119864 minus 5 6768191198641

FI3BEBDF(3) 718859119864 minus 5 1358621198642

DI2BBDF(2) 360130119864 minus 9 4000101198641

DI2BBDF(3) 526673119864 minus 9 4001921198641

DI2BBDF(4) 685529119864 minus 9 4103221198641

Table 5 The accuracy for problem 5

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 525481119864 minus 3 578083119864 minus 4

FI3BEBDF(3) 388475119864 minus 3 916583119864 minus 4

DI2BBDF(2) 856253119864 minus 5 229836119864 minus 4

DI2BBDF(3) 121629119864 minus 4 189066119864 minus 4

DI2BBDF(4) 165094119864 minus 4 230074119864 minus 4

10minus4

FI3BBDF(3) 551558119864 minus 5 483860119864 minus 2

FI3BEBDF(3) 427506119864 minus 5 774877119864 minus 2

DI2BBDF(2) 889803119864 minus 9 117438119864 minus 2

DI2BBDF(3) 130124119864 minus 8 131578119864 minus 2

DI2BBDF(4) 169456119864 minus 8 126071119864 minus 2

10minus6

FI3BBDF(3) 551803119864 minus 7 547768119864 + 0

FI3BEBDF(3) 427924119864 minus 7 777643119864 + 0

DI2BBDF(2) 126115119864 minus 11 119318119864 + 0

DI2BBDF(3) 627106119864 minus 11 133892119864 + 0

DI2BBDF(4) 104222119864 minus 11 126146119864 + 0

Mathematical Problems in Engineering 11

Table 6 The accuracy for problem 6

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 218555119864 minus 3 283084119864 minus 4

FI3BEBDF(3) 152719119864 minus 3 428500119864 minus 4

DI2BBDF(2) 387041119864 minus 5 538826119864 minus 5

DI2BBDF(3) 547958119864 minus 5 531673119864 minus 5

DI2BBDF(4) 747696119864 minus 5 618877119864 minus 5

10minus4

FI3BBDF(3) 229812119864 minus 5 148627119864 minus 2

FI3BEBDF(3) 178708119864 minus 5 232184119864 minus 2

DI2BBDF(2) 401638119864 minus 9 349712119864 minus 3

DI2BBDF(3) 587347119864 minus 9 375295119864 minus 3

DI2BBDF(4) 764927119864 minus 9 364804119864 minus 3

10minus6

FI3BBDF(3) 229936119864 minus 7 146660119864 + 0

FI3BEBDF(3) 178973119864 minus 7 228775119864 + 0

DI2BBDF(2) 990807119864 minus 12 338157119864 minus 1

DI2BBDF(3) 377716119864 minus 11 369837119864 minus 1

DI2BBDF(4) 462796119864 minus 12 356802119864 minus 1

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

log(h)minus6 minus5 minus4 minus3 minus2

FI2BBDF(3)FI2BEBDF(3)

DI2BBDF(2)DI2BBDF(3)DI2BBDF(4)

Figure 5 Graph of log(MAXE) versus log(ℎ) for problem 1

DI2BBDF(2) diagonally implicit 2-point block back-ward differentiation formulas of order 2DI2BBDF(3) diagonally implicit 2-point block back-ward differentiation formulas of order 3DI2BBDF(4) diagonally implicit 2-point block back-ward differentiation formulas of order 4FI2BBDF(3) fully implicit 2-point block backwarddifferentiation formulas of order 3FI2BEBDF(3) fully implicit 2-point block extendedbackward differentiation formulas of order 3FI3BBDF(3) fully implicit 3-point block backwarddifferentiation formulas of order 3

log(h)minus6 minus5 minus4 minus3 minus2

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

FI2BBDF(3)FI2BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

Figure 6 Graph of log(MAXE) versus log(ℎ) for problem 2

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus11minus12

minus10minus9minus8minus7minus6minus5minus4minus3minus2minus10

log(

MA

XE)

Figure 7 Graph of log(MAXE) versus log(ℎ) for problem 3

FI3BEBDF(3) fully implicit 3-point block extendedbackward differentiation formulas of order 3

7 Discussion

This section is divided into discussion of maximum error andcomputational time

71 Maximum Error From the numerical results in Tables1ndash6 we observe that DI2BBDF(2) DI2BBDF(3) andDI2BBDF(4) outperformed the existing methods in termsof maximum error This is expected because the diagonally

12 Mathematical Problems in Engineering

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

Figure 8 Graph of log(MAXE) versus log(ℎ) for problem 4

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus11

minus12

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

Figure 9 Graph of log(MAXE) versus log(ℎ) for problem 5

implicit method has less differentiation coefficients in orderto prevent the cumulative error The graphs in Figures 5ndash10depict the scales of maximum error versus step size h for theproposed methods as compared to existing methods Thereis a slight drop in accuracy for the proposed methods as theorder increases This is due to the increase in cumulativeerrors during the computation as more interpolating pointsare used Among the methods of order three DI2BBDF(3)is more accurate compared with FI2BBDF(3) FI3BBDF(3)FI2BEBDF(3) and FI3BEBDF(3) For all test problems we

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus11minus12

minus10minus9minus8minus7minus6minus5minus4minus3minus2minus10

log(

MA

XE)

Figure 10 Graph of log(MAXE) versus log(ℎ) for problem 6

conclude that the accuracy of the proposedmethod increasesas the step size becomes smaller

72 Computational Time In Tables 1-2 it can be seen thatthe execution time taken by DI2BBDF(2) DI2BBDF(3) andDI2BBDF(4) is comparable with that of FI2BBDF(3) andFI2BEBDF(3) The computational time increases as the stepsize becomes smaller However Tables 3ndash6 show that theproposed methods compute faster than FI3BBDF(3) andFI3BEBDF(3) This could be justified by the fact that the3-point method has to perform extra computation on theJacobian matrix since it involves a 3 by 3 matrix while the 2-point method has to compute Jacobian matrix of dimension2 by 2 only Among the proposed methods DI2BBDF(2)computes faster than DI2BBDF(3) and DI2BBDF(4) In factthe number of back values involved in every formula willaffect the computational time Since DI2BBDF(4) has moreback values than DI2BBDF(3) and DI2BBDF(2) it requiresextra time to compute the approximated solutions

8 Conclusion

Research conducted in this paper shows the capability ofconstructing the diagonally implicit BBDF method with A-stable properties From the results obtained via the numericalexperiment we can conclude that the proposed methodserves the purpose of significant alternative numericalmethod for solving linear and nonlinear stiff IVPs occurringin the fields of engineering and physical sciences

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 13

Acknowledgment

This researchwas supported by the Institute forMathematicalResearch (INSPEM) Department of Mathematics UniversitiPutra Malaysia (UPM) under Fundamental Grant Scheme(Project code 02-02-13-1380FR)

References

[1] C F Curtiss and J O Hirschfelder ldquoIntegration of stiff equa-tionsrdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 38 no 3 pp 235ndash243 1952

[2] J D Lambert Numercial Methods for Ordinary DifferentialSystems The Initial Value Problem John Wiley amp Sons NewYork NY USA 1991

[3] W E Milne Numerical Solution of Differential Equations JohnWiley amp Sons New York NY USA 1953

[4] L F Shampine and H A Watts ldquoBlock implicit one-stepmethodsrdquo Mathematics of Computation vol 23 pp 731ndash7401969

[5] S O Fatunla ldquoBlock methods for second order ODEsrdquo Inter-national Journal of Computer Mathematics vol 41 pp 55ndash631990

[6] Z A Majid M B Suleiman F I Ismail and K I Othmanldquo2-Point 1 block diagonally and 2-point 1 block fully implicitmethod for solving first order ordinary differential equationsrdquoin Proceedings of the 12th National Symposium on MathematicalScience vol 12 2004

[7] Z Abdul Majid N Z Mokhtar and M Suleiman ldquoDirecttwo-point block one-step method for solving general second-order ordinary differential equationsrdquo Mathematical Problemsin Engineering vol 2012 Article ID 184253 16 pages 2012

[8] Z B Ibrahim Block method for multistep formulas for solvingordinary differential equations [PhD thesis] Universiti PutraMalaysia Seri Kembangan Malaysia 2006

[9] Z B Ibrahim M Suleiman and K I Othman ldquoFixed coeffi-cients block backward differentiation formulas for the numer-ical solution of stiff ordinary differential equationsrdquo EuropeanJournal of Scientific Research vol 21 no 3 pp 508ndash520 2008

[10] Z B Ibrahim M B Suleiman and K I Othman ldquoImplicit r-point block backward differentiation formula for solving first-order stiff ODEsrdquo Applied Mathematics and Computation vol186 no 1 pp 558ndash565 2007

[11] Z B Ibrahim R Johari and F Ismail ldquoOn the stability of blockbackward differentiation formulaerdquo Matematika vol 19 no 2pp 83ndash89 2003

[12] N A Nasir Z B Ibrahim M B Suleiman and K I OthmanldquoFifth order two-point block backward differentiation formulasfor solving ordinary differential equationsrdquo Applied Mathemat-ical Sciences vol 5 no 69ndash72 pp 3505ndash3518 2011

[13] H Musa M B Suleiman and N Senu ldquoA-stable 2-Point blockextended backward differentiation formulas for stiff ordinarydifferential equationsrdquo in Proceedings of the 5th InternationalConference on Research and education in Mathematics vol 1450of AIP Conference Proceedings pp 254ndash258 2012

[14] H Musa M B Suleiman and N Senu ldquoFully implicit 3-pointblock extended backward differentiation formula for stiff initialvalue problemsrdquo Applied Mathematical Sciences vol 6 no 85ndash88 pp 4211ndash4227 2012

[15] R Alexander ldquoDiagonally implicit Runge-kutta for stiff ordi-nary differential equationsrdquo SIAM Journal on Numerical Analy-sis vol 14 no 6 pp 1006ndash1021 1977

[16] O Y Ababneh R Ahmad and E S Ismail ldquoDesign of newdiagonally implicit Runge-Kutta methods for stiff problemsrdquoApplied Mathematical Sciences vol 3 no 45 pp 2241ndash22532009

[17] F Ismail A L I Raed Al-Khasawneh M Suleiman and A BU Malik Hassan ldquoEmbedded pair of diagonally implicit runge-kutta method for solving ordinary differential equationsrdquo SainsMalaysiana vol 39 no 6 pp 1049ndash1054 2010

[18] Z A Majid and M B Suleiman ldquoPerformance of four pointdiagonally implicit block method for solving first order stiffordinary differential equationsrdquo Department of MathematicsUTM vol 22 no 2 pp 137ndash146 2006

[19] G G Dahlquist ldquoA special stability problem for linearmultistepmethodsrdquo BIT Numerical Mathematics vol 3 no 1 pp 27ndash431963

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Mathematical Problems in Engineering

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

where

119868 = [1 0

0 1] 119884 = [

119910119899+1

119910119899+2

] 119861 = [12057310

0 1205732

]

119865 = [119891119899+1

119891119899+2

] 120583 = [1205831

1205832

]

(43)

Implementing Newtonrsquos method to (42) produces

119865 = (119868 minus 119860)119884 minus ℎ119861119865 + 120583 (44)

Iteration for (44) is given by

119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2= 119865 (119910

(119894)

119899+1119899+2) sdot 1198651015840(119910(119894)

119899+1119899+2)minus1

(45)

where 119910(119894+1)119899+1

denotes the (119894 + 1)th iteration Equation (45) canbe rearranged into the following equation

1198651015840(119910(119894)

119899+1119899+2) (119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2) = 119865 (119910

(119894)

119899+1119899+2) (46)

Replacing (44) into (46) yields

(119868 minus 119860)119884 minus ℎ119861120597119891

120597119910(119910(119894)

119899+1119899+2) (119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2)

= (119868 minus 119860) (119910(119894)

119899+1119899+2) minus ℎ119861 (119910

(119894)

119899+1119899+2) + 120583

(47)

where (120597119891120597119910)(119910(119894)119899+1119899+2

) is the Jacobian matrix of 119891 with re-spect to 119910

41 Newton Iteration of DI2BBDF(2) Formula (16) is writtenin the form of (41)

1198651= 119910119899+1

minus2

3ℎ119891119899+1

+ 1205831

1198652= 119910119899+2

minus18

11119910119899+1

minus6

11ℎ119891119899+2

+ 1205832

(48)

Applying (48) into (47) will yield matrix form as follows

[[[

[

1 minus2

3ℎ120597119891119899+1

120597119910119899+1

0

0 1 minus6

11ℎ120597119891119899+2

120597119910119899+2

]]]

]

[119890(119894+1)

119899+1

119890(119894+1)

119899+2

]

= [

[

minus1 0

18

11minus1

]

]

[119910(119894)

119899+1

119910(119894)

119899+2

] + ℎ[[

[

2

30

06

11

]]

]

[119891(119894)

119899+1

119891(119894)

119899+2

] + [1205831

1205832

]

(49)

with 119890(119894+1)119899+1119899+2

= 119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2being the increment

42 Newton Iteration of DI2BBDF(3) Rearranging formula(23) in the form of (41) we have

1198651= 119910119899+1

minus6

11ℎ119891119899+1

+ 1205831

1198652= 119910119899+2

minus48

25119910119899+1

minus12

25ℎ119891119899+2

+ 1205832

(50)

Replacing (50) into (47) this will yieldmatrix form as follows

[[[

[

1 minus6

11ℎ120597119891119899+1

120597119910119899+1

0

0 1 minus12

25ℎ120597119891119899+2

120597119910119899+2

]]]

]

[119890(119894+1)

119899+1

119890(119894+1)

119899+2

]

= [

[

minus1 0

48

25minus1

]

]

[119910(119894)

119899+1

119910(119894)

119899+2

] + ℎ[[

[

6

110

012

25

]]

]

[119891(119894)

119899+1

119891(119894)

119899+2

] + [1205831

1205832

]

(51)

with 119890(119894+1)119899+1119899+2

= 119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2being the increment

43 Newton Iteration of DI2BBDF(4) We rewrite formula(30) in the form of (41) and obtain

1198651= 119910119899+1

minus12

25ℎ119891119899+1

+ 1205831

1198652= 119910119899+2

minus300

137119910119899+1

minus60

137ℎ119891119899+2

+ 1205832

(52)

Substituting (52) into (47) will produce

[[[

[

1 minus12

25ℎ120597119891119899+1

120597119910119899+1

0

0 1 minus60

137ℎ120597119891119899+2

120597119910119899+2

]]]

]

[119890(119894+1)

119899+1

119890(119894+1)

119899+2

]

= [

[

minus1 0

300

137minus1

]

]

[119910(119894)

119899+1

119910(119894)

119899+2

] + ℎ[[

[

12

250

060

137

]]

]

[119891(119894)

119899+1

119891(119894)

119899+2

] + [1205831

1205832

]

(53)

with 119890(119894+1)119899+1119899+2

= 119910(119894+1)

119899+1119899+2minus 119910(119894)

119899+1119899+2being the increment

All the formulas derived are implemented in predictor-corrector computation which is symbolized as PECE modeP and C indicate one application of the predictor and thecorrector respectively and E indicates one evaluation of thefunction119891 given119909 and119910The approximation calculations for119910119899+1

and 119910119899+2

in PECE are as follows

(1) P (Predict) 119910(119901)119899+1119899+2

(2) E (Evaluate) 1199101015840119899+1119899+2

= 119891(119909119899+1119899+2

119910(119901)

119899+1119899+2)

(3) C (Correct) 119910(119888)119899+1119899+2

(4) E (Evaluate) 1199101015840119899+1119899+2

= 119891(119909119899+1119899+2

119910(119888)

119899+1119899+2)

To approximate the solution to 119910(119888)

119899+1119899+2 we apply the

two-stage Newton type iteration The iteration process forDI2BBDF(2) is done as follows

Mathematical Problems in Engineering 9

(1) Compute the values for 119890(119894+1)119899+1119899+2

= 119860minus1119861 where

119860 =[[[

[

1 minus2

3ℎ120597119891119899+1

120597119910119899+1

0

0 1 minus6

11ℎ120597119891119899+2

120597119910119899+2

]]]

]

119861 =[[

[

minus119910(119894)

119899+1+2

3ℎ119891(119894)

119899+1+ 1205831

0

18

11119910(119894)

119899+1minus119910(119894)

119899+2+6

11ℎ119891(119894)

119899+2+ 1205832

]]

]

(54)

(2) Calculate the corrected value for 119910(119894+1)119899+1119899+2

with thevalue 119890(119894+1)

119899+1119899+2from Step (1)

(3) Solve 119890(119894+1)119899+1119899+2

= 119860minus1119861 for second stage iteration

(4) The final values for 119910(119894+1)119899+1119899+2

are obtained from thesecond stage iteration of 119890(119894+1)

119899+1119899+2

The similar iteration process is applied for DI2BBDF(3) andDI2BBDF(4)

5 Test Problems

In this section linear and nonlinear stiff problems aretested using C programming to examine the efficiency andreliability of the proposed method The following problemsare commonly found in engineering and physical sciencesparticularly in the studies of vibrations electrical circuits andchemical reaction

Problem 1 (linear) One has

1199101015840= minus10119910 + 10 (55)

Exact solution is

119910 (119909) = 1 + 119890minus10119909

(56)

Initial condition is

119910 (0) = 2

0 le 119909 le 10

(57)

(source Ibrahim [8])

Problem 2 (linear) One has

1199101015840

1= minus5119910

1minus 41199102 119910

1015840

2= minus4119910

1minus 51199102 (58)

Exact solutions are

1199101(119909) =

1

2119890minus9119909

+1

2119890minus119909 119910

2(119909) =

1

2119890minus9119909

minus1

2119890minus119909 (59)

Initial conditions are

1199101(0) = 1 119910

2(0) = 0

0 le 119909 le 20(60)

(source Musa et al [13])

Problem 3 (linear) One has

1199101015840= minus10119909119910 (61)

Exact solution is

119910 (119909) = 119890minus51199092

(62)

Initial condition is119910 (0) = 1

0 le 119909 le 10

(63)

(source Ibrahim et al [10])

Problem 4 (linear) One has

1199101015840

1= 91199101+ 24119910

2+ 5 cos119909 minus 1

3sin119909

1199101015840

2= minus 24119910

1minus 51119910

2minus 9 cos119909 + 1

3sin119909

(64)

Exact solutions are

1199101(119909) = 2119890

minus3119909minus 119890minus39119909

+1

3cos119909

1199102(119909) = minus119890

minus3119909+ 2119890minus39119909

minus1

3cos119909

(65)

Initial conditions are

1199101(0) =

4

3 1199102(0) =

2

3 0 le 119909 le 10 (66)

(source Musa et al [14])

Problem 5 (nonlinear) One has

1199101015840= minus

1199103

2 (67)

Exact solution is

119910 (119909) =1

radic1 + 119909 (68)

Initial condition is119910 (0) = 1

0 le 119909 le 4

(69)

(source Musa et al [14])

Problem 6 (nonlinear) One has

1199101015840=119910 (1 minus 119910)

2119910 minus 1 (70)

Exact solution is

119910 (119909) =1

2+ radic

1

4minus5

36119890minus119909 (71)

Initial condition is

119910 (0) =5

6

0 le 119909 le 1

(72)

(source Musa et al [14])

10 Mathematical Problems in Engineering

Table 1 The accuracy for problem 1

ℎ Methods MAXE TIME

10minus2

FI2BBDF(3) 597499119864 minus 2 136773119864 minus 3

FI2BEBDF(3) 567155119864 minus 2 145525119864 minus 3

DI2BBDF(2) 110568119864 minus 2 472498119864 minus 3

DI2BBDF(3) 166455119864 minus 2 477409119864 minus 3

DI2BBDF(4) 216342119864 minus 2 483298119864 minus 3

10minus4

FI2BBDF(3) 734012119864 minus 4 125797119864 minus 1

FI2BEBDF(3) 323640119864 minus 5 136259119864 minus 1

DI2BBDF(2) 118355119864 minus 6 167260119864 minus 1

DI2BBDF(3) 173430119864 minus 6 169325119864 minus 1

DI2BBDF(4) 225966119864 minus 6 172995119864 minus 1

10minus6

FI2BBDF(3) 735741119864 minus 6 1257001198641

FI2BEBDF(3) 350090119864 minus 7 1362301198641

DI2BBDF(2) 118742119864 minus 10 1491591198641

DI2BBDF(3) 173634119864 minus 10 1544931198641

DI2BBDF(4) 226000119864 minus 10 1546201198641

Table 2 The accuracy for problem 2

ℎ Methods MAXE TIME

10minus2

FI2BBDF(3) 271597119864 minus 2 596467119864 minus 3

FI2BEBDF(3) 149811119864 minus 2 661025119864 minus 3

DI2BBDF(2) 456073119864 minus 3 104480119864 minus 2

DI2BBDF(3) 686126119864 minus 3 108421119864 minus 2

DI2BBDF(4) 893279119864 minus 3 111089119864 minus 2

10minus4

FI2BBDF(3) 340437119864 minus 4 590430119864 minus 1

FI2BEBDF(3) 262649119864 minus 5 654305119864 minus 1

DI2BBDF(2) 485404119864 minus 7 373210119864 minus 1

DI2BBDF(3) 711155119864 minus 7 386556119864 minus 1

DI2BBDF(4) 926476119864 minus 7 387070119864 minus 1

10minus6

FI2BBDF(3) 341146119864 minus 6 5901981198641

FI2BEBDF(3) 244097119864 minus 7 6541061198641

DI2BBDF(2) 486843119864 minus 11 3484511198641

DI2BBDF(3) 711903119864 minus 11 3604901198641

DI2BBDF(4) 926598119864 minus 11 3606491198641

6 Numerical Results

The performance of the derived methods is compared withthe existing methods in terms of maximum error and exe-cution time We consider 10minus2 10minus4 and 10minus6 as the stepsize ℎ Tables 1 and 2 present the performance compar-ison of DI2BBDF(2) DI2BBDF(3) and DI2BBDF(4) withFI2BBDF(3) and FI2BEBDF(3) whereas Tables 3ndash6 exhibitthe comparison of proposed methods with FI3BBDF(3) andFI3BEBDF(3) In addition the graphs of log(MAXE) againstlog(ℎ) are illustrated as shown in Figures 5ndash10 The followingnotations are used in Tables 1ndash6

ℎ step sizeMAXE maximum errorTIME time execution using high performance com-puter (HPC)

Table 3 The accuracy for problem 3

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 356692119864 minus 2 141642119864 minus 3

FI3BEBDF(3) 124084119864 minus 2 302167119864 minus 3

DI2BBDF(2) 117256119864 minus 3 448489119864 minus 3

DI2BBDF(3) 172278119864 minus 3 481200119864 minus 3

DI2BBDF(4) 225003119864 minus 3 485301119864 minus 3

10minus4

FI3BBDF(3) 435640119864 minus 4 130103119864 minus 1

FI3BEBDF(3) 706516119864 minus 5 208115119864 minus 1

DI2BBDF(2) 118749119864 minus 7 160304119864 minus 1

DI2BBDF(3) 173636119864 minus 7 177445119864 minus 1

DI2BBDF(4) 226000119864 minus 7 178064119864 minus 1

10minus6

FI3BBDF(3) 434625119864 minus 6 1297751198641

FI3BEBDF(3) 703257119864 minus 7 2076091198640

DI2BBDF(2) 124989119864 minus 11 1507391198641

DI2BBDF(3) 173636119864 minus 11 1550731198641

DI2BBDF(4) 225999119864 minus 11 1602161198641

Table 4 The accuracy for problem 4

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 66269411986499 701775119864 minus 3

FI3BEBDF(3) 168449119864 minus 1 137847119864 minus 2

DI2BBDF(2) 280830119864 minus 1 106871119864 minus 2

DI2BBDF(3) 400389119864 minus 1 124969119864 minus 2

DI2BBDF(4) 477342119864 minus 1 126502119864 minus 2

10minus4

FI3BBDF(3) 845376119864 minus 3 676737119864 minus 1

FI3BEBDF(3) 695725119864 minus 3 135671119864 + 0

DI2BBDF(2) 356255119864 minus 5 432359119864 minus 1

DI2BBDF(3) 524241119864 minus 5 442599119864 minus 1

DI2BBDF(4) 685105119864 minus 5 456207119864 minus 1

10minus6

FI3BBDF(3) 854545119864 minus 5 6768191198641

FI3BEBDF(3) 718859119864 minus 5 1358621198642

DI2BBDF(2) 360130119864 minus 9 4000101198641

DI2BBDF(3) 526673119864 minus 9 4001921198641

DI2BBDF(4) 685529119864 minus 9 4103221198641

Table 5 The accuracy for problem 5

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 525481119864 minus 3 578083119864 minus 4

FI3BEBDF(3) 388475119864 minus 3 916583119864 minus 4

DI2BBDF(2) 856253119864 minus 5 229836119864 minus 4

DI2BBDF(3) 121629119864 minus 4 189066119864 minus 4

DI2BBDF(4) 165094119864 minus 4 230074119864 minus 4

10minus4

FI3BBDF(3) 551558119864 minus 5 483860119864 minus 2

FI3BEBDF(3) 427506119864 minus 5 774877119864 minus 2

DI2BBDF(2) 889803119864 minus 9 117438119864 minus 2

DI2BBDF(3) 130124119864 minus 8 131578119864 minus 2

DI2BBDF(4) 169456119864 minus 8 126071119864 minus 2

10minus6

FI3BBDF(3) 551803119864 minus 7 547768119864 + 0

FI3BEBDF(3) 427924119864 minus 7 777643119864 + 0

DI2BBDF(2) 126115119864 minus 11 119318119864 + 0

DI2BBDF(3) 627106119864 minus 11 133892119864 + 0

DI2BBDF(4) 104222119864 minus 11 126146119864 + 0

Mathematical Problems in Engineering 11

Table 6 The accuracy for problem 6

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 218555119864 minus 3 283084119864 minus 4

FI3BEBDF(3) 152719119864 minus 3 428500119864 minus 4

DI2BBDF(2) 387041119864 minus 5 538826119864 minus 5

DI2BBDF(3) 547958119864 minus 5 531673119864 minus 5

DI2BBDF(4) 747696119864 minus 5 618877119864 minus 5

10minus4

FI3BBDF(3) 229812119864 minus 5 148627119864 minus 2

FI3BEBDF(3) 178708119864 minus 5 232184119864 minus 2

DI2BBDF(2) 401638119864 minus 9 349712119864 minus 3

DI2BBDF(3) 587347119864 minus 9 375295119864 minus 3

DI2BBDF(4) 764927119864 minus 9 364804119864 minus 3

10minus6

FI3BBDF(3) 229936119864 minus 7 146660119864 + 0

FI3BEBDF(3) 178973119864 minus 7 228775119864 + 0

DI2BBDF(2) 990807119864 minus 12 338157119864 minus 1

DI2BBDF(3) 377716119864 minus 11 369837119864 minus 1

DI2BBDF(4) 462796119864 minus 12 356802119864 minus 1

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

log(h)minus6 minus5 minus4 minus3 minus2

FI2BBDF(3)FI2BEBDF(3)

DI2BBDF(2)DI2BBDF(3)DI2BBDF(4)

Figure 5 Graph of log(MAXE) versus log(ℎ) for problem 1

DI2BBDF(2) diagonally implicit 2-point block back-ward differentiation formulas of order 2DI2BBDF(3) diagonally implicit 2-point block back-ward differentiation formulas of order 3DI2BBDF(4) diagonally implicit 2-point block back-ward differentiation formulas of order 4FI2BBDF(3) fully implicit 2-point block backwarddifferentiation formulas of order 3FI2BEBDF(3) fully implicit 2-point block extendedbackward differentiation formulas of order 3FI3BBDF(3) fully implicit 3-point block backwarddifferentiation formulas of order 3

log(h)minus6 minus5 minus4 minus3 minus2

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

FI2BBDF(3)FI2BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

Figure 6 Graph of log(MAXE) versus log(ℎ) for problem 2

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus11minus12

minus10minus9minus8minus7minus6minus5minus4minus3minus2minus10

log(

MA

XE)

Figure 7 Graph of log(MAXE) versus log(ℎ) for problem 3

FI3BEBDF(3) fully implicit 3-point block extendedbackward differentiation formulas of order 3

7 Discussion

This section is divided into discussion of maximum error andcomputational time

71 Maximum Error From the numerical results in Tables1ndash6 we observe that DI2BBDF(2) DI2BBDF(3) andDI2BBDF(4) outperformed the existing methods in termsof maximum error This is expected because the diagonally

12 Mathematical Problems in Engineering

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

Figure 8 Graph of log(MAXE) versus log(ℎ) for problem 4

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus11

minus12

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

Figure 9 Graph of log(MAXE) versus log(ℎ) for problem 5

implicit method has less differentiation coefficients in orderto prevent the cumulative error The graphs in Figures 5ndash10depict the scales of maximum error versus step size h for theproposed methods as compared to existing methods Thereis a slight drop in accuracy for the proposed methods as theorder increases This is due to the increase in cumulativeerrors during the computation as more interpolating pointsare used Among the methods of order three DI2BBDF(3)is more accurate compared with FI2BBDF(3) FI3BBDF(3)FI2BEBDF(3) and FI3BEBDF(3) For all test problems we

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus11minus12

minus10minus9minus8minus7minus6minus5minus4minus3minus2minus10

log(

MA

XE)

Figure 10 Graph of log(MAXE) versus log(ℎ) for problem 6

conclude that the accuracy of the proposedmethod increasesas the step size becomes smaller

72 Computational Time In Tables 1-2 it can be seen thatthe execution time taken by DI2BBDF(2) DI2BBDF(3) andDI2BBDF(4) is comparable with that of FI2BBDF(3) andFI2BEBDF(3) The computational time increases as the stepsize becomes smaller However Tables 3ndash6 show that theproposed methods compute faster than FI3BBDF(3) andFI3BEBDF(3) This could be justified by the fact that the3-point method has to perform extra computation on theJacobian matrix since it involves a 3 by 3 matrix while the 2-point method has to compute Jacobian matrix of dimension2 by 2 only Among the proposed methods DI2BBDF(2)computes faster than DI2BBDF(3) and DI2BBDF(4) In factthe number of back values involved in every formula willaffect the computational time Since DI2BBDF(4) has moreback values than DI2BBDF(3) and DI2BBDF(2) it requiresextra time to compute the approximated solutions

8 Conclusion

Research conducted in this paper shows the capability ofconstructing the diagonally implicit BBDF method with A-stable properties From the results obtained via the numericalexperiment we can conclude that the proposed methodserves the purpose of significant alternative numericalmethod for solving linear and nonlinear stiff IVPs occurringin the fields of engineering and physical sciences

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 13

Acknowledgment

This researchwas supported by the Institute forMathematicalResearch (INSPEM) Department of Mathematics UniversitiPutra Malaysia (UPM) under Fundamental Grant Scheme(Project code 02-02-13-1380FR)

References

[1] C F Curtiss and J O Hirschfelder ldquoIntegration of stiff equa-tionsrdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 38 no 3 pp 235ndash243 1952

[2] J D Lambert Numercial Methods for Ordinary DifferentialSystems The Initial Value Problem John Wiley amp Sons NewYork NY USA 1991

[3] W E Milne Numerical Solution of Differential Equations JohnWiley amp Sons New York NY USA 1953

[4] L F Shampine and H A Watts ldquoBlock implicit one-stepmethodsrdquo Mathematics of Computation vol 23 pp 731ndash7401969

[5] S O Fatunla ldquoBlock methods for second order ODEsrdquo Inter-national Journal of Computer Mathematics vol 41 pp 55ndash631990

[6] Z A Majid M B Suleiman F I Ismail and K I Othmanldquo2-Point 1 block diagonally and 2-point 1 block fully implicitmethod for solving first order ordinary differential equationsrdquoin Proceedings of the 12th National Symposium on MathematicalScience vol 12 2004

[7] Z Abdul Majid N Z Mokhtar and M Suleiman ldquoDirecttwo-point block one-step method for solving general second-order ordinary differential equationsrdquo Mathematical Problemsin Engineering vol 2012 Article ID 184253 16 pages 2012

[8] Z B Ibrahim Block method for multistep formulas for solvingordinary differential equations [PhD thesis] Universiti PutraMalaysia Seri Kembangan Malaysia 2006

[9] Z B Ibrahim M Suleiman and K I Othman ldquoFixed coeffi-cients block backward differentiation formulas for the numer-ical solution of stiff ordinary differential equationsrdquo EuropeanJournal of Scientific Research vol 21 no 3 pp 508ndash520 2008

[10] Z B Ibrahim M B Suleiman and K I Othman ldquoImplicit r-point block backward differentiation formula for solving first-order stiff ODEsrdquo Applied Mathematics and Computation vol186 no 1 pp 558ndash565 2007

[11] Z B Ibrahim R Johari and F Ismail ldquoOn the stability of blockbackward differentiation formulaerdquo Matematika vol 19 no 2pp 83ndash89 2003

[12] N A Nasir Z B Ibrahim M B Suleiman and K I OthmanldquoFifth order two-point block backward differentiation formulasfor solving ordinary differential equationsrdquo Applied Mathemat-ical Sciences vol 5 no 69ndash72 pp 3505ndash3518 2011

[13] H Musa M B Suleiman and N Senu ldquoA-stable 2-Point blockextended backward differentiation formulas for stiff ordinarydifferential equationsrdquo in Proceedings of the 5th InternationalConference on Research and education in Mathematics vol 1450of AIP Conference Proceedings pp 254ndash258 2012

[14] H Musa M B Suleiman and N Senu ldquoFully implicit 3-pointblock extended backward differentiation formula for stiff initialvalue problemsrdquo Applied Mathematical Sciences vol 6 no 85ndash88 pp 4211ndash4227 2012

[15] R Alexander ldquoDiagonally implicit Runge-kutta for stiff ordi-nary differential equationsrdquo SIAM Journal on Numerical Analy-sis vol 14 no 6 pp 1006ndash1021 1977

[16] O Y Ababneh R Ahmad and E S Ismail ldquoDesign of newdiagonally implicit Runge-Kutta methods for stiff problemsrdquoApplied Mathematical Sciences vol 3 no 45 pp 2241ndash22532009

[17] F Ismail A L I Raed Al-Khasawneh M Suleiman and A BU Malik Hassan ldquoEmbedded pair of diagonally implicit runge-kutta method for solving ordinary differential equationsrdquo SainsMalaysiana vol 39 no 6 pp 1049ndash1054 2010

[18] Z A Majid and M B Suleiman ldquoPerformance of four pointdiagonally implicit block method for solving first order stiffordinary differential equationsrdquo Department of MathematicsUTM vol 22 no 2 pp 137ndash146 2006

[19] G G Dahlquist ldquoA special stability problem for linearmultistepmethodsrdquo BIT Numerical Mathematics vol 3 no 1 pp 27ndash431963

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

(1) Compute the values for 119890(119894+1)119899+1119899+2

= 119860minus1119861 where

119860 =[[[

[

1 minus2

3ℎ120597119891119899+1

120597119910119899+1

0

0 1 minus6

11ℎ120597119891119899+2

120597119910119899+2

]]]

]

119861 =[[

[

minus119910(119894)

119899+1+2

3ℎ119891(119894)

119899+1+ 1205831

0

18

11119910(119894)

119899+1minus119910(119894)

119899+2+6

11ℎ119891(119894)

119899+2+ 1205832

]]

]

(54)

(2) Calculate the corrected value for 119910(119894+1)119899+1119899+2

with thevalue 119890(119894+1)

119899+1119899+2from Step (1)

(3) Solve 119890(119894+1)119899+1119899+2

= 119860minus1119861 for second stage iteration

(4) The final values for 119910(119894+1)119899+1119899+2

are obtained from thesecond stage iteration of 119890(119894+1)

119899+1119899+2

The similar iteration process is applied for DI2BBDF(3) andDI2BBDF(4)

5 Test Problems

In this section linear and nonlinear stiff problems aretested using C programming to examine the efficiency andreliability of the proposed method The following problemsare commonly found in engineering and physical sciencesparticularly in the studies of vibrations electrical circuits andchemical reaction

Problem 1 (linear) One has

1199101015840= minus10119910 + 10 (55)

Exact solution is

119910 (119909) = 1 + 119890minus10119909

(56)

Initial condition is

119910 (0) = 2

0 le 119909 le 10

(57)

(source Ibrahim [8])

Problem 2 (linear) One has

1199101015840

1= minus5119910

1minus 41199102 119910

1015840

2= minus4119910

1minus 51199102 (58)

Exact solutions are

1199101(119909) =

1

2119890minus9119909

+1

2119890minus119909 119910

2(119909) =

1

2119890minus9119909

minus1

2119890minus119909 (59)

Initial conditions are

1199101(0) = 1 119910

2(0) = 0

0 le 119909 le 20(60)

(source Musa et al [13])

Problem 3 (linear) One has

1199101015840= minus10119909119910 (61)

Exact solution is

119910 (119909) = 119890minus51199092

(62)

Initial condition is119910 (0) = 1

0 le 119909 le 10

(63)

(source Ibrahim et al [10])

Problem 4 (linear) One has

1199101015840

1= 91199101+ 24119910

2+ 5 cos119909 minus 1

3sin119909

1199101015840

2= minus 24119910

1minus 51119910

2minus 9 cos119909 + 1

3sin119909

(64)

Exact solutions are

1199101(119909) = 2119890

minus3119909minus 119890minus39119909

+1

3cos119909

1199102(119909) = minus119890

minus3119909+ 2119890minus39119909

minus1

3cos119909

(65)

Initial conditions are

1199101(0) =

4

3 1199102(0) =

2

3 0 le 119909 le 10 (66)

(source Musa et al [14])

Problem 5 (nonlinear) One has

1199101015840= minus

1199103

2 (67)

Exact solution is

119910 (119909) =1

radic1 + 119909 (68)

Initial condition is119910 (0) = 1

0 le 119909 le 4

(69)

(source Musa et al [14])

Problem 6 (nonlinear) One has

1199101015840=119910 (1 minus 119910)

2119910 minus 1 (70)

Exact solution is

119910 (119909) =1

2+ radic

1

4minus5

36119890minus119909 (71)

Initial condition is

119910 (0) =5

6

0 le 119909 le 1

(72)

(source Musa et al [14])

10 Mathematical Problems in Engineering

Table 1 The accuracy for problem 1

ℎ Methods MAXE TIME

10minus2

FI2BBDF(3) 597499119864 minus 2 136773119864 minus 3

FI2BEBDF(3) 567155119864 minus 2 145525119864 minus 3

DI2BBDF(2) 110568119864 minus 2 472498119864 minus 3

DI2BBDF(3) 166455119864 minus 2 477409119864 minus 3

DI2BBDF(4) 216342119864 minus 2 483298119864 minus 3

10minus4

FI2BBDF(3) 734012119864 minus 4 125797119864 minus 1

FI2BEBDF(3) 323640119864 minus 5 136259119864 minus 1

DI2BBDF(2) 118355119864 minus 6 167260119864 minus 1

DI2BBDF(3) 173430119864 minus 6 169325119864 minus 1

DI2BBDF(4) 225966119864 minus 6 172995119864 minus 1

10minus6

FI2BBDF(3) 735741119864 minus 6 1257001198641

FI2BEBDF(3) 350090119864 minus 7 1362301198641

DI2BBDF(2) 118742119864 minus 10 1491591198641

DI2BBDF(3) 173634119864 minus 10 1544931198641

DI2BBDF(4) 226000119864 minus 10 1546201198641

Table 2 The accuracy for problem 2

ℎ Methods MAXE TIME

10minus2

FI2BBDF(3) 271597119864 minus 2 596467119864 minus 3

FI2BEBDF(3) 149811119864 minus 2 661025119864 minus 3

DI2BBDF(2) 456073119864 minus 3 104480119864 minus 2

DI2BBDF(3) 686126119864 minus 3 108421119864 minus 2

DI2BBDF(4) 893279119864 minus 3 111089119864 minus 2

10minus4

FI2BBDF(3) 340437119864 minus 4 590430119864 minus 1

FI2BEBDF(3) 262649119864 minus 5 654305119864 minus 1

DI2BBDF(2) 485404119864 minus 7 373210119864 minus 1

DI2BBDF(3) 711155119864 minus 7 386556119864 minus 1

DI2BBDF(4) 926476119864 minus 7 387070119864 minus 1

10minus6

FI2BBDF(3) 341146119864 minus 6 5901981198641

FI2BEBDF(3) 244097119864 minus 7 6541061198641

DI2BBDF(2) 486843119864 minus 11 3484511198641

DI2BBDF(3) 711903119864 minus 11 3604901198641

DI2BBDF(4) 926598119864 minus 11 3606491198641

6 Numerical Results

The performance of the derived methods is compared withthe existing methods in terms of maximum error and exe-cution time We consider 10minus2 10minus4 and 10minus6 as the stepsize ℎ Tables 1 and 2 present the performance compar-ison of DI2BBDF(2) DI2BBDF(3) and DI2BBDF(4) withFI2BBDF(3) and FI2BEBDF(3) whereas Tables 3ndash6 exhibitthe comparison of proposed methods with FI3BBDF(3) andFI3BEBDF(3) In addition the graphs of log(MAXE) againstlog(ℎ) are illustrated as shown in Figures 5ndash10 The followingnotations are used in Tables 1ndash6

ℎ step sizeMAXE maximum errorTIME time execution using high performance com-puter (HPC)

Table 3 The accuracy for problem 3

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 356692119864 minus 2 141642119864 minus 3

FI3BEBDF(3) 124084119864 minus 2 302167119864 minus 3

DI2BBDF(2) 117256119864 minus 3 448489119864 minus 3

DI2BBDF(3) 172278119864 minus 3 481200119864 minus 3

DI2BBDF(4) 225003119864 minus 3 485301119864 minus 3

10minus4

FI3BBDF(3) 435640119864 minus 4 130103119864 minus 1

FI3BEBDF(3) 706516119864 minus 5 208115119864 minus 1

DI2BBDF(2) 118749119864 minus 7 160304119864 minus 1

DI2BBDF(3) 173636119864 minus 7 177445119864 minus 1

DI2BBDF(4) 226000119864 minus 7 178064119864 minus 1

10minus6

FI3BBDF(3) 434625119864 minus 6 1297751198641

FI3BEBDF(3) 703257119864 minus 7 2076091198640

DI2BBDF(2) 124989119864 minus 11 1507391198641

DI2BBDF(3) 173636119864 minus 11 1550731198641

DI2BBDF(4) 225999119864 minus 11 1602161198641

Table 4 The accuracy for problem 4

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 66269411986499 701775119864 minus 3

FI3BEBDF(3) 168449119864 minus 1 137847119864 minus 2

DI2BBDF(2) 280830119864 minus 1 106871119864 minus 2

DI2BBDF(3) 400389119864 minus 1 124969119864 minus 2

DI2BBDF(4) 477342119864 minus 1 126502119864 minus 2

10minus4

FI3BBDF(3) 845376119864 minus 3 676737119864 minus 1

FI3BEBDF(3) 695725119864 minus 3 135671119864 + 0

DI2BBDF(2) 356255119864 minus 5 432359119864 minus 1

DI2BBDF(3) 524241119864 minus 5 442599119864 minus 1

DI2BBDF(4) 685105119864 minus 5 456207119864 minus 1

10minus6

FI3BBDF(3) 854545119864 minus 5 6768191198641

FI3BEBDF(3) 718859119864 minus 5 1358621198642

DI2BBDF(2) 360130119864 minus 9 4000101198641

DI2BBDF(3) 526673119864 minus 9 4001921198641

DI2BBDF(4) 685529119864 minus 9 4103221198641

Table 5 The accuracy for problem 5

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 525481119864 minus 3 578083119864 minus 4

FI3BEBDF(3) 388475119864 minus 3 916583119864 minus 4

DI2BBDF(2) 856253119864 minus 5 229836119864 minus 4

DI2BBDF(3) 121629119864 minus 4 189066119864 minus 4

DI2BBDF(4) 165094119864 minus 4 230074119864 minus 4

10minus4

FI3BBDF(3) 551558119864 minus 5 483860119864 minus 2

FI3BEBDF(3) 427506119864 minus 5 774877119864 minus 2

DI2BBDF(2) 889803119864 minus 9 117438119864 minus 2

DI2BBDF(3) 130124119864 minus 8 131578119864 minus 2

DI2BBDF(4) 169456119864 minus 8 126071119864 minus 2

10minus6

FI3BBDF(3) 551803119864 minus 7 547768119864 + 0

FI3BEBDF(3) 427924119864 minus 7 777643119864 + 0

DI2BBDF(2) 126115119864 minus 11 119318119864 + 0

DI2BBDF(3) 627106119864 minus 11 133892119864 + 0

DI2BBDF(4) 104222119864 minus 11 126146119864 + 0

Mathematical Problems in Engineering 11

Table 6 The accuracy for problem 6

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 218555119864 minus 3 283084119864 minus 4

FI3BEBDF(3) 152719119864 minus 3 428500119864 minus 4

DI2BBDF(2) 387041119864 minus 5 538826119864 minus 5

DI2BBDF(3) 547958119864 minus 5 531673119864 minus 5

DI2BBDF(4) 747696119864 minus 5 618877119864 minus 5

10minus4

FI3BBDF(3) 229812119864 minus 5 148627119864 minus 2

FI3BEBDF(3) 178708119864 minus 5 232184119864 minus 2

DI2BBDF(2) 401638119864 minus 9 349712119864 minus 3

DI2BBDF(3) 587347119864 minus 9 375295119864 minus 3

DI2BBDF(4) 764927119864 minus 9 364804119864 minus 3

10minus6

FI3BBDF(3) 229936119864 minus 7 146660119864 + 0

FI3BEBDF(3) 178973119864 minus 7 228775119864 + 0

DI2BBDF(2) 990807119864 minus 12 338157119864 minus 1

DI2BBDF(3) 377716119864 minus 11 369837119864 minus 1

DI2BBDF(4) 462796119864 minus 12 356802119864 minus 1

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

log(h)minus6 minus5 minus4 minus3 minus2

FI2BBDF(3)FI2BEBDF(3)

DI2BBDF(2)DI2BBDF(3)DI2BBDF(4)

Figure 5 Graph of log(MAXE) versus log(ℎ) for problem 1

DI2BBDF(2) diagonally implicit 2-point block back-ward differentiation formulas of order 2DI2BBDF(3) diagonally implicit 2-point block back-ward differentiation formulas of order 3DI2BBDF(4) diagonally implicit 2-point block back-ward differentiation formulas of order 4FI2BBDF(3) fully implicit 2-point block backwarddifferentiation formulas of order 3FI2BEBDF(3) fully implicit 2-point block extendedbackward differentiation formulas of order 3FI3BBDF(3) fully implicit 3-point block backwarddifferentiation formulas of order 3

log(h)minus6 minus5 minus4 minus3 minus2

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

FI2BBDF(3)FI2BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

Figure 6 Graph of log(MAXE) versus log(ℎ) for problem 2

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus11minus12

minus10minus9minus8minus7minus6minus5minus4minus3minus2minus10

log(

MA

XE)

Figure 7 Graph of log(MAXE) versus log(ℎ) for problem 3

FI3BEBDF(3) fully implicit 3-point block extendedbackward differentiation formulas of order 3

7 Discussion

This section is divided into discussion of maximum error andcomputational time

71 Maximum Error From the numerical results in Tables1ndash6 we observe that DI2BBDF(2) DI2BBDF(3) andDI2BBDF(4) outperformed the existing methods in termsof maximum error This is expected because the diagonally

12 Mathematical Problems in Engineering

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

Figure 8 Graph of log(MAXE) versus log(ℎ) for problem 4

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus11

minus12

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

Figure 9 Graph of log(MAXE) versus log(ℎ) for problem 5

implicit method has less differentiation coefficients in orderto prevent the cumulative error The graphs in Figures 5ndash10depict the scales of maximum error versus step size h for theproposed methods as compared to existing methods Thereis a slight drop in accuracy for the proposed methods as theorder increases This is due to the increase in cumulativeerrors during the computation as more interpolating pointsare used Among the methods of order three DI2BBDF(3)is more accurate compared with FI2BBDF(3) FI3BBDF(3)FI2BEBDF(3) and FI3BEBDF(3) For all test problems we

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus11minus12

minus10minus9minus8minus7minus6minus5minus4minus3minus2minus10

log(

MA

XE)

Figure 10 Graph of log(MAXE) versus log(ℎ) for problem 6

conclude that the accuracy of the proposedmethod increasesas the step size becomes smaller

72 Computational Time In Tables 1-2 it can be seen thatthe execution time taken by DI2BBDF(2) DI2BBDF(3) andDI2BBDF(4) is comparable with that of FI2BBDF(3) andFI2BEBDF(3) The computational time increases as the stepsize becomes smaller However Tables 3ndash6 show that theproposed methods compute faster than FI3BBDF(3) andFI3BEBDF(3) This could be justified by the fact that the3-point method has to perform extra computation on theJacobian matrix since it involves a 3 by 3 matrix while the 2-point method has to compute Jacobian matrix of dimension2 by 2 only Among the proposed methods DI2BBDF(2)computes faster than DI2BBDF(3) and DI2BBDF(4) In factthe number of back values involved in every formula willaffect the computational time Since DI2BBDF(4) has moreback values than DI2BBDF(3) and DI2BBDF(2) it requiresextra time to compute the approximated solutions

8 Conclusion

Research conducted in this paper shows the capability ofconstructing the diagonally implicit BBDF method with A-stable properties From the results obtained via the numericalexperiment we can conclude that the proposed methodserves the purpose of significant alternative numericalmethod for solving linear and nonlinear stiff IVPs occurringin the fields of engineering and physical sciences

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 13

Acknowledgment

This researchwas supported by the Institute forMathematicalResearch (INSPEM) Department of Mathematics UniversitiPutra Malaysia (UPM) under Fundamental Grant Scheme(Project code 02-02-13-1380FR)

References

[1] C F Curtiss and J O Hirschfelder ldquoIntegration of stiff equa-tionsrdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 38 no 3 pp 235ndash243 1952

[2] J D Lambert Numercial Methods for Ordinary DifferentialSystems The Initial Value Problem John Wiley amp Sons NewYork NY USA 1991

[3] W E Milne Numerical Solution of Differential Equations JohnWiley amp Sons New York NY USA 1953

[4] L F Shampine and H A Watts ldquoBlock implicit one-stepmethodsrdquo Mathematics of Computation vol 23 pp 731ndash7401969

[5] S O Fatunla ldquoBlock methods for second order ODEsrdquo Inter-national Journal of Computer Mathematics vol 41 pp 55ndash631990

[6] Z A Majid M B Suleiman F I Ismail and K I Othmanldquo2-Point 1 block diagonally and 2-point 1 block fully implicitmethod for solving first order ordinary differential equationsrdquoin Proceedings of the 12th National Symposium on MathematicalScience vol 12 2004

[7] Z Abdul Majid N Z Mokhtar and M Suleiman ldquoDirecttwo-point block one-step method for solving general second-order ordinary differential equationsrdquo Mathematical Problemsin Engineering vol 2012 Article ID 184253 16 pages 2012

[8] Z B Ibrahim Block method for multistep formulas for solvingordinary differential equations [PhD thesis] Universiti PutraMalaysia Seri Kembangan Malaysia 2006

[9] Z B Ibrahim M Suleiman and K I Othman ldquoFixed coeffi-cients block backward differentiation formulas for the numer-ical solution of stiff ordinary differential equationsrdquo EuropeanJournal of Scientific Research vol 21 no 3 pp 508ndash520 2008

[10] Z B Ibrahim M B Suleiman and K I Othman ldquoImplicit r-point block backward differentiation formula for solving first-order stiff ODEsrdquo Applied Mathematics and Computation vol186 no 1 pp 558ndash565 2007

[11] Z B Ibrahim R Johari and F Ismail ldquoOn the stability of blockbackward differentiation formulaerdquo Matematika vol 19 no 2pp 83ndash89 2003

[12] N A Nasir Z B Ibrahim M B Suleiman and K I OthmanldquoFifth order two-point block backward differentiation formulasfor solving ordinary differential equationsrdquo Applied Mathemat-ical Sciences vol 5 no 69ndash72 pp 3505ndash3518 2011

[13] H Musa M B Suleiman and N Senu ldquoA-stable 2-Point blockextended backward differentiation formulas for stiff ordinarydifferential equationsrdquo in Proceedings of the 5th InternationalConference on Research and education in Mathematics vol 1450of AIP Conference Proceedings pp 254ndash258 2012

[14] H Musa M B Suleiman and N Senu ldquoFully implicit 3-pointblock extended backward differentiation formula for stiff initialvalue problemsrdquo Applied Mathematical Sciences vol 6 no 85ndash88 pp 4211ndash4227 2012

[15] R Alexander ldquoDiagonally implicit Runge-kutta for stiff ordi-nary differential equationsrdquo SIAM Journal on Numerical Analy-sis vol 14 no 6 pp 1006ndash1021 1977

[16] O Y Ababneh R Ahmad and E S Ismail ldquoDesign of newdiagonally implicit Runge-Kutta methods for stiff problemsrdquoApplied Mathematical Sciences vol 3 no 45 pp 2241ndash22532009

[17] F Ismail A L I Raed Al-Khasawneh M Suleiman and A BU Malik Hassan ldquoEmbedded pair of diagonally implicit runge-kutta method for solving ordinary differential equationsrdquo SainsMalaysiana vol 39 no 6 pp 1049ndash1054 2010

[18] Z A Majid and M B Suleiman ldquoPerformance of four pointdiagonally implicit block method for solving first order stiffordinary differential equationsrdquo Department of MathematicsUTM vol 22 no 2 pp 137ndash146 2006

[19] G G Dahlquist ldquoA special stability problem for linearmultistepmethodsrdquo BIT Numerical Mathematics vol 3 no 1 pp 27ndash431963

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

Table 1 The accuracy for problem 1

ℎ Methods MAXE TIME

10minus2

FI2BBDF(3) 597499119864 minus 2 136773119864 minus 3

FI2BEBDF(3) 567155119864 minus 2 145525119864 minus 3

DI2BBDF(2) 110568119864 minus 2 472498119864 minus 3

DI2BBDF(3) 166455119864 minus 2 477409119864 minus 3

DI2BBDF(4) 216342119864 minus 2 483298119864 minus 3

10minus4

FI2BBDF(3) 734012119864 minus 4 125797119864 minus 1

FI2BEBDF(3) 323640119864 minus 5 136259119864 minus 1

DI2BBDF(2) 118355119864 minus 6 167260119864 minus 1

DI2BBDF(3) 173430119864 minus 6 169325119864 minus 1

DI2BBDF(4) 225966119864 minus 6 172995119864 minus 1

10minus6

FI2BBDF(3) 735741119864 minus 6 1257001198641

FI2BEBDF(3) 350090119864 minus 7 1362301198641

DI2BBDF(2) 118742119864 minus 10 1491591198641

DI2BBDF(3) 173634119864 minus 10 1544931198641

DI2BBDF(4) 226000119864 minus 10 1546201198641

Table 2 The accuracy for problem 2

ℎ Methods MAXE TIME

10minus2

FI2BBDF(3) 271597119864 minus 2 596467119864 minus 3

FI2BEBDF(3) 149811119864 minus 2 661025119864 minus 3

DI2BBDF(2) 456073119864 minus 3 104480119864 minus 2

DI2BBDF(3) 686126119864 minus 3 108421119864 minus 2

DI2BBDF(4) 893279119864 minus 3 111089119864 minus 2

10minus4

FI2BBDF(3) 340437119864 minus 4 590430119864 minus 1

FI2BEBDF(3) 262649119864 minus 5 654305119864 minus 1

DI2BBDF(2) 485404119864 minus 7 373210119864 minus 1

DI2BBDF(3) 711155119864 minus 7 386556119864 minus 1

DI2BBDF(4) 926476119864 minus 7 387070119864 minus 1

10minus6

FI2BBDF(3) 341146119864 minus 6 5901981198641

FI2BEBDF(3) 244097119864 minus 7 6541061198641

DI2BBDF(2) 486843119864 minus 11 3484511198641

DI2BBDF(3) 711903119864 minus 11 3604901198641

DI2BBDF(4) 926598119864 minus 11 3606491198641

6 Numerical Results

The performance of the derived methods is compared withthe existing methods in terms of maximum error and exe-cution time We consider 10minus2 10minus4 and 10minus6 as the stepsize ℎ Tables 1 and 2 present the performance compar-ison of DI2BBDF(2) DI2BBDF(3) and DI2BBDF(4) withFI2BBDF(3) and FI2BEBDF(3) whereas Tables 3ndash6 exhibitthe comparison of proposed methods with FI3BBDF(3) andFI3BEBDF(3) In addition the graphs of log(MAXE) againstlog(ℎ) are illustrated as shown in Figures 5ndash10 The followingnotations are used in Tables 1ndash6

ℎ step sizeMAXE maximum errorTIME time execution using high performance com-puter (HPC)

Table 3 The accuracy for problem 3

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 356692119864 minus 2 141642119864 minus 3

FI3BEBDF(3) 124084119864 minus 2 302167119864 minus 3

DI2BBDF(2) 117256119864 minus 3 448489119864 minus 3

DI2BBDF(3) 172278119864 minus 3 481200119864 minus 3

DI2BBDF(4) 225003119864 minus 3 485301119864 minus 3

10minus4

FI3BBDF(3) 435640119864 minus 4 130103119864 minus 1

FI3BEBDF(3) 706516119864 minus 5 208115119864 minus 1

DI2BBDF(2) 118749119864 minus 7 160304119864 minus 1

DI2BBDF(3) 173636119864 minus 7 177445119864 minus 1

DI2BBDF(4) 226000119864 minus 7 178064119864 minus 1

10minus6

FI3BBDF(3) 434625119864 minus 6 1297751198641

FI3BEBDF(3) 703257119864 minus 7 2076091198640

DI2BBDF(2) 124989119864 minus 11 1507391198641

DI2BBDF(3) 173636119864 minus 11 1550731198641

DI2BBDF(4) 225999119864 minus 11 1602161198641

Table 4 The accuracy for problem 4

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 66269411986499 701775119864 minus 3

FI3BEBDF(3) 168449119864 minus 1 137847119864 minus 2

DI2BBDF(2) 280830119864 minus 1 106871119864 minus 2

DI2BBDF(3) 400389119864 minus 1 124969119864 minus 2

DI2BBDF(4) 477342119864 minus 1 126502119864 minus 2

10minus4

FI3BBDF(3) 845376119864 minus 3 676737119864 minus 1

FI3BEBDF(3) 695725119864 minus 3 135671119864 + 0

DI2BBDF(2) 356255119864 minus 5 432359119864 minus 1

DI2BBDF(3) 524241119864 minus 5 442599119864 minus 1

DI2BBDF(4) 685105119864 minus 5 456207119864 minus 1

10minus6

FI3BBDF(3) 854545119864 minus 5 6768191198641

FI3BEBDF(3) 718859119864 minus 5 1358621198642

DI2BBDF(2) 360130119864 minus 9 4000101198641

DI2BBDF(3) 526673119864 minus 9 4001921198641

DI2BBDF(4) 685529119864 minus 9 4103221198641

Table 5 The accuracy for problem 5

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 525481119864 minus 3 578083119864 minus 4

FI3BEBDF(3) 388475119864 minus 3 916583119864 minus 4

DI2BBDF(2) 856253119864 minus 5 229836119864 minus 4

DI2BBDF(3) 121629119864 minus 4 189066119864 minus 4

DI2BBDF(4) 165094119864 minus 4 230074119864 minus 4

10minus4

FI3BBDF(3) 551558119864 minus 5 483860119864 minus 2

FI3BEBDF(3) 427506119864 minus 5 774877119864 minus 2

DI2BBDF(2) 889803119864 minus 9 117438119864 minus 2

DI2BBDF(3) 130124119864 minus 8 131578119864 minus 2

DI2BBDF(4) 169456119864 minus 8 126071119864 minus 2

10minus6

FI3BBDF(3) 551803119864 minus 7 547768119864 + 0

FI3BEBDF(3) 427924119864 minus 7 777643119864 + 0

DI2BBDF(2) 126115119864 minus 11 119318119864 + 0

DI2BBDF(3) 627106119864 minus 11 133892119864 + 0

DI2BBDF(4) 104222119864 minus 11 126146119864 + 0

Mathematical Problems in Engineering 11

Table 6 The accuracy for problem 6

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 218555119864 minus 3 283084119864 minus 4

FI3BEBDF(3) 152719119864 minus 3 428500119864 minus 4

DI2BBDF(2) 387041119864 minus 5 538826119864 minus 5

DI2BBDF(3) 547958119864 minus 5 531673119864 minus 5

DI2BBDF(4) 747696119864 minus 5 618877119864 minus 5

10minus4

FI3BBDF(3) 229812119864 minus 5 148627119864 minus 2

FI3BEBDF(3) 178708119864 minus 5 232184119864 minus 2

DI2BBDF(2) 401638119864 minus 9 349712119864 minus 3

DI2BBDF(3) 587347119864 minus 9 375295119864 minus 3

DI2BBDF(4) 764927119864 minus 9 364804119864 minus 3

10minus6

FI3BBDF(3) 229936119864 minus 7 146660119864 + 0

FI3BEBDF(3) 178973119864 minus 7 228775119864 + 0

DI2BBDF(2) 990807119864 minus 12 338157119864 minus 1

DI2BBDF(3) 377716119864 minus 11 369837119864 minus 1

DI2BBDF(4) 462796119864 minus 12 356802119864 minus 1

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

log(h)minus6 minus5 minus4 minus3 minus2

FI2BBDF(3)FI2BEBDF(3)

DI2BBDF(2)DI2BBDF(3)DI2BBDF(4)

Figure 5 Graph of log(MAXE) versus log(ℎ) for problem 1

DI2BBDF(2) diagonally implicit 2-point block back-ward differentiation formulas of order 2DI2BBDF(3) diagonally implicit 2-point block back-ward differentiation formulas of order 3DI2BBDF(4) diagonally implicit 2-point block back-ward differentiation formulas of order 4FI2BBDF(3) fully implicit 2-point block backwarddifferentiation formulas of order 3FI2BEBDF(3) fully implicit 2-point block extendedbackward differentiation formulas of order 3FI3BBDF(3) fully implicit 3-point block backwarddifferentiation formulas of order 3

log(h)minus6 minus5 minus4 minus3 minus2

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

FI2BBDF(3)FI2BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

Figure 6 Graph of log(MAXE) versus log(ℎ) for problem 2

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus11minus12

minus10minus9minus8minus7minus6minus5minus4minus3minus2minus10

log(

MA

XE)

Figure 7 Graph of log(MAXE) versus log(ℎ) for problem 3

FI3BEBDF(3) fully implicit 3-point block extendedbackward differentiation formulas of order 3

7 Discussion

This section is divided into discussion of maximum error andcomputational time

71 Maximum Error From the numerical results in Tables1ndash6 we observe that DI2BBDF(2) DI2BBDF(3) andDI2BBDF(4) outperformed the existing methods in termsof maximum error This is expected because the diagonally

12 Mathematical Problems in Engineering

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

Figure 8 Graph of log(MAXE) versus log(ℎ) for problem 4

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus11

minus12

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

Figure 9 Graph of log(MAXE) versus log(ℎ) for problem 5

implicit method has less differentiation coefficients in orderto prevent the cumulative error The graphs in Figures 5ndash10depict the scales of maximum error versus step size h for theproposed methods as compared to existing methods Thereis a slight drop in accuracy for the proposed methods as theorder increases This is due to the increase in cumulativeerrors during the computation as more interpolating pointsare used Among the methods of order three DI2BBDF(3)is more accurate compared with FI2BBDF(3) FI3BBDF(3)FI2BEBDF(3) and FI3BEBDF(3) For all test problems we

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus11minus12

minus10minus9minus8minus7minus6minus5minus4minus3minus2minus10

log(

MA

XE)

Figure 10 Graph of log(MAXE) versus log(ℎ) for problem 6

conclude that the accuracy of the proposedmethod increasesas the step size becomes smaller

72 Computational Time In Tables 1-2 it can be seen thatthe execution time taken by DI2BBDF(2) DI2BBDF(3) andDI2BBDF(4) is comparable with that of FI2BBDF(3) andFI2BEBDF(3) The computational time increases as the stepsize becomes smaller However Tables 3ndash6 show that theproposed methods compute faster than FI3BBDF(3) andFI3BEBDF(3) This could be justified by the fact that the3-point method has to perform extra computation on theJacobian matrix since it involves a 3 by 3 matrix while the 2-point method has to compute Jacobian matrix of dimension2 by 2 only Among the proposed methods DI2BBDF(2)computes faster than DI2BBDF(3) and DI2BBDF(4) In factthe number of back values involved in every formula willaffect the computational time Since DI2BBDF(4) has moreback values than DI2BBDF(3) and DI2BBDF(2) it requiresextra time to compute the approximated solutions

8 Conclusion

Research conducted in this paper shows the capability ofconstructing the diagonally implicit BBDF method with A-stable properties From the results obtained via the numericalexperiment we can conclude that the proposed methodserves the purpose of significant alternative numericalmethod for solving linear and nonlinear stiff IVPs occurringin the fields of engineering and physical sciences

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 13

Acknowledgment

This researchwas supported by the Institute forMathematicalResearch (INSPEM) Department of Mathematics UniversitiPutra Malaysia (UPM) under Fundamental Grant Scheme(Project code 02-02-13-1380FR)

References

[1] C F Curtiss and J O Hirschfelder ldquoIntegration of stiff equa-tionsrdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 38 no 3 pp 235ndash243 1952

[2] J D Lambert Numercial Methods for Ordinary DifferentialSystems The Initial Value Problem John Wiley amp Sons NewYork NY USA 1991

[3] W E Milne Numerical Solution of Differential Equations JohnWiley amp Sons New York NY USA 1953

[4] L F Shampine and H A Watts ldquoBlock implicit one-stepmethodsrdquo Mathematics of Computation vol 23 pp 731ndash7401969

[5] S O Fatunla ldquoBlock methods for second order ODEsrdquo Inter-national Journal of Computer Mathematics vol 41 pp 55ndash631990

[6] Z A Majid M B Suleiman F I Ismail and K I Othmanldquo2-Point 1 block diagonally and 2-point 1 block fully implicitmethod for solving first order ordinary differential equationsrdquoin Proceedings of the 12th National Symposium on MathematicalScience vol 12 2004

[7] Z Abdul Majid N Z Mokhtar and M Suleiman ldquoDirecttwo-point block one-step method for solving general second-order ordinary differential equationsrdquo Mathematical Problemsin Engineering vol 2012 Article ID 184253 16 pages 2012

[8] Z B Ibrahim Block method for multistep formulas for solvingordinary differential equations [PhD thesis] Universiti PutraMalaysia Seri Kembangan Malaysia 2006

[9] Z B Ibrahim M Suleiman and K I Othman ldquoFixed coeffi-cients block backward differentiation formulas for the numer-ical solution of stiff ordinary differential equationsrdquo EuropeanJournal of Scientific Research vol 21 no 3 pp 508ndash520 2008

[10] Z B Ibrahim M B Suleiman and K I Othman ldquoImplicit r-point block backward differentiation formula for solving first-order stiff ODEsrdquo Applied Mathematics and Computation vol186 no 1 pp 558ndash565 2007

[11] Z B Ibrahim R Johari and F Ismail ldquoOn the stability of blockbackward differentiation formulaerdquo Matematika vol 19 no 2pp 83ndash89 2003

[12] N A Nasir Z B Ibrahim M B Suleiman and K I OthmanldquoFifth order two-point block backward differentiation formulasfor solving ordinary differential equationsrdquo Applied Mathemat-ical Sciences vol 5 no 69ndash72 pp 3505ndash3518 2011

[13] H Musa M B Suleiman and N Senu ldquoA-stable 2-Point blockextended backward differentiation formulas for stiff ordinarydifferential equationsrdquo in Proceedings of the 5th InternationalConference on Research and education in Mathematics vol 1450of AIP Conference Proceedings pp 254ndash258 2012

[14] H Musa M B Suleiman and N Senu ldquoFully implicit 3-pointblock extended backward differentiation formula for stiff initialvalue problemsrdquo Applied Mathematical Sciences vol 6 no 85ndash88 pp 4211ndash4227 2012

[15] R Alexander ldquoDiagonally implicit Runge-kutta for stiff ordi-nary differential equationsrdquo SIAM Journal on Numerical Analy-sis vol 14 no 6 pp 1006ndash1021 1977

[16] O Y Ababneh R Ahmad and E S Ismail ldquoDesign of newdiagonally implicit Runge-Kutta methods for stiff problemsrdquoApplied Mathematical Sciences vol 3 no 45 pp 2241ndash22532009

[17] F Ismail A L I Raed Al-Khasawneh M Suleiman and A BU Malik Hassan ldquoEmbedded pair of diagonally implicit runge-kutta method for solving ordinary differential equationsrdquo SainsMalaysiana vol 39 no 6 pp 1049ndash1054 2010

[18] Z A Majid and M B Suleiman ldquoPerformance of four pointdiagonally implicit block method for solving first order stiffordinary differential equationsrdquo Department of MathematicsUTM vol 22 no 2 pp 137ndash146 2006

[19] G G Dahlquist ldquoA special stability problem for linearmultistepmethodsrdquo BIT Numerical Mathematics vol 3 no 1 pp 27ndash431963

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

Table 6 The accuracy for problem 6

ℎ Methods MAXE TIME

10minus2

FI3BBDF(3) 218555119864 minus 3 283084119864 minus 4

FI3BEBDF(3) 152719119864 minus 3 428500119864 minus 4

DI2BBDF(2) 387041119864 minus 5 538826119864 minus 5

DI2BBDF(3) 547958119864 minus 5 531673119864 minus 5

DI2BBDF(4) 747696119864 minus 5 618877119864 minus 5

10minus4

FI3BBDF(3) 229812119864 minus 5 148627119864 minus 2

FI3BEBDF(3) 178708119864 minus 5 232184119864 minus 2

DI2BBDF(2) 401638119864 minus 9 349712119864 minus 3

DI2BBDF(3) 587347119864 minus 9 375295119864 minus 3

DI2BBDF(4) 764927119864 minus 9 364804119864 minus 3

10minus6

FI3BBDF(3) 229936119864 minus 7 146660119864 + 0

FI3BEBDF(3) 178973119864 minus 7 228775119864 + 0

DI2BBDF(2) 990807119864 minus 12 338157119864 minus 1

DI2BBDF(3) 377716119864 minus 11 369837119864 minus 1

DI2BBDF(4) 462796119864 minus 12 356802119864 minus 1

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

log(h)minus6 minus5 minus4 minus3 minus2

FI2BBDF(3)FI2BEBDF(3)

DI2BBDF(2)DI2BBDF(3)DI2BBDF(4)

Figure 5 Graph of log(MAXE) versus log(ℎ) for problem 1

DI2BBDF(2) diagonally implicit 2-point block back-ward differentiation formulas of order 2DI2BBDF(3) diagonally implicit 2-point block back-ward differentiation formulas of order 3DI2BBDF(4) diagonally implicit 2-point block back-ward differentiation formulas of order 4FI2BBDF(3) fully implicit 2-point block backwarddifferentiation formulas of order 3FI2BEBDF(3) fully implicit 2-point block extendedbackward differentiation formulas of order 3FI3BBDF(3) fully implicit 3-point block backwarddifferentiation formulas of order 3

log(h)minus6 minus5 minus4 minus3 minus2

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

FI2BBDF(3)FI2BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

Figure 6 Graph of log(MAXE) versus log(ℎ) for problem 2

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus11minus12

minus10minus9minus8minus7minus6minus5minus4minus3minus2minus10

log(

MA

XE)

Figure 7 Graph of log(MAXE) versus log(ℎ) for problem 3

FI3BEBDF(3) fully implicit 3-point block extendedbackward differentiation formulas of order 3

7 Discussion

This section is divided into discussion of maximum error andcomputational time

71 Maximum Error From the numerical results in Tables1ndash6 we observe that DI2BBDF(2) DI2BBDF(3) andDI2BBDF(4) outperformed the existing methods in termsof maximum error This is expected because the diagonally

12 Mathematical Problems in Engineering

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

Figure 8 Graph of log(MAXE) versus log(ℎ) for problem 4

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus11

minus12

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

Figure 9 Graph of log(MAXE) versus log(ℎ) for problem 5

implicit method has less differentiation coefficients in orderto prevent the cumulative error The graphs in Figures 5ndash10depict the scales of maximum error versus step size h for theproposed methods as compared to existing methods Thereis a slight drop in accuracy for the proposed methods as theorder increases This is due to the increase in cumulativeerrors during the computation as more interpolating pointsare used Among the methods of order three DI2BBDF(3)is more accurate compared with FI2BBDF(3) FI3BBDF(3)FI2BEBDF(3) and FI3BEBDF(3) For all test problems we

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus11minus12

minus10minus9minus8minus7minus6minus5minus4minus3minus2minus10

log(

MA

XE)

Figure 10 Graph of log(MAXE) versus log(ℎ) for problem 6

conclude that the accuracy of the proposedmethod increasesas the step size becomes smaller

72 Computational Time In Tables 1-2 it can be seen thatthe execution time taken by DI2BBDF(2) DI2BBDF(3) andDI2BBDF(4) is comparable with that of FI2BBDF(3) andFI2BEBDF(3) The computational time increases as the stepsize becomes smaller However Tables 3ndash6 show that theproposed methods compute faster than FI3BBDF(3) andFI3BEBDF(3) This could be justified by the fact that the3-point method has to perform extra computation on theJacobian matrix since it involves a 3 by 3 matrix while the 2-point method has to compute Jacobian matrix of dimension2 by 2 only Among the proposed methods DI2BBDF(2)computes faster than DI2BBDF(3) and DI2BBDF(4) In factthe number of back values involved in every formula willaffect the computational time Since DI2BBDF(4) has moreback values than DI2BBDF(3) and DI2BBDF(2) it requiresextra time to compute the approximated solutions

8 Conclusion

Research conducted in this paper shows the capability ofconstructing the diagonally implicit BBDF method with A-stable properties From the results obtained via the numericalexperiment we can conclude that the proposed methodserves the purpose of significant alternative numericalmethod for solving linear and nonlinear stiff IVPs occurringin the fields of engineering and physical sciences

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 13

Acknowledgment

This researchwas supported by the Institute forMathematicalResearch (INSPEM) Department of Mathematics UniversitiPutra Malaysia (UPM) under Fundamental Grant Scheme(Project code 02-02-13-1380FR)

References

[1] C F Curtiss and J O Hirschfelder ldquoIntegration of stiff equa-tionsrdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 38 no 3 pp 235ndash243 1952

[2] J D Lambert Numercial Methods for Ordinary DifferentialSystems The Initial Value Problem John Wiley amp Sons NewYork NY USA 1991

[3] W E Milne Numerical Solution of Differential Equations JohnWiley amp Sons New York NY USA 1953

[4] L F Shampine and H A Watts ldquoBlock implicit one-stepmethodsrdquo Mathematics of Computation vol 23 pp 731ndash7401969

[5] S O Fatunla ldquoBlock methods for second order ODEsrdquo Inter-national Journal of Computer Mathematics vol 41 pp 55ndash631990

[6] Z A Majid M B Suleiman F I Ismail and K I Othmanldquo2-Point 1 block diagonally and 2-point 1 block fully implicitmethod for solving first order ordinary differential equationsrdquoin Proceedings of the 12th National Symposium on MathematicalScience vol 12 2004

[7] Z Abdul Majid N Z Mokhtar and M Suleiman ldquoDirecttwo-point block one-step method for solving general second-order ordinary differential equationsrdquo Mathematical Problemsin Engineering vol 2012 Article ID 184253 16 pages 2012

[8] Z B Ibrahim Block method for multistep formulas for solvingordinary differential equations [PhD thesis] Universiti PutraMalaysia Seri Kembangan Malaysia 2006

[9] Z B Ibrahim M Suleiman and K I Othman ldquoFixed coeffi-cients block backward differentiation formulas for the numer-ical solution of stiff ordinary differential equationsrdquo EuropeanJournal of Scientific Research vol 21 no 3 pp 508ndash520 2008

[10] Z B Ibrahim M B Suleiman and K I Othman ldquoImplicit r-point block backward differentiation formula for solving first-order stiff ODEsrdquo Applied Mathematics and Computation vol186 no 1 pp 558ndash565 2007

[11] Z B Ibrahim R Johari and F Ismail ldquoOn the stability of blockbackward differentiation formulaerdquo Matematika vol 19 no 2pp 83ndash89 2003

[12] N A Nasir Z B Ibrahim M B Suleiman and K I OthmanldquoFifth order two-point block backward differentiation formulasfor solving ordinary differential equationsrdquo Applied Mathemat-ical Sciences vol 5 no 69ndash72 pp 3505ndash3518 2011

[13] H Musa M B Suleiman and N Senu ldquoA-stable 2-Point blockextended backward differentiation formulas for stiff ordinarydifferential equationsrdquo in Proceedings of the 5th InternationalConference on Research and education in Mathematics vol 1450of AIP Conference Proceedings pp 254ndash258 2012

[14] H Musa M B Suleiman and N Senu ldquoFully implicit 3-pointblock extended backward differentiation formula for stiff initialvalue problemsrdquo Applied Mathematical Sciences vol 6 no 85ndash88 pp 4211ndash4227 2012

[15] R Alexander ldquoDiagonally implicit Runge-kutta for stiff ordi-nary differential equationsrdquo SIAM Journal on Numerical Analy-sis vol 14 no 6 pp 1006ndash1021 1977

[16] O Y Ababneh R Ahmad and E S Ismail ldquoDesign of newdiagonally implicit Runge-Kutta methods for stiff problemsrdquoApplied Mathematical Sciences vol 3 no 45 pp 2241ndash22532009

[17] F Ismail A L I Raed Al-Khasawneh M Suleiman and A BU Malik Hassan ldquoEmbedded pair of diagonally implicit runge-kutta method for solving ordinary differential equationsrdquo SainsMalaysiana vol 39 no 6 pp 1049ndash1054 2010

[18] Z A Majid and M B Suleiman ldquoPerformance of four pointdiagonally implicit block method for solving first order stiffordinary differential equationsrdquo Department of MathematicsUTM vol 22 no 2 pp 137ndash146 2006

[19] G G Dahlquist ldquoA special stability problem for linearmultistepmethodsrdquo BIT Numerical Mathematics vol 3 no 1 pp 27ndash431963

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

Figure 8 Graph of log(MAXE) versus log(ℎ) for problem 4

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus11

minus12

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

log(

MA

XE)

Figure 9 Graph of log(MAXE) versus log(ℎ) for problem 5

implicit method has less differentiation coefficients in orderto prevent the cumulative error The graphs in Figures 5ndash10depict the scales of maximum error versus step size h for theproposed methods as compared to existing methods Thereis a slight drop in accuracy for the proposed methods as theorder increases This is due to the increase in cumulativeerrors during the computation as more interpolating pointsare used Among the methods of order three DI2BBDF(3)is more accurate compared with FI2BBDF(3) FI3BBDF(3)FI2BEBDF(3) and FI3BEBDF(3) For all test problems we

FI3BBDF(3)FI3BEBDF(3)DI2BBDF(2)

DI2BBDF(3)DI2BBDF(4)

log(h)minus6 minus5 minus4 minus3 minus2

minus11minus12

minus10minus9minus8minus7minus6minus5minus4minus3minus2minus10

log(

MA

XE)

Figure 10 Graph of log(MAXE) versus log(ℎ) for problem 6

conclude that the accuracy of the proposedmethod increasesas the step size becomes smaller

72 Computational Time In Tables 1-2 it can be seen thatthe execution time taken by DI2BBDF(2) DI2BBDF(3) andDI2BBDF(4) is comparable with that of FI2BBDF(3) andFI2BEBDF(3) The computational time increases as the stepsize becomes smaller However Tables 3ndash6 show that theproposed methods compute faster than FI3BBDF(3) andFI3BEBDF(3) This could be justified by the fact that the3-point method has to perform extra computation on theJacobian matrix since it involves a 3 by 3 matrix while the 2-point method has to compute Jacobian matrix of dimension2 by 2 only Among the proposed methods DI2BBDF(2)computes faster than DI2BBDF(3) and DI2BBDF(4) In factthe number of back values involved in every formula willaffect the computational time Since DI2BBDF(4) has moreback values than DI2BBDF(3) and DI2BBDF(2) it requiresextra time to compute the approximated solutions

8 Conclusion

Research conducted in this paper shows the capability ofconstructing the diagonally implicit BBDF method with A-stable properties From the results obtained via the numericalexperiment we can conclude that the proposed methodserves the purpose of significant alternative numericalmethod for solving linear and nonlinear stiff IVPs occurringin the fields of engineering and physical sciences

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 13

Acknowledgment

This researchwas supported by the Institute forMathematicalResearch (INSPEM) Department of Mathematics UniversitiPutra Malaysia (UPM) under Fundamental Grant Scheme(Project code 02-02-13-1380FR)

References

[1] C F Curtiss and J O Hirschfelder ldquoIntegration of stiff equa-tionsrdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 38 no 3 pp 235ndash243 1952

[2] J D Lambert Numercial Methods for Ordinary DifferentialSystems The Initial Value Problem John Wiley amp Sons NewYork NY USA 1991

[3] W E Milne Numerical Solution of Differential Equations JohnWiley amp Sons New York NY USA 1953

[4] L F Shampine and H A Watts ldquoBlock implicit one-stepmethodsrdquo Mathematics of Computation vol 23 pp 731ndash7401969

[5] S O Fatunla ldquoBlock methods for second order ODEsrdquo Inter-national Journal of Computer Mathematics vol 41 pp 55ndash631990

[6] Z A Majid M B Suleiman F I Ismail and K I Othmanldquo2-Point 1 block diagonally and 2-point 1 block fully implicitmethod for solving first order ordinary differential equationsrdquoin Proceedings of the 12th National Symposium on MathematicalScience vol 12 2004

[7] Z Abdul Majid N Z Mokhtar and M Suleiman ldquoDirecttwo-point block one-step method for solving general second-order ordinary differential equationsrdquo Mathematical Problemsin Engineering vol 2012 Article ID 184253 16 pages 2012

[8] Z B Ibrahim Block method for multistep formulas for solvingordinary differential equations [PhD thesis] Universiti PutraMalaysia Seri Kembangan Malaysia 2006

[9] Z B Ibrahim M Suleiman and K I Othman ldquoFixed coeffi-cients block backward differentiation formulas for the numer-ical solution of stiff ordinary differential equationsrdquo EuropeanJournal of Scientific Research vol 21 no 3 pp 508ndash520 2008

[10] Z B Ibrahim M B Suleiman and K I Othman ldquoImplicit r-point block backward differentiation formula for solving first-order stiff ODEsrdquo Applied Mathematics and Computation vol186 no 1 pp 558ndash565 2007

[11] Z B Ibrahim R Johari and F Ismail ldquoOn the stability of blockbackward differentiation formulaerdquo Matematika vol 19 no 2pp 83ndash89 2003

[12] N A Nasir Z B Ibrahim M B Suleiman and K I OthmanldquoFifth order two-point block backward differentiation formulasfor solving ordinary differential equationsrdquo Applied Mathemat-ical Sciences vol 5 no 69ndash72 pp 3505ndash3518 2011

[13] H Musa M B Suleiman and N Senu ldquoA-stable 2-Point blockextended backward differentiation formulas for stiff ordinarydifferential equationsrdquo in Proceedings of the 5th InternationalConference on Research and education in Mathematics vol 1450of AIP Conference Proceedings pp 254ndash258 2012

[14] H Musa M B Suleiman and N Senu ldquoFully implicit 3-pointblock extended backward differentiation formula for stiff initialvalue problemsrdquo Applied Mathematical Sciences vol 6 no 85ndash88 pp 4211ndash4227 2012

[15] R Alexander ldquoDiagonally implicit Runge-kutta for stiff ordi-nary differential equationsrdquo SIAM Journal on Numerical Analy-sis vol 14 no 6 pp 1006ndash1021 1977

[16] O Y Ababneh R Ahmad and E S Ismail ldquoDesign of newdiagonally implicit Runge-Kutta methods for stiff problemsrdquoApplied Mathematical Sciences vol 3 no 45 pp 2241ndash22532009

[17] F Ismail A L I Raed Al-Khasawneh M Suleiman and A BU Malik Hassan ldquoEmbedded pair of diagonally implicit runge-kutta method for solving ordinary differential equationsrdquo SainsMalaysiana vol 39 no 6 pp 1049ndash1054 2010

[18] Z A Majid and M B Suleiman ldquoPerformance of four pointdiagonally implicit block method for solving first order stiffordinary differential equationsrdquo Department of MathematicsUTM vol 22 no 2 pp 137ndash146 2006

[19] G G Dahlquist ldquoA special stability problem for linearmultistepmethodsrdquo BIT Numerical Mathematics vol 3 no 1 pp 27ndash431963

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

Acknowledgment

This researchwas supported by the Institute forMathematicalResearch (INSPEM) Department of Mathematics UniversitiPutra Malaysia (UPM) under Fundamental Grant Scheme(Project code 02-02-13-1380FR)

References

[1] C F Curtiss and J O Hirschfelder ldquoIntegration of stiff equa-tionsrdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 38 no 3 pp 235ndash243 1952

[2] J D Lambert Numercial Methods for Ordinary DifferentialSystems The Initial Value Problem John Wiley amp Sons NewYork NY USA 1991

[3] W E Milne Numerical Solution of Differential Equations JohnWiley amp Sons New York NY USA 1953

[4] L F Shampine and H A Watts ldquoBlock implicit one-stepmethodsrdquo Mathematics of Computation vol 23 pp 731ndash7401969

[5] S O Fatunla ldquoBlock methods for second order ODEsrdquo Inter-national Journal of Computer Mathematics vol 41 pp 55ndash631990

[6] Z A Majid M B Suleiman F I Ismail and K I Othmanldquo2-Point 1 block diagonally and 2-point 1 block fully implicitmethod for solving first order ordinary differential equationsrdquoin Proceedings of the 12th National Symposium on MathematicalScience vol 12 2004

[7] Z Abdul Majid N Z Mokhtar and M Suleiman ldquoDirecttwo-point block one-step method for solving general second-order ordinary differential equationsrdquo Mathematical Problemsin Engineering vol 2012 Article ID 184253 16 pages 2012

[8] Z B Ibrahim Block method for multistep formulas for solvingordinary differential equations [PhD thesis] Universiti PutraMalaysia Seri Kembangan Malaysia 2006

[9] Z B Ibrahim M Suleiman and K I Othman ldquoFixed coeffi-cients block backward differentiation formulas for the numer-ical solution of stiff ordinary differential equationsrdquo EuropeanJournal of Scientific Research vol 21 no 3 pp 508ndash520 2008

[10] Z B Ibrahim M B Suleiman and K I Othman ldquoImplicit r-point block backward differentiation formula for solving first-order stiff ODEsrdquo Applied Mathematics and Computation vol186 no 1 pp 558ndash565 2007

[11] Z B Ibrahim R Johari and F Ismail ldquoOn the stability of blockbackward differentiation formulaerdquo Matematika vol 19 no 2pp 83ndash89 2003

[12] N A Nasir Z B Ibrahim M B Suleiman and K I OthmanldquoFifth order two-point block backward differentiation formulasfor solving ordinary differential equationsrdquo Applied Mathemat-ical Sciences vol 5 no 69ndash72 pp 3505ndash3518 2011

[13] H Musa M B Suleiman and N Senu ldquoA-stable 2-Point blockextended backward differentiation formulas for stiff ordinarydifferential equationsrdquo in Proceedings of the 5th InternationalConference on Research and education in Mathematics vol 1450of AIP Conference Proceedings pp 254ndash258 2012

[14] H Musa M B Suleiman and N Senu ldquoFully implicit 3-pointblock extended backward differentiation formula for stiff initialvalue problemsrdquo Applied Mathematical Sciences vol 6 no 85ndash88 pp 4211ndash4227 2012

[15] R Alexander ldquoDiagonally implicit Runge-kutta for stiff ordi-nary differential equationsrdquo SIAM Journal on Numerical Analy-sis vol 14 no 6 pp 1006ndash1021 1977

[16] O Y Ababneh R Ahmad and E S Ismail ldquoDesign of newdiagonally implicit Runge-Kutta methods for stiff problemsrdquoApplied Mathematical Sciences vol 3 no 45 pp 2241ndash22532009

[17] F Ismail A L I Raed Al-Khasawneh M Suleiman and A BU Malik Hassan ldquoEmbedded pair of diagonally implicit runge-kutta method for solving ordinary differential equationsrdquo SainsMalaysiana vol 39 no 6 pp 1049ndash1054 2010

[18] Z A Majid and M B Suleiman ldquoPerformance of four pointdiagonally implicit block method for solving first order stiffordinary differential equationsrdquo Department of MathematicsUTM vol 22 no 2 pp 137ndash146 2006

[19] G G Dahlquist ldquoA special stability problem for linearmultistepmethodsrdquo BIT Numerical Mathematics vol 3 no 1 pp 27ndash431963

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of