Research ArticleTwo Embedded Pairs of Runge-Kutta Type Methodsfor Direct Solution of Special Fourth-Order OrdinaryDifferential Equations

Kasim Hussain12 Fudziah Ismail13 and Norazak Senu13

1Department of Mathematics Faculty of Science Universiti Putra Malaysia (UPM) 43400 SerdangSelangor Malaysia2Department of Mathematics College of Science Al-Mustansiriyah University Baghdad Iraq3Institute for Mathematical Research Universiti Putra Malaysia (UPM) 43400 Serdang Selangor Malaysia

Correspondence should be addressed to Fudziah Ismail fudziah iyahoocommy

Received 14 August 2015 Revised 3 November 2015 Accepted 8 November 2015

Academic Editor Tarek Ahmed-Ali

Copyright copy 2015 Kasim Hussain et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We present two pairs of embedded Runge-Kutta type methods for direct solution of fourth-order ordinary differential equations(ODEs) of the form 119910

(119894V)= 119891(119909 119910) denoted as RKFD methods The first pair which we will call RKFD5(4) has orders 5 and 4

and the second one has orders 6 and 5 and we will call it RKFD6(5) The techniques used in the derivation of the methods arethat the higher order methods are very precise and the lower order methods give the best error estimate Based on these pairswe have developed variable step codes and we have used them to solve a set of special fourth-order problems Numerical resultsshow the robustness and the efficiency of the new RKFD pairs as compared with the well-known embedded Runge-Kutta pairs inthe scientific literature after reducing the problems into a system of first-order ordinary differential equations (ODEs) and solvingthem

1 Introduction

This paper deals with embedded RKFD methods for directlysolving special fourth-order ordinary differential equations(ODEs) of the form

119910(119894V)

(119909) = 119891 (119909 119910) 119909 ge 1199090 (1)

with initial conditions

119910 (1199090) = 1199100

1199101015840(1199090) = 119910

1015840

0

11991010158401015840

(1199090) = 11991010158401015840

0

119910101584010158401015840

(1199090) = 119910101584010158401015840

0

(2)

in which the first second and third derivatives do not appearexplicitly This type of problems can be found in variousfields of applied science and engineering such as beam theory[1 2] fluid dynamics [3] neural networks [4] and electric cir-cuits [5] Traditionally the fourth-order ordinary differentialequations are transformed to a first-order system of ordinarydifferential equations so that standard numerical methodscan be applied (see [6ndash11]) However several researchers (see[1 12 13]) observed the drawback of this technique as it wastesa lot of computing time and human effort Therefore directintegrationmethods have attracted significant attention fromseveral authors for solving higher order ODEs because thesedirect methods demonstrated the features in accuracy andspeed (see [14ndash23]) However all the methods discussedabove are multistep methods in nature This paper primarilyaims to construct a one-step method to solve special fourth-order ODEs directly this new method is self-starting innature

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 196595 12 pageshttpdxdoiorg1011552015196595

2 Mathematical Problems in Engineering

The general form of RKFD method with 119904-stage forsolving special fourth-order ODEs (1) can be expressed asfollows [24]

119910119899+1 = 119910119899 + ℎ1199101015840

119899 +ℎ2

211991010158401015840

119899 +ℎ3

6119910101584010158401015840

119899 + ℎ4119904

sum

119894=1

119887119894119896119894

1199101015840

119899+1 = 1199101015840

119899 + ℎ11991010158401015840

119899 +ℎ2

2119910101584010158401015840

119899 + ℎ3119904

sum

119894=1

1198871015840

119894 119896119894

11991010158401015840

119899+1 = 11991010158401015840

119899 + ℎ119910101584010158401015840

119899 + ℎ2119904

sum

119894=1

11988710158401015840

119894 119896119894

119910101584010158401015840

119899+1 = 119910101584010158401015840

119899 + ℎ

119904

sum

119894=1

119887101584010158401015840

119894 119896119894

(3)

where

1198961 = 119891 (119909119899 119910119899)

119896119894 = 119891(119909119899 + 119888119894ℎ 119910119899 + ℎ1198881198941199101015840

119899 +ℎ2

21198882

119894 11991010158401015840

119899 +ℎ3

61198883

119894 119910101584010158401015840

119899

+ ℎ4119904

sum

119895=1

119886119894119895119896119895)

(4)

for 119894 = 2 3 119904The parameters 119887119894 119887

1015840119894 11988710158401015840119894 119887101584010158401015840119894 119886119894119895 and 119888119894 of the RKFD

method are to be determined for 119894 = 1 2 119904 and 119895 =

1 2 119904 and supposed to be real The RKFD method is anexplicit method if 119886119894119895 = 0 for 119894 le 119895 and is an implicit methodif 119886119894119895 = 0 for some 119894 such that 119894 le 119895

To determine the parameters of the RKFD method givenin (3)-(4) the RKFD method expression is expanded usingthe Taylor series expansion After doing some algebraic sim-plifications this expansion is equated to the true solution thatis given by the Taylor series expansion The direct expansionof the truncation error is used to derive the order conditionsfor the RKFD method [25] A good deal of algebraic andnumerical calculations are required for the above operationwhich were carried out using algebra package Maple [26]Algebraic order conditions for the RKFD method can beobtained from the direct expansion of the local truncationerror

In this paper we will derive embedded Runge-Kuttapairs for direct integration of special fourth-order ODEsEmbedded pairs of RK type methods have a built-in localtruncation error estimate as a result the step size can becontrolled at virtually no extra cost and hence an efficientvariable step size code can be developed

In recent years the construction of embedded Runge-Kuttamethod is an effective research area yielding continuousdevelopment to the existing codes The present paper isprimarily dedicated as an extra work in this research areaThis technique involves two Runge-Kutta formulae of orders119903 and V (119903 gt V usually 119903 = V + 1) (see [25 27ndash33])We are interested in deriving the effective embedded 119903(V)

pairs of RKFDmethods that provide a cheap error estimationfor variable step size codes They depend on the methods(119888 119860 119887 119887

1015840 11988710158401015840 119887101584010158401015840

) of order 119903 and (119888 119860 1015840 10158401015840 101584010158401015840

) of orderV Butcher tableau of embedded RKFD pair can be written asfollows

119888 119860

119887119879

1198871015840119879

11988710158401015840119879

119887101584010158401015840119879

119879

1015840119879

10158401015840119879

101584010158401015840119879

(5)

Themethodwill compute119910119899+11199101015840119899+111991010158401015840119899+1 and119910

101584010158401015840119899+1 to approx-

imate 119910(119909119899+1) 1199101015840(119909119899+1) 119910

10158401015840(119909119899+1) and 119910

101584010158401015840(119909119899+1) where 119910119899+1

is the computed solution and 119910(119909119899+1) is the exact solutionThe remainder of this paper is organized as follows

In Section 2 we present the order conditions of RKFDmethod as well as the basic concepts and notations whichare used for embedded method In Section 3 we present theconstruction of the new embedded RKFD pairs of orders5(4) and 6(5) respectively In Section 4 we carry out thenumerical experiments to show the efficiency of the newembedded RKFD pairs when compared with the well-knownRunge-Kutta pairs from the scientific literature Conclusionsof the paper are given in Section 5

2 The Order Conditions of RKFD Method

The order conditions of RKFD method up to fifth order havebeen derived using Taylor series expansion by Hussain et al[24]We use the same approach to derive the order conditionsup to seventh order and for convenience we will present thealgebraic order conditions of the RKFDmethod given in [24]together with the seventh order method in this paper Nextwe give the order conditions up to order 7

The order conditions for 119910 are as follows

Fourth order

119904

sum

119894=1

119887119894 =1

24 (6)

Fifth order

119904

sum

119894=1

119887119894119888119894 =1

120 (7)

Mathematical Problems in Engineering 3

Sixth order

119904

sum

119894=1

1198871198941198882

119894 =1

360 (8)

Seventh order

119904

sum

119894=1

1198871198941198883

119894 =1

840 (9)

The order conditions for 1199101015840 are as follows

Third order

119904

sum

119894=1

1198871015840

119894 =1

6 (10)

Fourth order

119904

sum

119894=1

1198871015840

119894 119888119894 =1

24 (11)

Fifth order

119904

sum

119894=1

1198871015840

119894 1198882

119894 =1

60 (12)

Sixth order

119904

sum

119894=1

1198871015840

119894 1198883

119894 =1

120 (13)

Seventh order

119904

sum

119894=1

1198871015840

119894 1198884

119894 =1

210 (14)

119904

sum

119894119895=1

1198871015840

119894 119886119894119895 =1

5040 (15)

The order conditions for 11991010158401015840 are as follows

Second order

119904

sum

119894=1

11988710158401015840

119894 =1

2 (16)

Third order

119904

sum

119894=1

11988710158401015840

119894 119888119894 =1

6 (17)

Fourth order

119904

sum

119894=1

11988710158401015840

119894 1198882

119894 =1

12 (18)

Fifth order

119904

sum

119894=1

11988710158401015840

119894 1198883

119894 =1

20 (19)

Sixth order

119904

sum

119894=1

11988710158401015840

119894 1198884

119894 =1

30 (20)

119904

sum

119894119895=1

11988710158401015840

119894 119886119894119895 =1

720 (21)

Seventh order

119904

sum

119894=1

11988710158401015840

119894 1198885

119894 =1

42

119904

sum

119894=1

11988710158401015840

119894 119886119894119895119888119895 =1

5040

119904

sum

119894=1

11988710158401015840

119894 119888119894119886119894119895 =1

1008

(22)

The order conditions for 119910101584010158401015840 are as follows

First order

119904

sum

119894=1

119887101584010158401015840

119894 = 1 (23)

Second order

119904

sum

119894=1

119887101584010158401015840

119894 119888119894 =1

2 (24)

Third order

119904

sum

119894=1

119887101584010158401015840

119894 1198882

119894 =1

3 (25)

Fourth order

119904

sum

119894=1

119887101584010158401015840

119894 1198883

119894 =1

4 (26)

4 Mathematical Problems in Engineering

Fifth order119904

sum

119894=1

119887101584010158401015840

119894 1198884

119894 =1

5 (27)

119904

sum

119894119895=1

119887101584010158401015840

119894 119886119894119895 =1

120 (28)

Sixth order

119904

sum

119894=1

119887101584010158401015840

119894 1198885

119894 =1

6 (29)

119904

sum

119894119895=1

119887101584010158401015840

119894 119886119894119895119888119895 =1

720

119904

sum

119894=1

119887101584010158401015840

119894 119888119894119886119894119895 =1

144

(30)

Seventh order

119904

sum

119894=1

119887101584010158401015840

119894 1198886

119894 =1

7

119904

sum

119894=1

119887101584010158401015840

119894 1198882

119895 119886119894119895 =1

168

119904

sum

119894=1

119887101584010158401015840

119894 1198861198941198951198882

119895 =1

2520

119904

sum

119894=1

119887101584010158401015840

119894 119888119894119886119894119895119888119895 =1

840

(31)

The following strategies are utilized for developing efficientembedded pairs

(1) The quantities of 120591(119903+1)

2 and 120591(V+1)

2 should be assmall as possible for higher and lower order RKFDmethod respectively where

10038171003817100381710038171003817120591(119903+1)100381710038171003817100381710038172

= radic

1198991

sum

119894=1

(120591(119903+1)

119894)2+

1198992

sum

119894=1

(1205911015840(119903+1)

119894)2+

1198993

sum

119894=1

(12059110158401015840(119903+1)

119894)2+

1198994

sum

119894=1

(120591101584010158401015840(119903+1)

119894)2

10038171003817100381710038171003817120591(V+1)100381710038171003817100381710038172

= radic

1198991

sum

119894=1

(120591(V+1)119894

)2+

1198992

sum

119894=1

(1205911015840(V+1)119894

)2+

1198993

sum

119894=1

(12059110158401015840(V+1)119894

)2+

1198994

sum

119894=1

(120591101584010158401015840(V+1)119894

)2

(32)

where 120591(119903+1)

119894 1205911015840(119903+1)119894

12059110158401015840(119903+1)119894

and 120591101584010158401015840(119903+1)

119894are called the

error terms for 119910 1199101015840 11991010158401015840 and 119910101584010158401015840 respectively

(2) The following quantities given in [34] should be assmall as possible

(i)

119862(V+2)

=

10038171003817100381710038171003817120591(V+2)

minus 120591(V+2)100381710038171003817100381710038172

1003817100381710038171003817120591(V+1)10038171003817100381710038172

1198621015840(V+2)

=

100381710038171003817100381710038171205911015840(V+2)

minus 1205911015840(V+2)100381710038171003817100381710038172

10038171003817100381710038171205911015840(V+1)10038171003817100381710038172

11986210158401015840(V+2)

=

1003817100381710038171003817100381712059110158401015840(V+2)

minus 12059110158401015840(V+2)100381710038171003817100381710038172

100381710038171003817100381712059110158401015840(V+1)10038171003817100381710038172

119862101584010158401015840(V+2)

=

10038171003817100381710038171003817120591101584010158401015840(V+2)

minus 120591101584010158401015840(V+2)100381710038171003817100381710038172

1003817100381710038171003817120591101584010158401015840(V+1)10038171003817100381710038172

(33)

(ii)

119861(V+2)

=

10038171003817100381710038171003817120591(V+2)100381710038171003817100381710038172

1003817100381710038171003817120591(V+1)10038171003817100381710038172

1198611015840(V+2)

=

100381710038171003817100381710038171205911015840(V+2)100381710038171003817100381710038172

10038171003817100381710038171205911015840(V+1)10038171003817100381710038172

11986110158401015840(V+2)

=

1003817100381710038171003817100381712059110158401015840(V+2)100381710038171003817100381710038172

100381710038171003817100381712059110158401015840(V+1)10038171003817100381710038172

119861101584010158401015840(V+2)

=

10038171003817100381710038171003817120591101584010158401015840(V+2)100381710038171003817100381710038172

1003817100381710038171003817120591101584010158401015840(V+1)10038171003817100381710038172

(34)

where 120591(V+1) 120591

1015840(V+1) 12059110158401015840(V+1) and 120591

101584010158401015840(V+1) are calledthe error coefficients for 119910 119910

1015840 11991010158401015840 and 119910

101584010158401015840 of theembedded RKFD pairs respectively

(3) We defined the local error estimation at the point 119905119899+1by the following formula

EST = max 1003817100381710038171003817120573119899+1

1003817100381710038171003817infin 100381710038171003817100381710038171205731015840

119899+1

10038171003817100381710038171003817infin1003817100381710038171003817100381712057310158401015840

119899+1

10038171003817100381710038171003817infin10038171003817100381710038171003817120573101584010158401015840

119899+1

10038171003817100381710038171003817infin (35)

where120573119899+1 = 119910119899+1 minus 119910119899+1

1205731015840

119899+1 = 1199101015840

119899+1 minus 1199101015840

119899+1

12057310158401015840

119899+1 = 11991010158401015840

119899+1 minus 11991010158401015840

119899+1

120573101584010158401015840

119899+1 = 119910101584010158401015840

119899+1 minus 119910101584010158401015840

119899+1

(36)

where 119910 1199101015840 11991010158401015840 and 119910

101584010158401015840 and 119910 1199101015840 11991010158401015840 and 119910

101584010158401015840

are solutions using the higher order formula and thelower order formula respectively

The local error estimation EST can be used to control thestep size ℎ by the standard formula as given in [35ndash38]

ℎ119899+1 = 09ℎ119899 (TOLEST

)

1(V+1) (37)

where 09 is a safety factor the local error estimation ateach step is represented by EST and TOL is the maximumallowable local error which is the precision required

Mathematical Problems in Engineering 5

If EST le TOL then the step is accepted and we appliedthe procedure of performing local extrapolation (or higherorder mode) meaning that the more accurate approximationwill be used to advance the integration If EST gt TOL thenthe step is rejected and the step size ℎ will be updated usingformula (37)

3 Construction of Embedded ExplicitRKFD Pairs

The construction of embedded explicit RKFD pairs will bediscussed in this section In specific we will derive twoembedded RKFD pairs of orders 5(4) and 6(5) with three andfour stages per step respectively

31The Derivation of Embedded RKFD5(4) Pair This sectionwill focus on the derivation of embedded RKFD5(4) pair withthree stages The authors in [24] derived three-stage fifth-order RKFD method and the solution is given as follows

1198882 =3

5+

radic6

10

1198883 =3

5minus

radic6

10

119887101584010158401015840

1 =1

9

119887101584010158401015840

2 =4

9minus

radic6

36

119887101584010158401015840

3 =4

9+

radic6

36

11988710158401015840

1 =1

9

11988710158401015840

2 =7

36minus

radic6

18

11988710158401015840

3 =7

36+

radic6

18

1198871015840

1 =1

18

1198871015840

2 =1

18minus

radic6

48

1198871015840

3 =1

18+

radic6

48

1198871 =19

1080

1198872 =13

1080minus

11radic6

2160

1198873 =13

1080+

11radic6

2160

11988621 =4059

187793

120591998400(5) 2

0

0005

001

0015

002

0025

003

005 01 015 02 025 030b998400

3

Figure 1 The graph of 1205911015840(5)2 versus 10158403

11988631 = minus1502

532215

11988632 =1826

569317

(38)

Now based on the above solution for values of 119860 and 119888 wederive a three-stage fourth-order embedded formula Solvingthe algebraic conditions (6) (10)-(11) (16)ndash(18) and (23)ndash(26)simultaneously gives a solution for 119894 in terms of 1 and 3 andthe solution of 1015840119894 in terms of 10158403 which are given as follows

2 = minus1 minus 3 +1

24

1015840

1 =1

12+

1

72radic6 + (

2

5minus

2

5radic6)

1015840

3

1015840

2 =1

12minus

1

72radic6 minus (

7

5minus

2

5radic6)

1015840

3

(39)

while the values of 10158401015840119894 and 101584010158401015840119894 119894 = 1 2 3 are the same as the

fifth-order method Our objective now is to select the valuesof the free parameters 1 3 and

10158403 and thus the values of119862

(6)119861(6) 1198621015840(6) 1198611015840(6) 120591(5)2 and 120591

1015840(5)2 are as small as possible

Using the above solution we get 1198621015840(6)

= 1200 is a constantvalue however 119861

1015840(6) and 1205911015840(5)

2 are functions in terms of10158403 by plotting the graph of 120591

1015840(5)2 versus

10158403 (see Figure 1)

From the numerical experiment and for the accurate resultwe choose

10158403 = 110 using Minimize command in Maple

Software which gives the error norm 1205911015840(5)

2 = 114566368times

10minus3 and 119861

1015840(6)= 04726195824

The quantities119862(6)119861(6) and 120591(5)

2 are functions in termsof 1 3 Choose 1 = 1100 and plot the graph of 119862

(6)119861(6) and 120591

(5)2 against 3 in the interval [0 008] From

numerical experiment and for the optimized pair therefore3 = 3100 is chosen which gives 119861

(6)= 05887532657

119862(6)

= 05887532657 and 120591(5)

2 = 3726445732 times 10minus3

6 Mathematical Problems in Engineering

Consequently we denote this pair as RKFD5(4) method andit can be written in Butcher tableau as follows

3

5+

radic6

10

4059

187793

3

5minus

radic6

10minus

1502

532215

1826

569317

19

1080

13

1080minus

11radic6

2160

13

1080+

11radic6

2160

1

18

1

18minus

radic6

48

1

18+

radic6

48

1

9

7

36minus

radic6

18

7

36+

radic6

18

1

9

4

9minus

radic6

36

4

9+

radic6

36

1

100

1

600

3

100

37

300minus

47radic6

1800minus

17

300+

47radic6

1800

1

10

1

9

7

36minus

radic6

18

7

36+

radic6

18

1

9

4

9minus

radic6

36

4

9+

radic6

36

(40)

32 The Derivation of Embedded RKFD6(5) Pair In this sec-tion a four-stage embedded RKFD6(5) pair will be derivedFor the sixth-order (119903 = 6) method the order conditions ofRKFD method up to order six need to be solved to derivea four-stage sixth-order RKFD method namely RKFD6method To determine the parameters of RKFD6method wechoose (10)ndash(13) from order conditions for 119910

1015840 (16)ndash(20) fromorder conditions for 119910

10158401015840 and (23)ndash(27) and (29) from orderconditions for 119910

101584010158401015840 It involved 15 nonlinear equations with 15unknowns and then was solved simultaneously which resultsin a unique solution as follows

1198871015840

1 =1

24

1198871015840

2 =1

16minus

radic5

48

1198871015840

3 =1

16+

radic5

48

1198871015840

4 = 0

11988710158401015840

1 =1

12

11988710158401015840

2 =5

24minus

radic5

24

11988710158401015840

3 =5

24+

radic5

24

11988710158401015840

4 = 0

119887101584010158401015840

1 =1

12

119887101584010158401015840

2 =5

12

119887101584010158401015840

3 =5

12

119887101584010158401015840

4 =1

12

1198882 =1

2+

radic5

10

1198883 =1

2minus

radic5

10

1198884 = 1

(41)

Substituting the values of the above 1198882 1198883 and 1198884 into the orderconditions for 119910 we get

1198872 =1

72minus

7radic5

360+ radic51198871

1198873 =1

72+

7radic5

360minus radic51198871

1198874 = minus1198871 +1

72

(42)

Here we have one free parameter 1198871 which can be chosen byminimizing the error norm of the seventh order conditionsfor 119910 according to Dormand et al [34] The error norms andthe global error of the seventh order conditions are definedas follows

10038171003817100381710038171003817120591(7)100381710038171003817100381710038172

= radic

119899119901+1

sum

119894=1

(120591(7)

119894)2

100381710038171003817100381710038171205911015840(7)100381710038171003817100381710038172

= radic

1198991015840119901+1

sum

119894=1

(1205911015840(7)

119894)2

1003817100381710038171003817100381712059110158401015840(7)100381710038171003817100381710038172

= radic

11989910158401015840119901+1

sum

119894=1

(12059110158401015840(7)

119894)2

10038171003817100381710038171003817120591101584010158401015840(7)100381710038171003817100381710038172

= radic

119899101584010158401015840119901+1

sum

119894=1

(120591101584010158401015840(7)

119894)2

10038171003817100381710038171003817120591(7)

119892

100381710038171003817100381710038172

= radic

119899119901+1

sum

119894=1

(120591(7)

119894)2+

1198991015840119901+1

sum

119894=1

(1205911015840(7)

119894)2+

11989910158401015840119901+1

sum

119894=1

(12059110158401015840(7)

119894)2+

119899101584010158401015840119901+1

sum

119894=1

(120591101584010158401015840(7)

119894)2

(43)

Mathematical Problems in Engineering 7

where 120591(7) 1205911015840(7) 120591

10158401015840(7) and 120591101584010158401015840(7) are the local truncation

errors norms for 119910 1199101015840 11991010158401015840 and 119910101584010158401015840 of the RKFD6 method

respectively and 120591(7)119892 is the global error As a result we find

the error equations of 119910 as

10038171003817100381710038171003817120591(7)100381710038171003817100381710038172

=1

6300

radic(minus17 + 12601198871)2 (44)

By minimizing 120591(7)

2 with respect to the free parameter 1198871we obtain 1198871 = 171260 that is the optimal value and gives120591(7)

2 = 0 and

1198872 =1

72minus

radic5

168

1198873 =1

72+

radic5

168

1198874 =1

2520

(45)

To find the coefficients 11988621 11988631 11988632 11988641 11988642 and 11988643substituting the values of 1198871 1198872 1198873 1198874 119887

10158401 11988710158402 11988710158403 11988710158404 119887101584010158401 119887101584010158402

119887101584010158403 119887101584010158404 1198871015840101584010158401 1198871015840101584010158402 1198871015840101584010158403 and 119887

1015840101584010158404 into (28) (30) and (31) of the

order conditions of 119910101584010158401015840 and (15) of the order conditions of

1199101015840 the following system of nonlinear equations needs to be

solved

1

1211988632 +

1

24(11988642 + 11988643) +

radic5

120(11988642 minus 11988643) =

1

840 (46)

(1

8+

radic5

24) 11988632 +

1

40(11988642 + 11988643)

+radic5

120(11988642 minus 11988643) =

1

2520

(47)

(1

16minus

radic5

48) 11988621 + (

1

16+

radic5

48) (11988631 + 11988632) =

1

5040 (48)

5

12(11988621 + 11988631 + 11988632) +

1

12(11988641 + 11988642 + 11988643) =

1

120 (49)

(5

24+

5radic5

120) 11988632 + (

1

24+

radic5

120) 11988642

+ (1

24minus

radic5

120) 11988643 =

1

720

(50)

(5

24+

5radic5

120) 11988621 + (

5

24minus

5radic5

120) (11988631 + 11988632)

+1

12(11988641 + 11988642 + 11988643) =

1

144

(51)

Solving (46)ndash(51) simultaneously we get

11988621 =1

168+

11radic5

4200

11988631 =1

420minus

radic5

700

11988632 =1

280minus

radic5

840

11988641 = minus1

840minus

radic5

168

11988642 =1

84minus

radic5

140

11988643 =5

168minus

11radic5

840

(52)

Consequently the local truncation error norms and theglobal error of the seventh order conditions of RKFD6method are calculated and given as follows

10038171003817100381710038171003817120591(7)100381710038171003817100381710038172

= 0

100381710038171003817100381710038171205911015840(7)100381710038171003817100381710038172

= 2380952381 times 10minus4

1003817100381710038171003817100381712059110158401015840(7)100381710038171003817100381710038172

= 4761904762 times 10minus4

10038171003817100381710038171003817120591101584010158401015840(7)100381710038171003817100381710038172

= 4761904762 times 10minus4

10038171003817100381710038171003817120591(7)

119892

100381710038171003817100381710038172= 7142857143 times 10

minus4

(53)

Finally all the parameters of sixth-order RKFDmethod withfour-stage and denoted as RKFD6 are written in Butchertableau as follows

1

2+

radic5

10

1

168+

11radic5

4200

1

2minus

radic5

10

1

420minus

radic5

700

1

280minus

radic5

840

1 minus1

840minus

radic5

168

1

84minus

radic5

140

5

168+

11radic5

840

17

1260

1

72minus

radic5

168

1

72+

radic5

168

1

2520

1

24

1

16minus

radic5

48

1

16+

radic5

480

1

12

5

24minus

radic5

24

5

24+

radic5

240

1

12

5

12

5

12

1

12

(54)

Now depending on the above values of 119860 and 119888 of RKFD6method together with the equations up to order five that is(6)-(7) (10)ndash(12) (16)ndash(19) and (23)ndash(28) solving the system

8 Mathematical Problems in Engineering

simultaneously produces a solution of an embedded RKFDformula of order five which is given by

1 =1

30minus (

1

2minus

radic5

10) 2 minus (

1

2+

radic5

10) 3

4 =1

120minus (

1

2+

radic5

10) 2 minus (

1

2minus

radic5

10) 3

1015840

1 =1

16+

radic5

80minus

radic5

51015840

3

1015840

2 =1

8minus 1015840

3

1015840

4 = minus1

48minus

radic5

80+

radic5

51015840

3

(55)

while the values of 10158401015840119894 and

101584010158401015840119894 119894 = 1 2 3 4 are the same

as the RKFD6 method Now our goal is to determine the

values of the free parameters 2 3 and 10158403 such that the

values of 119862(7) 119861(7) 1198621015840(7) 119861

1015840(7) 1205911015840(6)

2 and 120591(6)

2 are assmall as possible The quantities 119862

(7) 119861(7) and 120591

(6)2 are

functions depending on 2 and 3 Picking 2 = 140 andplotting the graph of119862(7)119861(7) and 120591

(6)2 against 2 and from

the numerical experiment therefore we choose 3 = 120

that is the optimal value which gives 119862(7)

= 1507670351119861(7)

= 1507670351 and the local truncation error 120591(6)

2 =

9444444448times10minus3The quantities1198621015840(7) 1198611015840(7) and 120591

1015840(6)2 are

functions in terms of 10158403 Plotting the graph of these functionsand for the optimized pair we found that

10158403 = 110 is the

optimal value using Minimize command in Maple Softwarewhich yields 119862

1015840(7)= 1683161973 1198611015840(7) = 1708476855 and

1205911015840(6)

2 = 81256471 times 10minus4 (see Figure 2)

Finally all the parameters of the embedded pair arerepresented in the Butcher tableau and denoted as RKFD6(5)method as follows

1

2+

radic5

10

1

168+

11radic5

4200

1

2minus

radic5

10

1

420minus

radic5

700

1

280minus

radic5

840

1 minus1

840minus

radic5

168

1

84minus

radic5

140

5

168+

11radic5

840

17

1260

1

72minus

radic5

168

1

72+

radic5

168

1

2520

1

24

1

16minus

radic5

48

1

16+

radic5

480

1

12

5

24minus

radic5

24

5

24+

radic5

240

1

12

5

12

5

12

1

12

minus1

240minus

radic5

400

1

40

1

20minus

7

240+

radic5

400

1

16minus

3radic5

400

1

40

1

10minus

1

48+

3radic5

400

1

12

5

24minus

radic5

24

5

24+

radic5

240

1

12

5

12

5

12

1

12

(56)

4 Numerical Experiments

In this section we present some problems which involvespecial fourth-order ODEs of the form 119910

(119894V)(119909) = 119891(119909 119910(119909))

in order to test the performance of the two new embeddedpairs derived in Section 3 The numerical results obtainedare compared with the well-known embedded Runge-Kuttamethods which are chosen from the scientific literature after

Mathematical Problems in Engineering 9120591

998400(6) 2

005 01 015 02 025 030b998400

3

0

0002

0004

0006

0008

001

0012

0014

0016

Figure 2 The graph of 1205911015840(6)2 versus 10158403

transforming the same problems to a system of first-orderdifferential equations and solving themThemethods chosenin the numerical experiments are as follows

(i) RKFD6(5) the new embedded Runge-Kutta pair oforders 6(5) derived in Section 3 in this paper

(ii) RKFD5(4) the new embedded Runge-Kutta pair oforders 5(4) derived in Section 3 in this paper

(iii) RK6(5)V the embedded Runge-Kutta pair of orders6(5) derived by Verner as given in [39]

(iv) RK5(4)D the embedded Runge-Kutta pair of orders5(4) with FSAL property derived by Dormand andPrince [28]

Problem 1 We consider that the homogeneous linear prob-lem is as follows

119910(119894V)

= minus4119910

119910 (0) = 0 1199101015840(0) = 1 119910

10158401015840(0) = 2 119910

101584010158401015840(0) = 2

(57)

The exact solution is given by 119910(119909) = e119909 sin(119909) The problemis integrated in the interval [0 5]

Problem 2 We consider that the homogeneous nonlinearproblem is as follows

119910(119894V)

= minus15

161199107

119910 (0) = 1 1199101015840(0) =

1

2 11991010158401015840

(0) = minus1

4 119910101584010158401015840

(0) =3

8

(58)

The exact solution is given by 119910(119909) = radic1 + 119909 The problem isintegrated in the interval [0 10]

Problem 3 We consider that the inhomogeneous nonlinearproblem is as follows

119910(119894V)

= 1199102+ cos2 (119909) + sin (119909) minus 1

119910 (0) = 0 1199101015840(0) = 1 119910

10158401015840(0) = 0 119910

101584010158401015840(0) = minus1

(59)

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

35325215

log10(M

AXE

RR)

log10(function evaluations)

minus10

minus8

minus6

minus4

minus2

Figure 3 The efficiency curves for Problem 1

The exact solution is given by 119910(119909) = sin(119909) The problem isintegrated in the interval [0 8]

Problem 4 We consider that the linear system is as follows

119910(119894V)

= e3119909119906

119910 (0) = 1 1199101015840(0) = minus1 119910

10158401015840(0) = 1 119910

101584010158401015840(0) = minus1

119911(119894V)

= 16eminus119909119910

119911 (0) = 1 1199111015840(0) = minus2 119911

10158401015840(0) = 4 119911

101584010158401015840(0) = minus8

119908(119894V)

= 81eminus119909119911

119908 (0) = 1 1199081015840(0) = minus3 119908

10158401015840(0) = 9 119908

101584010158401015840(0) = minus27

119906(119894V)

= 256eminus119909119908

119906 (0) = 1 1199061015840(0) = minus4 119906

10158401015840(0) = 16 119906

101584010158401015840(0) = minus64

(60)

The exact solution is given by

119910 = eminus119909

119911 = eminus2119909

119908 = eminus3119909

119906 = eminus4119909

(61)

The problem is integrated in the interval [0 3]

In the numerical experiments we have computed foreach method and problem the maximum global error andthe number of function evaluations used in the integrationIn Figures 3ndash6 we plot these values in double logarithmic

10 Mathematical Problems in Engineering

18 2 22 24 26 28 314 16

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

log10(M

AXE

RR)

log10(function evaluations)

Figure 4 The efficiency curves for Problem 2

325215

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

log10(function evaluations)

log10(M

AXE

RR)

minus10

minus8

minus6

minus4

minus2

0

Figure 5 The efficiency curves for Problem 3

325215

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

log10(function evaluations)

log10(M

AXE

RR)

minus10

minus8

minus6

minus4

minus2

0

Figure 6 The efficiency curves for Problem 4

Table 1 The comparison results of the total number of successfulsteps and failure steps betweenRKFD6(5) RKFD5(4) RK6(5)V andRK5(4)D methods for all problems

Problems Methods Successful steps Failure steps

1

RKFD6(5) 440 4RKFD5(4) 705 0RK6(5)V 461 9RK5(4)D 768 0

2

RKFD6(5) 151 0RKFD5(4) 200 0RK6(5)V 208 0RK5(4)D 253 0

3

RKFD6(5) 330 4RKFD5(4) 466 0RK6(5)V 348 8RK5(4)D 497 0

4

RKFD6(5) 264 5RKFD5(4) 340 6RK6(5)V 339 5RK5(4)D 484 5

scale From Table 1 and Figures 3ndash6 we observed that thenew RKFD6(5) and RKFD5(4) methods are more efficient tosolve directly fourth-order differential equations comparedwith RK6(5)V and RK5(4)D methods in terms of functionevaluations and the total number of steps This is due tothe fact that when using RK6(5)V and RK5(4)D methodsthe fourth-order ODEs need to be transformed into a first-order system and hence the dimension will increase fourtimes

5 Conclusion

We have constructed the two pairs of embedded RKFDmethods for directly solving special fourth-order ODEs ofthe form 119910

(119894V)(119909) = 119891(119909 119910(119909)) using variable step size

codes in this paperThese methods are denoted as RKFD5(4)and RKFD6(5) respectively Based on the methods thevariables step size codes are developed and used to solvespecial fourth-order ODEs Numerical results show thatthe two new embedded RKFD pairs are more effective ascompared with the existing embedded Runge-Kutta pairs ofthe same order in the literature From the numerical resultswe conclude that the two new embedded RKFD methodsare computationally more efficient in solving special fourth-order ODEs and outperformed the well-known embeddedRunge-Kutta methods

Conflict of Interests

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

Mathematical Problems in Engineering 11

References

[1] S N Jator ldquoNumerical integrators for fourth order initial andboundary value problemsrdquo International Journal of Pure andApplied Mathematics vol 47 no 4 pp 563ndash576 2008

[2] Z Bai and H Wang ldquoOn positive solutions of some nonlinearfourth-order beam equationsrdquo Journal ofMathematical Analysisand Applications vol 270 no 2 pp 357ndash368 2002

[3] A K Alomari N Ratib Anakira A S Bataineh and I HashimldquoApproximate solution of nonlinear system of BVP arising influid flow problemrdquoMathematical Problems in Engineering vol2013 Article ID 136043 7 pages 2013

[4] A Malek and R Shekari Beidokhti ldquoNumerical solution forhigh order differential equations using a hybrid neuralnetworkmdashoptimization methodrdquo Applied Mathematics andComputation vol 183 no 1 pp 260ndash271 2006

[5] A Boutayeb and A Chetouani ldquoA mini-review of numericalmethods for high-order problemsrdquo International Journal ofComputer Mathematics vol 84 no 4 pp 563ndash579 2007

[6] P Onumanyi U W Sirisena and S N Jator ldquoContinuous finitedifference approximations for solving differential equationsrdquoInternational Journal of Computer Mathematics vol 72 no 1pp 15ndash27 1999

[7] D Sarafyan ldquoNew algorithms for the continuous approximatesolutions of ordinary differential equations and the upgradingof the order of the processesrdquo Computers amp Mathematics withApplications vol 20 no 1 pp 77ndash100 1990

[8] GDahlquist ldquoOn accuracy and unconditional stability of linearmultistep methods for second order differential equationsrdquo BITNumerical Mathematics vol 18 no 2 pp 133ndash136 1978

[9] R P K Chan P Leone and A Tsai ldquoOrder conditions andsymmetry for two-step hybrid methodsrdquo International Journalof Computer Mathematics vol 81 no 12 pp 1519ndash1536 2004

[10] C E Abhulimen and F O Otunta ldquoA sixth order multideriva-tivemultistepmethods for stiff system of differential equationsrdquoInternational Journal of Numerical Mathematics vol 1 no 2 pp248ndash268 2006

[11] J D Lambert Computational Methods in Ordinary DifferentialEquations Wiley London UK 1973

[12] D O Awoyemi ldquoAlgorithmic collocation approach for directsolution of fourth-order initial-value problems of ordinarydifferential equationsrdquo International Journal of ComputerMath-ematics vol 82 no 3 pp 321ndash329 2005

[13] NWaeleh Z A Majid F Ismail andM Suleiman ldquoNumericalsolution of higher order ordinary differential equations bydirect block coderdquo Journal of Mathematics and Statistics vol 8no 1 pp 77ndash81 2011

[14] S J Kayode ldquoAn efficient zero-stable numerical method forfourth-order differential equationsrdquo International Journal ofMathematics and Mathematical Sciences vol 2008 Article ID364021 10 pages 2008

[15] S J Kayode ldquoAnorder six zero-stablemethod for direct solutionof fourth order ordinary differential equationsrdquo AmericanJournal of Applied Sciences vol 5 no 11 pp 1461ndash1466 2008

[16] B T Olabode and T J Alabi ldquoDirect block predictor-correctormethod for the solution of general fourth order ODEsrdquo Journalof Mathematics Research vol 5 no 1 pp 26ndash33 2013

[17] L K Yap F Ismail and N Senu ldquoAn accurate block hybrid col-locationmethod for third order ordinary differential equationsrdquoJournal of Applied Mathematics vol 2014 Article ID 549597 9pages 2014

[18] S Mehrkanoon ldquoA direct variable step block multistep methodfor solving general third-order ODEsrdquo Numerical Algorithmsvol 57 no 1 pp 53ndash66 2011

[19] Z A Majid and M B Suleiman ldquoDirect integration implicitvariable steps method for solving higher order systems ofordinary differential equations directlyrdquo Sains Malaysiana vol35 no 2 pp 63ndash68 2006

[20] D O Awoyemi ldquoA P-stable linear multistep method for solvinggeneral third order ordinary differential equationsrdquo Interna-tional Journal of Computer Mathematics vol 80 no 8 pp 985ndash991 2003

[21] N Waeleh Z A Majid and F Ismail ldquoA new algorithm forsolving higher order IVPs of ODEsrdquo Applied MathematicalSciences vol 5 no 56 pp 2795ndash2805 2011

[22] D O Awoyemi and O M Idowu ldquoA class of hybrid collocationmethods for third-order ordinary differential equationsrdquo Inter-national Journal of Computer Mathematics vol 82 no 10 pp1287ndash1293 2005

[23] S N Jator ldquoSolving second order initial value problems by ahybrid multistep method without predictorsrdquo Applied Mathe-matics and Computation vol 217 no 8 pp 4036ndash4046 2010

[24] K Hussain F Ismail and N Senu ldquoRunge-Kutta type methodsfor directly solving special fourth-order ordinary differentialequationsrdquo Mathematical Problems in Engineering vol 2015Article ID 893763 11 pages 2015

[25] J R DormandNumerical Methods for Differential Equations AComputational Approach Library of Engineering MathematicsCRC Press Boca Raton Fla USA 1996

[26] W Gander and D Gruntz ldquoDerivation of numerical methodsusing computer algebrardquo SIAM Review vol 41 no 3 pp 577ndash593 1999

[27] F B Ismail and M B Suleiman ldquoEmbedded singly diagonallyimplicit Runge-Kutta methods (45) in (56) For the integrationof stiff systems of ODEsrdquo International Journal of ComputerMathematics vol 66 no 3-4 pp 325ndash341 1998

[28] J R Dormand and P J Prince ldquoA family of embeddedRunge-Kutta formulaerdquo Journal of Computational and AppliedMathematics vol 6 no 1 pp 19ndash26 1980

[29] P J Prince and J R Dormand ldquoHigh order embedded Runge-Kutta formulaerdquo Journal of Computational and Applied Mathe-matics vol 7 no 1 pp 67ndash75 1981

[30] R A Al-Khasawneh F Ismail and M Suleiman ldquoEmbeddeddiagonally implicit Runge-Kutta-Nystrom 4(3) pair for solvingspecial second-order IVPsrdquo Applied Mathematics and Compu-tation vol 190 no 2 pp 1803ndash1814 2007

[31] E Fehlberg ldquoLow-order classical Runge-Kutta formulas withstepsize control and their application to some heat transferproblemsrdquo NASA TR R-315 1969

[32] E Fehlberg ldquoClassical fifth sixth seventh and eighth orderRunge-Kutta formulas with stepsize controlrdquo NASA TR R-287NASA 1968

[33] M Calvo J I Montijano and L Randez ldquoA new embeddedpair of Runge-Kutta formulas of orders 5 and 6rdquo Computers ampMathematics with Applications vol 20 no 1 pp 15ndash24 1990

[34] J R Dormand M E A El-Mikkawy and P J Prince ldquoFamiliesof Runge-Kutta-Nystrom formulaerdquo IMA Journal of NumericalAnalysis vol 7 no 2 pp 235ndash250 1987

[35] L F Shampine Numerical Solution of Ordinary DifferentialEquations Chapman amp Hall New York NY USA 1994

[36] J C Butcher Numerical Methods for Ordinary DifferentialEquations John Wiley amp Sons Chichester UK 2008

12 Mathematical Problems in Engineering

[37] E Hairer S P Noslashrsett and G Wanner Solving Ordinary Dif-ferential Equations I Nonstiff Problems vol 8 Springer BerlinGermany 2nd edition 1993

[38] N Senu M Mechee F Ismail and Z Siri ldquoEmbedded explicitRungendashKutta type methods for directly solving special thirdorder differential equations 119910

101584010158401015840= 119891(119909 119910)rdquo Applied Mathemat-

ics and Computation vol 240 pp 281ndash293 2014[39] J H Verner ldquoExplicit Runge-Kutta pairs with lower stage-

orderrdquo Numerical Algorithms vol 65 no 3 pp 555ndash577 2014

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

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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

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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

2 Mathematical Problems in Engineering

The general form of RKFD method with 119904-stage forsolving special fourth-order ODEs (1) can be expressed asfollows [24]

119910119899+1 = 119910119899 + ℎ1199101015840

119899 +ℎ2

211991010158401015840

119899 +ℎ3

6119910101584010158401015840

119899 + ℎ4119904

sum

119894=1

119887119894119896119894

1199101015840

119899+1 = 1199101015840

119899 + ℎ11991010158401015840

119899 +ℎ2

2119910101584010158401015840

119899 + ℎ3119904

sum

119894=1

1198871015840

119894 119896119894

11991010158401015840

119899+1 = 11991010158401015840

119899 + ℎ119910101584010158401015840

119899 + ℎ2119904

sum

119894=1

11988710158401015840

119894 119896119894

119910101584010158401015840

119899+1 = 119910101584010158401015840

119899 + ℎ

119904

sum

119894=1

119887101584010158401015840

119894 119896119894

(3)

where

1198961 = 119891 (119909119899 119910119899)

119896119894 = 119891(119909119899 + 119888119894ℎ 119910119899 + ℎ1198881198941199101015840

119899 +ℎ2

21198882

119894 11991010158401015840

119899 +ℎ3

61198883

119894 119910101584010158401015840

119899

+ ℎ4119904

sum

119895=1

119886119894119895119896119895)

(4)

for 119894 = 2 3 119904The parameters 119887119894 119887

1015840119894 11988710158401015840119894 119887101584010158401015840119894 119886119894119895 and 119888119894 of the RKFD

method are to be determined for 119894 = 1 2 119904 and 119895 =

1 2 119904 and supposed to be real The RKFD method is anexplicit method if 119886119894119895 = 0 for 119894 le 119895 and is an implicit methodif 119886119894119895 = 0 for some 119894 such that 119894 le 119895

To determine the parameters of the RKFD method givenin (3)-(4) the RKFD method expression is expanded usingthe Taylor series expansion After doing some algebraic sim-plifications this expansion is equated to the true solution thatis given by the Taylor series expansion The direct expansionof the truncation error is used to derive the order conditionsfor the RKFD method [25] A good deal of algebraic andnumerical calculations are required for the above operationwhich were carried out using algebra package Maple [26]Algebraic order conditions for the RKFD method can beobtained from the direct expansion of the local truncationerror

In this paper we will derive embedded Runge-Kuttapairs for direct integration of special fourth-order ODEsEmbedded pairs of RK type methods have a built-in localtruncation error estimate as a result the step size can becontrolled at virtually no extra cost and hence an efficientvariable step size code can be developed

In recent years the construction of embedded Runge-Kuttamethod is an effective research area yielding continuousdevelopment to the existing codes The present paper isprimarily dedicated as an extra work in this research areaThis technique involves two Runge-Kutta formulae of orders119903 and V (119903 gt V usually 119903 = V + 1) (see [25 27ndash33])We are interested in deriving the effective embedded 119903(V)

pairs of RKFDmethods that provide a cheap error estimationfor variable step size codes They depend on the methods(119888 119860 119887 119887

1015840 11988710158401015840 119887101584010158401015840

) of order 119903 and (119888 119860 1015840 10158401015840 101584010158401015840

) of orderV Butcher tableau of embedded RKFD pair can be written asfollows

119888 119860

119887119879

1198871015840119879

11988710158401015840119879

119887101584010158401015840119879

119879

1015840119879

10158401015840119879

101584010158401015840119879

(5)

Themethodwill compute119910119899+11199101015840119899+111991010158401015840119899+1 and119910

101584010158401015840119899+1 to approx-

imate 119910(119909119899+1) 1199101015840(119909119899+1) 119910

10158401015840(119909119899+1) and 119910

101584010158401015840(119909119899+1) where 119910119899+1

is the computed solution and 119910(119909119899+1) is the exact solutionThe remainder of this paper is organized as follows

In Section 2 we present the order conditions of RKFDmethod as well as the basic concepts and notations whichare used for embedded method In Section 3 we present theconstruction of the new embedded RKFD pairs of orders5(4) and 6(5) respectively In Section 4 we carry out thenumerical experiments to show the efficiency of the newembedded RKFD pairs when compared with the well-knownRunge-Kutta pairs from the scientific literature Conclusionsof the paper are given in Section 5

2 The Order Conditions of RKFD Method

The order conditions of RKFD method up to fifth order havebeen derived using Taylor series expansion by Hussain et al[24]We use the same approach to derive the order conditionsup to seventh order and for convenience we will present thealgebraic order conditions of the RKFDmethod given in [24]together with the seventh order method in this paper Nextwe give the order conditions up to order 7

The order conditions for 119910 are as follows

Fourth order

119904

sum

119894=1

119887119894 =1

24 (6)

Fifth order

119904

sum

119894=1

119887119894119888119894 =1

120 (7)

Mathematical Problems in Engineering 3

Sixth order

119904

sum

119894=1

1198871198941198882

119894 =1

360 (8)

Seventh order

119904

sum

119894=1

1198871198941198883

119894 =1

840 (9)

The order conditions for 1199101015840 are as follows

Third order

119904

sum

119894=1

1198871015840

119894 =1

6 (10)

Fourth order

119904

sum

119894=1

1198871015840

119894 119888119894 =1

24 (11)

Fifth order

119904

sum

119894=1

1198871015840

119894 1198882

119894 =1

60 (12)

Sixth order

119904

sum

119894=1

1198871015840

119894 1198883

119894 =1

120 (13)

Seventh order

119904

sum

119894=1

1198871015840

119894 1198884

119894 =1

210 (14)

119904

sum

119894119895=1

1198871015840

119894 119886119894119895 =1

5040 (15)

The order conditions for 11991010158401015840 are as follows

Second order

119904

sum

119894=1

11988710158401015840

119894 =1

2 (16)

Third order

119904

sum

119894=1

11988710158401015840

119894 119888119894 =1

6 (17)

Fourth order

119904

sum

119894=1

11988710158401015840

119894 1198882

119894 =1

12 (18)

Fifth order

119904

sum

119894=1

11988710158401015840

119894 1198883

119894 =1

20 (19)

Sixth order

119904

sum

119894=1

11988710158401015840

119894 1198884

119894 =1

30 (20)

119904

sum

119894119895=1

11988710158401015840

119894 119886119894119895 =1

720 (21)

Seventh order

119904

sum

119894=1

11988710158401015840

119894 1198885

119894 =1

42

119904

sum

119894=1

11988710158401015840

119894 119886119894119895119888119895 =1

5040

119904

sum

119894=1

11988710158401015840

119894 119888119894119886119894119895 =1

1008

(22)

The order conditions for 119910101584010158401015840 are as follows

First order

119904

sum

119894=1

119887101584010158401015840

119894 = 1 (23)

Second order

119904

sum

119894=1

119887101584010158401015840

119894 119888119894 =1

2 (24)

Third order

119904

sum

119894=1

119887101584010158401015840

119894 1198882

119894 =1

3 (25)

Fourth order

119904

sum

119894=1

119887101584010158401015840

119894 1198883

119894 =1

4 (26)

4 Mathematical Problems in Engineering

Fifth order119904

sum

119894=1

119887101584010158401015840

119894 1198884

119894 =1

5 (27)

119904

sum

119894119895=1

119887101584010158401015840

119894 119886119894119895 =1

120 (28)

Sixth order

119904

sum

119894=1

119887101584010158401015840

119894 1198885

119894 =1

6 (29)

119904

sum

119894119895=1

119887101584010158401015840

119894 119886119894119895119888119895 =1

720

119904

sum

119894=1

119887101584010158401015840

119894 119888119894119886119894119895 =1

144

(30)

Seventh order

119904

sum

119894=1

119887101584010158401015840

119894 1198886

119894 =1

7

119904

sum

119894=1

119887101584010158401015840

119894 1198882

119895 119886119894119895 =1

168

119904

sum

119894=1

119887101584010158401015840

119894 1198861198941198951198882

119895 =1

2520

119904

sum

119894=1

119887101584010158401015840

119894 119888119894119886119894119895119888119895 =1

840

(31)

The following strategies are utilized for developing efficientembedded pairs

(1) The quantities of 120591(119903+1)

2 and 120591(V+1)

2 should be assmall as possible for higher and lower order RKFDmethod respectively where

10038171003817100381710038171003817120591(119903+1)100381710038171003817100381710038172

= radic

1198991

sum

119894=1

(120591(119903+1)

119894)2+

1198992

sum

119894=1

(1205911015840(119903+1)

119894)2+

1198993

sum

119894=1

(12059110158401015840(119903+1)

119894)2+

1198994

sum

119894=1

(120591101584010158401015840(119903+1)

119894)2

10038171003817100381710038171003817120591(V+1)100381710038171003817100381710038172

= radic

1198991

sum

119894=1

(120591(V+1)119894

)2+

1198992

sum

119894=1

(1205911015840(V+1)119894

)2+

1198993

sum

119894=1

(12059110158401015840(V+1)119894

)2+

1198994

sum

119894=1

(120591101584010158401015840(V+1)119894

)2

(32)

where 120591(119903+1)

119894 1205911015840(119903+1)119894

12059110158401015840(119903+1)119894

and 120591101584010158401015840(119903+1)

119894are called the

error terms for 119910 1199101015840 11991010158401015840 and 119910101584010158401015840 respectively

(2) The following quantities given in [34] should be assmall as possible

(i)

119862(V+2)

=

10038171003817100381710038171003817120591(V+2)

minus 120591(V+2)100381710038171003817100381710038172

1003817100381710038171003817120591(V+1)10038171003817100381710038172

1198621015840(V+2)

=

100381710038171003817100381710038171205911015840(V+2)

minus 1205911015840(V+2)100381710038171003817100381710038172

10038171003817100381710038171205911015840(V+1)10038171003817100381710038172

11986210158401015840(V+2)

=

1003817100381710038171003817100381712059110158401015840(V+2)

minus 12059110158401015840(V+2)100381710038171003817100381710038172

100381710038171003817100381712059110158401015840(V+1)10038171003817100381710038172

119862101584010158401015840(V+2)

=

10038171003817100381710038171003817120591101584010158401015840(V+2)

minus 120591101584010158401015840(V+2)100381710038171003817100381710038172

1003817100381710038171003817120591101584010158401015840(V+1)10038171003817100381710038172

(33)

(ii)

119861(V+2)

=

10038171003817100381710038171003817120591(V+2)100381710038171003817100381710038172

1003817100381710038171003817120591(V+1)10038171003817100381710038172

1198611015840(V+2)

=

100381710038171003817100381710038171205911015840(V+2)100381710038171003817100381710038172

10038171003817100381710038171205911015840(V+1)10038171003817100381710038172

11986110158401015840(V+2)

=

1003817100381710038171003817100381712059110158401015840(V+2)100381710038171003817100381710038172

100381710038171003817100381712059110158401015840(V+1)10038171003817100381710038172

119861101584010158401015840(V+2)

=

10038171003817100381710038171003817120591101584010158401015840(V+2)100381710038171003817100381710038172

1003817100381710038171003817120591101584010158401015840(V+1)10038171003817100381710038172

(34)

where 120591(V+1) 120591

1015840(V+1) 12059110158401015840(V+1) and 120591

101584010158401015840(V+1) are calledthe error coefficients for 119910 119910

1015840 11991010158401015840 and 119910

101584010158401015840 of theembedded RKFD pairs respectively

(3) We defined the local error estimation at the point 119905119899+1by the following formula

EST = max 1003817100381710038171003817120573119899+1

1003817100381710038171003817infin 100381710038171003817100381710038171205731015840

119899+1

10038171003817100381710038171003817infin1003817100381710038171003817100381712057310158401015840

119899+1

10038171003817100381710038171003817infin10038171003817100381710038171003817120573101584010158401015840

119899+1

10038171003817100381710038171003817infin (35)

where120573119899+1 = 119910119899+1 minus 119910119899+1

1205731015840

119899+1 = 1199101015840

119899+1 minus 1199101015840

119899+1

12057310158401015840

119899+1 = 11991010158401015840

119899+1 minus 11991010158401015840

119899+1

120573101584010158401015840

119899+1 = 119910101584010158401015840

119899+1 minus 119910101584010158401015840

119899+1

(36)

where 119910 1199101015840 11991010158401015840 and 119910

101584010158401015840 and 119910 1199101015840 11991010158401015840 and 119910

101584010158401015840

are solutions using the higher order formula and thelower order formula respectively

The local error estimation EST can be used to control thestep size ℎ by the standard formula as given in [35ndash38]

ℎ119899+1 = 09ℎ119899 (TOLEST

)

1(V+1) (37)

where 09 is a safety factor the local error estimation ateach step is represented by EST and TOL is the maximumallowable local error which is the precision required

Mathematical Problems in Engineering 5

If EST le TOL then the step is accepted and we appliedthe procedure of performing local extrapolation (or higherorder mode) meaning that the more accurate approximationwill be used to advance the integration If EST gt TOL thenthe step is rejected and the step size ℎ will be updated usingformula (37)

3 Construction of Embedded ExplicitRKFD Pairs

The construction of embedded explicit RKFD pairs will bediscussed in this section In specific we will derive twoembedded RKFD pairs of orders 5(4) and 6(5) with three andfour stages per step respectively

31The Derivation of Embedded RKFD5(4) Pair This sectionwill focus on the derivation of embedded RKFD5(4) pair withthree stages The authors in [24] derived three-stage fifth-order RKFD method and the solution is given as follows

1198882 =3

5+

radic6

10

1198883 =3

5minus

radic6

10

119887101584010158401015840

1 =1

9

119887101584010158401015840

2 =4

9minus

radic6

36

119887101584010158401015840

3 =4

9+

radic6

36

11988710158401015840

1 =1

9

11988710158401015840

2 =7

36minus

radic6

18

11988710158401015840

3 =7

36+

radic6

18

1198871015840

1 =1

18

1198871015840

2 =1

18minus

radic6

48

1198871015840

3 =1

18+

radic6

48

1198871 =19

1080

1198872 =13

1080minus

11radic6

2160

1198873 =13

1080+

11radic6

2160

11988621 =4059

187793

120591998400(5) 2

0

0005

001

0015

002

0025

003

005 01 015 02 025 030b998400

3

Figure 1 The graph of 1205911015840(5)2 versus 10158403

11988631 = minus1502

532215

11988632 =1826

569317

(38)

Now based on the above solution for values of 119860 and 119888 wederive a three-stage fourth-order embedded formula Solvingthe algebraic conditions (6) (10)-(11) (16)ndash(18) and (23)ndash(26)simultaneously gives a solution for 119894 in terms of 1 and 3 andthe solution of 1015840119894 in terms of 10158403 which are given as follows

2 = minus1 minus 3 +1

24

1015840

1 =1

12+

1

72radic6 + (

2

5minus

2

5radic6)

1015840

3

1015840

2 =1

12minus

1

72radic6 minus (

7

5minus

2

5radic6)

1015840

3

(39)

while the values of 10158401015840119894 and 101584010158401015840119894 119894 = 1 2 3 are the same as the

fifth-order method Our objective now is to select the valuesof the free parameters 1 3 and

10158403 and thus the values of119862

(6)119861(6) 1198621015840(6) 1198611015840(6) 120591(5)2 and 120591

1015840(5)2 are as small as possible

Using the above solution we get 1198621015840(6)

= 1200 is a constantvalue however 119861

1015840(6) and 1205911015840(5)

2 are functions in terms of10158403 by plotting the graph of 120591

1015840(5)2 versus

10158403 (see Figure 1)

From the numerical experiment and for the accurate resultwe choose

10158403 = 110 using Minimize command in Maple

Software which gives the error norm 1205911015840(5)

2 = 114566368times

10minus3 and 119861

1015840(6)= 04726195824

The quantities119862(6)119861(6) and 120591(5)

2 are functions in termsof 1 3 Choose 1 = 1100 and plot the graph of 119862

(6)119861(6) and 120591

(5)2 against 3 in the interval [0 008] From

numerical experiment and for the optimized pair therefore3 = 3100 is chosen which gives 119861

(6)= 05887532657

119862(6)

= 05887532657 and 120591(5)

2 = 3726445732 times 10minus3

6 Mathematical Problems in Engineering

Consequently we denote this pair as RKFD5(4) method andit can be written in Butcher tableau as follows

3

5+

radic6

10

4059

187793

3

5minus

radic6

10minus

1502

532215

1826

569317

19

1080

13

1080minus

11radic6

2160

13

1080+

11radic6

2160

1

18

1

18minus

radic6

48

1

18+

radic6

48

1

9

7

36minus

radic6

18

7

36+

radic6

18

1

9

4

9minus

radic6

36

4

9+

radic6

36

1

100

1

600

3

100

37

300minus

47radic6

1800minus

17

300+

47radic6

1800

1

10

1

9

7

36minus

radic6

18

7

36+

radic6

18

1

9

4

9minus

radic6

36

4

9+

radic6

36

(40)

32 The Derivation of Embedded RKFD6(5) Pair In this sec-tion a four-stage embedded RKFD6(5) pair will be derivedFor the sixth-order (119903 = 6) method the order conditions ofRKFD method up to order six need to be solved to derivea four-stage sixth-order RKFD method namely RKFD6method To determine the parameters of RKFD6method wechoose (10)ndash(13) from order conditions for 119910

1015840 (16)ndash(20) fromorder conditions for 119910

10158401015840 and (23)ndash(27) and (29) from orderconditions for 119910

101584010158401015840 It involved 15 nonlinear equations with 15unknowns and then was solved simultaneously which resultsin a unique solution as follows

1198871015840

1 =1

24

1198871015840

2 =1

16minus

radic5

48

1198871015840

3 =1

16+

radic5

48

1198871015840

4 = 0

11988710158401015840

1 =1

12

11988710158401015840

2 =5

24minus

radic5

24

11988710158401015840

3 =5

24+

radic5

24

11988710158401015840

4 = 0

119887101584010158401015840

1 =1

12

119887101584010158401015840

2 =5

12

119887101584010158401015840

3 =5

12

119887101584010158401015840

4 =1

12

1198882 =1

2+

radic5

10

1198883 =1

2minus

radic5

10

1198884 = 1

(41)

Substituting the values of the above 1198882 1198883 and 1198884 into the orderconditions for 119910 we get

1198872 =1

72minus

7radic5

360+ radic51198871

1198873 =1

72+

7radic5

360minus radic51198871

1198874 = minus1198871 +1

72

(42)

Here we have one free parameter 1198871 which can be chosen byminimizing the error norm of the seventh order conditionsfor 119910 according to Dormand et al [34] The error norms andthe global error of the seventh order conditions are definedas follows

10038171003817100381710038171003817120591(7)100381710038171003817100381710038172

= radic

119899119901+1

sum

119894=1

(120591(7)

119894)2

100381710038171003817100381710038171205911015840(7)100381710038171003817100381710038172

= radic

1198991015840119901+1

sum

119894=1

(1205911015840(7)

119894)2

1003817100381710038171003817100381712059110158401015840(7)100381710038171003817100381710038172

= radic

11989910158401015840119901+1

sum

119894=1

(12059110158401015840(7)

119894)2

10038171003817100381710038171003817120591101584010158401015840(7)100381710038171003817100381710038172

= radic

119899101584010158401015840119901+1

sum

119894=1

(120591101584010158401015840(7)

119894)2

10038171003817100381710038171003817120591(7)

119892

100381710038171003817100381710038172

= radic

119899119901+1

sum

119894=1

(120591(7)

119894)2+

1198991015840119901+1

sum

119894=1

(1205911015840(7)

119894)2+

11989910158401015840119901+1

sum

119894=1

(12059110158401015840(7)

119894)2+

119899101584010158401015840119901+1

sum

119894=1

(120591101584010158401015840(7)

119894)2

(43)

Mathematical Problems in Engineering 7

where 120591(7) 1205911015840(7) 120591

10158401015840(7) and 120591101584010158401015840(7) are the local truncation

errors norms for 119910 1199101015840 11991010158401015840 and 119910101584010158401015840 of the RKFD6 method

respectively and 120591(7)119892 is the global error As a result we find

the error equations of 119910 as

10038171003817100381710038171003817120591(7)100381710038171003817100381710038172

=1

6300

radic(minus17 + 12601198871)2 (44)

By minimizing 120591(7)

2 with respect to the free parameter 1198871we obtain 1198871 = 171260 that is the optimal value and gives120591(7)

2 = 0 and

1198872 =1

72minus

radic5

168

1198873 =1

72+

radic5

168

1198874 =1

2520

(45)

To find the coefficients 11988621 11988631 11988632 11988641 11988642 and 11988643substituting the values of 1198871 1198872 1198873 1198874 119887

10158401 11988710158402 11988710158403 11988710158404 119887101584010158401 119887101584010158402

119887101584010158403 119887101584010158404 1198871015840101584010158401 1198871015840101584010158402 1198871015840101584010158403 and 119887

1015840101584010158404 into (28) (30) and (31) of the

order conditions of 119910101584010158401015840 and (15) of the order conditions of

1199101015840 the following system of nonlinear equations needs to be

solved

1

1211988632 +

1

24(11988642 + 11988643) +

radic5

120(11988642 minus 11988643) =

1

840 (46)

(1

8+

radic5

24) 11988632 +

1

40(11988642 + 11988643)

+radic5

120(11988642 minus 11988643) =

1

2520

(47)

(1

16minus

radic5

48) 11988621 + (

1

16+

radic5

48) (11988631 + 11988632) =

1

5040 (48)

5

12(11988621 + 11988631 + 11988632) +

1

12(11988641 + 11988642 + 11988643) =

1

120 (49)

(5

24+

5radic5

120) 11988632 + (

1

24+

radic5

120) 11988642

+ (1

24minus

radic5

120) 11988643 =

1

720

(50)

(5

24+

5radic5

120) 11988621 + (

5

24minus

5radic5

120) (11988631 + 11988632)

+1

12(11988641 + 11988642 + 11988643) =

1

144

(51)

Solving (46)ndash(51) simultaneously we get

11988621 =1

168+

11radic5

4200

11988631 =1

420minus

radic5

700

11988632 =1

280minus

radic5

840

11988641 = minus1

840minus

radic5

168

11988642 =1

84minus

radic5

140

11988643 =5

168minus

11radic5

840

(52)

Consequently the local truncation error norms and theglobal error of the seventh order conditions of RKFD6method are calculated and given as follows

10038171003817100381710038171003817120591(7)100381710038171003817100381710038172

= 0

100381710038171003817100381710038171205911015840(7)100381710038171003817100381710038172

= 2380952381 times 10minus4

1003817100381710038171003817100381712059110158401015840(7)100381710038171003817100381710038172

= 4761904762 times 10minus4

10038171003817100381710038171003817120591101584010158401015840(7)100381710038171003817100381710038172

= 4761904762 times 10minus4

10038171003817100381710038171003817120591(7)

119892

100381710038171003817100381710038172= 7142857143 times 10

minus4

(53)

Finally all the parameters of sixth-order RKFDmethod withfour-stage and denoted as RKFD6 are written in Butchertableau as follows

1

2+

radic5

10

1

168+

11radic5

4200

1

2minus

radic5

10

1

420minus

radic5

700

1

280minus

radic5

840

1 minus1

840minus

radic5

168

1

84minus

radic5

140

5

168+

11radic5

840

17

1260

1

72minus

radic5

168

1

72+

radic5

168

1

2520

1

24

1

16minus

radic5

48

1

16+

radic5

480

1

12

5

24minus

radic5

24

5

24+

radic5

240

1

12

5

12

5

12

1

12

(54)

Now depending on the above values of 119860 and 119888 of RKFD6method together with the equations up to order five that is(6)-(7) (10)ndash(12) (16)ndash(19) and (23)ndash(28) solving the system

8 Mathematical Problems in Engineering

simultaneously produces a solution of an embedded RKFDformula of order five which is given by

1 =1

30minus (

1

2minus

radic5

10) 2 minus (

1

2+

radic5

10) 3

4 =1

120minus (

1

2+

radic5

10) 2 minus (

1

2minus

radic5

10) 3

1015840

1 =1

16+

radic5

80minus

radic5

51015840

3

1015840

2 =1

8minus 1015840

3

1015840

4 = minus1

48minus

radic5

80+

radic5

51015840

3

(55)

while the values of 10158401015840119894 and

101584010158401015840119894 119894 = 1 2 3 4 are the same

as the RKFD6 method Now our goal is to determine the

values of the free parameters 2 3 and 10158403 such that the

values of 119862(7) 119861(7) 1198621015840(7) 119861

1015840(7) 1205911015840(6)

2 and 120591(6)

2 are assmall as possible The quantities 119862

(7) 119861(7) and 120591

(6)2 are

functions depending on 2 and 3 Picking 2 = 140 andplotting the graph of119862(7)119861(7) and 120591

(6)2 against 2 and from

the numerical experiment therefore we choose 3 = 120

that is the optimal value which gives 119862(7)

= 1507670351119861(7)

= 1507670351 and the local truncation error 120591(6)

2 =

9444444448times10minus3The quantities1198621015840(7) 1198611015840(7) and 120591

1015840(6)2 are

functions in terms of 10158403 Plotting the graph of these functionsand for the optimized pair we found that

10158403 = 110 is the

optimal value using Minimize command in Maple Softwarewhich yields 119862

1015840(7)= 1683161973 1198611015840(7) = 1708476855 and

1205911015840(6)

2 = 81256471 times 10minus4 (see Figure 2)

Finally all the parameters of the embedded pair arerepresented in the Butcher tableau and denoted as RKFD6(5)method as follows

1

2+

radic5

10

1

168+

11radic5

4200

1

2minus

radic5

10

1

420minus

radic5

700

1

280minus

radic5

840

1 minus1

840minus

radic5

168

1

84minus

radic5

140

5

168+

11radic5

840

17

1260

1

72minus

radic5

168

1

72+

radic5

168

1

2520

1

24

1

16minus

radic5

48

1

16+

radic5

480

1

12

5

24minus

radic5

24

5

24+

radic5

240

1

12

5

12

5

12

1

12

minus1

240minus

radic5

400

1

40

1

20minus

7

240+

radic5

400

1

16minus

3radic5

400

1

40

1

10minus

1

48+

3radic5

400

1

12

5

24minus

radic5

24

5

24+

radic5

240

1

12

5

12

5

12

1

12

(56)

4 Numerical Experiments

In this section we present some problems which involvespecial fourth-order ODEs of the form 119910

(119894V)(119909) = 119891(119909 119910(119909))

in order to test the performance of the two new embeddedpairs derived in Section 3 The numerical results obtainedare compared with the well-known embedded Runge-Kuttamethods which are chosen from the scientific literature after

Mathematical Problems in Engineering 9120591

998400(6) 2

005 01 015 02 025 030b998400

3

0

0002

0004

0006

0008

001

0012

0014

0016

Figure 2 The graph of 1205911015840(6)2 versus 10158403

transforming the same problems to a system of first-orderdifferential equations and solving themThemethods chosenin the numerical experiments are as follows

(i) RKFD6(5) the new embedded Runge-Kutta pair oforders 6(5) derived in Section 3 in this paper

(ii) RKFD5(4) the new embedded Runge-Kutta pair oforders 5(4) derived in Section 3 in this paper

(iii) RK6(5)V the embedded Runge-Kutta pair of orders6(5) derived by Verner as given in [39]

(iv) RK5(4)D the embedded Runge-Kutta pair of orders5(4) with FSAL property derived by Dormand andPrince [28]

Problem 1 We consider that the homogeneous linear prob-lem is as follows

119910(119894V)

= minus4119910

119910 (0) = 0 1199101015840(0) = 1 119910

10158401015840(0) = 2 119910

101584010158401015840(0) = 2

(57)

The exact solution is given by 119910(119909) = e119909 sin(119909) The problemis integrated in the interval [0 5]

Problem 2 We consider that the homogeneous nonlinearproblem is as follows

119910(119894V)

= minus15

161199107

119910 (0) = 1 1199101015840(0) =

1

2 11991010158401015840

(0) = minus1

4 119910101584010158401015840

(0) =3

8

(58)

The exact solution is given by 119910(119909) = radic1 + 119909 The problem isintegrated in the interval [0 10]

Problem 3 We consider that the inhomogeneous nonlinearproblem is as follows

119910(119894V)

= 1199102+ cos2 (119909) + sin (119909) minus 1

119910 (0) = 0 1199101015840(0) = 1 119910

10158401015840(0) = 0 119910

101584010158401015840(0) = minus1

(59)

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

35325215

log10(M

AXE

RR)

log10(function evaluations)

minus10

minus8

minus6

minus4

minus2

Figure 3 The efficiency curves for Problem 1

The exact solution is given by 119910(119909) = sin(119909) The problem isintegrated in the interval [0 8]

Problem 4 We consider that the linear system is as follows

119910(119894V)

= e3119909119906

119910 (0) = 1 1199101015840(0) = minus1 119910

10158401015840(0) = 1 119910

101584010158401015840(0) = minus1

119911(119894V)

= 16eminus119909119910

119911 (0) = 1 1199111015840(0) = minus2 119911

10158401015840(0) = 4 119911

101584010158401015840(0) = minus8

119908(119894V)

= 81eminus119909119911

119908 (0) = 1 1199081015840(0) = minus3 119908

10158401015840(0) = 9 119908

101584010158401015840(0) = minus27

119906(119894V)

= 256eminus119909119908

119906 (0) = 1 1199061015840(0) = minus4 119906

10158401015840(0) = 16 119906

101584010158401015840(0) = minus64

(60)

The exact solution is given by

119910 = eminus119909

119911 = eminus2119909

119908 = eminus3119909

119906 = eminus4119909

(61)

The problem is integrated in the interval [0 3]

In the numerical experiments we have computed foreach method and problem the maximum global error andthe number of function evaluations used in the integrationIn Figures 3ndash6 we plot these values in double logarithmic

10 Mathematical Problems in Engineering

18 2 22 24 26 28 314 16

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

log10(M

AXE

RR)

log10(function evaluations)

Figure 4 The efficiency curves for Problem 2

325215

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

log10(function evaluations)

log10(M

AXE

RR)

minus10

minus8

minus6

minus4

minus2

0

Figure 5 The efficiency curves for Problem 3

325215

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

log10(function evaluations)

log10(M

AXE

RR)

minus10

minus8

minus6

minus4

minus2

0

Figure 6 The efficiency curves for Problem 4

Table 1 The comparison results of the total number of successfulsteps and failure steps betweenRKFD6(5) RKFD5(4) RK6(5)V andRK5(4)D methods for all problems

Problems Methods Successful steps Failure steps

1

RKFD6(5) 440 4RKFD5(4) 705 0RK6(5)V 461 9RK5(4)D 768 0

2

RKFD6(5) 151 0RKFD5(4) 200 0RK6(5)V 208 0RK5(4)D 253 0

3

RKFD6(5) 330 4RKFD5(4) 466 0RK6(5)V 348 8RK5(4)D 497 0

4

RKFD6(5) 264 5RKFD5(4) 340 6RK6(5)V 339 5RK5(4)D 484 5

scale From Table 1 and Figures 3ndash6 we observed that thenew RKFD6(5) and RKFD5(4) methods are more efficient tosolve directly fourth-order differential equations comparedwith RK6(5)V and RK5(4)D methods in terms of functionevaluations and the total number of steps This is due tothe fact that when using RK6(5)V and RK5(4)D methodsthe fourth-order ODEs need to be transformed into a first-order system and hence the dimension will increase fourtimes

5 Conclusion

We have constructed the two pairs of embedded RKFDmethods for directly solving special fourth-order ODEs ofthe form 119910

(119894V)(119909) = 119891(119909 119910(119909)) using variable step size

codes in this paperThese methods are denoted as RKFD5(4)and RKFD6(5) respectively Based on the methods thevariables step size codes are developed and used to solvespecial fourth-order ODEs Numerical results show thatthe two new embedded RKFD pairs are more effective ascompared with the existing embedded Runge-Kutta pairs ofthe same order in the literature From the numerical resultswe conclude that the two new embedded RKFD methodsare computationally more efficient in solving special fourth-order ODEs and outperformed the well-known embeddedRunge-Kutta methods

Conflict of Interests

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

Mathematical Problems in Engineering 11

References

[1] S N Jator ldquoNumerical integrators for fourth order initial andboundary value problemsrdquo International Journal of Pure andApplied Mathematics vol 47 no 4 pp 563ndash576 2008

[2] Z Bai and H Wang ldquoOn positive solutions of some nonlinearfourth-order beam equationsrdquo Journal ofMathematical Analysisand Applications vol 270 no 2 pp 357ndash368 2002

[3] A K Alomari N Ratib Anakira A S Bataineh and I HashimldquoApproximate solution of nonlinear system of BVP arising influid flow problemrdquoMathematical Problems in Engineering vol2013 Article ID 136043 7 pages 2013

[4] A Malek and R Shekari Beidokhti ldquoNumerical solution forhigh order differential equations using a hybrid neuralnetworkmdashoptimization methodrdquo Applied Mathematics andComputation vol 183 no 1 pp 260ndash271 2006

[5] A Boutayeb and A Chetouani ldquoA mini-review of numericalmethods for high-order problemsrdquo International Journal ofComputer Mathematics vol 84 no 4 pp 563ndash579 2007

[6] P Onumanyi U W Sirisena and S N Jator ldquoContinuous finitedifference approximations for solving differential equationsrdquoInternational Journal of Computer Mathematics vol 72 no 1pp 15ndash27 1999

[7] D Sarafyan ldquoNew algorithms for the continuous approximatesolutions of ordinary differential equations and the upgradingof the order of the processesrdquo Computers amp Mathematics withApplications vol 20 no 1 pp 77ndash100 1990

[8] GDahlquist ldquoOn accuracy and unconditional stability of linearmultistep methods for second order differential equationsrdquo BITNumerical Mathematics vol 18 no 2 pp 133ndash136 1978

[9] R P K Chan P Leone and A Tsai ldquoOrder conditions andsymmetry for two-step hybrid methodsrdquo International Journalof Computer Mathematics vol 81 no 12 pp 1519ndash1536 2004

[10] C E Abhulimen and F O Otunta ldquoA sixth order multideriva-tivemultistepmethods for stiff system of differential equationsrdquoInternational Journal of Numerical Mathematics vol 1 no 2 pp248ndash268 2006

[11] J D Lambert Computational Methods in Ordinary DifferentialEquations Wiley London UK 1973

[12] D O Awoyemi ldquoAlgorithmic collocation approach for directsolution of fourth-order initial-value problems of ordinarydifferential equationsrdquo International Journal of ComputerMath-ematics vol 82 no 3 pp 321ndash329 2005

[13] NWaeleh Z A Majid F Ismail andM Suleiman ldquoNumericalsolution of higher order ordinary differential equations bydirect block coderdquo Journal of Mathematics and Statistics vol 8no 1 pp 77ndash81 2011

[14] S J Kayode ldquoAn efficient zero-stable numerical method forfourth-order differential equationsrdquo International Journal ofMathematics and Mathematical Sciences vol 2008 Article ID364021 10 pages 2008

[15] S J Kayode ldquoAnorder six zero-stablemethod for direct solutionof fourth order ordinary differential equationsrdquo AmericanJournal of Applied Sciences vol 5 no 11 pp 1461ndash1466 2008

[16] B T Olabode and T J Alabi ldquoDirect block predictor-correctormethod for the solution of general fourth order ODEsrdquo Journalof Mathematics Research vol 5 no 1 pp 26ndash33 2013

[17] L K Yap F Ismail and N Senu ldquoAn accurate block hybrid col-locationmethod for third order ordinary differential equationsrdquoJournal of Applied Mathematics vol 2014 Article ID 549597 9pages 2014

[18] S Mehrkanoon ldquoA direct variable step block multistep methodfor solving general third-order ODEsrdquo Numerical Algorithmsvol 57 no 1 pp 53ndash66 2011

[19] Z A Majid and M B Suleiman ldquoDirect integration implicitvariable steps method for solving higher order systems ofordinary differential equations directlyrdquo Sains Malaysiana vol35 no 2 pp 63ndash68 2006

[20] D O Awoyemi ldquoA P-stable linear multistep method for solvinggeneral third order ordinary differential equationsrdquo Interna-tional Journal of Computer Mathematics vol 80 no 8 pp 985ndash991 2003

[21] N Waeleh Z A Majid and F Ismail ldquoA new algorithm forsolving higher order IVPs of ODEsrdquo Applied MathematicalSciences vol 5 no 56 pp 2795ndash2805 2011

[22] D O Awoyemi and O M Idowu ldquoA class of hybrid collocationmethods for third-order ordinary differential equationsrdquo Inter-national Journal of Computer Mathematics vol 82 no 10 pp1287ndash1293 2005

[23] S N Jator ldquoSolving second order initial value problems by ahybrid multistep method without predictorsrdquo Applied Mathe-matics and Computation vol 217 no 8 pp 4036ndash4046 2010

[24] K Hussain F Ismail and N Senu ldquoRunge-Kutta type methodsfor directly solving special fourth-order ordinary differentialequationsrdquo Mathematical Problems in Engineering vol 2015Article ID 893763 11 pages 2015

[25] J R DormandNumerical Methods for Differential Equations AComputational Approach Library of Engineering MathematicsCRC Press Boca Raton Fla USA 1996

[26] W Gander and D Gruntz ldquoDerivation of numerical methodsusing computer algebrardquo SIAM Review vol 41 no 3 pp 577ndash593 1999

[27] F B Ismail and M B Suleiman ldquoEmbedded singly diagonallyimplicit Runge-Kutta methods (45) in (56) For the integrationof stiff systems of ODEsrdquo International Journal of ComputerMathematics vol 66 no 3-4 pp 325ndash341 1998

[28] J R Dormand and P J Prince ldquoA family of embeddedRunge-Kutta formulaerdquo Journal of Computational and AppliedMathematics vol 6 no 1 pp 19ndash26 1980

[29] P J Prince and J R Dormand ldquoHigh order embedded Runge-Kutta formulaerdquo Journal of Computational and Applied Mathe-matics vol 7 no 1 pp 67ndash75 1981

[30] R A Al-Khasawneh F Ismail and M Suleiman ldquoEmbeddeddiagonally implicit Runge-Kutta-Nystrom 4(3) pair for solvingspecial second-order IVPsrdquo Applied Mathematics and Compu-tation vol 190 no 2 pp 1803ndash1814 2007

[31] E Fehlberg ldquoLow-order classical Runge-Kutta formulas withstepsize control and their application to some heat transferproblemsrdquo NASA TR R-315 1969

[32] E Fehlberg ldquoClassical fifth sixth seventh and eighth orderRunge-Kutta formulas with stepsize controlrdquo NASA TR R-287NASA 1968

[33] M Calvo J I Montijano and L Randez ldquoA new embeddedpair of Runge-Kutta formulas of orders 5 and 6rdquo Computers ampMathematics with Applications vol 20 no 1 pp 15ndash24 1990

[34] J R Dormand M E A El-Mikkawy and P J Prince ldquoFamiliesof Runge-Kutta-Nystrom formulaerdquo IMA Journal of NumericalAnalysis vol 7 no 2 pp 235ndash250 1987

[35] L F Shampine Numerical Solution of Ordinary DifferentialEquations Chapman amp Hall New York NY USA 1994

[36] J C Butcher Numerical Methods for Ordinary DifferentialEquations John Wiley amp Sons Chichester UK 2008

12 Mathematical Problems in Engineering

[37] E Hairer S P Noslashrsett and G Wanner Solving Ordinary Dif-ferential Equations I Nonstiff Problems vol 8 Springer BerlinGermany 2nd edition 1993

[38] N Senu M Mechee F Ismail and Z Siri ldquoEmbedded explicitRungendashKutta type methods for directly solving special thirdorder differential equations 119910

101584010158401015840= 119891(119909 119910)rdquo Applied Mathemat-

ics and Computation vol 240 pp 281ndash293 2014[39] J H Verner ldquoExplicit Runge-Kutta pairs with lower stage-

orderrdquo Numerical Algorithms vol 65 no 3 pp 555ndash577 2014

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 3

Sixth order

119904

sum

119894=1

1198871198941198882

119894 =1

360 (8)

Seventh order

119904

sum

119894=1

1198871198941198883

119894 =1

840 (9)

The order conditions for 1199101015840 are as follows

Third order

119904

sum

119894=1

1198871015840

119894 =1

6 (10)

Fourth order

119904

sum

119894=1

1198871015840

119894 119888119894 =1

24 (11)

Fifth order

119904

sum

119894=1

1198871015840

119894 1198882

119894 =1

60 (12)

Sixth order

119904

sum

119894=1

1198871015840

119894 1198883

119894 =1

120 (13)

Seventh order

119904

sum

119894=1

1198871015840

119894 1198884

119894 =1

210 (14)

119904

sum

119894119895=1

1198871015840

119894 119886119894119895 =1

5040 (15)

The order conditions for 11991010158401015840 are as follows

Second order

119904

sum

119894=1

11988710158401015840

119894 =1

2 (16)

Third order

119904

sum

119894=1

11988710158401015840

119894 119888119894 =1

6 (17)

Fourth order

119904

sum

119894=1

11988710158401015840

119894 1198882

119894 =1

12 (18)

Fifth order

119904

sum

119894=1

11988710158401015840

119894 1198883

119894 =1

20 (19)

Sixth order

119904

sum

119894=1

11988710158401015840

119894 1198884

119894 =1

30 (20)

119904

sum

119894119895=1

11988710158401015840

119894 119886119894119895 =1

720 (21)

Seventh order

119904

sum

119894=1

11988710158401015840

119894 1198885

119894 =1

42

119904

sum

119894=1

11988710158401015840

119894 119886119894119895119888119895 =1

5040

119904

sum

119894=1

11988710158401015840

119894 119888119894119886119894119895 =1

1008

(22)

The order conditions for 119910101584010158401015840 are as follows

First order

119904

sum

119894=1

119887101584010158401015840

119894 = 1 (23)

Second order

119904

sum

119894=1

119887101584010158401015840

119894 119888119894 =1

2 (24)

Third order

119904

sum

119894=1

119887101584010158401015840

119894 1198882

119894 =1

3 (25)

Fourth order

119904

sum

119894=1

119887101584010158401015840

119894 1198883

119894 =1

4 (26)

4 Mathematical Problems in Engineering

Fifth order119904

sum

119894=1

119887101584010158401015840

119894 1198884

119894 =1

5 (27)

119904

sum

119894119895=1

119887101584010158401015840

119894 119886119894119895 =1

120 (28)

Sixth order

119904

sum

119894=1

119887101584010158401015840

119894 1198885

119894 =1

6 (29)

119904

sum

119894119895=1

119887101584010158401015840

119894 119886119894119895119888119895 =1

720

119904

sum

119894=1

119887101584010158401015840

119894 119888119894119886119894119895 =1

144

(30)

Seventh order

119904

sum

119894=1

119887101584010158401015840

119894 1198886

119894 =1

7

119904

sum

119894=1

119887101584010158401015840

119894 1198882

119895 119886119894119895 =1

168

119904

sum

119894=1

119887101584010158401015840

119894 1198861198941198951198882

119895 =1

2520

119904

sum

119894=1

119887101584010158401015840

119894 119888119894119886119894119895119888119895 =1

840

(31)

The following strategies are utilized for developing efficientembedded pairs

(1) The quantities of 120591(119903+1)

2 and 120591(V+1)

2 should be assmall as possible for higher and lower order RKFDmethod respectively where

10038171003817100381710038171003817120591(119903+1)100381710038171003817100381710038172

= radic

1198991

sum

119894=1

(120591(119903+1)

119894)2+

1198992

sum

119894=1

(1205911015840(119903+1)

119894)2+

1198993

sum

119894=1

(12059110158401015840(119903+1)

119894)2+

1198994

sum

119894=1

(120591101584010158401015840(119903+1)

119894)2

10038171003817100381710038171003817120591(V+1)100381710038171003817100381710038172

= radic

1198991

sum

119894=1

(120591(V+1)119894

)2+

1198992

sum

119894=1

(1205911015840(V+1)119894

)2+

1198993

sum

119894=1

(12059110158401015840(V+1)119894

)2+

1198994

sum

119894=1

(120591101584010158401015840(V+1)119894

)2

(32)

where 120591(119903+1)

119894 1205911015840(119903+1)119894

12059110158401015840(119903+1)119894

and 120591101584010158401015840(119903+1)

119894are called the

error terms for 119910 1199101015840 11991010158401015840 and 119910101584010158401015840 respectively

(2) The following quantities given in [34] should be assmall as possible

(i)

119862(V+2)

=

10038171003817100381710038171003817120591(V+2)

minus 120591(V+2)100381710038171003817100381710038172

1003817100381710038171003817120591(V+1)10038171003817100381710038172

1198621015840(V+2)

=

100381710038171003817100381710038171205911015840(V+2)

minus 1205911015840(V+2)100381710038171003817100381710038172

10038171003817100381710038171205911015840(V+1)10038171003817100381710038172

11986210158401015840(V+2)

=

1003817100381710038171003817100381712059110158401015840(V+2)

minus 12059110158401015840(V+2)100381710038171003817100381710038172

100381710038171003817100381712059110158401015840(V+1)10038171003817100381710038172

119862101584010158401015840(V+2)

=

10038171003817100381710038171003817120591101584010158401015840(V+2)

minus 120591101584010158401015840(V+2)100381710038171003817100381710038172

1003817100381710038171003817120591101584010158401015840(V+1)10038171003817100381710038172

(33)

(ii)

119861(V+2)

=

10038171003817100381710038171003817120591(V+2)100381710038171003817100381710038172

1003817100381710038171003817120591(V+1)10038171003817100381710038172

1198611015840(V+2)

=

100381710038171003817100381710038171205911015840(V+2)100381710038171003817100381710038172

10038171003817100381710038171205911015840(V+1)10038171003817100381710038172

11986110158401015840(V+2)

=

1003817100381710038171003817100381712059110158401015840(V+2)100381710038171003817100381710038172

100381710038171003817100381712059110158401015840(V+1)10038171003817100381710038172

119861101584010158401015840(V+2)

=

10038171003817100381710038171003817120591101584010158401015840(V+2)100381710038171003817100381710038172

1003817100381710038171003817120591101584010158401015840(V+1)10038171003817100381710038172

(34)

where 120591(V+1) 120591

1015840(V+1) 12059110158401015840(V+1) and 120591

101584010158401015840(V+1) are calledthe error coefficients for 119910 119910

1015840 11991010158401015840 and 119910

101584010158401015840 of theembedded RKFD pairs respectively

(3) We defined the local error estimation at the point 119905119899+1by the following formula

EST = max 1003817100381710038171003817120573119899+1

1003817100381710038171003817infin 100381710038171003817100381710038171205731015840

119899+1

10038171003817100381710038171003817infin1003817100381710038171003817100381712057310158401015840

119899+1

10038171003817100381710038171003817infin10038171003817100381710038171003817120573101584010158401015840

119899+1

10038171003817100381710038171003817infin (35)

where120573119899+1 = 119910119899+1 minus 119910119899+1

1205731015840

119899+1 = 1199101015840

119899+1 minus 1199101015840

119899+1

12057310158401015840

119899+1 = 11991010158401015840

119899+1 minus 11991010158401015840

119899+1

120573101584010158401015840

119899+1 = 119910101584010158401015840

119899+1 minus 119910101584010158401015840

119899+1

(36)

where 119910 1199101015840 11991010158401015840 and 119910

101584010158401015840 and 119910 1199101015840 11991010158401015840 and 119910

101584010158401015840

are solutions using the higher order formula and thelower order formula respectively

The local error estimation EST can be used to control thestep size ℎ by the standard formula as given in [35ndash38]

ℎ119899+1 = 09ℎ119899 (TOLEST

)

1(V+1) (37)

where 09 is a safety factor the local error estimation ateach step is represented by EST and TOL is the maximumallowable local error which is the precision required

Mathematical Problems in Engineering 5

If EST le TOL then the step is accepted and we appliedthe procedure of performing local extrapolation (or higherorder mode) meaning that the more accurate approximationwill be used to advance the integration If EST gt TOL thenthe step is rejected and the step size ℎ will be updated usingformula (37)

3 Construction of Embedded ExplicitRKFD Pairs

The construction of embedded explicit RKFD pairs will bediscussed in this section In specific we will derive twoembedded RKFD pairs of orders 5(4) and 6(5) with three andfour stages per step respectively

31The Derivation of Embedded RKFD5(4) Pair This sectionwill focus on the derivation of embedded RKFD5(4) pair withthree stages The authors in [24] derived three-stage fifth-order RKFD method and the solution is given as follows

1198882 =3

5+

radic6

10

1198883 =3

5minus

radic6

10

119887101584010158401015840

1 =1

9

119887101584010158401015840

2 =4

9minus

radic6

36

119887101584010158401015840

3 =4

9+

radic6

36

11988710158401015840

1 =1

9

11988710158401015840

2 =7

36minus

radic6

18

11988710158401015840

3 =7

36+

radic6

18

1198871015840

1 =1

18

1198871015840

2 =1

18minus

radic6

48

1198871015840

3 =1

18+

radic6

48

1198871 =19

1080

1198872 =13

1080minus

11radic6

2160

1198873 =13

1080+

11radic6

2160

11988621 =4059

187793

120591998400(5) 2

0

0005

001

0015

002

0025

003

005 01 015 02 025 030b998400

3

Figure 1 The graph of 1205911015840(5)2 versus 10158403

11988631 = minus1502

532215

11988632 =1826

569317

(38)

Now based on the above solution for values of 119860 and 119888 wederive a three-stage fourth-order embedded formula Solvingthe algebraic conditions (6) (10)-(11) (16)ndash(18) and (23)ndash(26)simultaneously gives a solution for 119894 in terms of 1 and 3 andthe solution of 1015840119894 in terms of 10158403 which are given as follows

2 = minus1 minus 3 +1

24

1015840

1 =1

12+

1

72radic6 + (

2

5minus

2

5radic6)

1015840

3

1015840

2 =1

12minus

1

72radic6 minus (

7

5minus

2

5radic6)

1015840

3

(39)

while the values of 10158401015840119894 and 101584010158401015840119894 119894 = 1 2 3 are the same as the

fifth-order method Our objective now is to select the valuesof the free parameters 1 3 and

10158403 and thus the values of119862

(6)119861(6) 1198621015840(6) 1198611015840(6) 120591(5)2 and 120591

1015840(5)2 are as small as possible

Using the above solution we get 1198621015840(6)

= 1200 is a constantvalue however 119861

1015840(6) and 1205911015840(5)

2 are functions in terms of10158403 by plotting the graph of 120591

1015840(5)2 versus

10158403 (see Figure 1)

From the numerical experiment and for the accurate resultwe choose

10158403 = 110 using Minimize command in Maple

Software which gives the error norm 1205911015840(5)

2 = 114566368times

10minus3 and 119861

1015840(6)= 04726195824

The quantities119862(6)119861(6) and 120591(5)

2 are functions in termsof 1 3 Choose 1 = 1100 and plot the graph of 119862

(6)119861(6) and 120591

(5)2 against 3 in the interval [0 008] From

numerical experiment and for the optimized pair therefore3 = 3100 is chosen which gives 119861

(6)= 05887532657

119862(6)

= 05887532657 and 120591(5)

2 = 3726445732 times 10minus3

6 Mathematical Problems in Engineering

Consequently we denote this pair as RKFD5(4) method andit can be written in Butcher tableau as follows

3

5+

radic6

10

4059

187793

3

5minus

radic6

10minus

1502

532215

1826

569317

19

1080

13

1080minus

11radic6

2160

13

1080+

11radic6

2160

1

18

1

18minus

radic6

48

1

18+

radic6

48

1

9

7

36minus

radic6

18

7

36+

radic6

18

1

9

4

9minus

radic6

36

4

9+

radic6

36

1

100

1

600

3

100

37

300minus

47radic6

1800minus

17

300+

47radic6

1800

1

10

1

9

7

36minus

radic6

18

7

36+

radic6

18

1

9

4

9minus

radic6

36

4

9+

radic6

36

(40)

32 The Derivation of Embedded RKFD6(5) Pair In this sec-tion a four-stage embedded RKFD6(5) pair will be derivedFor the sixth-order (119903 = 6) method the order conditions ofRKFD method up to order six need to be solved to derivea four-stage sixth-order RKFD method namely RKFD6method To determine the parameters of RKFD6method wechoose (10)ndash(13) from order conditions for 119910

1015840 (16)ndash(20) fromorder conditions for 119910

10158401015840 and (23)ndash(27) and (29) from orderconditions for 119910

101584010158401015840 It involved 15 nonlinear equations with 15unknowns and then was solved simultaneously which resultsin a unique solution as follows

1198871015840

1 =1

24

1198871015840

2 =1

16minus

radic5

48

1198871015840

3 =1

16+

radic5

48

1198871015840

4 = 0

11988710158401015840

1 =1

12

11988710158401015840

2 =5

24minus

radic5

24

11988710158401015840

3 =5

24+

radic5

24

11988710158401015840

4 = 0

119887101584010158401015840

1 =1

12

119887101584010158401015840

2 =5

12

119887101584010158401015840

3 =5

12

119887101584010158401015840

4 =1

12

1198882 =1

2+

radic5

10

1198883 =1

2minus

radic5

10

1198884 = 1

(41)

Substituting the values of the above 1198882 1198883 and 1198884 into the orderconditions for 119910 we get

1198872 =1

72minus

7radic5

360+ radic51198871

1198873 =1

72+

7radic5

360minus radic51198871

1198874 = minus1198871 +1

72

(42)

Here we have one free parameter 1198871 which can be chosen byminimizing the error norm of the seventh order conditionsfor 119910 according to Dormand et al [34] The error norms andthe global error of the seventh order conditions are definedas follows

10038171003817100381710038171003817120591(7)100381710038171003817100381710038172

= radic

119899119901+1

sum

119894=1

(120591(7)

119894)2

100381710038171003817100381710038171205911015840(7)100381710038171003817100381710038172

= radic

1198991015840119901+1

sum

119894=1

(1205911015840(7)

119894)2

1003817100381710038171003817100381712059110158401015840(7)100381710038171003817100381710038172

= radic

11989910158401015840119901+1

sum

119894=1

(12059110158401015840(7)

119894)2

10038171003817100381710038171003817120591101584010158401015840(7)100381710038171003817100381710038172

= radic

119899101584010158401015840119901+1

sum

119894=1

(120591101584010158401015840(7)

119894)2

10038171003817100381710038171003817120591(7)

119892

100381710038171003817100381710038172

= radic

119899119901+1

sum

119894=1

(120591(7)

119894)2+

1198991015840119901+1

sum

119894=1

(1205911015840(7)

119894)2+

11989910158401015840119901+1

sum

119894=1

(12059110158401015840(7)

119894)2+

119899101584010158401015840119901+1

sum

119894=1

(120591101584010158401015840(7)

119894)2

(43)

Mathematical Problems in Engineering 7

where 120591(7) 1205911015840(7) 120591

10158401015840(7) and 120591101584010158401015840(7) are the local truncation

errors norms for 119910 1199101015840 11991010158401015840 and 119910101584010158401015840 of the RKFD6 method

respectively and 120591(7)119892 is the global error As a result we find

the error equations of 119910 as

10038171003817100381710038171003817120591(7)100381710038171003817100381710038172

=1

6300

radic(minus17 + 12601198871)2 (44)

By minimizing 120591(7)

2 with respect to the free parameter 1198871we obtain 1198871 = 171260 that is the optimal value and gives120591(7)

2 = 0 and

1198872 =1

72minus

radic5

168

1198873 =1

72+

radic5

168

1198874 =1

2520

(45)

To find the coefficients 11988621 11988631 11988632 11988641 11988642 and 11988643substituting the values of 1198871 1198872 1198873 1198874 119887

10158401 11988710158402 11988710158403 11988710158404 119887101584010158401 119887101584010158402

119887101584010158403 119887101584010158404 1198871015840101584010158401 1198871015840101584010158402 1198871015840101584010158403 and 119887

1015840101584010158404 into (28) (30) and (31) of the

order conditions of 119910101584010158401015840 and (15) of the order conditions of

1199101015840 the following system of nonlinear equations needs to be

solved

1

1211988632 +

1

24(11988642 + 11988643) +

radic5

120(11988642 minus 11988643) =

1

840 (46)

(1

8+

radic5

24) 11988632 +

1

40(11988642 + 11988643)

+radic5

120(11988642 minus 11988643) =

1

2520

(47)

(1

16minus

radic5

48) 11988621 + (

1

16+

radic5

48) (11988631 + 11988632) =

1

5040 (48)

5

12(11988621 + 11988631 + 11988632) +

1

12(11988641 + 11988642 + 11988643) =

1

120 (49)

(5

24+

5radic5

120) 11988632 + (

1

24+

radic5

120) 11988642

+ (1

24minus

radic5

120) 11988643 =

1

720

(50)

(5

24+

5radic5

120) 11988621 + (

5

24minus

5radic5

120) (11988631 + 11988632)

+1

12(11988641 + 11988642 + 11988643) =

1

144

(51)

Solving (46)ndash(51) simultaneously we get

11988621 =1

168+

11radic5

4200

11988631 =1

420minus

radic5

700

11988632 =1

280minus

radic5

840

11988641 = minus1

840minus

radic5

168

11988642 =1

84minus

radic5

140

11988643 =5

168minus

11radic5

840

(52)

Consequently the local truncation error norms and theglobal error of the seventh order conditions of RKFD6method are calculated and given as follows

10038171003817100381710038171003817120591(7)100381710038171003817100381710038172

= 0

100381710038171003817100381710038171205911015840(7)100381710038171003817100381710038172

= 2380952381 times 10minus4

1003817100381710038171003817100381712059110158401015840(7)100381710038171003817100381710038172

= 4761904762 times 10minus4

10038171003817100381710038171003817120591101584010158401015840(7)100381710038171003817100381710038172

= 4761904762 times 10minus4

10038171003817100381710038171003817120591(7)

119892

100381710038171003817100381710038172= 7142857143 times 10

minus4

(53)

Finally all the parameters of sixth-order RKFDmethod withfour-stage and denoted as RKFD6 are written in Butchertableau as follows

1

2+

radic5

10

1

168+

11radic5

4200

1

2minus

radic5

10

1

420minus

radic5

700

1

280minus

radic5

840

1 minus1

840minus

radic5

168

1

84minus

radic5

140

5

168+

11radic5

840

17

1260

1

72minus

radic5

168

1

72+

radic5

168

1

2520

1

24

1

16minus

radic5

48

1

16+

radic5

480

1

12

5

24minus

radic5

24

5

24+

radic5

240

1

12

5

12

5

12

1

12

(54)

Now depending on the above values of 119860 and 119888 of RKFD6method together with the equations up to order five that is(6)-(7) (10)ndash(12) (16)ndash(19) and (23)ndash(28) solving the system

8 Mathematical Problems in Engineering

simultaneously produces a solution of an embedded RKFDformula of order five which is given by

1 =1

30minus (

1

2minus

radic5

10) 2 minus (

1

2+

radic5

10) 3

4 =1

120minus (

1

2+

radic5

10) 2 minus (

1

2minus

radic5

10) 3

1015840

1 =1

16+

radic5

80minus

radic5

51015840

3

1015840

2 =1

8minus 1015840

3

1015840

4 = minus1

48minus

radic5

80+

radic5

51015840

3

(55)

while the values of 10158401015840119894 and

101584010158401015840119894 119894 = 1 2 3 4 are the same

as the RKFD6 method Now our goal is to determine the

values of the free parameters 2 3 and 10158403 such that the

values of 119862(7) 119861(7) 1198621015840(7) 119861

1015840(7) 1205911015840(6)

2 and 120591(6)

2 are assmall as possible The quantities 119862

(7) 119861(7) and 120591

(6)2 are

functions depending on 2 and 3 Picking 2 = 140 andplotting the graph of119862(7)119861(7) and 120591

(6)2 against 2 and from

the numerical experiment therefore we choose 3 = 120

that is the optimal value which gives 119862(7)

= 1507670351119861(7)

= 1507670351 and the local truncation error 120591(6)

2 =

9444444448times10minus3The quantities1198621015840(7) 1198611015840(7) and 120591

1015840(6)2 are

functions in terms of 10158403 Plotting the graph of these functionsand for the optimized pair we found that

10158403 = 110 is the

optimal value using Minimize command in Maple Softwarewhich yields 119862

1015840(7)= 1683161973 1198611015840(7) = 1708476855 and

1205911015840(6)

2 = 81256471 times 10minus4 (see Figure 2)

Finally all the parameters of the embedded pair arerepresented in the Butcher tableau and denoted as RKFD6(5)method as follows

1

2+

radic5

10

1

168+

11radic5

4200

1

2minus

radic5

10

1

420minus

radic5

700

1

280minus

radic5

840

1 minus1

840minus

radic5

168

1

84minus

radic5

140

5

168+

11radic5

840

17

1260

1

72minus

radic5

168

1

72+

radic5

168

1

2520

1

24

1

16minus

radic5

48

1

16+

radic5

480

1

12

5

24minus

radic5

24

5

24+

radic5

240

1

12

5

12

5

12

1

12

minus1

240minus

radic5

400

1

40

1

20minus

7

240+

radic5

400

1

16minus

3radic5

400

1

40

1

10minus

1

48+

3radic5

400

1

12

5

24minus

radic5

24

5

24+

radic5

240

1

12

5

12

5

12

1

12

(56)

4 Numerical Experiments

In this section we present some problems which involvespecial fourth-order ODEs of the form 119910

(119894V)(119909) = 119891(119909 119910(119909))

in order to test the performance of the two new embeddedpairs derived in Section 3 The numerical results obtainedare compared with the well-known embedded Runge-Kuttamethods which are chosen from the scientific literature after

Mathematical Problems in Engineering 9120591

998400(6) 2

005 01 015 02 025 030b998400

3

0

0002

0004

0006

0008

001

0012

0014

0016

Figure 2 The graph of 1205911015840(6)2 versus 10158403

transforming the same problems to a system of first-orderdifferential equations and solving themThemethods chosenin the numerical experiments are as follows

(i) RKFD6(5) the new embedded Runge-Kutta pair oforders 6(5) derived in Section 3 in this paper

(ii) RKFD5(4) the new embedded Runge-Kutta pair oforders 5(4) derived in Section 3 in this paper

(iii) RK6(5)V the embedded Runge-Kutta pair of orders6(5) derived by Verner as given in [39]

(iv) RK5(4)D the embedded Runge-Kutta pair of orders5(4) with FSAL property derived by Dormand andPrince [28]

Problem 1 We consider that the homogeneous linear prob-lem is as follows

119910(119894V)

= minus4119910

119910 (0) = 0 1199101015840(0) = 1 119910

10158401015840(0) = 2 119910

101584010158401015840(0) = 2

(57)

The exact solution is given by 119910(119909) = e119909 sin(119909) The problemis integrated in the interval [0 5]

Problem 2 We consider that the homogeneous nonlinearproblem is as follows

119910(119894V)

= minus15

161199107

119910 (0) = 1 1199101015840(0) =

1

2 11991010158401015840

(0) = minus1

4 119910101584010158401015840

(0) =3

8

(58)

The exact solution is given by 119910(119909) = radic1 + 119909 The problem isintegrated in the interval [0 10]

Problem 3 We consider that the inhomogeneous nonlinearproblem is as follows

119910(119894V)

= 1199102+ cos2 (119909) + sin (119909) minus 1

119910 (0) = 0 1199101015840(0) = 1 119910

10158401015840(0) = 0 119910

101584010158401015840(0) = minus1

(59)

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

35325215

log10(M

AXE

RR)

log10(function evaluations)

minus10

minus8

minus6

minus4

minus2

Figure 3 The efficiency curves for Problem 1

The exact solution is given by 119910(119909) = sin(119909) The problem isintegrated in the interval [0 8]

Problem 4 We consider that the linear system is as follows

119910(119894V)

= e3119909119906

119910 (0) = 1 1199101015840(0) = minus1 119910

10158401015840(0) = 1 119910

101584010158401015840(0) = minus1

119911(119894V)

= 16eminus119909119910

119911 (0) = 1 1199111015840(0) = minus2 119911

10158401015840(0) = 4 119911

101584010158401015840(0) = minus8

119908(119894V)

= 81eminus119909119911

119908 (0) = 1 1199081015840(0) = minus3 119908

10158401015840(0) = 9 119908

101584010158401015840(0) = minus27

119906(119894V)

= 256eminus119909119908

119906 (0) = 1 1199061015840(0) = minus4 119906

10158401015840(0) = 16 119906

101584010158401015840(0) = minus64

(60)

The exact solution is given by

119910 = eminus119909

119911 = eminus2119909

119908 = eminus3119909

119906 = eminus4119909

(61)

The problem is integrated in the interval [0 3]

In the numerical experiments we have computed foreach method and problem the maximum global error andthe number of function evaluations used in the integrationIn Figures 3ndash6 we plot these values in double logarithmic

10 Mathematical Problems in Engineering

18 2 22 24 26 28 314 16

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

log10(M

AXE

RR)

log10(function evaluations)

Figure 4 The efficiency curves for Problem 2

325215

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

log10(function evaluations)

log10(M

AXE

RR)

minus10

minus8

minus6

minus4

minus2

0

Figure 5 The efficiency curves for Problem 3

325215

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

log10(function evaluations)

log10(M

AXE

RR)

minus10

minus8

minus6

minus4

minus2

0

Figure 6 The efficiency curves for Problem 4

Table 1 The comparison results of the total number of successfulsteps and failure steps betweenRKFD6(5) RKFD5(4) RK6(5)V andRK5(4)D methods for all problems

Problems Methods Successful steps Failure steps

1

RKFD6(5) 440 4RKFD5(4) 705 0RK6(5)V 461 9RK5(4)D 768 0

2

RKFD6(5) 151 0RKFD5(4) 200 0RK6(5)V 208 0RK5(4)D 253 0

3

RKFD6(5) 330 4RKFD5(4) 466 0RK6(5)V 348 8RK5(4)D 497 0

4

RKFD6(5) 264 5RKFD5(4) 340 6RK6(5)V 339 5RK5(4)D 484 5

scale From Table 1 and Figures 3ndash6 we observed that thenew RKFD6(5) and RKFD5(4) methods are more efficient tosolve directly fourth-order differential equations comparedwith RK6(5)V and RK5(4)D methods in terms of functionevaluations and the total number of steps This is due tothe fact that when using RK6(5)V and RK5(4)D methodsthe fourth-order ODEs need to be transformed into a first-order system and hence the dimension will increase fourtimes

5 Conclusion

We have constructed the two pairs of embedded RKFDmethods for directly solving special fourth-order ODEs ofthe form 119910

(119894V)(119909) = 119891(119909 119910(119909)) using variable step size

codes in this paperThese methods are denoted as RKFD5(4)and RKFD6(5) respectively Based on the methods thevariables step size codes are developed and used to solvespecial fourth-order ODEs Numerical results show thatthe two new embedded RKFD pairs are more effective ascompared with the existing embedded Runge-Kutta pairs ofthe same order in the literature From the numerical resultswe conclude that the two new embedded RKFD methodsare computationally more efficient in solving special fourth-order ODEs and outperformed the well-known embeddedRunge-Kutta methods

Conflict of Interests

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

Mathematical Problems in Engineering 11

References

[1] S N Jator ldquoNumerical integrators for fourth order initial andboundary value problemsrdquo International Journal of Pure andApplied Mathematics vol 47 no 4 pp 563ndash576 2008

[2] Z Bai and H Wang ldquoOn positive solutions of some nonlinearfourth-order beam equationsrdquo Journal ofMathematical Analysisand Applications vol 270 no 2 pp 357ndash368 2002

[3] A K Alomari N Ratib Anakira A S Bataineh and I HashimldquoApproximate solution of nonlinear system of BVP arising influid flow problemrdquoMathematical Problems in Engineering vol2013 Article ID 136043 7 pages 2013

[4] A Malek and R Shekari Beidokhti ldquoNumerical solution forhigh order differential equations using a hybrid neuralnetworkmdashoptimization methodrdquo Applied Mathematics andComputation vol 183 no 1 pp 260ndash271 2006

[5] A Boutayeb and A Chetouani ldquoA mini-review of numericalmethods for high-order problemsrdquo International Journal ofComputer Mathematics vol 84 no 4 pp 563ndash579 2007

[6] P Onumanyi U W Sirisena and S N Jator ldquoContinuous finitedifference approximations for solving differential equationsrdquoInternational Journal of Computer Mathematics vol 72 no 1pp 15ndash27 1999

[7] D Sarafyan ldquoNew algorithms for the continuous approximatesolutions of ordinary differential equations and the upgradingof the order of the processesrdquo Computers amp Mathematics withApplications vol 20 no 1 pp 77ndash100 1990

[8] GDahlquist ldquoOn accuracy and unconditional stability of linearmultistep methods for second order differential equationsrdquo BITNumerical Mathematics vol 18 no 2 pp 133ndash136 1978

[9] R P K Chan P Leone and A Tsai ldquoOrder conditions andsymmetry for two-step hybrid methodsrdquo International Journalof Computer Mathematics vol 81 no 12 pp 1519ndash1536 2004

[10] C E Abhulimen and F O Otunta ldquoA sixth order multideriva-tivemultistepmethods for stiff system of differential equationsrdquoInternational Journal of Numerical Mathematics vol 1 no 2 pp248ndash268 2006

[11] J D Lambert Computational Methods in Ordinary DifferentialEquations Wiley London UK 1973

[12] D O Awoyemi ldquoAlgorithmic collocation approach for directsolution of fourth-order initial-value problems of ordinarydifferential equationsrdquo International Journal of ComputerMath-ematics vol 82 no 3 pp 321ndash329 2005

[13] NWaeleh Z A Majid F Ismail andM Suleiman ldquoNumericalsolution of higher order ordinary differential equations bydirect block coderdquo Journal of Mathematics and Statistics vol 8no 1 pp 77ndash81 2011

[14] S J Kayode ldquoAn efficient zero-stable numerical method forfourth-order differential equationsrdquo International Journal ofMathematics and Mathematical Sciences vol 2008 Article ID364021 10 pages 2008

[15] S J Kayode ldquoAnorder six zero-stablemethod for direct solutionof fourth order ordinary differential equationsrdquo AmericanJournal of Applied Sciences vol 5 no 11 pp 1461ndash1466 2008

[16] B T Olabode and T J Alabi ldquoDirect block predictor-correctormethod for the solution of general fourth order ODEsrdquo Journalof Mathematics Research vol 5 no 1 pp 26ndash33 2013

[17] L K Yap F Ismail and N Senu ldquoAn accurate block hybrid col-locationmethod for third order ordinary differential equationsrdquoJournal of Applied Mathematics vol 2014 Article ID 549597 9pages 2014

[18] S Mehrkanoon ldquoA direct variable step block multistep methodfor solving general third-order ODEsrdquo Numerical Algorithmsvol 57 no 1 pp 53ndash66 2011

[19] Z A Majid and M B Suleiman ldquoDirect integration implicitvariable steps method for solving higher order systems ofordinary differential equations directlyrdquo Sains Malaysiana vol35 no 2 pp 63ndash68 2006

[20] D O Awoyemi ldquoA P-stable linear multistep method for solvinggeneral third order ordinary differential equationsrdquo Interna-tional Journal of Computer Mathematics vol 80 no 8 pp 985ndash991 2003

[21] N Waeleh Z A Majid and F Ismail ldquoA new algorithm forsolving higher order IVPs of ODEsrdquo Applied MathematicalSciences vol 5 no 56 pp 2795ndash2805 2011

[22] D O Awoyemi and O M Idowu ldquoA class of hybrid collocationmethods for third-order ordinary differential equationsrdquo Inter-national Journal of Computer Mathematics vol 82 no 10 pp1287ndash1293 2005

[23] S N Jator ldquoSolving second order initial value problems by ahybrid multistep method without predictorsrdquo Applied Mathe-matics and Computation vol 217 no 8 pp 4036ndash4046 2010

[24] K Hussain F Ismail and N Senu ldquoRunge-Kutta type methodsfor directly solving special fourth-order ordinary differentialequationsrdquo Mathematical Problems in Engineering vol 2015Article ID 893763 11 pages 2015

[25] J R DormandNumerical Methods for Differential Equations AComputational Approach Library of Engineering MathematicsCRC Press Boca Raton Fla USA 1996

[26] W Gander and D Gruntz ldquoDerivation of numerical methodsusing computer algebrardquo SIAM Review vol 41 no 3 pp 577ndash593 1999

[27] F B Ismail and M B Suleiman ldquoEmbedded singly diagonallyimplicit Runge-Kutta methods (45) in (56) For the integrationof stiff systems of ODEsrdquo International Journal of ComputerMathematics vol 66 no 3-4 pp 325ndash341 1998

[28] J R Dormand and P J Prince ldquoA family of embeddedRunge-Kutta formulaerdquo Journal of Computational and AppliedMathematics vol 6 no 1 pp 19ndash26 1980

[29] P J Prince and J R Dormand ldquoHigh order embedded Runge-Kutta formulaerdquo Journal of Computational and Applied Mathe-matics vol 7 no 1 pp 67ndash75 1981

[30] R A Al-Khasawneh F Ismail and M Suleiman ldquoEmbeddeddiagonally implicit Runge-Kutta-Nystrom 4(3) pair for solvingspecial second-order IVPsrdquo Applied Mathematics and Compu-tation vol 190 no 2 pp 1803ndash1814 2007

[31] E Fehlberg ldquoLow-order classical Runge-Kutta formulas withstepsize control and their application to some heat transferproblemsrdquo NASA TR R-315 1969

[32] E Fehlberg ldquoClassical fifth sixth seventh and eighth orderRunge-Kutta formulas with stepsize controlrdquo NASA TR R-287NASA 1968

[33] M Calvo J I Montijano and L Randez ldquoA new embeddedpair of Runge-Kutta formulas of orders 5 and 6rdquo Computers ampMathematics with Applications vol 20 no 1 pp 15ndash24 1990

[34] J R Dormand M E A El-Mikkawy and P J Prince ldquoFamiliesof Runge-Kutta-Nystrom formulaerdquo IMA Journal of NumericalAnalysis vol 7 no 2 pp 235ndash250 1987

[35] L F Shampine Numerical Solution of Ordinary DifferentialEquations Chapman amp Hall New York NY USA 1994

[36] J C Butcher Numerical Methods for Ordinary DifferentialEquations John Wiley amp Sons Chichester UK 2008

12 Mathematical Problems in Engineering

[37] E Hairer S P Noslashrsett and G Wanner Solving Ordinary Dif-ferential Equations I Nonstiff Problems vol 8 Springer BerlinGermany 2nd edition 1993

[38] N Senu M Mechee F Ismail and Z Siri ldquoEmbedded explicitRungendashKutta type methods for directly solving special thirdorder differential equations 119910

101584010158401015840= 119891(119909 119910)rdquo Applied Mathemat-

ics and Computation vol 240 pp 281ndash293 2014[39] J H Verner ldquoExplicit Runge-Kutta pairs with lower stage-

orderrdquo Numerical Algorithms vol 65 no 3 pp 555ndash577 2014

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

4 Mathematical Problems in Engineering

Fifth order119904

sum

119894=1

119887101584010158401015840

119894 1198884

119894 =1

5 (27)

119904

sum

119894119895=1

119887101584010158401015840

119894 119886119894119895 =1

120 (28)

Sixth order

119904

sum

119894=1

119887101584010158401015840

119894 1198885

119894 =1

6 (29)

119904

sum

119894119895=1

119887101584010158401015840

119894 119886119894119895119888119895 =1

720

119904

sum

119894=1

119887101584010158401015840

119894 119888119894119886119894119895 =1

144

(30)

Seventh order

119904

sum

119894=1

119887101584010158401015840

119894 1198886

119894 =1

7

119904

sum

119894=1

119887101584010158401015840

119894 1198882

119895 119886119894119895 =1

168

119904

sum

119894=1

119887101584010158401015840

119894 1198861198941198951198882

119895 =1

2520

119904

sum

119894=1

119887101584010158401015840

119894 119888119894119886119894119895119888119895 =1

840

(31)

The following strategies are utilized for developing efficientembedded pairs

(1) The quantities of 120591(119903+1)

2 and 120591(V+1)

2 should be assmall as possible for higher and lower order RKFDmethod respectively where

10038171003817100381710038171003817120591(119903+1)100381710038171003817100381710038172

= radic

1198991

sum

119894=1

(120591(119903+1)

119894)2+

1198992

sum

119894=1

(1205911015840(119903+1)

119894)2+

1198993

sum

119894=1

(12059110158401015840(119903+1)

119894)2+

1198994

sum

119894=1

(120591101584010158401015840(119903+1)

119894)2

10038171003817100381710038171003817120591(V+1)100381710038171003817100381710038172

= radic

1198991

sum

119894=1

(120591(V+1)119894

)2+

1198992

sum

119894=1

(1205911015840(V+1)119894

)2+

1198993

sum

119894=1

(12059110158401015840(V+1)119894

)2+

1198994

sum

119894=1

(120591101584010158401015840(V+1)119894

)2

(32)

where 120591(119903+1)

119894 1205911015840(119903+1)119894

12059110158401015840(119903+1)119894

and 120591101584010158401015840(119903+1)

119894are called the

error terms for 119910 1199101015840 11991010158401015840 and 119910101584010158401015840 respectively

(2) The following quantities given in [34] should be assmall as possible

(i)

119862(V+2)

=

10038171003817100381710038171003817120591(V+2)

minus 120591(V+2)100381710038171003817100381710038172

1003817100381710038171003817120591(V+1)10038171003817100381710038172

1198621015840(V+2)

=

100381710038171003817100381710038171205911015840(V+2)

minus 1205911015840(V+2)100381710038171003817100381710038172

10038171003817100381710038171205911015840(V+1)10038171003817100381710038172

11986210158401015840(V+2)

=

1003817100381710038171003817100381712059110158401015840(V+2)

minus 12059110158401015840(V+2)100381710038171003817100381710038172

100381710038171003817100381712059110158401015840(V+1)10038171003817100381710038172

119862101584010158401015840(V+2)

=

10038171003817100381710038171003817120591101584010158401015840(V+2)

minus 120591101584010158401015840(V+2)100381710038171003817100381710038172

1003817100381710038171003817120591101584010158401015840(V+1)10038171003817100381710038172

(33)

(ii)

119861(V+2)

=

10038171003817100381710038171003817120591(V+2)100381710038171003817100381710038172

1003817100381710038171003817120591(V+1)10038171003817100381710038172

1198611015840(V+2)

=

100381710038171003817100381710038171205911015840(V+2)100381710038171003817100381710038172

10038171003817100381710038171205911015840(V+1)10038171003817100381710038172

11986110158401015840(V+2)

=

1003817100381710038171003817100381712059110158401015840(V+2)100381710038171003817100381710038172

100381710038171003817100381712059110158401015840(V+1)10038171003817100381710038172

119861101584010158401015840(V+2)

=

10038171003817100381710038171003817120591101584010158401015840(V+2)100381710038171003817100381710038172

1003817100381710038171003817120591101584010158401015840(V+1)10038171003817100381710038172

(34)

where 120591(V+1) 120591

1015840(V+1) 12059110158401015840(V+1) and 120591

101584010158401015840(V+1) are calledthe error coefficients for 119910 119910

1015840 11991010158401015840 and 119910

101584010158401015840 of theembedded RKFD pairs respectively

(3) We defined the local error estimation at the point 119905119899+1by the following formula

EST = max 1003817100381710038171003817120573119899+1

1003817100381710038171003817infin 100381710038171003817100381710038171205731015840

119899+1

10038171003817100381710038171003817infin1003817100381710038171003817100381712057310158401015840

119899+1

10038171003817100381710038171003817infin10038171003817100381710038171003817120573101584010158401015840

119899+1

10038171003817100381710038171003817infin (35)

where120573119899+1 = 119910119899+1 minus 119910119899+1

1205731015840

119899+1 = 1199101015840

119899+1 minus 1199101015840

119899+1

12057310158401015840

119899+1 = 11991010158401015840

119899+1 minus 11991010158401015840

119899+1

120573101584010158401015840

119899+1 = 119910101584010158401015840

119899+1 minus 119910101584010158401015840

119899+1

(36)

where 119910 1199101015840 11991010158401015840 and 119910

101584010158401015840 and 119910 1199101015840 11991010158401015840 and 119910

101584010158401015840

are solutions using the higher order formula and thelower order formula respectively

The local error estimation EST can be used to control thestep size ℎ by the standard formula as given in [35ndash38]

ℎ119899+1 = 09ℎ119899 (TOLEST

)

1(V+1) (37)

where 09 is a safety factor the local error estimation ateach step is represented by EST and TOL is the maximumallowable local error which is the precision required

Mathematical Problems in Engineering 5

If EST le TOL then the step is accepted and we appliedthe procedure of performing local extrapolation (or higherorder mode) meaning that the more accurate approximationwill be used to advance the integration If EST gt TOL thenthe step is rejected and the step size ℎ will be updated usingformula (37)

3 Construction of Embedded ExplicitRKFD Pairs

The construction of embedded explicit RKFD pairs will bediscussed in this section In specific we will derive twoembedded RKFD pairs of orders 5(4) and 6(5) with three andfour stages per step respectively

31The Derivation of Embedded RKFD5(4) Pair This sectionwill focus on the derivation of embedded RKFD5(4) pair withthree stages The authors in [24] derived three-stage fifth-order RKFD method and the solution is given as follows

1198882 =3

5+

radic6

10

1198883 =3

5minus

radic6

10

119887101584010158401015840

1 =1

9

119887101584010158401015840

2 =4

9minus

radic6

36

119887101584010158401015840

3 =4

9+

radic6

36

11988710158401015840

1 =1

9

11988710158401015840

2 =7

36minus

radic6

18

11988710158401015840

3 =7

36+

radic6

18

1198871015840

1 =1

18

1198871015840

2 =1

18minus

radic6

48

1198871015840

3 =1

18+

radic6

48

1198871 =19

1080

1198872 =13

1080minus

11radic6

2160

1198873 =13

1080+

11radic6

2160

11988621 =4059

187793

120591998400(5) 2

0

0005

001

0015

002

0025

003

005 01 015 02 025 030b998400

3

Figure 1 The graph of 1205911015840(5)2 versus 10158403

11988631 = minus1502

532215

11988632 =1826

569317

(38)

Now based on the above solution for values of 119860 and 119888 wederive a three-stage fourth-order embedded formula Solvingthe algebraic conditions (6) (10)-(11) (16)ndash(18) and (23)ndash(26)simultaneously gives a solution for 119894 in terms of 1 and 3 andthe solution of 1015840119894 in terms of 10158403 which are given as follows

2 = minus1 minus 3 +1

24

1015840

1 =1

12+

1

72radic6 + (

2

5minus

2

5radic6)

1015840

3

1015840

2 =1

12minus

1

72radic6 minus (

7

5minus

2

5radic6)

1015840

3

(39)

while the values of 10158401015840119894 and 101584010158401015840119894 119894 = 1 2 3 are the same as the

fifth-order method Our objective now is to select the valuesof the free parameters 1 3 and

10158403 and thus the values of119862

(6)119861(6) 1198621015840(6) 1198611015840(6) 120591(5)2 and 120591

1015840(5)2 are as small as possible

Using the above solution we get 1198621015840(6)

= 1200 is a constantvalue however 119861

1015840(6) and 1205911015840(5)

2 are functions in terms of10158403 by plotting the graph of 120591

1015840(5)2 versus

10158403 (see Figure 1)

From the numerical experiment and for the accurate resultwe choose

10158403 = 110 using Minimize command in Maple

Software which gives the error norm 1205911015840(5)

2 = 114566368times

10minus3 and 119861

1015840(6)= 04726195824

The quantities119862(6)119861(6) and 120591(5)

2 are functions in termsof 1 3 Choose 1 = 1100 and plot the graph of 119862

(6)119861(6) and 120591

(5)2 against 3 in the interval [0 008] From

numerical experiment and for the optimized pair therefore3 = 3100 is chosen which gives 119861

(6)= 05887532657

119862(6)

= 05887532657 and 120591(5)

2 = 3726445732 times 10minus3

6 Mathematical Problems in Engineering

Consequently we denote this pair as RKFD5(4) method andit can be written in Butcher tableau as follows

3

5+

radic6

10

4059

187793

3

5minus

radic6

10minus

1502

532215

1826

569317

19

1080

13

1080minus

11radic6

2160

13

1080+

11radic6

2160

1

18

1

18minus

radic6

48

1

18+

radic6

48

1

9

7

36minus

radic6

18

7

36+

radic6

18

1

9

4

9minus

radic6

36

4

9+

radic6

36

1

100

1

600

3

100

37

300minus

47radic6

1800minus

17

300+

47radic6

1800

1

10

1

9

7

36minus

radic6

18

7

36+

radic6

18

1

9

4

9minus

radic6

36

4

9+

radic6

36

(40)

32 The Derivation of Embedded RKFD6(5) Pair In this sec-tion a four-stage embedded RKFD6(5) pair will be derivedFor the sixth-order (119903 = 6) method the order conditions ofRKFD method up to order six need to be solved to derivea four-stage sixth-order RKFD method namely RKFD6method To determine the parameters of RKFD6method wechoose (10)ndash(13) from order conditions for 119910

1015840 (16)ndash(20) fromorder conditions for 119910

10158401015840 and (23)ndash(27) and (29) from orderconditions for 119910

101584010158401015840 It involved 15 nonlinear equations with 15unknowns and then was solved simultaneously which resultsin a unique solution as follows

1198871015840

1 =1

24

1198871015840

2 =1

16minus

radic5

48

1198871015840

3 =1

16+

radic5

48

1198871015840

4 = 0

11988710158401015840

1 =1

12

11988710158401015840

2 =5

24minus

radic5

24

11988710158401015840

3 =5

24+

radic5

24

11988710158401015840

4 = 0

119887101584010158401015840

1 =1

12

119887101584010158401015840

2 =5

12

119887101584010158401015840

3 =5

12

119887101584010158401015840

4 =1

12

1198882 =1

2+

radic5

10

1198883 =1

2minus

radic5

10

1198884 = 1

(41)

Substituting the values of the above 1198882 1198883 and 1198884 into the orderconditions for 119910 we get

1198872 =1

72minus

7radic5

360+ radic51198871

1198873 =1

72+

7radic5

360minus radic51198871

1198874 = minus1198871 +1

72

(42)

Here we have one free parameter 1198871 which can be chosen byminimizing the error norm of the seventh order conditionsfor 119910 according to Dormand et al [34] The error norms andthe global error of the seventh order conditions are definedas follows

10038171003817100381710038171003817120591(7)100381710038171003817100381710038172

= radic

119899119901+1

sum

119894=1

(120591(7)

119894)2

100381710038171003817100381710038171205911015840(7)100381710038171003817100381710038172

= radic

1198991015840119901+1

sum

119894=1

(1205911015840(7)

119894)2

1003817100381710038171003817100381712059110158401015840(7)100381710038171003817100381710038172

= radic

11989910158401015840119901+1

sum

119894=1

(12059110158401015840(7)

119894)2

10038171003817100381710038171003817120591101584010158401015840(7)100381710038171003817100381710038172

= radic

119899101584010158401015840119901+1

sum

119894=1

(120591101584010158401015840(7)

119894)2

10038171003817100381710038171003817120591(7)

119892

100381710038171003817100381710038172

= radic

119899119901+1

sum

119894=1

(120591(7)

119894)2+

1198991015840119901+1

sum

119894=1

(1205911015840(7)

119894)2+

11989910158401015840119901+1

sum

119894=1

(12059110158401015840(7)

119894)2+

119899101584010158401015840119901+1

sum

119894=1

(120591101584010158401015840(7)

119894)2

(43)

Mathematical Problems in Engineering 7

where 120591(7) 1205911015840(7) 120591

10158401015840(7) and 120591101584010158401015840(7) are the local truncation

errors norms for 119910 1199101015840 11991010158401015840 and 119910101584010158401015840 of the RKFD6 method

respectively and 120591(7)119892 is the global error As a result we find

the error equations of 119910 as

10038171003817100381710038171003817120591(7)100381710038171003817100381710038172

=1

6300

radic(minus17 + 12601198871)2 (44)

By minimizing 120591(7)

2 with respect to the free parameter 1198871we obtain 1198871 = 171260 that is the optimal value and gives120591(7)

2 = 0 and

1198872 =1

72minus

radic5

168

1198873 =1

72+

radic5

168

1198874 =1

2520

(45)

To find the coefficients 11988621 11988631 11988632 11988641 11988642 and 11988643substituting the values of 1198871 1198872 1198873 1198874 119887

10158401 11988710158402 11988710158403 11988710158404 119887101584010158401 119887101584010158402

119887101584010158403 119887101584010158404 1198871015840101584010158401 1198871015840101584010158402 1198871015840101584010158403 and 119887

1015840101584010158404 into (28) (30) and (31) of the

order conditions of 119910101584010158401015840 and (15) of the order conditions of

1199101015840 the following system of nonlinear equations needs to be

solved

1

1211988632 +

1

24(11988642 + 11988643) +

radic5

120(11988642 minus 11988643) =

1

840 (46)

(1

8+

radic5

24) 11988632 +

1

40(11988642 + 11988643)

+radic5

120(11988642 minus 11988643) =

1

2520

(47)

(1

16minus

radic5

48) 11988621 + (

1

16+

radic5

48) (11988631 + 11988632) =

1

5040 (48)

5

12(11988621 + 11988631 + 11988632) +

1

12(11988641 + 11988642 + 11988643) =

1

120 (49)

(5

24+

5radic5

120) 11988632 + (

1

24+

radic5

120) 11988642

+ (1

24minus

radic5

120) 11988643 =

1

720

(50)

(5

24+

5radic5

120) 11988621 + (

5

24minus

5radic5

120) (11988631 + 11988632)

+1

12(11988641 + 11988642 + 11988643) =

1

144

(51)

Solving (46)ndash(51) simultaneously we get

11988621 =1

168+

11radic5

4200

11988631 =1

420minus

radic5

700

11988632 =1

280minus

radic5

840

11988641 = minus1

840minus

radic5

168

11988642 =1

84minus

radic5

140

11988643 =5

168minus

11radic5

840

(52)

Consequently the local truncation error norms and theglobal error of the seventh order conditions of RKFD6method are calculated and given as follows

10038171003817100381710038171003817120591(7)100381710038171003817100381710038172

= 0

100381710038171003817100381710038171205911015840(7)100381710038171003817100381710038172

= 2380952381 times 10minus4

1003817100381710038171003817100381712059110158401015840(7)100381710038171003817100381710038172

= 4761904762 times 10minus4

10038171003817100381710038171003817120591101584010158401015840(7)100381710038171003817100381710038172

= 4761904762 times 10minus4

10038171003817100381710038171003817120591(7)

119892

100381710038171003817100381710038172= 7142857143 times 10

minus4

(53)

Finally all the parameters of sixth-order RKFDmethod withfour-stage and denoted as RKFD6 are written in Butchertableau as follows

1

2+

radic5

10

1

168+

11radic5

4200

1

2minus

radic5

10

1

420minus

radic5

700

1

280minus

radic5

840

1 minus1

840minus

radic5

168

1

84minus

radic5

140

5

168+

11radic5

840

17

1260

1

72minus

radic5

168

1

72+

radic5

168

1

2520

1

24

1

16minus

radic5

48

1

16+

radic5

480

1

12

5

24minus

radic5

24

5

24+

radic5

240

1

12

5

12

5

12

1

12

(54)

Now depending on the above values of 119860 and 119888 of RKFD6method together with the equations up to order five that is(6)-(7) (10)ndash(12) (16)ndash(19) and (23)ndash(28) solving the system

8 Mathematical Problems in Engineering

simultaneously produces a solution of an embedded RKFDformula of order five which is given by

1 =1

30minus (

1

2minus

radic5

10) 2 minus (

1

2+

radic5

10) 3

4 =1

120minus (

1

2+

radic5

10) 2 minus (

1

2minus

radic5

10) 3

1015840

1 =1

16+

radic5

80minus

radic5

51015840

3

1015840

2 =1

8minus 1015840

3

1015840

4 = minus1

48minus

radic5

80+

radic5

51015840

3

(55)

while the values of 10158401015840119894 and

101584010158401015840119894 119894 = 1 2 3 4 are the same

as the RKFD6 method Now our goal is to determine the

values of the free parameters 2 3 and 10158403 such that the

values of 119862(7) 119861(7) 1198621015840(7) 119861

1015840(7) 1205911015840(6)

2 and 120591(6)

2 are assmall as possible The quantities 119862

(7) 119861(7) and 120591

(6)2 are

functions depending on 2 and 3 Picking 2 = 140 andplotting the graph of119862(7)119861(7) and 120591

(6)2 against 2 and from

the numerical experiment therefore we choose 3 = 120

that is the optimal value which gives 119862(7)

= 1507670351119861(7)

= 1507670351 and the local truncation error 120591(6)

2 =

9444444448times10minus3The quantities1198621015840(7) 1198611015840(7) and 120591

1015840(6)2 are

functions in terms of 10158403 Plotting the graph of these functionsand for the optimized pair we found that

10158403 = 110 is the

optimal value using Minimize command in Maple Softwarewhich yields 119862

1015840(7)= 1683161973 1198611015840(7) = 1708476855 and

1205911015840(6)

2 = 81256471 times 10minus4 (see Figure 2)

Finally all the parameters of the embedded pair arerepresented in the Butcher tableau and denoted as RKFD6(5)method as follows

1

2+

radic5

10

1

168+

11radic5

4200

1

2minus

radic5

10

1

420minus

radic5

700

1

280minus

radic5

840

1 minus1

840minus

radic5

168

1

84minus

radic5

140

5

168+

11radic5

840

17

1260

1

72minus

radic5

168

1

72+

radic5

168

1

2520

1

24

1

16minus

radic5

48

1

16+

radic5

480

1

12

5

24minus

radic5

24

5

24+

radic5

240

1

12

5

12

5

12

1

12

minus1

240minus

radic5

400

1

40

1

20minus

7

240+

radic5

400

1

16minus

3radic5

400

1

40

1

10minus

1

48+

3radic5

400

1

12

5

24minus

radic5

24

5

24+

radic5

240

1

12

5

12

5

12

1

12

(56)

4 Numerical Experiments

In this section we present some problems which involvespecial fourth-order ODEs of the form 119910

(119894V)(119909) = 119891(119909 119910(119909))

in order to test the performance of the two new embeddedpairs derived in Section 3 The numerical results obtainedare compared with the well-known embedded Runge-Kuttamethods which are chosen from the scientific literature after

Mathematical Problems in Engineering 9120591

998400(6) 2

005 01 015 02 025 030b998400

3

0

0002

0004

0006

0008

001

0012

0014

0016

Figure 2 The graph of 1205911015840(6)2 versus 10158403

transforming the same problems to a system of first-orderdifferential equations and solving themThemethods chosenin the numerical experiments are as follows

(i) RKFD6(5) the new embedded Runge-Kutta pair oforders 6(5) derived in Section 3 in this paper

(ii) RKFD5(4) the new embedded Runge-Kutta pair oforders 5(4) derived in Section 3 in this paper

(iii) RK6(5)V the embedded Runge-Kutta pair of orders6(5) derived by Verner as given in [39]

(iv) RK5(4)D the embedded Runge-Kutta pair of orders5(4) with FSAL property derived by Dormand andPrince [28]

Problem 1 We consider that the homogeneous linear prob-lem is as follows

119910(119894V)

= minus4119910

119910 (0) = 0 1199101015840(0) = 1 119910

10158401015840(0) = 2 119910

101584010158401015840(0) = 2

(57)

The exact solution is given by 119910(119909) = e119909 sin(119909) The problemis integrated in the interval [0 5]

Problem 2 We consider that the homogeneous nonlinearproblem is as follows

119910(119894V)

= minus15

161199107

119910 (0) = 1 1199101015840(0) =

1

2 11991010158401015840

(0) = minus1

4 119910101584010158401015840

(0) =3

8

(58)

The exact solution is given by 119910(119909) = radic1 + 119909 The problem isintegrated in the interval [0 10]

Problem 3 We consider that the inhomogeneous nonlinearproblem is as follows

119910(119894V)

= 1199102+ cos2 (119909) + sin (119909) minus 1

119910 (0) = 0 1199101015840(0) = 1 119910

10158401015840(0) = 0 119910

101584010158401015840(0) = minus1

(59)

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

35325215

log10(M

AXE

RR)

log10(function evaluations)

minus10

minus8

minus6

minus4

minus2

Figure 3 The efficiency curves for Problem 1

The exact solution is given by 119910(119909) = sin(119909) The problem isintegrated in the interval [0 8]

Problem 4 We consider that the linear system is as follows

119910(119894V)

= e3119909119906

119910 (0) = 1 1199101015840(0) = minus1 119910

10158401015840(0) = 1 119910

101584010158401015840(0) = minus1

119911(119894V)

= 16eminus119909119910

119911 (0) = 1 1199111015840(0) = minus2 119911

10158401015840(0) = 4 119911

101584010158401015840(0) = minus8

119908(119894V)

= 81eminus119909119911

119908 (0) = 1 1199081015840(0) = minus3 119908

10158401015840(0) = 9 119908

101584010158401015840(0) = minus27

119906(119894V)

= 256eminus119909119908

119906 (0) = 1 1199061015840(0) = minus4 119906

10158401015840(0) = 16 119906

101584010158401015840(0) = minus64

(60)

The exact solution is given by

119910 = eminus119909

119911 = eminus2119909

119908 = eminus3119909

119906 = eminus4119909

(61)

The problem is integrated in the interval [0 3]

In the numerical experiments we have computed foreach method and problem the maximum global error andthe number of function evaluations used in the integrationIn Figures 3ndash6 we plot these values in double logarithmic

10 Mathematical Problems in Engineering

18 2 22 24 26 28 314 16

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

log10(M

AXE

RR)

log10(function evaluations)

Figure 4 The efficiency curves for Problem 2

325215

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

log10(function evaluations)

log10(M

AXE

RR)

minus10

minus8

minus6

minus4

minus2

0

Figure 5 The efficiency curves for Problem 3

325215

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

log10(function evaluations)

log10(M

AXE

RR)

minus10

minus8

minus6

minus4

minus2

0

Figure 6 The efficiency curves for Problem 4

Table 1 The comparison results of the total number of successfulsteps and failure steps betweenRKFD6(5) RKFD5(4) RK6(5)V andRK5(4)D methods for all problems

Problems Methods Successful steps Failure steps

1

RKFD6(5) 440 4RKFD5(4) 705 0RK6(5)V 461 9RK5(4)D 768 0

2

RKFD6(5) 151 0RKFD5(4) 200 0RK6(5)V 208 0RK5(4)D 253 0

3

RKFD6(5) 330 4RKFD5(4) 466 0RK6(5)V 348 8RK5(4)D 497 0

4

RKFD6(5) 264 5RKFD5(4) 340 6RK6(5)V 339 5RK5(4)D 484 5

scale From Table 1 and Figures 3ndash6 we observed that thenew RKFD6(5) and RKFD5(4) methods are more efficient tosolve directly fourth-order differential equations comparedwith RK6(5)V and RK5(4)D methods in terms of functionevaluations and the total number of steps This is due tothe fact that when using RK6(5)V and RK5(4)D methodsthe fourth-order ODEs need to be transformed into a first-order system and hence the dimension will increase fourtimes

5 Conclusion

We have constructed the two pairs of embedded RKFDmethods for directly solving special fourth-order ODEs ofthe form 119910

(119894V)(119909) = 119891(119909 119910(119909)) using variable step size

codes in this paperThese methods are denoted as RKFD5(4)and RKFD6(5) respectively Based on the methods thevariables step size codes are developed and used to solvespecial fourth-order ODEs Numerical results show thatthe two new embedded RKFD pairs are more effective ascompared with the existing embedded Runge-Kutta pairs ofthe same order in the literature From the numerical resultswe conclude that the two new embedded RKFD methodsare computationally more efficient in solving special fourth-order ODEs and outperformed the well-known embeddedRunge-Kutta methods

Conflict of Interests

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

Mathematical Problems in Engineering 11

References

[1] S N Jator ldquoNumerical integrators for fourth order initial andboundary value problemsrdquo International Journal of Pure andApplied Mathematics vol 47 no 4 pp 563ndash576 2008

[2] Z Bai and H Wang ldquoOn positive solutions of some nonlinearfourth-order beam equationsrdquo Journal ofMathematical Analysisand Applications vol 270 no 2 pp 357ndash368 2002

[3] A K Alomari N Ratib Anakira A S Bataineh and I HashimldquoApproximate solution of nonlinear system of BVP arising influid flow problemrdquoMathematical Problems in Engineering vol2013 Article ID 136043 7 pages 2013

[4] A Malek and R Shekari Beidokhti ldquoNumerical solution forhigh order differential equations using a hybrid neuralnetworkmdashoptimization methodrdquo Applied Mathematics andComputation vol 183 no 1 pp 260ndash271 2006

[5] A Boutayeb and A Chetouani ldquoA mini-review of numericalmethods for high-order problemsrdquo International Journal ofComputer Mathematics vol 84 no 4 pp 563ndash579 2007

[6] P Onumanyi U W Sirisena and S N Jator ldquoContinuous finitedifference approximations for solving differential equationsrdquoInternational Journal of Computer Mathematics vol 72 no 1pp 15ndash27 1999

[7] D Sarafyan ldquoNew algorithms for the continuous approximatesolutions of ordinary differential equations and the upgradingof the order of the processesrdquo Computers amp Mathematics withApplications vol 20 no 1 pp 77ndash100 1990

[8] GDahlquist ldquoOn accuracy and unconditional stability of linearmultistep methods for second order differential equationsrdquo BITNumerical Mathematics vol 18 no 2 pp 133ndash136 1978

[9] R P K Chan P Leone and A Tsai ldquoOrder conditions andsymmetry for two-step hybrid methodsrdquo International Journalof Computer Mathematics vol 81 no 12 pp 1519ndash1536 2004

[10] C E Abhulimen and F O Otunta ldquoA sixth order multideriva-tivemultistepmethods for stiff system of differential equationsrdquoInternational Journal of Numerical Mathematics vol 1 no 2 pp248ndash268 2006

[11] J D Lambert Computational Methods in Ordinary DifferentialEquations Wiley London UK 1973

[12] D O Awoyemi ldquoAlgorithmic collocation approach for directsolution of fourth-order initial-value problems of ordinarydifferential equationsrdquo International Journal of ComputerMath-ematics vol 82 no 3 pp 321ndash329 2005

[13] NWaeleh Z A Majid F Ismail andM Suleiman ldquoNumericalsolution of higher order ordinary differential equations bydirect block coderdquo Journal of Mathematics and Statistics vol 8no 1 pp 77ndash81 2011

[14] S J Kayode ldquoAn efficient zero-stable numerical method forfourth-order differential equationsrdquo International Journal ofMathematics and Mathematical Sciences vol 2008 Article ID364021 10 pages 2008

[15] S J Kayode ldquoAnorder six zero-stablemethod for direct solutionof fourth order ordinary differential equationsrdquo AmericanJournal of Applied Sciences vol 5 no 11 pp 1461ndash1466 2008

[16] B T Olabode and T J Alabi ldquoDirect block predictor-correctormethod for the solution of general fourth order ODEsrdquo Journalof Mathematics Research vol 5 no 1 pp 26ndash33 2013

[17] L K Yap F Ismail and N Senu ldquoAn accurate block hybrid col-locationmethod for third order ordinary differential equationsrdquoJournal of Applied Mathematics vol 2014 Article ID 549597 9pages 2014

[18] S Mehrkanoon ldquoA direct variable step block multistep methodfor solving general third-order ODEsrdquo Numerical Algorithmsvol 57 no 1 pp 53ndash66 2011

[19] Z A Majid and M B Suleiman ldquoDirect integration implicitvariable steps method for solving higher order systems ofordinary differential equations directlyrdquo Sains Malaysiana vol35 no 2 pp 63ndash68 2006

[20] D O Awoyemi ldquoA P-stable linear multistep method for solvinggeneral third order ordinary differential equationsrdquo Interna-tional Journal of Computer Mathematics vol 80 no 8 pp 985ndash991 2003

[21] N Waeleh Z A Majid and F Ismail ldquoA new algorithm forsolving higher order IVPs of ODEsrdquo Applied MathematicalSciences vol 5 no 56 pp 2795ndash2805 2011

[22] D O Awoyemi and O M Idowu ldquoA class of hybrid collocationmethods for third-order ordinary differential equationsrdquo Inter-national Journal of Computer Mathematics vol 82 no 10 pp1287ndash1293 2005

[23] S N Jator ldquoSolving second order initial value problems by ahybrid multistep method without predictorsrdquo Applied Mathe-matics and Computation vol 217 no 8 pp 4036ndash4046 2010

[24] K Hussain F Ismail and N Senu ldquoRunge-Kutta type methodsfor directly solving special fourth-order ordinary differentialequationsrdquo Mathematical Problems in Engineering vol 2015Article ID 893763 11 pages 2015

[25] J R DormandNumerical Methods for Differential Equations AComputational Approach Library of Engineering MathematicsCRC Press Boca Raton Fla USA 1996

[26] W Gander and D Gruntz ldquoDerivation of numerical methodsusing computer algebrardquo SIAM Review vol 41 no 3 pp 577ndash593 1999

[27] F B Ismail and M B Suleiman ldquoEmbedded singly diagonallyimplicit Runge-Kutta methods (45) in (56) For the integrationof stiff systems of ODEsrdquo International Journal of ComputerMathematics vol 66 no 3-4 pp 325ndash341 1998

[28] J R Dormand and P J Prince ldquoA family of embeddedRunge-Kutta formulaerdquo Journal of Computational and AppliedMathematics vol 6 no 1 pp 19ndash26 1980

[29] P J Prince and J R Dormand ldquoHigh order embedded Runge-Kutta formulaerdquo Journal of Computational and Applied Mathe-matics vol 7 no 1 pp 67ndash75 1981

[30] R A Al-Khasawneh F Ismail and M Suleiman ldquoEmbeddeddiagonally implicit Runge-Kutta-Nystrom 4(3) pair for solvingspecial second-order IVPsrdquo Applied Mathematics and Compu-tation vol 190 no 2 pp 1803ndash1814 2007

[31] E Fehlberg ldquoLow-order classical Runge-Kutta formulas withstepsize control and their application to some heat transferproblemsrdquo NASA TR R-315 1969

[32] E Fehlberg ldquoClassical fifth sixth seventh and eighth orderRunge-Kutta formulas with stepsize controlrdquo NASA TR R-287NASA 1968

[33] M Calvo J I Montijano and L Randez ldquoA new embeddedpair of Runge-Kutta formulas of orders 5 and 6rdquo Computers ampMathematics with Applications vol 20 no 1 pp 15ndash24 1990

[34] J R Dormand M E A El-Mikkawy and P J Prince ldquoFamiliesof Runge-Kutta-Nystrom formulaerdquo IMA Journal of NumericalAnalysis vol 7 no 2 pp 235ndash250 1987

[35] L F Shampine Numerical Solution of Ordinary DifferentialEquations Chapman amp Hall New York NY USA 1994

[36] J C Butcher Numerical Methods for Ordinary DifferentialEquations John Wiley amp Sons Chichester UK 2008

12 Mathematical Problems in Engineering

[37] E Hairer S P Noslashrsett and G Wanner Solving Ordinary Dif-ferential Equations I Nonstiff Problems vol 8 Springer BerlinGermany 2nd edition 1993

[38] N Senu M Mechee F Ismail and Z Siri ldquoEmbedded explicitRungendashKutta type methods for directly solving special thirdorder differential equations 119910

101584010158401015840= 119891(119909 119910)rdquo Applied Mathemat-

ics and Computation vol 240 pp 281ndash293 2014[39] J H Verner ldquoExplicit Runge-Kutta pairs with lower stage-

orderrdquo Numerical Algorithms vol 65 no 3 pp 555ndash577 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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 5

If EST le TOL then the step is accepted and we appliedthe procedure of performing local extrapolation (or higherorder mode) meaning that the more accurate approximationwill be used to advance the integration If EST gt TOL thenthe step is rejected and the step size ℎ will be updated usingformula (37)

3 Construction of Embedded ExplicitRKFD Pairs

The construction of embedded explicit RKFD pairs will bediscussed in this section In specific we will derive twoembedded RKFD pairs of orders 5(4) and 6(5) with three andfour stages per step respectively

31The Derivation of Embedded RKFD5(4) Pair This sectionwill focus on the derivation of embedded RKFD5(4) pair withthree stages The authors in [24] derived three-stage fifth-order RKFD method and the solution is given as follows

1198882 =3

5+

radic6

10

1198883 =3

5minus

radic6

10

119887101584010158401015840

1 =1

9

119887101584010158401015840

2 =4

9minus

radic6

36

119887101584010158401015840

3 =4

9+

radic6

36

11988710158401015840

1 =1

9

11988710158401015840

2 =7

36minus

radic6

18

11988710158401015840

3 =7

36+

radic6

18

1198871015840

1 =1

18

1198871015840

2 =1

18minus

radic6

48

1198871015840

3 =1

18+

radic6

48

1198871 =19

1080

1198872 =13

1080minus

11radic6

2160

1198873 =13

1080+

11radic6

2160

11988621 =4059

187793

120591998400(5) 2

0

0005

001

0015

002

0025

003

005 01 015 02 025 030b998400

3

Figure 1 The graph of 1205911015840(5)2 versus 10158403

11988631 = minus1502

532215

11988632 =1826

569317

(38)

Now based on the above solution for values of 119860 and 119888 wederive a three-stage fourth-order embedded formula Solvingthe algebraic conditions (6) (10)-(11) (16)ndash(18) and (23)ndash(26)simultaneously gives a solution for 119894 in terms of 1 and 3 andthe solution of 1015840119894 in terms of 10158403 which are given as follows

2 = minus1 minus 3 +1

24

1015840

1 =1

12+

1

72radic6 + (

2

5minus

2

5radic6)

1015840

3

1015840

2 =1

12minus

1

72radic6 minus (

7

5minus

2

5radic6)

1015840

3

(39)

while the values of 10158401015840119894 and 101584010158401015840119894 119894 = 1 2 3 are the same as the

fifth-order method Our objective now is to select the valuesof the free parameters 1 3 and

10158403 and thus the values of119862

(6)119861(6) 1198621015840(6) 1198611015840(6) 120591(5)2 and 120591

1015840(5)2 are as small as possible

Using the above solution we get 1198621015840(6)

= 1200 is a constantvalue however 119861

1015840(6) and 1205911015840(5)

2 are functions in terms of10158403 by plotting the graph of 120591

1015840(5)2 versus

10158403 (see Figure 1)

From the numerical experiment and for the accurate resultwe choose

10158403 = 110 using Minimize command in Maple

Software which gives the error norm 1205911015840(5)

2 = 114566368times

10minus3 and 119861

1015840(6)= 04726195824

The quantities119862(6)119861(6) and 120591(5)

2 are functions in termsof 1 3 Choose 1 = 1100 and plot the graph of 119862

(6)119861(6) and 120591

(5)2 against 3 in the interval [0 008] From

numerical experiment and for the optimized pair therefore3 = 3100 is chosen which gives 119861

(6)= 05887532657

119862(6)

= 05887532657 and 120591(5)

2 = 3726445732 times 10minus3

6 Mathematical Problems in Engineering

Consequently we denote this pair as RKFD5(4) method andit can be written in Butcher tableau as follows

3

5+

radic6

10

4059

187793

3

5minus

radic6

10minus

1502

532215

1826

569317

19

1080

13

1080minus

11radic6

2160

13

1080+

11radic6

2160

1

18

1

18minus

radic6

48

1

18+

radic6

48

1

9

7

36minus

radic6

18

7

36+

radic6

18

1

9

4

9minus

radic6

36

4

9+

radic6

36

1

100

1

600

3

100

37

300minus

47radic6

1800minus

17

300+

47radic6

1800

1

10

1

9

7

36minus

radic6

18

7

36+

radic6

18

1

9

4

9minus

radic6

36

4

9+

radic6

36

(40)

32 The Derivation of Embedded RKFD6(5) Pair In this sec-tion a four-stage embedded RKFD6(5) pair will be derivedFor the sixth-order (119903 = 6) method the order conditions ofRKFD method up to order six need to be solved to derivea four-stage sixth-order RKFD method namely RKFD6method To determine the parameters of RKFD6method wechoose (10)ndash(13) from order conditions for 119910

1015840 (16)ndash(20) fromorder conditions for 119910

10158401015840 and (23)ndash(27) and (29) from orderconditions for 119910

101584010158401015840 It involved 15 nonlinear equations with 15unknowns and then was solved simultaneously which resultsin a unique solution as follows

1198871015840

1 =1

24

1198871015840

2 =1

16minus

radic5

48

1198871015840

3 =1

16+

radic5

48

1198871015840

4 = 0

11988710158401015840

1 =1

12

11988710158401015840

2 =5

24minus

radic5

24

11988710158401015840

3 =5

24+

radic5

24

11988710158401015840

4 = 0

119887101584010158401015840

1 =1

12

119887101584010158401015840

2 =5

12

119887101584010158401015840

3 =5

12

119887101584010158401015840

4 =1

12

1198882 =1

2+

radic5

10

1198883 =1

2minus

radic5

10

1198884 = 1

(41)

Substituting the values of the above 1198882 1198883 and 1198884 into the orderconditions for 119910 we get

1198872 =1

72minus

7radic5

360+ radic51198871

1198873 =1

72+

7radic5

360minus radic51198871

1198874 = minus1198871 +1

72

(42)

Here we have one free parameter 1198871 which can be chosen byminimizing the error norm of the seventh order conditionsfor 119910 according to Dormand et al [34] The error norms andthe global error of the seventh order conditions are definedas follows

10038171003817100381710038171003817120591(7)100381710038171003817100381710038172

= radic

119899119901+1

sum

119894=1

(120591(7)

119894)2

100381710038171003817100381710038171205911015840(7)100381710038171003817100381710038172

= radic

1198991015840119901+1

sum

119894=1

(1205911015840(7)

119894)2

1003817100381710038171003817100381712059110158401015840(7)100381710038171003817100381710038172

= radic

11989910158401015840119901+1

sum

119894=1

(12059110158401015840(7)

119894)2

10038171003817100381710038171003817120591101584010158401015840(7)100381710038171003817100381710038172

= radic

119899101584010158401015840119901+1

sum

119894=1

(120591101584010158401015840(7)

119894)2

10038171003817100381710038171003817120591(7)

119892

100381710038171003817100381710038172

= radic

119899119901+1

sum

119894=1

(120591(7)

119894)2+

1198991015840119901+1

sum

119894=1

(1205911015840(7)

119894)2+

11989910158401015840119901+1

sum

119894=1

(12059110158401015840(7)

119894)2+

119899101584010158401015840119901+1

sum

119894=1

(120591101584010158401015840(7)

119894)2

(43)

Mathematical Problems in Engineering 7

where 120591(7) 1205911015840(7) 120591

10158401015840(7) and 120591101584010158401015840(7) are the local truncation

errors norms for 119910 1199101015840 11991010158401015840 and 119910101584010158401015840 of the RKFD6 method

respectively and 120591(7)119892 is the global error As a result we find

the error equations of 119910 as

10038171003817100381710038171003817120591(7)100381710038171003817100381710038172

=1

6300

radic(minus17 + 12601198871)2 (44)

By minimizing 120591(7)

2 with respect to the free parameter 1198871we obtain 1198871 = 171260 that is the optimal value and gives120591(7)

2 = 0 and

1198872 =1

72minus

radic5

168

1198873 =1

72+

radic5

168

1198874 =1

2520

(45)

To find the coefficients 11988621 11988631 11988632 11988641 11988642 and 11988643substituting the values of 1198871 1198872 1198873 1198874 119887

10158401 11988710158402 11988710158403 11988710158404 119887101584010158401 119887101584010158402

119887101584010158403 119887101584010158404 1198871015840101584010158401 1198871015840101584010158402 1198871015840101584010158403 and 119887

1015840101584010158404 into (28) (30) and (31) of the

order conditions of 119910101584010158401015840 and (15) of the order conditions of

1199101015840 the following system of nonlinear equations needs to be

solved

1

1211988632 +

1

24(11988642 + 11988643) +

radic5

120(11988642 minus 11988643) =

1

840 (46)

(1

8+

radic5

24) 11988632 +

1

40(11988642 + 11988643)

+radic5

120(11988642 minus 11988643) =

1

2520

(47)

(1

16minus

radic5

48) 11988621 + (

1

16+

radic5

48) (11988631 + 11988632) =

1

5040 (48)

5

12(11988621 + 11988631 + 11988632) +

1

12(11988641 + 11988642 + 11988643) =

1

120 (49)

(5

24+

5radic5

120) 11988632 + (

1

24+

radic5

120) 11988642

+ (1

24minus

radic5

120) 11988643 =

1

720

(50)

(5

24+

5radic5

120) 11988621 + (

5

24minus

5radic5

120) (11988631 + 11988632)

+1

12(11988641 + 11988642 + 11988643) =

1

144

(51)

Solving (46)ndash(51) simultaneously we get

11988621 =1

168+

11radic5

4200

11988631 =1

420minus

radic5

700

11988632 =1

280minus

radic5

840

11988641 = minus1

840minus

radic5

168

11988642 =1

84minus

radic5

140

11988643 =5

168minus

11radic5

840

(52)

Consequently the local truncation error norms and theglobal error of the seventh order conditions of RKFD6method are calculated and given as follows

10038171003817100381710038171003817120591(7)100381710038171003817100381710038172

= 0

100381710038171003817100381710038171205911015840(7)100381710038171003817100381710038172

= 2380952381 times 10minus4

1003817100381710038171003817100381712059110158401015840(7)100381710038171003817100381710038172

= 4761904762 times 10minus4

10038171003817100381710038171003817120591101584010158401015840(7)100381710038171003817100381710038172

= 4761904762 times 10minus4

10038171003817100381710038171003817120591(7)

119892

100381710038171003817100381710038172= 7142857143 times 10

minus4

(53)

Finally all the parameters of sixth-order RKFDmethod withfour-stage and denoted as RKFD6 are written in Butchertableau as follows

1

2+

radic5

10

1

168+

11radic5

4200

1

2minus

radic5

10

1

420minus

radic5

700

1

280minus

radic5

840

1 minus1

840minus

radic5

168

1

84minus

radic5

140

5

168+

11radic5

840

17

1260

1

72minus

radic5

168

1

72+

radic5

168

1

2520

1

24

1

16minus

radic5

48

1

16+

radic5

480

1

12

5

24minus

radic5

24

5

24+

radic5

240

1

12

5

12

5

12

1

12

(54)

Now depending on the above values of 119860 and 119888 of RKFD6method together with the equations up to order five that is(6)-(7) (10)ndash(12) (16)ndash(19) and (23)ndash(28) solving the system

8 Mathematical Problems in Engineering

simultaneously produces a solution of an embedded RKFDformula of order five which is given by

1 =1

30minus (

1

2minus

radic5

10) 2 minus (

1

2+

radic5

10) 3

4 =1

120minus (

1

2+

radic5

10) 2 minus (

1

2minus

radic5

10) 3

1015840

1 =1

16+

radic5

80minus

radic5

51015840

3

1015840

2 =1

8minus 1015840

3

1015840

4 = minus1

48minus

radic5

80+

radic5

51015840

3

(55)

while the values of 10158401015840119894 and

101584010158401015840119894 119894 = 1 2 3 4 are the same

as the RKFD6 method Now our goal is to determine the

values of the free parameters 2 3 and 10158403 such that the

values of 119862(7) 119861(7) 1198621015840(7) 119861

1015840(7) 1205911015840(6)

2 and 120591(6)

2 are assmall as possible The quantities 119862

(7) 119861(7) and 120591

(6)2 are

functions depending on 2 and 3 Picking 2 = 140 andplotting the graph of119862(7)119861(7) and 120591

(6)2 against 2 and from

the numerical experiment therefore we choose 3 = 120

that is the optimal value which gives 119862(7)

= 1507670351119861(7)

= 1507670351 and the local truncation error 120591(6)

2 =

9444444448times10minus3The quantities1198621015840(7) 1198611015840(7) and 120591

1015840(6)2 are

functions in terms of 10158403 Plotting the graph of these functionsand for the optimized pair we found that

10158403 = 110 is the

optimal value using Minimize command in Maple Softwarewhich yields 119862

1015840(7)= 1683161973 1198611015840(7) = 1708476855 and

1205911015840(6)

2 = 81256471 times 10minus4 (see Figure 2)

Finally all the parameters of the embedded pair arerepresented in the Butcher tableau and denoted as RKFD6(5)method as follows

1

2+

radic5

10

1

168+

11radic5

4200

1

2minus

radic5

10

1

420minus

radic5

700

1

280minus

radic5

840

1 minus1

840minus

radic5

168

1

84minus

radic5

140

5

168+

11radic5

840

17

1260

1

72minus

radic5

168

1

72+

radic5

168

1

2520

1

24

1

16minus

radic5

48

1

16+

radic5

480

1

12

5

24minus

radic5

24

5

24+

radic5

240

1

12

5

12

5

12

1

12

minus1

240minus

radic5

400

1

40

1

20minus

7

240+

radic5

400

1

16minus

3radic5

400

1

40

1

10minus

1

48+

3radic5

400

1

12

5

24minus

radic5

24

5

24+

radic5

240

1

12

5

12

5

12

1

12

(56)

4 Numerical Experiments

In this section we present some problems which involvespecial fourth-order ODEs of the form 119910

(119894V)(119909) = 119891(119909 119910(119909))

in order to test the performance of the two new embeddedpairs derived in Section 3 The numerical results obtainedare compared with the well-known embedded Runge-Kuttamethods which are chosen from the scientific literature after

Mathematical Problems in Engineering 9120591

998400(6) 2

005 01 015 02 025 030b998400

3

0

0002

0004

0006

0008

001

0012

0014

0016

Figure 2 The graph of 1205911015840(6)2 versus 10158403

transforming the same problems to a system of first-orderdifferential equations and solving themThemethods chosenin the numerical experiments are as follows

(i) RKFD6(5) the new embedded Runge-Kutta pair oforders 6(5) derived in Section 3 in this paper

(ii) RKFD5(4) the new embedded Runge-Kutta pair oforders 5(4) derived in Section 3 in this paper

(iii) RK6(5)V the embedded Runge-Kutta pair of orders6(5) derived by Verner as given in [39]

(iv) RK5(4)D the embedded Runge-Kutta pair of orders5(4) with FSAL property derived by Dormand andPrince [28]

Problem 1 We consider that the homogeneous linear prob-lem is as follows

119910(119894V)

= minus4119910

119910 (0) = 0 1199101015840(0) = 1 119910

10158401015840(0) = 2 119910

101584010158401015840(0) = 2

(57)

The exact solution is given by 119910(119909) = e119909 sin(119909) The problemis integrated in the interval [0 5]

Problem 2 We consider that the homogeneous nonlinearproblem is as follows

119910(119894V)

= minus15

161199107

119910 (0) = 1 1199101015840(0) =

1

2 11991010158401015840

(0) = minus1

4 119910101584010158401015840

(0) =3

8

(58)

The exact solution is given by 119910(119909) = radic1 + 119909 The problem isintegrated in the interval [0 10]

Problem 3 We consider that the inhomogeneous nonlinearproblem is as follows

119910(119894V)

= 1199102+ cos2 (119909) + sin (119909) minus 1

119910 (0) = 0 1199101015840(0) = 1 119910

10158401015840(0) = 0 119910

101584010158401015840(0) = minus1

(59)

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

35325215

log10(M

AXE

RR)

log10(function evaluations)

minus10

minus8

minus6

minus4

minus2

Figure 3 The efficiency curves for Problem 1

The exact solution is given by 119910(119909) = sin(119909) The problem isintegrated in the interval [0 8]

Problem 4 We consider that the linear system is as follows

119910(119894V)

= e3119909119906

119910 (0) = 1 1199101015840(0) = minus1 119910

10158401015840(0) = 1 119910

101584010158401015840(0) = minus1

119911(119894V)

= 16eminus119909119910

119911 (0) = 1 1199111015840(0) = minus2 119911

10158401015840(0) = 4 119911

101584010158401015840(0) = minus8

119908(119894V)

= 81eminus119909119911

119908 (0) = 1 1199081015840(0) = minus3 119908

10158401015840(0) = 9 119908

101584010158401015840(0) = minus27

119906(119894V)

= 256eminus119909119908

119906 (0) = 1 1199061015840(0) = minus4 119906

10158401015840(0) = 16 119906

101584010158401015840(0) = minus64

(60)

The exact solution is given by

119910 = eminus119909

119911 = eminus2119909

119908 = eminus3119909

119906 = eminus4119909

(61)

The problem is integrated in the interval [0 3]

In the numerical experiments we have computed foreach method and problem the maximum global error andthe number of function evaluations used in the integrationIn Figures 3ndash6 we plot these values in double logarithmic

10 Mathematical Problems in Engineering

18 2 22 24 26 28 314 16

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

log10(M

AXE

RR)

log10(function evaluations)

Figure 4 The efficiency curves for Problem 2

325215

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

log10(function evaluations)

log10(M

AXE

RR)

minus10

minus8

minus6

minus4

minus2

0

Figure 5 The efficiency curves for Problem 3

325215

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

log10(function evaluations)

log10(M

AXE

RR)

minus10

minus8

minus6

minus4

minus2

0

Figure 6 The efficiency curves for Problem 4

Table 1 The comparison results of the total number of successfulsteps and failure steps betweenRKFD6(5) RKFD5(4) RK6(5)V andRK5(4)D methods for all problems

Problems Methods Successful steps Failure steps

1

RKFD6(5) 440 4RKFD5(4) 705 0RK6(5)V 461 9RK5(4)D 768 0

2

RKFD6(5) 151 0RKFD5(4) 200 0RK6(5)V 208 0RK5(4)D 253 0

3

RKFD6(5) 330 4RKFD5(4) 466 0RK6(5)V 348 8RK5(4)D 497 0

4

RKFD6(5) 264 5RKFD5(4) 340 6RK6(5)V 339 5RK5(4)D 484 5

scale From Table 1 and Figures 3ndash6 we observed that thenew RKFD6(5) and RKFD5(4) methods are more efficient tosolve directly fourth-order differential equations comparedwith RK6(5)V and RK5(4)D methods in terms of functionevaluations and the total number of steps This is due tothe fact that when using RK6(5)V and RK5(4)D methodsthe fourth-order ODEs need to be transformed into a first-order system and hence the dimension will increase fourtimes

5 Conclusion

We have constructed the two pairs of embedded RKFDmethods for directly solving special fourth-order ODEs ofthe form 119910

(119894V)(119909) = 119891(119909 119910(119909)) using variable step size

codes in this paperThese methods are denoted as RKFD5(4)and RKFD6(5) respectively Based on the methods thevariables step size codes are developed and used to solvespecial fourth-order ODEs Numerical results show thatthe two new embedded RKFD pairs are more effective ascompared with the existing embedded Runge-Kutta pairs ofthe same order in the literature From the numerical resultswe conclude that the two new embedded RKFD methodsare computationally more efficient in solving special fourth-order ODEs and outperformed the well-known embeddedRunge-Kutta methods

Conflict of Interests

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

Mathematical Problems in Engineering 11

References

[1] S N Jator ldquoNumerical integrators for fourth order initial andboundary value problemsrdquo International Journal of Pure andApplied Mathematics vol 47 no 4 pp 563ndash576 2008

[2] Z Bai and H Wang ldquoOn positive solutions of some nonlinearfourth-order beam equationsrdquo Journal ofMathematical Analysisand Applications vol 270 no 2 pp 357ndash368 2002

[3] A K Alomari N Ratib Anakira A S Bataineh and I HashimldquoApproximate solution of nonlinear system of BVP arising influid flow problemrdquoMathematical Problems in Engineering vol2013 Article ID 136043 7 pages 2013

[4] A Malek and R Shekari Beidokhti ldquoNumerical solution forhigh order differential equations using a hybrid neuralnetworkmdashoptimization methodrdquo Applied Mathematics andComputation vol 183 no 1 pp 260ndash271 2006

[5] A Boutayeb and A Chetouani ldquoA mini-review of numericalmethods for high-order problemsrdquo International Journal ofComputer Mathematics vol 84 no 4 pp 563ndash579 2007

[6] P Onumanyi U W Sirisena and S N Jator ldquoContinuous finitedifference approximations for solving differential equationsrdquoInternational Journal of Computer Mathematics vol 72 no 1pp 15ndash27 1999

[7] D Sarafyan ldquoNew algorithms for the continuous approximatesolutions of ordinary differential equations and the upgradingof the order of the processesrdquo Computers amp Mathematics withApplications vol 20 no 1 pp 77ndash100 1990

[8] GDahlquist ldquoOn accuracy and unconditional stability of linearmultistep methods for second order differential equationsrdquo BITNumerical Mathematics vol 18 no 2 pp 133ndash136 1978

[9] R P K Chan P Leone and A Tsai ldquoOrder conditions andsymmetry for two-step hybrid methodsrdquo International Journalof Computer Mathematics vol 81 no 12 pp 1519ndash1536 2004

[10] C E Abhulimen and F O Otunta ldquoA sixth order multideriva-tivemultistepmethods for stiff system of differential equationsrdquoInternational Journal of Numerical Mathematics vol 1 no 2 pp248ndash268 2006

[11] J D Lambert Computational Methods in Ordinary DifferentialEquations Wiley London UK 1973

[12] D O Awoyemi ldquoAlgorithmic collocation approach for directsolution of fourth-order initial-value problems of ordinarydifferential equationsrdquo International Journal of ComputerMath-ematics vol 82 no 3 pp 321ndash329 2005

[13] NWaeleh Z A Majid F Ismail andM Suleiman ldquoNumericalsolution of higher order ordinary differential equations bydirect block coderdquo Journal of Mathematics and Statistics vol 8no 1 pp 77ndash81 2011

[14] S J Kayode ldquoAn efficient zero-stable numerical method forfourth-order differential equationsrdquo International Journal ofMathematics and Mathematical Sciences vol 2008 Article ID364021 10 pages 2008

[15] S J Kayode ldquoAnorder six zero-stablemethod for direct solutionof fourth order ordinary differential equationsrdquo AmericanJournal of Applied Sciences vol 5 no 11 pp 1461ndash1466 2008

[16] B T Olabode and T J Alabi ldquoDirect block predictor-correctormethod for the solution of general fourth order ODEsrdquo Journalof Mathematics Research vol 5 no 1 pp 26ndash33 2013

[17] L K Yap F Ismail and N Senu ldquoAn accurate block hybrid col-locationmethod for third order ordinary differential equationsrdquoJournal of Applied Mathematics vol 2014 Article ID 549597 9pages 2014

[18] S Mehrkanoon ldquoA direct variable step block multistep methodfor solving general third-order ODEsrdquo Numerical Algorithmsvol 57 no 1 pp 53ndash66 2011

[19] Z A Majid and M B Suleiman ldquoDirect integration implicitvariable steps method for solving higher order systems ofordinary differential equations directlyrdquo Sains Malaysiana vol35 no 2 pp 63ndash68 2006

[20] D O Awoyemi ldquoA P-stable linear multistep method for solvinggeneral third order ordinary differential equationsrdquo Interna-tional Journal of Computer Mathematics vol 80 no 8 pp 985ndash991 2003

[21] N Waeleh Z A Majid and F Ismail ldquoA new algorithm forsolving higher order IVPs of ODEsrdquo Applied MathematicalSciences vol 5 no 56 pp 2795ndash2805 2011

[22] D O Awoyemi and O M Idowu ldquoA class of hybrid collocationmethods for third-order ordinary differential equationsrdquo Inter-national Journal of Computer Mathematics vol 82 no 10 pp1287ndash1293 2005

[23] S N Jator ldquoSolving second order initial value problems by ahybrid multistep method without predictorsrdquo Applied Mathe-matics and Computation vol 217 no 8 pp 4036ndash4046 2010

[24] K Hussain F Ismail and N Senu ldquoRunge-Kutta type methodsfor directly solving special fourth-order ordinary differentialequationsrdquo Mathematical Problems in Engineering vol 2015Article ID 893763 11 pages 2015

[25] J R DormandNumerical Methods for Differential Equations AComputational Approach Library of Engineering MathematicsCRC Press Boca Raton Fla USA 1996

[26] W Gander and D Gruntz ldquoDerivation of numerical methodsusing computer algebrardquo SIAM Review vol 41 no 3 pp 577ndash593 1999

[27] F B Ismail and M B Suleiman ldquoEmbedded singly diagonallyimplicit Runge-Kutta methods (45) in (56) For the integrationof stiff systems of ODEsrdquo International Journal of ComputerMathematics vol 66 no 3-4 pp 325ndash341 1998

[28] J R Dormand and P J Prince ldquoA family of embeddedRunge-Kutta formulaerdquo Journal of Computational and AppliedMathematics vol 6 no 1 pp 19ndash26 1980

[29] P J Prince and J R Dormand ldquoHigh order embedded Runge-Kutta formulaerdquo Journal of Computational and Applied Mathe-matics vol 7 no 1 pp 67ndash75 1981

[30] R A Al-Khasawneh F Ismail and M Suleiman ldquoEmbeddeddiagonally implicit Runge-Kutta-Nystrom 4(3) pair for solvingspecial second-order IVPsrdquo Applied Mathematics and Compu-tation vol 190 no 2 pp 1803ndash1814 2007

[31] E Fehlberg ldquoLow-order classical Runge-Kutta formulas withstepsize control and their application to some heat transferproblemsrdquo NASA TR R-315 1969

[32] E Fehlberg ldquoClassical fifth sixth seventh and eighth orderRunge-Kutta formulas with stepsize controlrdquo NASA TR R-287NASA 1968

[33] M Calvo J I Montijano and L Randez ldquoA new embeddedpair of Runge-Kutta formulas of orders 5 and 6rdquo Computers ampMathematics with Applications vol 20 no 1 pp 15ndash24 1990

[34] J R Dormand M E A El-Mikkawy and P J Prince ldquoFamiliesof Runge-Kutta-Nystrom formulaerdquo IMA Journal of NumericalAnalysis vol 7 no 2 pp 235ndash250 1987

[35] L F Shampine Numerical Solution of Ordinary DifferentialEquations Chapman amp Hall New York NY USA 1994

[36] J C Butcher Numerical Methods for Ordinary DifferentialEquations John Wiley amp Sons Chichester UK 2008

12 Mathematical Problems in Engineering

[37] E Hairer S P Noslashrsett and G Wanner Solving Ordinary Dif-ferential Equations I Nonstiff Problems vol 8 Springer BerlinGermany 2nd edition 1993

[38] N Senu M Mechee F Ismail and Z Siri ldquoEmbedded explicitRungendashKutta type methods for directly solving special thirdorder differential equations 119910

101584010158401015840= 119891(119909 119910)rdquo Applied Mathemat-

ics and Computation vol 240 pp 281ndash293 2014[39] J H Verner ldquoExplicit Runge-Kutta pairs with lower stage-

orderrdquo Numerical Algorithms vol 65 no 3 pp 555ndash577 2014

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

6 Mathematical Problems in Engineering

Consequently we denote this pair as RKFD5(4) method andit can be written in Butcher tableau as follows

3

5+

radic6

10

4059

187793

3

5minus

radic6

10minus

1502

532215

1826

569317

19

1080

13

1080minus

11radic6

2160

13

1080+

11radic6

2160

1

18

1

18minus

radic6

48

1

18+

radic6

48

1

9

7

36minus

radic6

18

7

36+

radic6

18

1

9

4

9minus

radic6

36

4

9+

radic6

36

1

100

1

600

3

100

37

300minus

47radic6

1800minus

17

300+

47radic6

1800

1

10

1

9

7

36minus

radic6

18

7

36+

radic6

18

1

9

4

9minus

radic6

36

4

9+

radic6

36

(40)

32 The Derivation of Embedded RKFD6(5) Pair In this sec-tion a four-stage embedded RKFD6(5) pair will be derivedFor the sixth-order (119903 = 6) method the order conditions ofRKFD method up to order six need to be solved to derivea four-stage sixth-order RKFD method namely RKFD6method To determine the parameters of RKFD6method wechoose (10)ndash(13) from order conditions for 119910

1015840 (16)ndash(20) fromorder conditions for 119910

10158401015840 and (23)ndash(27) and (29) from orderconditions for 119910

101584010158401015840 It involved 15 nonlinear equations with 15unknowns and then was solved simultaneously which resultsin a unique solution as follows

1198871015840

1 =1

24

1198871015840

2 =1

16minus

radic5

48

1198871015840

3 =1

16+

radic5

48

1198871015840

4 = 0

11988710158401015840

1 =1

12

11988710158401015840

2 =5

24minus

radic5

24

11988710158401015840

3 =5

24+

radic5

24

11988710158401015840

4 = 0

119887101584010158401015840

1 =1

12

119887101584010158401015840

2 =5

12

119887101584010158401015840

3 =5

12

119887101584010158401015840

4 =1

12

1198882 =1

2+

radic5

10

1198883 =1

2minus

radic5

10

1198884 = 1

(41)

Substituting the values of the above 1198882 1198883 and 1198884 into the orderconditions for 119910 we get

1198872 =1

72minus

7radic5

360+ radic51198871

1198873 =1

72+

7radic5

360minus radic51198871

1198874 = minus1198871 +1

72

(42)

Here we have one free parameter 1198871 which can be chosen byminimizing the error norm of the seventh order conditionsfor 119910 according to Dormand et al [34] The error norms andthe global error of the seventh order conditions are definedas follows

10038171003817100381710038171003817120591(7)100381710038171003817100381710038172

= radic

119899119901+1

sum

119894=1

(120591(7)

119894)2

100381710038171003817100381710038171205911015840(7)100381710038171003817100381710038172

= radic

1198991015840119901+1

sum

119894=1

(1205911015840(7)

119894)2

1003817100381710038171003817100381712059110158401015840(7)100381710038171003817100381710038172

= radic

11989910158401015840119901+1

sum

119894=1

(12059110158401015840(7)

119894)2

10038171003817100381710038171003817120591101584010158401015840(7)100381710038171003817100381710038172

= radic

119899101584010158401015840119901+1

sum

119894=1

(120591101584010158401015840(7)

119894)2

10038171003817100381710038171003817120591(7)

119892

100381710038171003817100381710038172

= radic

119899119901+1

sum

119894=1

(120591(7)

119894)2+

1198991015840119901+1

sum

119894=1

(1205911015840(7)

119894)2+

11989910158401015840119901+1

sum

119894=1

(12059110158401015840(7)

119894)2+

119899101584010158401015840119901+1

sum

119894=1

(120591101584010158401015840(7)

119894)2

(43)

Mathematical Problems in Engineering 7

where 120591(7) 1205911015840(7) 120591

10158401015840(7) and 120591101584010158401015840(7) are the local truncation

errors norms for 119910 1199101015840 11991010158401015840 and 119910101584010158401015840 of the RKFD6 method

respectively and 120591(7)119892 is the global error As a result we find

the error equations of 119910 as

10038171003817100381710038171003817120591(7)100381710038171003817100381710038172

=1

6300

radic(minus17 + 12601198871)2 (44)

By minimizing 120591(7)

2 with respect to the free parameter 1198871we obtain 1198871 = 171260 that is the optimal value and gives120591(7)

2 = 0 and

1198872 =1

72minus

radic5

168

1198873 =1

72+

radic5

168

1198874 =1

2520

(45)

To find the coefficients 11988621 11988631 11988632 11988641 11988642 and 11988643substituting the values of 1198871 1198872 1198873 1198874 119887

10158401 11988710158402 11988710158403 11988710158404 119887101584010158401 119887101584010158402

119887101584010158403 119887101584010158404 1198871015840101584010158401 1198871015840101584010158402 1198871015840101584010158403 and 119887

1015840101584010158404 into (28) (30) and (31) of the

order conditions of 119910101584010158401015840 and (15) of the order conditions of

1199101015840 the following system of nonlinear equations needs to be

solved

1

1211988632 +

1

24(11988642 + 11988643) +

radic5

120(11988642 minus 11988643) =

1

840 (46)

(1

8+

radic5

24) 11988632 +

1

40(11988642 + 11988643)

+radic5

120(11988642 minus 11988643) =

1

2520

(47)

(1

16minus

radic5

48) 11988621 + (

1

16+

radic5

48) (11988631 + 11988632) =

1

5040 (48)

5

12(11988621 + 11988631 + 11988632) +

1

12(11988641 + 11988642 + 11988643) =

1

120 (49)

(5

24+

5radic5

120) 11988632 + (

1

24+

radic5

120) 11988642

+ (1

24minus

radic5

120) 11988643 =

1

720

(50)

(5

24+

5radic5

120) 11988621 + (

5

24minus

5radic5

120) (11988631 + 11988632)

+1

12(11988641 + 11988642 + 11988643) =

1

144

(51)

Solving (46)ndash(51) simultaneously we get

11988621 =1

168+

11radic5

4200

11988631 =1

420minus

radic5

700

11988632 =1

280minus

radic5

840

11988641 = minus1

840minus

radic5

168

11988642 =1

84minus

radic5

140

11988643 =5

168minus

11radic5

840

(52)

Consequently the local truncation error norms and theglobal error of the seventh order conditions of RKFD6method are calculated and given as follows

10038171003817100381710038171003817120591(7)100381710038171003817100381710038172

= 0

100381710038171003817100381710038171205911015840(7)100381710038171003817100381710038172

= 2380952381 times 10minus4

1003817100381710038171003817100381712059110158401015840(7)100381710038171003817100381710038172

= 4761904762 times 10minus4

10038171003817100381710038171003817120591101584010158401015840(7)100381710038171003817100381710038172

= 4761904762 times 10minus4

10038171003817100381710038171003817120591(7)

119892

100381710038171003817100381710038172= 7142857143 times 10

minus4

(53)

Finally all the parameters of sixth-order RKFDmethod withfour-stage and denoted as RKFD6 are written in Butchertableau as follows

1

2+

radic5

10

1

168+

11radic5

4200

1

2minus

radic5

10

1

420minus

radic5

700

1

280minus

radic5

840

1 minus1

840minus

radic5

168

1

84minus

radic5

140

5

168+

11radic5

840

17

1260

1

72minus

radic5

168

1

72+

radic5

168

1

2520

1

24

1

16minus

radic5

48

1

16+

radic5

480

1

12

5

24minus

radic5

24

5

24+

radic5

240

1

12

5

12

5

12

1

12

(54)

Now depending on the above values of 119860 and 119888 of RKFD6method together with the equations up to order five that is(6)-(7) (10)ndash(12) (16)ndash(19) and (23)ndash(28) solving the system

8 Mathematical Problems in Engineering

simultaneously produces a solution of an embedded RKFDformula of order five which is given by

1 =1

30minus (

1

2minus

radic5

10) 2 minus (

1

2+

radic5

10) 3

4 =1

120minus (

1

2+

radic5

10) 2 minus (

1

2minus

radic5

10) 3

1015840

1 =1

16+

radic5

80minus

radic5

51015840

3

1015840

2 =1

8minus 1015840

3

1015840

4 = minus1

48minus

radic5

80+

radic5

51015840

3

(55)

while the values of 10158401015840119894 and

101584010158401015840119894 119894 = 1 2 3 4 are the same

as the RKFD6 method Now our goal is to determine the

values of the free parameters 2 3 and 10158403 such that the

values of 119862(7) 119861(7) 1198621015840(7) 119861

1015840(7) 1205911015840(6)

2 and 120591(6)

2 are assmall as possible The quantities 119862

(7) 119861(7) and 120591

(6)2 are

functions depending on 2 and 3 Picking 2 = 140 andplotting the graph of119862(7)119861(7) and 120591

(6)2 against 2 and from

the numerical experiment therefore we choose 3 = 120

that is the optimal value which gives 119862(7)

= 1507670351119861(7)

= 1507670351 and the local truncation error 120591(6)

2 =

9444444448times10minus3The quantities1198621015840(7) 1198611015840(7) and 120591

1015840(6)2 are

functions in terms of 10158403 Plotting the graph of these functionsand for the optimized pair we found that

10158403 = 110 is the

optimal value using Minimize command in Maple Softwarewhich yields 119862

1015840(7)= 1683161973 1198611015840(7) = 1708476855 and

1205911015840(6)

2 = 81256471 times 10minus4 (see Figure 2)

Finally all the parameters of the embedded pair arerepresented in the Butcher tableau and denoted as RKFD6(5)method as follows

1

2+

radic5

10

1

168+

11radic5

4200

1

2minus

radic5

10

1

420minus

radic5

700

1

280minus

radic5

840

1 minus1

840minus

radic5

168

1

84minus

radic5

140

5

168+

11radic5

840

17

1260

1

72minus

radic5

168

1

72+

radic5

168

1

2520

1

24

1

16minus

radic5

48

1

16+

radic5

480

1

12

5

24minus

radic5

24

5

24+

radic5

240

1

12

5

12

5

12

1

12

minus1

240minus

radic5

400

1

40

1

20minus

7

240+

radic5

400

1

16minus

3radic5

400

1

40

1

10minus

1

48+

3radic5

400

1

12

5

24minus

radic5

24

5

24+

radic5

240

1

12

5

12

5

12

1

12

(56)

4 Numerical Experiments

In this section we present some problems which involvespecial fourth-order ODEs of the form 119910

(119894V)(119909) = 119891(119909 119910(119909))

in order to test the performance of the two new embeddedpairs derived in Section 3 The numerical results obtainedare compared with the well-known embedded Runge-Kuttamethods which are chosen from the scientific literature after

Mathematical Problems in Engineering 9120591

998400(6) 2

005 01 015 02 025 030b998400

3

0

0002

0004

0006

0008

001

0012

0014

0016

Figure 2 The graph of 1205911015840(6)2 versus 10158403

transforming the same problems to a system of first-orderdifferential equations and solving themThemethods chosenin the numerical experiments are as follows

(i) RKFD6(5) the new embedded Runge-Kutta pair oforders 6(5) derived in Section 3 in this paper

(ii) RKFD5(4) the new embedded Runge-Kutta pair oforders 5(4) derived in Section 3 in this paper

(iii) RK6(5)V the embedded Runge-Kutta pair of orders6(5) derived by Verner as given in [39]

(iv) RK5(4)D the embedded Runge-Kutta pair of orders5(4) with FSAL property derived by Dormand andPrince [28]

Problem 1 We consider that the homogeneous linear prob-lem is as follows

119910(119894V)

= minus4119910

119910 (0) = 0 1199101015840(0) = 1 119910

10158401015840(0) = 2 119910

101584010158401015840(0) = 2

(57)

The exact solution is given by 119910(119909) = e119909 sin(119909) The problemis integrated in the interval [0 5]

Problem 2 We consider that the homogeneous nonlinearproblem is as follows

119910(119894V)

= minus15

161199107

119910 (0) = 1 1199101015840(0) =

1

2 11991010158401015840

(0) = minus1

4 119910101584010158401015840

(0) =3

8

(58)

The exact solution is given by 119910(119909) = radic1 + 119909 The problem isintegrated in the interval [0 10]

Problem 3 We consider that the inhomogeneous nonlinearproblem is as follows

119910(119894V)

= 1199102+ cos2 (119909) + sin (119909) minus 1

119910 (0) = 0 1199101015840(0) = 1 119910

10158401015840(0) = 0 119910

101584010158401015840(0) = minus1

(59)

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

35325215

log10(M

AXE

RR)

log10(function evaluations)

minus10

minus8

minus6

minus4

minus2

Figure 3 The efficiency curves for Problem 1

The exact solution is given by 119910(119909) = sin(119909) The problem isintegrated in the interval [0 8]

Problem 4 We consider that the linear system is as follows

119910(119894V)

= e3119909119906

119910 (0) = 1 1199101015840(0) = minus1 119910

10158401015840(0) = 1 119910

101584010158401015840(0) = minus1

119911(119894V)

= 16eminus119909119910

119911 (0) = 1 1199111015840(0) = minus2 119911

10158401015840(0) = 4 119911

101584010158401015840(0) = minus8

119908(119894V)

= 81eminus119909119911

119908 (0) = 1 1199081015840(0) = minus3 119908

10158401015840(0) = 9 119908

101584010158401015840(0) = minus27

119906(119894V)

= 256eminus119909119908

119906 (0) = 1 1199061015840(0) = minus4 119906

10158401015840(0) = 16 119906

101584010158401015840(0) = minus64

(60)

The exact solution is given by

119910 = eminus119909

119911 = eminus2119909

119908 = eminus3119909

119906 = eminus4119909

(61)

The problem is integrated in the interval [0 3]

In the numerical experiments we have computed foreach method and problem the maximum global error andthe number of function evaluations used in the integrationIn Figures 3ndash6 we plot these values in double logarithmic

10 Mathematical Problems in Engineering

18 2 22 24 26 28 314 16

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

log10(M

AXE

RR)

log10(function evaluations)

Figure 4 The efficiency curves for Problem 2

325215

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

log10(function evaluations)

log10(M

AXE

RR)

minus10

minus8

minus6

minus4

minus2

0

Figure 5 The efficiency curves for Problem 3

325215

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

log10(function evaluations)

log10(M

AXE

RR)

minus10

minus8

minus6

minus4

minus2

0

Figure 6 The efficiency curves for Problem 4

Table 1 The comparison results of the total number of successfulsteps and failure steps betweenRKFD6(5) RKFD5(4) RK6(5)V andRK5(4)D methods for all problems

Problems Methods Successful steps Failure steps

1

RKFD6(5) 440 4RKFD5(4) 705 0RK6(5)V 461 9RK5(4)D 768 0

2

RKFD6(5) 151 0RKFD5(4) 200 0RK6(5)V 208 0RK5(4)D 253 0

3

RKFD6(5) 330 4RKFD5(4) 466 0RK6(5)V 348 8RK5(4)D 497 0

4

RKFD6(5) 264 5RKFD5(4) 340 6RK6(5)V 339 5RK5(4)D 484 5

scale From Table 1 and Figures 3ndash6 we observed that thenew RKFD6(5) and RKFD5(4) methods are more efficient tosolve directly fourth-order differential equations comparedwith RK6(5)V and RK5(4)D methods in terms of functionevaluations and the total number of steps This is due tothe fact that when using RK6(5)V and RK5(4)D methodsthe fourth-order ODEs need to be transformed into a first-order system and hence the dimension will increase fourtimes

5 Conclusion

We have constructed the two pairs of embedded RKFDmethods for directly solving special fourth-order ODEs ofthe form 119910

(119894V)(119909) = 119891(119909 119910(119909)) using variable step size

codes in this paperThese methods are denoted as RKFD5(4)and RKFD6(5) respectively Based on the methods thevariables step size codes are developed and used to solvespecial fourth-order ODEs Numerical results show thatthe two new embedded RKFD pairs are more effective ascompared with the existing embedded Runge-Kutta pairs ofthe same order in the literature From the numerical resultswe conclude that the two new embedded RKFD methodsare computationally more efficient in solving special fourth-order ODEs and outperformed the well-known embeddedRunge-Kutta methods

Conflict of Interests

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

Mathematical Problems in Engineering 11

References

[1] S N Jator ldquoNumerical integrators for fourth order initial andboundary value problemsrdquo International Journal of Pure andApplied Mathematics vol 47 no 4 pp 563ndash576 2008

[2] Z Bai and H Wang ldquoOn positive solutions of some nonlinearfourth-order beam equationsrdquo Journal ofMathematical Analysisand Applications vol 270 no 2 pp 357ndash368 2002

[3] A K Alomari N Ratib Anakira A S Bataineh and I HashimldquoApproximate solution of nonlinear system of BVP arising influid flow problemrdquoMathematical Problems in Engineering vol2013 Article ID 136043 7 pages 2013

[4] A Malek and R Shekari Beidokhti ldquoNumerical solution forhigh order differential equations using a hybrid neuralnetworkmdashoptimization methodrdquo Applied Mathematics andComputation vol 183 no 1 pp 260ndash271 2006

[5] A Boutayeb and A Chetouani ldquoA mini-review of numericalmethods for high-order problemsrdquo International Journal ofComputer Mathematics vol 84 no 4 pp 563ndash579 2007

[6] P Onumanyi U W Sirisena and S N Jator ldquoContinuous finitedifference approximations for solving differential equationsrdquoInternational Journal of Computer Mathematics vol 72 no 1pp 15ndash27 1999

[7] D Sarafyan ldquoNew algorithms for the continuous approximatesolutions of ordinary differential equations and the upgradingof the order of the processesrdquo Computers amp Mathematics withApplications vol 20 no 1 pp 77ndash100 1990

[8] GDahlquist ldquoOn accuracy and unconditional stability of linearmultistep methods for second order differential equationsrdquo BITNumerical Mathematics vol 18 no 2 pp 133ndash136 1978

[9] R P K Chan P Leone and A Tsai ldquoOrder conditions andsymmetry for two-step hybrid methodsrdquo International Journalof Computer Mathematics vol 81 no 12 pp 1519ndash1536 2004

[10] C E Abhulimen and F O Otunta ldquoA sixth order multideriva-tivemultistepmethods for stiff system of differential equationsrdquoInternational Journal of Numerical Mathematics vol 1 no 2 pp248ndash268 2006

[11] J D Lambert Computational Methods in Ordinary DifferentialEquations Wiley London UK 1973

[12] D O Awoyemi ldquoAlgorithmic collocation approach for directsolution of fourth-order initial-value problems of ordinarydifferential equationsrdquo International Journal of ComputerMath-ematics vol 82 no 3 pp 321ndash329 2005

[13] NWaeleh Z A Majid F Ismail andM Suleiman ldquoNumericalsolution of higher order ordinary differential equations bydirect block coderdquo Journal of Mathematics and Statistics vol 8no 1 pp 77ndash81 2011

[14] S J Kayode ldquoAn efficient zero-stable numerical method forfourth-order differential equationsrdquo International Journal ofMathematics and Mathematical Sciences vol 2008 Article ID364021 10 pages 2008

[15] S J Kayode ldquoAnorder six zero-stablemethod for direct solutionof fourth order ordinary differential equationsrdquo AmericanJournal of Applied Sciences vol 5 no 11 pp 1461ndash1466 2008

[16] B T Olabode and T J Alabi ldquoDirect block predictor-correctormethod for the solution of general fourth order ODEsrdquo Journalof Mathematics Research vol 5 no 1 pp 26ndash33 2013

[17] L K Yap F Ismail and N Senu ldquoAn accurate block hybrid col-locationmethod for third order ordinary differential equationsrdquoJournal of Applied Mathematics vol 2014 Article ID 549597 9pages 2014

[18] S Mehrkanoon ldquoA direct variable step block multistep methodfor solving general third-order ODEsrdquo Numerical Algorithmsvol 57 no 1 pp 53ndash66 2011

[19] Z A Majid and M B Suleiman ldquoDirect integration implicitvariable steps method for solving higher order systems ofordinary differential equations directlyrdquo Sains Malaysiana vol35 no 2 pp 63ndash68 2006

[20] D O Awoyemi ldquoA P-stable linear multistep method for solvinggeneral third order ordinary differential equationsrdquo Interna-tional Journal of Computer Mathematics vol 80 no 8 pp 985ndash991 2003

[21] N Waeleh Z A Majid and F Ismail ldquoA new algorithm forsolving higher order IVPs of ODEsrdquo Applied MathematicalSciences vol 5 no 56 pp 2795ndash2805 2011

[22] D O Awoyemi and O M Idowu ldquoA class of hybrid collocationmethods for third-order ordinary differential equationsrdquo Inter-national Journal of Computer Mathematics vol 82 no 10 pp1287ndash1293 2005

[23] S N Jator ldquoSolving second order initial value problems by ahybrid multistep method without predictorsrdquo Applied Mathe-matics and Computation vol 217 no 8 pp 4036ndash4046 2010

[24] K Hussain F Ismail and N Senu ldquoRunge-Kutta type methodsfor directly solving special fourth-order ordinary differentialequationsrdquo Mathematical Problems in Engineering vol 2015Article ID 893763 11 pages 2015

[25] J R DormandNumerical Methods for Differential Equations AComputational Approach Library of Engineering MathematicsCRC Press Boca Raton Fla USA 1996

[26] W Gander and D Gruntz ldquoDerivation of numerical methodsusing computer algebrardquo SIAM Review vol 41 no 3 pp 577ndash593 1999

[27] F B Ismail and M B Suleiman ldquoEmbedded singly diagonallyimplicit Runge-Kutta methods (45) in (56) For the integrationof stiff systems of ODEsrdquo International Journal of ComputerMathematics vol 66 no 3-4 pp 325ndash341 1998

[28] J R Dormand and P J Prince ldquoA family of embeddedRunge-Kutta formulaerdquo Journal of Computational and AppliedMathematics vol 6 no 1 pp 19ndash26 1980

[29] P J Prince and J R Dormand ldquoHigh order embedded Runge-Kutta formulaerdquo Journal of Computational and Applied Mathe-matics vol 7 no 1 pp 67ndash75 1981

[30] R A Al-Khasawneh F Ismail and M Suleiman ldquoEmbeddeddiagonally implicit Runge-Kutta-Nystrom 4(3) pair for solvingspecial second-order IVPsrdquo Applied Mathematics and Compu-tation vol 190 no 2 pp 1803ndash1814 2007

[31] E Fehlberg ldquoLow-order classical Runge-Kutta formulas withstepsize control and their application to some heat transferproblemsrdquo NASA TR R-315 1969

[32] E Fehlberg ldquoClassical fifth sixth seventh and eighth orderRunge-Kutta formulas with stepsize controlrdquo NASA TR R-287NASA 1968

[33] M Calvo J I Montijano and L Randez ldquoA new embeddedpair of Runge-Kutta formulas of orders 5 and 6rdquo Computers ampMathematics with Applications vol 20 no 1 pp 15ndash24 1990

[34] J R Dormand M E A El-Mikkawy and P J Prince ldquoFamiliesof Runge-Kutta-Nystrom formulaerdquo IMA Journal of NumericalAnalysis vol 7 no 2 pp 235ndash250 1987

[35] L F Shampine Numerical Solution of Ordinary DifferentialEquations Chapman amp Hall New York NY USA 1994

[36] J C Butcher Numerical Methods for Ordinary DifferentialEquations John Wiley amp Sons Chichester UK 2008

12 Mathematical Problems in Engineering

[37] E Hairer S P Noslashrsett and G Wanner Solving Ordinary Dif-ferential Equations I Nonstiff Problems vol 8 Springer BerlinGermany 2nd edition 1993

[38] N Senu M Mechee F Ismail and Z Siri ldquoEmbedded explicitRungendashKutta type methods for directly solving special thirdorder differential equations 119910

101584010158401015840= 119891(119909 119910)rdquo Applied Mathemat-

ics and Computation vol 240 pp 281ndash293 2014[39] J H Verner ldquoExplicit Runge-Kutta pairs with lower stage-

orderrdquo Numerical Algorithms vol 65 no 3 pp 555ndash577 2014

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 7

where 120591(7) 1205911015840(7) 120591

10158401015840(7) and 120591101584010158401015840(7) are the local truncation

errors norms for 119910 1199101015840 11991010158401015840 and 119910101584010158401015840 of the RKFD6 method

respectively and 120591(7)119892 is the global error As a result we find

the error equations of 119910 as

10038171003817100381710038171003817120591(7)100381710038171003817100381710038172

=1

6300

radic(minus17 + 12601198871)2 (44)

By minimizing 120591(7)

2 with respect to the free parameter 1198871we obtain 1198871 = 171260 that is the optimal value and gives120591(7)

2 = 0 and

1198872 =1

72minus

radic5

168

1198873 =1

72+

radic5

168

1198874 =1

2520

(45)

To find the coefficients 11988621 11988631 11988632 11988641 11988642 and 11988643substituting the values of 1198871 1198872 1198873 1198874 119887

10158401 11988710158402 11988710158403 11988710158404 119887101584010158401 119887101584010158402

119887101584010158403 119887101584010158404 1198871015840101584010158401 1198871015840101584010158402 1198871015840101584010158403 and 119887

1015840101584010158404 into (28) (30) and (31) of the

order conditions of 119910101584010158401015840 and (15) of the order conditions of

1199101015840 the following system of nonlinear equations needs to be

solved

1

1211988632 +

1

24(11988642 + 11988643) +

radic5

120(11988642 minus 11988643) =

1

840 (46)

(1

8+

radic5

24) 11988632 +

1

40(11988642 + 11988643)

+radic5

120(11988642 minus 11988643) =

1

2520

(47)

(1

16minus

radic5

48) 11988621 + (

1

16+

radic5

48) (11988631 + 11988632) =

1

5040 (48)

5

12(11988621 + 11988631 + 11988632) +

1

12(11988641 + 11988642 + 11988643) =

1

120 (49)

(5

24+

5radic5

120) 11988632 + (

1

24+

radic5

120) 11988642

+ (1

24minus

radic5

120) 11988643 =

1

720

(50)

(5

24+

5radic5

120) 11988621 + (

5

24minus

5radic5

120) (11988631 + 11988632)

+1

12(11988641 + 11988642 + 11988643) =

1

144

(51)

Solving (46)ndash(51) simultaneously we get

11988621 =1

168+

11radic5

4200

11988631 =1

420minus

radic5

700

11988632 =1

280minus

radic5

840

11988641 = minus1

840minus

radic5

168

11988642 =1

84minus

radic5

140

11988643 =5

168minus

11radic5

840

(52)

Consequently the local truncation error norms and theglobal error of the seventh order conditions of RKFD6method are calculated and given as follows

10038171003817100381710038171003817120591(7)100381710038171003817100381710038172

= 0

100381710038171003817100381710038171205911015840(7)100381710038171003817100381710038172

= 2380952381 times 10minus4

1003817100381710038171003817100381712059110158401015840(7)100381710038171003817100381710038172

= 4761904762 times 10minus4

10038171003817100381710038171003817120591101584010158401015840(7)100381710038171003817100381710038172

= 4761904762 times 10minus4

10038171003817100381710038171003817120591(7)

119892

100381710038171003817100381710038172= 7142857143 times 10

minus4

(53)

Finally all the parameters of sixth-order RKFDmethod withfour-stage and denoted as RKFD6 are written in Butchertableau as follows

1

2+

radic5

10

1

168+

11radic5

4200

1

2minus

radic5

10

1

420minus

radic5

700

1

280minus

radic5

840

1 minus1

840minus

radic5

168

1

84minus

radic5

140

5

168+

11radic5

840

17

1260

1

72minus

radic5

168

1

72+

radic5

168

1

2520

1

24

1

16minus

radic5

48

1

16+

radic5

480

1

12

5

24minus

radic5

24

5

24+

radic5

240

1

12

5

12

5

12

1

12

(54)

Now depending on the above values of 119860 and 119888 of RKFD6method together with the equations up to order five that is(6)-(7) (10)ndash(12) (16)ndash(19) and (23)ndash(28) solving the system

8 Mathematical Problems in Engineering

simultaneously produces a solution of an embedded RKFDformula of order five which is given by

1 =1

30minus (

1

2minus

radic5

10) 2 minus (

1

2+

radic5

10) 3

4 =1

120minus (

1

2+

radic5

10) 2 minus (

1

2minus

radic5

10) 3

1015840

1 =1

16+

radic5

80minus

radic5

51015840

3

1015840

2 =1

8minus 1015840

3

1015840

4 = minus1

48minus

radic5

80+

radic5

51015840

3

(55)

while the values of 10158401015840119894 and

101584010158401015840119894 119894 = 1 2 3 4 are the same

as the RKFD6 method Now our goal is to determine the

values of the free parameters 2 3 and 10158403 such that the

values of 119862(7) 119861(7) 1198621015840(7) 119861

1015840(7) 1205911015840(6)

2 and 120591(6)

2 are assmall as possible The quantities 119862

(7) 119861(7) and 120591

(6)2 are

functions depending on 2 and 3 Picking 2 = 140 andplotting the graph of119862(7)119861(7) and 120591

(6)2 against 2 and from

the numerical experiment therefore we choose 3 = 120

that is the optimal value which gives 119862(7)

= 1507670351119861(7)

= 1507670351 and the local truncation error 120591(6)

2 =

9444444448times10minus3The quantities1198621015840(7) 1198611015840(7) and 120591

1015840(6)2 are

functions in terms of 10158403 Plotting the graph of these functionsand for the optimized pair we found that

10158403 = 110 is the

optimal value using Minimize command in Maple Softwarewhich yields 119862

1015840(7)= 1683161973 1198611015840(7) = 1708476855 and

1205911015840(6)

2 = 81256471 times 10minus4 (see Figure 2)

Finally all the parameters of the embedded pair arerepresented in the Butcher tableau and denoted as RKFD6(5)method as follows

1

2+

radic5

10

1

168+

11radic5

4200

1

2minus

radic5

10

1

420minus

radic5

700

1

280minus

radic5

840

1 minus1

840minus

radic5

168

1

84minus

radic5

140

5

168+

11radic5

840

17

1260

1

72minus

radic5

168

1

72+

radic5

168

1

2520

1

24

1

16minus

radic5

48

1

16+

radic5

480

1

12

5

24minus

radic5

24

5

24+

radic5

240

1

12

5

12

5

12

1

12

minus1

240minus

radic5

400

1

40

1

20minus

7

240+

radic5

400

1

16minus

3radic5

400

1

40

1

10minus

1

48+

3radic5

400

1

12

5

24minus

radic5

24

5

24+

radic5

240

1

12

5

12

5

12

1

12

(56)

4 Numerical Experiments

In this section we present some problems which involvespecial fourth-order ODEs of the form 119910

(119894V)(119909) = 119891(119909 119910(119909))

in order to test the performance of the two new embeddedpairs derived in Section 3 The numerical results obtainedare compared with the well-known embedded Runge-Kuttamethods which are chosen from the scientific literature after

Mathematical Problems in Engineering 9120591

998400(6) 2

005 01 015 02 025 030b998400

3

0

0002

0004

0006

0008

001

0012

0014

0016

Figure 2 The graph of 1205911015840(6)2 versus 10158403

transforming the same problems to a system of first-orderdifferential equations and solving themThemethods chosenin the numerical experiments are as follows

(i) RKFD6(5) the new embedded Runge-Kutta pair oforders 6(5) derived in Section 3 in this paper

(ii) RKFD5(4) the new embedded Runge-Kutta pair oforders 5(4) derived in Section 3 in this paper

(iii) RK6(5)V the embedded Runge-Kutta pair of orders6(5) derived by Verner as given in [39]

(iv) RK5(4)D the embedded Runge-Kutta pair of orders5(4) with FSAL property derived by Dormand andPrince [28]

Problem 1 We consider that the homogeneous linear prob-lem is as follows

119910(119894V)

= minus4119910

119910 (0) = 0 1199101015840(0) = 1 119910

10158401015840(0) = 2 119910

101584010158401015840(0) = 2

(57)

The exact solution is given by 119910(119909) = e119909 sin(119909) The problemis integrated in the interval [0 5]

Problem 2 We consider that the homogeneous nonlinearproblem is as follows

119910(119894V)

= minus15

161199107

119910 (0) = 1 1199101015840(0) =

1

2 11991010158401015840

(0) = minus1

4 119910101584010158401015840

(0) =3

8

(58)

The exact solution is given by 119910(119909) = radic1 + 119909 The problem isintegrated in the interval [0 10]

Problem 3 We consider that the inhomogeneous nonlinearproblem is as follows

119910(119894V)

= 1199102+ cos2 (119909) + sin (119909) minus 1

119910 (0) = 0 1199101015840(0) = 1 119910

10158401015840(0) = 0 119910

101584010158401015840(0) = minus1

(59)

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

35325215

log10(M

AXE

RR)

log10(function evaluations)

minus10

minus8

minus6

minus4

minus2

Figure 3 The efficiency curves for Problem 1

The exact solution is given by 119910(119909) = sin(119909) The problem isintegrated in the interval [0 8]

Problem 4 We consider that the linear system is as follows

119910(119894V)

= e3119909119906

119910 (0) = 1 1199101015840(0) = minus1 119910

10158401015840(0) = 1 119910

101584010158401015840(0) = minus1

119911(119894V)

= 16eminus119909119910

119911 (0) = 1 1199111015840(0) = minus2 119911

10158401015840(0) = 4 119911

101584010158401015840(0) = minus8

119908(119894V)

= 81eminus119909119911

119908 (0) = 1 1199081015840(0) = minus3 119908

10158401015840(0) = 9 119908

101584010158401015840(0) = minus27

119906(119894V)

= 256eminus119909119908

119906 (0) = 1 1199061015840(0) = minus4 119906

10158401015840(0) = 16 119906

101584010158401015840(0) = minus64

(60)

The exact solution is given by

119910 = eminus119909

119911 = eminus2119909

119908 = eminus3119909

119906 = eminus4119909

(61)

The problem is integrated in the interval [0 3]

In the numerical experiments we have computed foreach method and problem the maximum global error andthe number of function evaluations used in the integrationIn Figures 3ndash6 we plot these values in double logarithmic

10 Mathematical Problems in Engineering

18 2 22 24 26 28 314 16

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

log10(M

AXE

RR)

log10(function evaluations)

Figure 4 The efficiency curves for Problem 2

325215

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

log10(function evaluations)

log10(M

AXE

RR)

minus10

minus8

minus6

minus4

minus2

0

Figure 5 The efficiency curves for Problem 3

325215

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

log10(function evaluations)

log10(M

AXE

RR)

minus10

minus8

minus6

minus4

minus2

0

Figure 6 The efficiency curves for Problem 4

Table 1 The comparison results of the total number of successfulsteps and failure steps betweenRKFD6(5) RKFD5(4) RK6(5)V andRK5(4)D methods for all problems

Problems Methods Successful steps Failure steps

1

RKFD6(5) 440 4RKFD5(4) 705 0RK6(5)V 461 9RK5(4)D 768 0

2

RKFD6(5) 151 0RKFD5(4) 200 0RK6(5)V 208 0RK5(4)D 253 0

3

RKFD6(5) 330 4RKFD5(4) 466 0RK6(5)V 348 8RK5(4)D 497 0

4

RKFD6(5) 264 5RKFD5(4) 340 6RK6(5)V 339 5RK5(4)D 484 5

scale From Table 1 and Figures 3ndash6 we observed that thenew RKFD6(5) and RKFD5(4) methods are more efficient tosolve directly fourth-order differential equations comparedwith RK6(5)V and RK5(4)D methods in terms of functionevaluations and the total number of steps This is due tothe fact that when using RK6(5)V and RK5(4)D methodsthe fourth-order ODEs need to be transformed into a first-order system and hence the dimension will increase fourtimes

5 Conclusion

We have constructed the two pairs of embedded RKFDmethods for directly solving special fourth-order ODEs ofthe form 119910

(119894V)(119909) = 119891(119909 119910(119909)) using variable step size

codes in this paperThese methods are denoted as RKFD5(4)and RKFD6(5) respectively Based on the methods thevariables step size codes are developed and used to solvespecial fourth-order ODEs Numerical results show thatthe two new embedded RKFD pairs are more effective ascompared with the existing embedded Runge-Kutta pairs ofthe same order in the literature From the numerical resultswe conclude that the two new embedded RKFD methodsare computationally more efficient in solving special fourth-order ODEs and outperformed the well-known embeddedRunge-Kutta methods

Conflict of Interests

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

Mathematical Problems in Engineering 11

References

[1] S N Jator ldquoNumerical integrators for fourth order initial andboundary value problemsrdquo International Journal of Pure andApplied Mathematics vol 47 no 4 pp 563ndash576 2008

[2] Z Bai and H Wang ldquoOn positive solutions of some nonlinearfourth-order beam equationsrdquo Journal ofMathematical Analysisand Applications vol 270 no 2 pp 357ndash368 2002

[3] A K Alomari N Ratib Anakira A S Bataineh and I HashimldquoApproximate solution of nonlinear system of BVP arising influid flow problemrdquoMathematical Problems in Engineering vol2013 Article ID 136043 7 pages 2013

[4] A Malek and R Shekari Beidokhti ldquoNumerical solution forhigh order differential equations using a hybrid neuralnetworkmdashoptimization methodrdquo Applied Mathematics andComputation vol 183 no 1 pp 260ndash271 2006

[5] A Boutayeb and A Chetouani ldquoA mini-review of numericalmethods for high-order problemsrdquo International Journal ofComputer Mathematics vol 84 no 4 pp 563ndash579 2007

[6] P Onumanyi U W Sirisena and S N Jator ldquoContinuous finitedifference approximations for solving differential equationsrdquoInternational Journal of Computer Mathematics vol 72 no 1pp 15ndash27 1999

[7] D Sarafyan ldquoNew algorithms for the continuous approximatesolutions of ordinary differential equations and the upgradingof the order of the processesrdquo Computers amp Mathematics withApplications vol 20 no 1 pp 77ndash100 1990

[8] GDahlquist ldquoOn accuracy and unconditional stability of linearmultistep methods for second order differential equationsrdquo BITNumerical Mathematics vol 18 no 2 pp 133ndash136 1978

[9] R P K Chan P Leone and A Tsai ldquoOrder conditions andsymmetry for two-step hybrid methodsrdquo International Journalof Computer Mathematics vol 81 no 12 pp 1519ndash1536 2004

[10] C E Abhulimen and F O Otunta ldquoA sixth order multideriva-tivemultistepmethods for stiff system of differential equationsrdquoInternational Journal of Numerical Mathematics vol 1 no 2 pp248ndash268 2006

[11] J D Lambert Computational Methods in Ordinary DifferentialEquations Wiley London UK 1973

[12] D O Awoyemi ldquoAlgorithmic collocation approach for directsolution of fourth-order initial-value problems of ordinarydifferential equationsrdquo International Journal of ComputerMath-ematics vol 82 no 3 pp 321ndash329 2005

[13] NWaeleh Z A Majid F Ismail andM Suleiman ldquoNumericalsolution of higher order ordinary differential equations bydirect block coderdquo Journal of Mathematics and Statistics vol 8no 1 pp 77ndash81 2011

[14] S J Kayode ldquoAn efficient zero-stable numerical method forfourth-order differential equationsrdquo International Journal ofMathematics and Mathematical Sciences vol 2008 Article ID364021 10 pages 2008

[15] S J Kayode ldquoAnorder six zero-stablemethod for direct solutionof fourth order ordinary differential equationsrdquo AmericanJournal of Applied Sciences vol 5 no 11 pp 1461ndash1466 2008

[16] B T Olabode and T J Alabi ldquoDirect block predictor-correctormethod for the solution of general fourth order ODEsrdquo Journalof Mathematics Research vol 5 no 1 pp 26ndash33 2013

[17] L K Yap F Ismail and N Senu ldquoAn accurate block hybrid col-locationmethod for third order ordinary differential equationsrdquoJournal of Applied Mathematics vol 2014 Article ID 549597 9pages 2014

[18] S Mehrkanoon ldquoA direct variable step block multistep methodfor solving general third-order ODEsrdquo Numerical Algorithmsvol 57 no 1 pp 53ndash66 2011

[19] Z A Majid and M B Suleiman ldquoDirect integration implicitvariable steps method for solving higher order systems ofordinary differential equations directlyrdquo Sains Malaysiana vol35 no 2 pp 63ndash68 2006

[20] D O Awoyemi ldquoA P-stable linear multistep method for solvinggeneral third order ordinary differential equationsrdquo Interna-tional Journal of Computer Mathematics vol 80 no 8 pp 985ndash991 2003

[21] N Waeleh Z A Majid and F Ismail ldquoA new algorithm forsolving higher order IVPs of ODEsrdquo Applied MathematicalSciences vol 5 no 56 pp 2795ndash2805 2011

[22] D O Awoyemi and O M Idowu ldquoA class of hybrid collocationmethods for third-order ordinary differential equationsrdquo Inter-national Journal of Computer Mathematics vol 82 no 10 pp1287ndash1293 2005

[23] S N Jator ldquoSolving second order initial value problems by ahybrid multistep method without predictorsrdquo Applied Mathe-matics and Computation vol 217 no 8 pp 4036ndash4046 2010

[24] K Hussain F Ismail and N Senu ldquoRunge-Kutta type methodsfor directly solving special fourth-order ordinary differentialequationsrdquo Mathematical Problems in Engineering vol 2015Article ID 893763 11 pages 2015

[25] J R DormandNumerical Methods for Differential Equations AComputational Approach Library of Engineering MathematicsCRC Press Boca Raton Fla USA 1996

[26] W Gander and D Gruntz ldquoDerivation of numerical methodsusing computer algebrardquo SIAM Review vol 41 no 3 pp 577ndash593 1999

[27] F B Ismail and M B Suleiman ldquoEmbedded singly diagonallyimplicit Runge-Kutta methods (45) in (56) For the integrationof stiff systems of ODEsrdquo International Journal of ComputerMathematics vol 66 no 3-4 pp 325ndash341 1998

[28] J R Dormand and P J Prince ldquoA family of embeddedRunge-Kutta formulaerdquo Journal of Computational and AppliedMathematics vol 6 no 1 pp 19ndash26 1980

[29] P J Prince and J R Dormand ldquoHigh order embedded Runge-Kutta formulaerdquo Journal of Computational and Applied Mathe-matics vol 7 no 1 pp 67ndash75 1981

[30] R A Al-Khasawneh F Ismail and M Suleiman ldquoEmbeddeddiagonally implicit Runge-Kutta-Nystrom 4(3) pair for solvingspecial second-order IVPsrdquo Applied Mathematics and Compu-tation vol 190 no 2 pp 1803ndash1814 2007

[31] E Fehlberg ldquoLow-order classical Runge-Kutta formulas withstepsize control and their application to some heat transferproblemsrdquo NASA TR R-315 1969

[32] E Fehlberg ldquoClassical fifth sixth seventh and eighth orderRunge-Kutta formulas with stepsize controlrdquo NASA TR R-287NASA 1968

[33] M Calvo J I Montijano and L Randez ldquoA new embeddedpair of Runge-Kutta formulas of orders 5 and 6rdquo Computers ampMathematics with Applications vol 20 no 1 pp 15ndash24 1990

[34] J R Dormand M E A El-Mikkawy and P J Prince ldquoFamiliesof Runge-Kutta-Nystrom formulaerdquo IMA Journal of NumericalAnalysis vol 7 no 2 pp 235ndash250 1987

[35] L F Shampine Numerical Solution of Ordinary DifferentialEquations Chapman amp Hall New York NY USA 1994

[36] J C Butcher Numerical Methods for Ordinary DifferentialEquations John Wiley amp Sons Chichester UK 2008

12 Mathematical Problems in Engineering

[37] E Hairer S P Noslashrsett and G Wanner Solving Ordinary Dif-ferential Equations I Nonstiff Problems vol 8 Springer BerlinGermany 2nd edition 1993

[38] N Senu M Mechee F Ismail and Z Siri ldquoEmbedded explicitRungendashKutta type methods for directly solving special thirdorder differential equations 119910

101584010158401015840= 119891(119909 119910)rdquo Applied Mathemat-

ics and Computation vol 240 pp 281ndash293 2014[39] J H Verner ldquoExplicit Runge-Kutta pairs with lower stage-

orderrdquo Numerical Algorithms vol 65 no 3 pp 555ndash577 2014

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

8 Mathematical Problems in Engineering

simultaneously produces a solution of an embedded RKFDformula of order five which is given by

1 =1

30minus (

1

2minus

radic5

10) 2 minus (

1

2+

radic5

10) 3

4 =1

120minus (

1

2+

radic5

10) 2 minus (

1

2minus

radic5

10) 3

1015840

1 =1

16+

radic5

80minus

radic5

51015840

3

1015840

2 =1

8minus 1015840

3

1015840

4 = minus1

48minus

radic5

80+

radic5

51015840

3

(55)

while the values of 10158401015840119894 and

101584010158401015840119894 119894 = 1 2 3 4 are the same

as the RKFD6 method Now our goal is to determine the

values of the free parameters 2 3 and 10158403 such that the

values of 119862(7) 119861(7) 1198621015840(7) 119861

1015840(7) 1205911015840(6)

2 and 120591(6)

2 are assmall as possible The quantities 119862

(7) 119861(7) and 120591

(6)2 are

functions depending on 2 and 3 Picking 2 = 140 andplotting the graph of119862(7)119861(7) and 120591

(6)2 against 2 and from

the numerical experiment therefore we choose 3 = 120

that is the optimal value which gives 119862(7)

= 1507670351119861(7)

= 1507670351 and the local truncation error 120591(6)

2 =

9444444448times10minus3The quantities1198621015840(7) 1198611015840(7) and 120591

1015840(6)2 are

functions in terms of 10158403 Plotting the graph of these functionsand for the optimized pair we found that

10158403 = 110 is the

optimal value using Minimize command in Maple Softwarewhich yields 119862

1015840(7)= 1683161973 1198611015840(7) = 1708476855 and

1205911015840(6)

2 = 81256471 times 10minus4 (see Figure 2)

Finally all the parameters of the embedded pair arerepresented in the Butcher tableau and denoted as RKFD6(5)method as follows

1

2+

radic5

10

1

168+

11radic5

4200

1

2minus

radic5

10

1

420minus

radic5

700

1

280minus

radic5

840

1 minus1

840minus

radic5

168

1

84minus

radic5

140

5

168+

11radic5

840

17

1260

1

72minus

radic5

168

1

72+

radic5

168

1

2520

1

24

1

16minus

radic5

48

1

16+

radic5

480

1

12

5

24minus

radic5

24

5

24+

radic5

240

1

12

5

12

5

12

1

12

minus1

240minus

radic5

400

1

40

1

20minus

7

240+

radic5

400

1

16minus

3radic5

400

1

40

1

10minus

1

48+

3radic5

400

1

12

5

24minus

radic5

24

5

24+

radic5

240

1

12

5

12

5

12

1

12

(56)

4 Numerical Experiments

In this section we present some problems which involvespecial fourth-order ODEs of the form 119910

(119894V)(119909) = 119891(119909 119910(119909))

in order to test the performance of the two new embeddedpairs derived in Section 3 The numerical results obtainedare compared with the well-known embedded Runge-Kuttamethods which are chosen from the scientific literature after

Mathematical Problems in Engineering 9120591

998400(6) 2

005 01 015 02 025 030b998400

3

0

0002

0004

0006

0008

001

0012

0014

0016

Figure 2 The graph of 1205911015840(6)2 versus 10158403

transforming the same problems to a system of first-orderdifferential equations and solving themThemethods chosenin the numerical experiments are as follows

(i) RKFD6(5) the new embedded Runge-Kutta pair oforders 6(5) derived in Section 3 in this paper

(ii) RKFD5(4) the new embedded Runge-Kutta pair oforders 5(4) derived in Section 3 in this paper

(iii) RK6(5)V the embedded Runge-Kutta pair of orders6(5) derived by Verner as given in [39]

(iv) RK5(4)D the embedded Runge-Kutta pair of orders5(4) with FSAL property derived by Dormand andPrince [28]

Problem 1 We consider that the homogeneous linear prob-lem is as follows

119910(119894V)

= minus4119910

119910 (0) = 0 1199101015840(0) = 1 119910

10158401015840(0) = 2 119910

101584010158401015840(0) = 2

(57)

The exact solution is given by 119910(119909) = e119909 sin(119909) The problemis integrated in the interval [0 5]

Problem 2 We consider that the homogeneous nonlinearproblem is as follows

119910(119894V)

= minus15

161199107

119910 (0) = 1 1199101015840(0) =

1

2 11991010158401015840

(0) = minus1

4 119910101584010158401015840

(0) =3

8

(58)

The exact solution is given by 119910(119909) = radic1 + 119909 The problem isintegrated in the interval [0 10]

Problem 3 We consider that the inhomogeneous nonlinearproblem is as follows

119910(119894V)

= 1199102+ cos2 (119909) + sin (119909) minus 1

119910 (0) = 0 1199101015840(0) = 1 119910

10158401015840(0) = 0 119910

101584010158401015840(0) = minus1

(59)

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

35325215

log10(M

AXE

RR)

log10(function evaluations)

minus10

minus8

minus6

minus4

minus2

Figure 3 The efficiency curves for Problem 1

The exact solution is given by 119910(119909) = sin(119909) The problem isintegrated in the interval [0 8]

Problem 4 We consider that the linear system is as follows

119910(119894V)

= e3119909119906

119910 (0) = 1 1199101015840(0) = minus1 119910

10158401015840(0) = 1 119910

101584010158401015840(0) = minus1

119911(119894V)

= 16eminus119909119910

119911 (0) = 1 1199111015840(0) = minus2 119911

10158401015840(0) = 4 119911

101584010158401015840(0) = minus8

119908(119894V)

= 81eminus119909119911

119908 (0) = 1 1199081015840(0) = minus3 119908

10158401015840(0) = 9 119908

101584010158401015840(0) = minus27

119906(119894V)

= 256eminus119909119908

119906 (0) = 1 1199061015840(0) = minus4 119906

10158401015840(0) = 16 119906

101584010158401015840(0) = minus64

(60)

The exact solution is given by

119910 = eminus119909

119911 = eminus2119909

119908 = eminus3119909

119906 = eminus4119909

(61)

The problem is integrated in the interval [0 3]

In the numerical experiments we have computed foreach method and problem the maximum global error andthe number of function evaluations used in the integrationIn Figures 3ndash6 we plot these values in double logarithmic

10 Mathematical Problems in Engineering

18 2 22 24 26 28 314 16

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

log10(M

AXE

RR)

log10(function evaluations)

Figure 4 The efficiency curves for Problem 2

325215

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

log10(function evaluations)

log10(M

AXE

RR)

minus10

minus8

minus6

minus4

minus2

0

Figure 5 The efficiency curves for Problem 3

325215

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

log10(function evaluations)

log10(M

AXE

RR)

minus10

minus8

minus6

minus4

minus2

0

Figure 6 The efficiency curves for Problem 4

Table 1 The comparison results of the total number of successfulsteps and failure steps betweenRKFD6(5) RKFD5(4) RK6(5)V andRK5(4)D methods for all problems

Problems Methods Successful steps Failure steps

1

RKFD6(5) 440 4RKFD5(4) 705 0RK6(5)V 461 9RK5(4)D 768 0

2

RKFD6(5) 151 0RKFD5(4) 200 0RK6(5)V 208 0RK5(4)D 253 0

3

RKFD6(5) 330 4RKFD5(4) 466 0RK6(5)V 348 8RK5(4)D 497 0

4

RKFD6(5) 264 5RKFD5(4) 340 6RK6(5)V 339 5RK5(4)D 484 5

scale From Table 1 and Figures 3ndash6 we observed that thenew RKFD6(5) and RKFD5(4) methods are more efficient tosolve directly fourth-order differential equations comparedwith RK6(5)V and RK5(4)D methods in terms of functionevaluations and the total number of steps This is due tothe fact that when using RK6(5)V and RK5(4)D methodsthe fourth-order ODEs need to be transformed into a first-order system and hence the dimension will increase fourtimes

5 Conclusion

We have constructed the two pairs of embedded RKFDmethods for directly solving special fourth-order ODEs ofthe form 119910

(119894V)(119909) = 119891(119909 119910(119909)) using variable step size

codes in this paperThese methods are denoted as RKFD5(4)and RKFD6(5) respectively Based on the methods thevariables step size codes are developed and used to solvespecial fourth-order ODEs Numerical results show thatthe two new embedded RKFD pairs are more effective ascompared with the existing embedded Runge-Kutta pairs ofthe same order in the literature From the numerical resultswe conclude that the two new embedded RKFD methodsare computationally more efficient in solving special fourth-order ODEs and outperformed the well-known embeddedRunge-Kutta methods

Conflict of Interests

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

Mathematical Problems in Engineering 11

References

[1] S N Jator ldquoNumerical integrators for fourth order initial andboundary value problemsrdquo International Journal of Pure andApplied Mathematics vol 47 no 4 pp 563ndash576 2008

[2] Z Bai and H Wang ldquoOn positive solutions of some nonlinearfourth-order beam equationsrdquo Journal ofMathematical Analysisand Applications vol 270 no 2 pp 357ndash368 2002

[3] A K Alomari N Ratib Anakira A S Bataineh and I HashimldquoApproximate solution of nonlinear system of BVP arising influid flow problemrdquoMathematical Problems in Engineering vol2013 Article ID 136043 7 pages 2013

[4] A Malek and R Shekari Beidokhti ldquoNumerical solution forhigh order differential equations using a hybrid neuralnetworkmdashoptimization methodrdquo Applied Mathematics andComputation vol 183 no 1 pp 260ndash271 2006

[5] A Boutayeb and A Chetouani ldquoA mini-review of numericalmethods for high-order problemsrdquo International Journal ofComputer Mathematics vol 84 no 4 pp 563ndash579 2007

[6] P Onumanyi U W Sirisena and S N Jator ldquoContinuous finitedifference approximations for solving differential equationsrdquoInternational Journal of Computer Mathematics vol 72 no 1pp 15ndash27 1999

[7] D Sarafyan ldquoNew algorithms for the continuous approximatesolutions of ordinary differential equations and the upgradingof the order of the processesrdquo Computers amp Mathematics withApplications vol 20 no 1 pp 77ndash100 1990

[8] GDahlquist ldquoOn accuracy and unconditional stability of linearmultistep methods for second order differential equationsrdquo BITNumerical Mathematics vol 18 no 2 pp 133ndash136 1978

[9] R P K Chan P Leone and A Tsai ldquoOrder conditions andsymmetry for two-step hybrid methodsrdquo International Journalof Computer Mathematics vol 81 no 12 pp 1519ndash1536 2004

[10] C E Abhulimen and F O Otunta ldquoA sixth order multideriva-tivemultistepmethods for stiff system of differential equationsrdquoInternational Journal of Numerical Mathematics vol 1 no 2 pp248ndash268 2006

[11] J D Lambert Computational Methods in Ordinary DifferentialEquations Wiley London UK 1973

[12] D O Awoyemi ldquoAlgorithmic collocation approach for directsolution of fourth-order initial-value problems of ordinarydifferential equationsrdquo International Journal of ComputerMath-ematics vol 82 no 3 pp 321ndash329 2005

[13] NWaeleh Z A Majid F Ismail andM Suleiman ldquoNumericalsolution of higher order ordinary differential equations bydirect block coderdquo Journal of Mathematics and Statistics vol 8no 1 pp 77ndash81 2011

[14] S J Kayode ldquoAn efficient zero-stable numerical method forfourth-order differential equationsrdquo International Journal ofMathematics and Mathematical Sciences vol 2008 Article ID364021 10 pages 2008

[15] S J Kayode ldquoAnorder six zero-stablemethod for direct solutionof fourth order ordinary differential equationsrdquo AmericanJournal of Applied Sciences vol 5 no 11 pp 1461ndash1466 2008

[16] B T Olabode and T J Alabi ldquoDirect block predictor-correctormethod for the solution of general fourth order ODEsrdquo Journalof Mathematics Research vol 5 no 1 pp 26ndash33 2013

[17] L K Yap F Ismail and N Senu ldquoAn accurate block hybrid col-locationmethod for third order ordinary differential equationsrdquoJournal of Applied Mathematics vol 2014 Article ID 549597 9pages 2014

[18] S Mehrkanoon ldquoA direct variable step block multistep methodfor solving general third-order ODEsrdquo Numerical Algorithmsvol 57 no 1 pp 53ndash66 2011

[19] Z A Majid and M B Suleiman ldquoDirect integration implicitvariable steps method for solving higher order systems ofordinary differential equations directlyrdquo Sains Malaysiana vol35 no 2 pp 63ndash68 2006

[20] D O Awoyemi ldquoA P-stable linear multistep method for solvinggeneral third order ordinary differential equationsrdquo Interna-tional Journal of Computer Mathematics vol 80 no 8 pp 985ndash991 2003

[21] N Waeleh Z A Majid and F Ismail ldquoA new algorithm forsolving higher order IVPs of ODEsrdquo Applied MathematicalSciences vol 5 no 56 pp 2795ndash2805 2011

[22] D O Awoyemi and O M Idowu ldquoA class of hybrid collocationmethods for third-order ordinary differential equationsrdquo Inter-national Journal of Computer Mathematics vol 82 no 10 pp1287ndash1293 2005

[23] S N Jator ldquoSolving second order initial value problems by ahybrid multistep method without predictorsrdquo Applied Mathe-matics and Computation vol 217 no 8 pp 4036ndash4046 2010

[24] K Hussain F Ismail and N Senu ldquoRunge-Kutta type methodsfor directly solving special fourth-order ordinary differentialequationsrdquo Mathematical Problems in Engineering vol 2015Article ID 893763 11 pages 2015

[25] J R DormandNumerical Methods for Differential Equations AComputational Approach Library of Engineering MathematicsCRC Press Boca Raton Fla USA 1996

[26] W Gander and D Gruntz ldquoDerivation of numerical methodsusing computer algebrardquo SIAM Review vol 41 no 3 pp 577ndash593 1999

[27] F B Ismail and M B Suleiman ldquoEmbedded singly diagonallyimplicit Runge-Kutta methods (45) in (56) For the integrationof stiff systems of ODEsrdquo International Journal of ComputerMathematics vol 66 no 3-4 pp 325ndash341 1998

[28] J R Dormand and P J Prince ldquoA family of embeddedRunge-Kutta formulaerdquo Journal of Computational and AppliedMathematics vol 6 no 1 pp 19ndash26 1980

[29] P J Prince and J R Dormand ldquoHigh order embedded Runge-Kutta formulaerdquo Journal of Computational and Applied Mathe-matics vol 7 no 1 pp 67ndash75 1981

[30] R A Al-Khasawneh F Ismail and M Suleiman ldquoEmbeddeddiagonally implicit Runge-Kutta-Nystrom 4(3) pair for solvingspecial second-order IVPsrdquo Applied Mathematics and Compu-tation vol 190 no 2 pp 1803ndash1814 2007

[31] E Fehlberg ldquoLow-order classical Runge-Kutta formulas withstepsize control and their application to some heat transferproblemsrdquo NASA TR R-315 1969

[32] E Fehlberg ldquoClassical fifth sixth seventh and eighth orderRunge-Kutta formulas with stepsize controlrdquo NASA TR R-287NASA 1968

[33] M Calvo J I Montijano and L Randez ldquoA new embeddedpair of Runge-Kutta formulas of orders 5 and 6rdquo Computers ampMathematics with Applications vol 20 no 1 pp 15ndash24 1990

[34] J R Dormand M E A El-Mikkawy and P J Prince ldquoFamiliesof Runge-Kutta-Nystrom formulaerdquo IMA Journal of NumericalAnalysis vol 7 no 2 pp 235ndash250 1987

[35] L F Shampine Numerical Solution of Ordinary DifferentialEquations Chapman amp Hall New York NY USA 1994

[36] J C Butcher Numerical Methods for Ordinary DifferentialEquations John Wiley amp Sons Chichester UK 2008

12 Mathematical Problems in Engineering

[37] E Hairer S P Noslashrsett and G Wanner Solving Ordinary Dif-ferential Equations I Nonstiff Problems vol 8 Springer BerlinGermany 2nd edition 1993

[38] N Senu M Mechee F Ismail and Z Siri ldquoEmbedded explicitRungendashKutta type methods for directly solving special thirdorder differential equations 119910

101584010158401015840= 119891(119909 119910)rdquo Applied Mathemat-

ics and Computation vol 240 pp 281ndash293 2014[39] J H Verner ldquoExplicit Runge-Kutta pairs with lower stage-

orderrdquo Numerical Algorithms vol 65 no 3 pp 555ndash577 2014

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 9120591

998400(6) 2

005 01 015 02 025 030b998400

3

0

0002

0004

0006

0008

001

0012

0014

0016

Figure 2 The graph of 1205911015840(6)2 versus 10158403

transforming the same problems to a system of first-orderdifferential equations and solving themThemethods chosenin the numerical experiments are as follows

(i) RKFD6(5) the new embedded Runge-Kutta pair oforders 6(5) derived in Section 3 in this paper

(ii) RKFD5(4) the new embedded Runge-Kutta pair oforders 5(4) derived in Section 3 in this paper

(iii) RK6(5)V the embedded Runge-Kutta pair of orders6(5) derived by Verner as given in [39]

(iv) RK5(4)D the embedded Runge-Kutta pair of orders5(4) with FSAL property derived by Dormand andPrince [28]

Problem 1 We consider that the homogeneous linear prob-lem is as follows

119910(119894V)

= minus4119910

119910 (0) = 0 1199101015840(0) = 1 119910

10158401015840(0) = 2 119910

101584010158401015840(0) = 2

(57)

The exact solution is given by 119910(119909) = e119909 sin(119909) The problemis integrated in the interval [0 5]

Problem 2 We consider that the homogeneous nonlinearproblem is as follows

119910(119894V)

= minus15

161199107

119910 (0) = 1 1199101015840(0) =

1

2 11991010158401015840

(0) = minus1

4 119910101584010158401015840

(0) =3

8

(58)

The exact solution is given by 119910(119909) = radic1 + 119909 The problem isintegrated in the interval [0 10]

Problem 3 We consider that the inhomogeneous nonlinearproblem is as follows

119910(119894V)

= 1199102+ cos2 (119909) + sin (119909) minus 1

119910 (0) = 0 1199101015840(0) = 1 119910

10158401015840(0) = 0 119910

101584010158401015840(0) = minus1

(59)

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

35325215

log10(M

AXE

RR)

log10(function evaluations)

minus10

minus8

minus6

minus4

minus2

Figure 3 The efficiency curves for Problem 1

The exact solution is given by 119910(119909) = sin(119909) The problem isintegrated in the interval [0 8]

Problem 4 We consider that the linear system is as follows

119910(119894V)

= e3119909119906

119910 (0) = 1 1199101015840(0) = minus1 119910

10158401015840(0) = 1 119910

101584010158401015840(0) = minus1

119911(119894V)

= 16eminus119909119910

119911 (0) = 1 1199111015840(0) = minus2 119911

10158401015840(0) = 4 119911

101584010158401015840(0) = minus8

119908(119894V)

= 81eminus119909119911

119908 (0) = 1 1199081015840(0) = minus3 119908

10158401015840(0) = 9 119908

101584010158401015840(0) = minus27

119906(119894V)

= 256eminus119909119908

119906 (0) = 1 1199061015840(0) = minus4 119906

10158401015840(0) = 16 119906

101584010158401015840(0) = minus64

(60)

The exact solution is given by

119910 = eminus119909

119911 = eminus2119909

119908 = eminus3119909

119906 = eminus4119909

(61)

The problem is integrated in the interval [0 3]

In the numerical experiments we have computed foreach method and problem the maximum global error andthe number of function evaluations used in the integrationIn Figures 3ndash6 we plot these values in double logarithmic

10 Mathematical Problems in Engineering

18 2 22 24 26 28 314 16

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

log10(M

AXE

RR)

log10(function evaluations)

Figure 4 The efficiency curves for Problem 2

325215

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

log10(function evaluations)

log10(M

AXE

RR)

minus10

minus8

minus6

minus4

minus2

0

Figure 5 The efficiency curves for Problem 3

325215

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

log10(function evaluations)

log10(M

AXE

RR)

minus10

minus8

minus6

minus4

minus2

0

Figure 6 The efficiency curves for Problem 4

Table 1 The comparison results of the total number of successfulsteps and failure steps betweenRKFD6(5) RKFD5(4) RK6(5)V andRK5(4)D methods for all problems

Problems Methods Successful steps Failure steps

1

RKFD6(5) 440 4RKFD5(4) 705 0RK6(5)V 461 9RK5(4)D 768 0

2

RKFD6(5) 151 0RKFD5(4) 200 0RK6(5)V 208 0RK5(4)D 253 0

3

RKFD6(5) 330 4RKFD5(4) 466 0RK6(5)V 348 8RK5(4)D 497 0

4

RKFD6(5) 264 5RKFD5(4) 340 6RK6(5)V 339 5RK5(4)D 484 5

scale From Table 1 and Figures 3ndash6 we observed that thenew RKFD6(5) and RKFD5(4) methods are more efficient tosolve directly fourth-order differential equations comparedwith RK6(5)V and RK5(4)D methods in terms of functionevaluations and the total number of steps This is due tothe fact that when using RK6(5)V and RK5(4)D methodsthe fourth-order ODEs need to be transformed into a first-order system and hence the dimension will increase fourtimes

5 Conclusion

We have constructed the two pairs of embedded RKFDmethods for directly solving special fourth-order ODEs ofthe form 119910

(119894V)(119909) = 119891(119909 119910(119909)) using variable step size

codes in this paperThese methods are denoted as RKFD5(4)and RKFD6(5) respectively Based on the methods thevariables step size codes are developed and used to solvespecial fourth-order ODEs Numerical results show thatthe two new embedded RKFD pairs are more effective ascompared with the existing embedded Runge-Kutta pairs ofthe same order in the literature From the numerical resultswe conclude that the two new embedded RKFD methodsare computationally more efficient in solving special fourth-order ODEs and outperformed the well-known embeddedRunge-Kutta methods

Conflict of Interests

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

Mathematical Problems in Engineering 11

References

[1] S N Jator ldquoNumerical integrators for fourth order initial andboundary value problemsrdquo International Journal of Pure andApplied Mathematics vol 47 no 4 pp 563ndash576 2008

[2] Z Bai and H Wang ldquoOn positive solutions of some nonlinearfourth-order beam equationsrdquo Journal ofMathematical Analysisand Applications vol 270 no 2 pp 357ndash368 2002

[3] A K Alomari N Ratib Anakira A S Bataineh and I HashimldquoApproximate solution of nonlinear system of BVP arising influid flow problemrdquoMathematical Problems in Engineering vol2013 Article ID 136043 7 pages 2013

[4] A Malek and R Shekari Beidokhti ldquoNumerical solution forhigh order differential equations using a hybrid neuralnetworkmdashoptimization methodrdquo Applied Mathematics andComputation vol 183 no 1 pp 260ndash271 2006

[5] A Boutayeb and A Chetouani ldquoA mini-review of numericalmethods for high-order problemsrdquo International Journal ofComputer Mathematics vol 84 no 4 pp 563ndash579 2007

[6] P Onumanyi U W Sirisena and S N Jator ldquoContinuous finitedifference approximations for solving differential equationsrdquoInternational Journal of Computer Mathematics vol 72 no 1pp 15ndash27 1999

[7] D Sarafyan ldquoNew algorithms for the continuous approximatesolutions of ordinary differential equations and the upgradingof the order of the processesrdquo Computers amp Mathematics withApplications vol 20 no 1 pp 77ndash100 1990

[8] GDahlquist ldquoOn accuracy and unconditional stability of linearmultistep methods for second order differential equationsrdquo BITNumerical Mathematics vol 18 no 2 pp 133ndash136 1978

[9] R P K Chan P Leone and A Tsai ldquoOrder conditions andsymmetry for two-step hybrid methodsrdquo International Journalof Computer Mathematics vol 81 no 12 pp 1519ndash1536 2004

[10] C E Abhulimen and F O Otunta ldquoA sixth order multideriva-tivemultistepmethods for stiff system of differential equationsrdquoInternational Journal of Numerical Mathematics vol 1 no 2 pp248ndash268 2006

[11] J D Lambert Computational Methods in Ordinary DifferentialEquations Wiley London UK 1973

[12] D O Awoyemi ldquoAlgorithmic collocation approach for directsolution of fourth-order initial-value problems of ordinarydifferential equationsrdquo International Journal of ComputerMath-ematics vol 82 no 3 pp 321ndash329 2005

[13] NWaeleh Z A Majid F Ismail andM Suleiman ldquoNumericalsolution of higher order ordinary differential equations bydirect block coderdquo Journal of Mathematics and Statistics vol 8no 1 pp 77ndash81 2011

[14] S J Kayode ldquoAn efficient zero-stable numerical method forfourth-order differential equationsrdquo International Journal ofMathematics and Mathematical Sciences vol 2008 Article ID364021 10 pages 2008

[15] S J Kayode ldquoAnorder six zero-stablemethod for direct solutionof fourth order ordinary differential equationsrdquo AmericanJournal of Applied Sciences vol 5 no 11 pp 1461ndash1466 2008

[16] B T Olabode and T J Alabi ldquoDirect block predictor-correctormethod for the solution of general fourth order ODEsrdquo Journalof Mathematics Research vol 5 no 1 pp 26ndash33 2013

[17] L K Yap F Ismail and N Senu ldquoAn accurate block hybrid col-locationmethod for third order ordinary differential equationsrdquoJournal of Applied Mathematics vol 2014 Article ID 549597 9pages 2014

[18] S Mehrkanoon ldquoA direct variable step block multistep methodfor solving general third-order ODEsrdquo Numerical Algorithmsvol 57 no 1 pp 53ndash66 2011

[19] Z A Majid and M B Suleiman ldquoDirect integration implicitvariable steps method for solving higher order systems ofordinary differential equations directlyrdquo Sains Malaysiana vol35 no 2 pp 63ndash68 2006

[20] D O Awoyemi ldquoA P-stable linear multistep method for solvinggeneral third order ordinary differential equationsrdquo Interna-tional Journal of Computer Mathematics vol 80 no 8 pp 985ndash991 2003

[21] N Waeleh Z A Majid and F Ismail ldquoA new algorithm forsolving higher order IVPs of ODEsrdquo Applied MathematicalSciences vol 5 no 56 pp 2795ndash2805 2011

[22] D O Awoyemi and O M Idowu ldquoA class of hybrid collocationmethods for third-order ordinary differential equationsrdquo Inter-national Journal of Computer Mathematics vol 82 no 10 pp1287ndash1293 2005

[23] S N Jator ldquoSolving second order initial value problems by ahybrid multistep method without predictorsrdquo Applied Mathe-matics and Computation vol 217 no 8 pp 4036ndash4046 2010

[24] K Hussain F Ismail and N Senu ldquoRunge-Kutta type methodsfor directly solving special fourth-order ordinary differentialequationsrdquo Mathematical Problems in Engineering vol 2015Article ID 893763 11 pages 2015

[25] J R DormandNumerical Methods for Differential Equations AComputational Approach Library of Engineering MathematicsCRC Press Boca Raton Fla USA 1996

[26] W Gander and D Gruntz ldquoDerivation of numerical methodsusing computer algebrardquo SIAM Review vol 41 no 3 pp 577ndash593 1999

[27] F B Ismail and M B Suleiman ldquoEmbedded singly diagonallyimplicit Runge-Kutta methods (45) in (56) For the integrationof stiff systems of ODEsrdquo International Journal of ComputerMathematics vol 66 no 3-4 pp 325ndash341 1998

[28] J R Dormand and P J Prince ldquoA family of embeddedRunge-Kutta formulaerdquo Journal of Computational and AppliedMathematics vol 6 no 1 pp 19ndash26 1980

[29] P J Prince and J R Dormand ldquoHigh order embedded Runge-Kutta formulaerdquo Journal of Computational and Applied Mathe-matics vol 7 no 1 pp 67ndash75 1981

[30] R A Al-Khasawneh F Ismail and M Suleiman ldquoEmbeddeddiagonally implicit Runge-Kutta-Nystrom 4(3) pair for solvingspecial second-order IVPsrdquo Applied Mathematics and Compu-tation vol 190 no 2 pp 1803ndash1814 2007

[31] E Fehlberg ldquoLow-order classical Runge-Kutta formulas withstepsize control and their application to some heat transferproblemsrdquo NASA TR R-315 1969

[32] E Fehlberg ldquoClassical fifth sixth seventh and eighth orderRunge-Kutta formulas with stepsize controlrdquo NASA TR R-287NASA 1968

[33] M Calvo J I Montijano and L Randez ldquoA new embeddedpair of Runge-Kutta formulas of orders 5 and 6rdquo Computers ampMathematics with Applications vol 20 no 1 pp 15ndash24 1990

[34] J R Dormand M E A El-Mikkawy and P J Prince ldquoFamiliesof Runge-Kutta-Nystrom formulaerdquo IMA Journal of NumericalAnalysis vol 7 no 2 pp 235ndash250 1987

[35] L F Shampine Numerical Solution of Ordinary DifferentialEquations Chapman amp Hall New York NY USA 1994

[36] J C Butcher Numerical Methods for Ordinary DifferentialEquations John Wiley amp Sons Chichester UK 2008

12 Mathematical Problems in Engineering

[37] E Hairer S P Noslashrsett and G Wanner Solving Ordinary Dif-ferential Equations I Nonstiff Problems vol 8 Springer BerlinGermany 2nd edition 1993

[38] N Senu M Mechee F Ismail and Z Siri ldquoEmbedded explicitRungendashKutta type methods for directly solving special thirdorder differential equations 119910

101584010158401015840= 119891(119909 119910)rdquo Applied Mathemat-

ics and Computation vol 240 pp 281ndash293 2014[39] J H Verner ldquoExplicit Runge-Kutta pairs with lower stage-

orderrdquo Numerical Algorithms vol 65 no 3 pp 555ndash577 2014

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

18 2 22 24 26 28 314 16

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

log10(M

AXE

RR)

log10(function evaluations)

Figure 4 The efficiency curves for Problem 2

325215

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

log10(function evaluations)

log10(M

AXE

RR)

minus10

minus8

minus6

minus4

minus2

0

Figure 5 The efficiency curves for Problem 3

325215

RK5(4)DRK6(5)V

RKFD5(4)RKFD6(5)

log10(function evaluations)

log10(M

AXE

RR)

minus10

minus8

minus6

minus4

minus2

0

Figure 6 The efficiency curves for Problem 4

Table 1 The comparison results of the total number of successfulsteps and failure steps betweenRKFD6(5) RKFD5(4) RK6(5)V andRK5(4)D methods for all problems

Problems Methods Successful steps Failure steps

1

RKFD6(5) 440 4RKFD5(4) 705 0RK6(5)V 461 9RK5(4)D 768 0

2

RKFD6(5) 151 0RKFD5(4) 200 0RK6(5)V 208 0RK5(4)D 253 0

3

RKFD6(5) 330 4RKFD5(4) 466 0RK6(5)V 348 8RK5(4)D 497 0

4

RKFD6(5) 264 5RKFD5(4) 340 6RK6(5)V 339 5RK5(4)D 484 5

scale From Table 1 and Figures 3ndash6 we observed that thenew RKFD6(5) and RKFD5(4) methods are more efficient tosolve directly fourth-order differential equations comparedwith RK6(5)V and RK5(4)D methods in terms of functionevaluations and the total number of steps This is due tothe fact that when using RK6(5)V and RK5(4)D methodsthe fourth-order ODEs need to be transformed into a first-order system and hence the dimension will increase fourtimes

5 Conclusion

We have constructed the two pairs of embedded RKFDmethods for directly solving special fourth-order ODEs ofthe form 119910

(119894V)(119909) = 119891(119909 119910(119909)) using variable step size

codes in this paperThese methods are denoted as RKFD5(4)and RKFD6(5) respectively Based on the methods thevariables step size codes are developed and used to solvespecial fourth-order ODEs Numerical results show thatthe two new embedded RKFD pairs are more effective ascompared with the existing embedded Runge-Kutta pairs ofthe same order in the literature From the numerical resultswe conclude that the two new embedded RKFD methodsare computationally more efficient in solving special fourth-order ODEs and outperformed the well-known embeddedRunge-Kutta methods

Conflict of Interests

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

Mathematical Problems in Engineering 11

References

[1] S N Jator ldquoNumerical integrators for fourth order initial andboundary value problemsrdquo International Journal of Pure andApplied Mathematics vol 47 no 4 pp 563ndash576 2008

[2] Z Bai and H Wang ldquoOn positive solutions of some nonlinearfourth-order beam equationsrdquo Journal ofMathematical Analysisand Applications vol 270 no 2 pp 357ndash368 2002

[3] A K Alomari N Ratib Anakira A S Bataineh and I HashimldquoApproximate solution of nonlinear system of BVP arising influid flow problemrdquoMathematical Problems in Engineering vol2013 Article ID 136043 7 pages 2013

[4] A Malek and R Shekari Beidokhti ldquoNumerical solution forhigh order differential equations using a hybrid neuralnetworkmdashoptimization methodrdquo Applied Mathematics andComputation vol 183 no 1 pp 260ndash271 2006

[5] A Boutayeb and A Chetouani ldquoA mini-review of numericalmethods for high-order problemsrdquo International Journal ofComputer Mathematics vol 84 no 4 pp 563ndash579 2007

[6] P Onumanyi U W Sirisena and S N Jator ldquoContinuous finitedifference approximations for solving differential equationsrdquoInternational Journal of Computer Mathematics vol 72 no 1pp 15ndash27 1999

[7] D Sarafyan ldquoNew algorithms for the continuous approximatesolutions of ordinary differential equations and the upgradingof the order of the processesrdquo Computers amp Mathematics withApplications vol 20 no 1 pp 77ndash100 1990

[8] GDahlquist ldquoOn accuracy and unconditional stability of linearmultistep methods for second order differential equationsrdquo BITNumerical Mathematics vol 18 no 2 pp 133ndash136 1978

[9] R P K Chan P Leone and A Tsai ldquoOrder conditions andsymmetry for two-step hybrid methodsrdquo International Journalof Computer Mathematics vol 81 no 12 pp 1519ndash1536 2004

[10] C E Abhulimen and F O Otunta ldquoA sixth order multideriva-tivemultistepmethods for stiff system of differential equationsrdquoInternational Journal of Numerical Mathematics vol 1 no 2 pp248ndash268 2006

[11] J D Lambert Computational Methods in Ordinary DifferentialEquations Wiley London UK 1973

[12] D O Awoyemi ldquoAlgorithmic collocation approach for directsolution of fourth-order initial-value problems of ordinarydifferential equationsrdquo International Journal of ComputerMath-ematics vol 82 no 3 pp 321ndash329 2005

[13] NWaeleh Z A Majid F Ismail andM Suleiman ldquoNumericalsolution of higher order ordinary differential equations bydirect block coderdquo Journal of Mathematics and Statistics vol 8no 1 pp 77ndash81 2011

[14] S J Kayode ldquoAn efficient zero-stable numerical method forfourth-order differential equationsrdquo International Journal ofMathematics and Mathematical Sciences vol 2008 Article ID364021 10 pages 2008

[15] S J Kayode ldquoAnorder six zero-stablemethod for direct solutionof fourth order ordinary differential equationsrdquo AmericanJournal of Applied Sciences vol 5 no 11 pp 1461ndash1466 2008

[16] B T Olabode and T J Alabi ldquoDirect block predictor-correctormethod for the solution of general fourth order ODEsrdquo Journalof Mathematics Research vol 5 no 1 pp 26ndash33 2013

[17] L K Yap F Ismail and N Senu ldquoAn accurate block hybrid col-locationmethod for third order ordinary differential equationsrdquoJournal of Applied Mathematics vol 2014 Article ID 549597 9pages 2014

[18] S Mehrkanoon ldquoA direct variable step block multistep methodfor solving general third-order ODEsrdquo Numerical Algorithmsvol 57 no 1 pp 53ndash66 2011

[19] Z A Majid and M B Suleiman ldquoDirect integration implicitvariable steps method for solving higher order systems ofordinary differential equations directlyrdquo Sains Malaysiana vol35 no 2 pp 63ndash68 2006

[20] D O Awoyemi ldquoA P-stable linear multistep method for solvinggeneral third order ordinary differential equationsrdquo Interna-tional Journal of Computer Mathematics vol 80 no 8 pp 985ndash991 2003

[21] N Waeleh Z A Majid and F Ismail ldquoA new algorithm forsolving higher order IVPs of ODEsrdquo Applied MathematicalSciences vol 5 no 56 pp 2795ndash2805 2011

[22] D O Awoyemi and O M Idowu ldquoA class of hybrid collocationmethods for third-order ordinary differential equationsrdquo Inter-national Journal of Computer Mathematics vol 82 no 10 pp1287ndash1293 2005

[23] S N Jator ldquoSolving second order initial value problems by ahybrid multistep method without predictorsrdquo Applied Mathe-matics and Computation vol 217 no 8 pp 4036ndash4046 2010

[24] K Hussain F Ismail and N Senu ldquoRunge-Kutta type methodsfor directly solving special fourth-order ordinary differentialequationsrdquo Mathematical Problems in Engineering vol 2015Article ID 893763 11 pages 2015

[25] J R DormandNumerical Methods for Differential Equations AComputational Approach Library of Engineering MathematicsCRC Press Boca Raton Fla USA 1996

[26] W Gander and D Gruntz ldquoDerivation of numerical methodsusing computer algebrardquo SIAM Review vol 41 no 3 pp 577ndash593 1999

[27] F B Ismail and M B Suleiman ldquoEmbedded singly diagonallyimplicit Runge-Kutta methods (45) in (56) For the integrationof stiff systems of ODEsrdquo International Journal of ComputerMathematics vol 66 no 3-4 pp 325ndash341 1998

[28] J R Dormand and P J Prince ldquoA family of embeddedRunge-Kutta formulaerdquo Journal of Computational and AppliedMathematics vol 6 no 1 pp 19ndash26 1980

[29] P J Prince and J R Dormand ldquoHigh order embedded Runge-Kutta formulaerdquo Journal of Computational and Applied Mathe-matics vol 7 no 1 pp 67ndash75 1981

[30] R A Al-Khasawneh F Ismail and M Suleiman ldquoEmbeddeddiagonally implicit Runge-Kutta-Nystrom 4(3) pair for solvingspecial second-order IVPsrdquo Applied Mathematics and Compu-tation vol 190 no 2 pp 1803ndash1814 2007

[31] E Fehlberg ldquoLow-order classical Runge-Kutta formulas withstepsize control and their application to some heat transferproblemsrdquo NASA TR R-315 1969

[32] E Fehlberg ldquoClassical fifth sixth seventh and eighth orderRunge-Kutta formulas with stepsize controlrdquo NASA TR R-287NASA 1968

[33] M Calvo J I Montijano and L Randez ldquoA new embeddedpair of Runge-Kutta formulas of orders 5 and 6rdquo Computers ampMathematics with Applications vol 20 no 1 pp 15ndash24 1990

[34] J R Dormand M E A El-Mikkawy and P J Prince ldquoFamiliesof Runge-Kutta-Nystrom formulaerdquo IMA Journal of NumericalAnalysis vol 7 no 2 pp 235ndash250 1987

[35] L F Shampine Numerical Solution of Ordinary DifferentialEquations Chapman amp Hall New York NY USA 1994

[36] J C Butcher Numerical Methods for Ordinary DifferentialEquations John Wiley amp Sons Chichester UK 2008

12 Mathematical Problems in Engineering

[37] E Hairer S P Noslashrsett and G Wanner Solving Ordinary Dif-ferential Equations I Nonstiff Problems vol 8 Springer BerlinGermany 2nd edition 1993

[38] N Senu M Mechee F Ismail and Z Siri ldquoEmbedded explicitRungendashKutta type methods for directly solving special thirdorder differential equations 119910

101584010158401015840= 119891(119909 119910)rdquo Applied Mathemat-

ics and Computation vol 240 pp 281ndash293 2014[39] J H Verner ldquoExplicit Runge-Kutta pairs with lower stage-

orderrdquo Numerical Algorithms vol 65 no 3 pp 555ndash577 2014

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 11

References

[1] S N Jator ldquoNumerical integrators for fourth order initial andboundary value problemsrdquo International Journal of Pure andApplied Mathematics vol 47 no 4 pp 563ndash576 2008

[2] Z Bai and H Wang ldquoOn positive solutions of some nonlinearfourth-order beam equationsrdquo Journal ofMathematical Analysisand Applications vol 270 no 2 pp 357ndash368 2002

[3] A K Alomari N Ratib Anakira A S Bataineh and I HashimldquoApproximate solution of nonlinear system of BVP arising influid flow problemrdquoMathematical Problems in Engineering vol2013 Article ID 136043 7 pages 2013

[4] A Malek and R Shekari Beidokhti ldquoNumerical solution forhigh order differential equations using a hybrid neuralnetworkmdashoptimization methodrdquo Applied Mathematics andComputation vol 183 no 1 pp 260ndash271 2006

[5] A Boutayeb and A Chetouani ldquoA mini-review of numericalmethods for high-order problemsrdquo International Journal ofComputer Mathematics vol 84 no 4 pp 563ndash579 2007

[6] P Onumanyi U W Sirisena and S N Jator ldquoContinuous finitedifference approximations for solving differential equationsrdquoInternational Journal of Computer Mathematics vol 72 no 1pp 15ndash27 1999

[7] D Sarafyan ldquoNew algorithms for the continuous approximatesolutions of ordinary differential equations and the upgradingof the order of the processesrdquo Computers amp Mathematics withApplications vol 20 no 1 pp 77ndash100 1990

[8] GDahlquist ldquoOn accuracy and unconditional stability of linearmultistep methods for second order differential equationsrdquo BITNumerical Mathematics vol 18 no 2 pp 133ndash136 1978

[9] R P K Chan P Leone and A Tsai ldquoOrder conditions andsymmetry for two-step hybrid methodsrdquo International Journalof Computer Mathematics vol 81 no 12 pp 1519ndash1536 2004

[10] C E Abhulimen and F O Otunta ldquoA sixth order multideriva-tivemultistepmethods for stiff system of differential equationsrdquoInternational Journal of Numerical Mathematics vol 1 no 2 pp248ndash268 2006

[11] J D Lambert Computational Methods in Ordinary DifferentialEquations Wiley London UK 1973

[12] D O Awoyemi ldquoAlgorithmic collocation approach for directsolution of fourth-order initial-value problems of ordinarydifferential equationsrdquo International Journal of ComputerMath-ematics vol 82 no 3 pp 321ndash329 2005

[13] NWaeleh Z A Majid F Ismail andM Suleiman ldquoNumericalsolution of higher order ordinary differential equations bydirect block coderdquo Journal of Mathematics and Statistics vol 8no 1 pp 77ndash81 2011

[14] S J Kayode ldquoAn efficient zero-stable numerical method forfourth-order differential equationsrdquo International Journal ofMathematics and Mathematical Sciences vol 2008 Article ID364021 10 pages 2008

[15] S J Kayode ldquoAnorder six zero-stablemethod for direct solutionof fourth order ordinary differential equationsrdquo AmericanJournal of Applied Sciences vol 5 no 11 pp 1461ndash1466 2008

[16] B T Olabode and T J Alabi ldquoDirect block predictor-correctormethod for the solution of general fourth order ODEsrdquo Journalof Mathematics Research vol 5 no 1 pp 26ndash33 2013

[17] L K Yap F Ismail and N Senu ldquoAn accurate block hybrid col-locationmethod for third order ordinary differential equationsrdquoJournal of Applied Mathematics vol 2014 Article ID 549597 9pages 2014

[18] S Mehrkanoon ldquoA direct variable step block multistep methodfor solving general third-order ODEsrdquo Numerical Algorithmsvol 57 no 1 pp 53ndash66 2011

[19] Z A Majid and M B Suleiman ldquoDirect integration implicitvariable steps method for solving higher order systems ofordinary differential equations directlyrdquo Sains Malaysiana vol35 no 2 pp 63ndash68 2006

[20] D O Awoyemi ldquoA P-stable linear multistep method for solvinggeneral third order ordinary differential equationsrdquo Interna-tional Journal of Computer Mathematics vol 80 no 8 pp 985ndash991 2003

[21] N Waeleh Z A Majid and F Ismail ldquoA new algorithm forsolving higher order IVPs of ODEsrdquo Applied MathematicalSciences vol 5 no 56 pp 2795ndash2805 2011

[22] D O Awoyemi and O M Idowu ldquoA class of hybrid collocationmethods for third-order ordinary differential equationsrdquo Inter-national Journal of Computer Mathematics vol 82 no 10 pp1287ndash1293 2005

[23] S N Jator ldquoSolving second order initial value problems by ahybrid multistep method without predictorsrdquo Applied Mathe-matics and Computation vol 217 no 8 pp 4036ndash4046 2010

[24] K Hussain F Ismail and N Senu ldquoRunge-Kutta type methodsfor directly solving special fourth-order ordinary differentialequationsrdquo Mathematical Problems in Engineering vol 2015Article ID 893763 11 pages 2015

[25] J R DormandNumerical Methods for Differential Equations AComputational Approach Library of Engineering MathematicsCRC Press Boca Raton Fla USA 1996

[26] W Gander and D Gruntz ldquoDerivation of numerical methodsusing computer algebrardquo SIAM Review vol 41 no 3 pp 577ndash593 1999

[27] F B Ismail and M B Suleiman ldquoEmbedded singly diagonallyimplicit Runge-Kutta methods (45) in (56) For the integrationof stiff systems of ODEsrdquo International Journal of ComputerMathematics vol 66 no 3-4 pp 325ndash341 1998

[28] J R Dormand and P J Prince ldquoA family of embeddedRunge-Kutta formulaerdquo Journal of Computational and AppliedMathematics vol 6 no 1 pp 19ndash26 1980

[29] P J Prince and J R Dormand ldquoHigh order embedded Runge-Kutta formulaerdquo Journal of Computational and Applied Mathe-matics vol 7 no 1 pp 67ndash75 1981

[30] R A Al-Khasawneh F Ismail and M Suleiman ldquoEmbeddeddiagonally implicit Runge-Kutta-Nystrom 4(3) pair for solvingspecial second-order IVPsrdquo Applied Mathematics and Compu-tation vol 190 no 2 pp 1803ndash1814 2007

[31] E Fehlberg ldquoLow-order classical Runge-Kutta formulas withstepsize control and their application to some heat transferproblemsrdquo NASA TR R-315 1969

[32] E Fehlberg ldquoClassical fifth sixth seventh and eighth orderRunge-Kutta formulas with stepsize controlrdquo NASA TR R-287NASA 1968

[33] M Calvo J I Montijano and L Randez ldquoA new embeddedpair of Runge-Kutta formulas of orders 5 and 6rdquo Computers ampMathematics with Applications vol 20 no 1 pp 15ndash24 1990

[34] J R Dormand M E A El-Mikkawy and P J Prince ldquoFamiliesof Runge-Kutta-Nystrom formulaerdquo IMA Journal of NumericalAnalysis vol 7 no 2 pp 235ndash250 1987

[35] L F Shampine Numerical Solution of Ordinary DifferentialEquations Chapman amp Hall New York NY USA 1994

[36] J C Butcher Numerical Methods for Ordinary DifferentialEquations John Wiley amp Sons Chichester UK 2008

12 Mathematical Problems in Engineering

[37] E Hairer S P Noslashrsett and G Wanner Solving Ordinary Dif-ferential Equations I Nonstiff Problems vol 8 Springer BerlinGermany 2nd edition 1993

[38] N Senu M Mechee F Ismail and Z Siri ldquoEmbedded explicitRungendashKutta type methods for directly solving special thirdorder differential equations 119910

101584010158401015840= 119891(119909 119910)rdquo Applied Mathemat-

ics and Computation vol 240 pp 281ndash293 2014[39] J H Verner ldquoExplicit Runge-Kutta pairs with lower stage-

orderrdquo Numerical Algorithms vol 65 no 3 pp 555ndash577 2014

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

[37] E Hairer S P Noslashrsett and G Wanner Solving Ordinary Dif-ferential Equations I Nonstiff Problems vol 8 Springer BerlinGermany 2nd edition 1993

[38] N Senu M Mechee F Ismail and Z Siri ldquoEmbedded explicitRungendashKutta type methods for directly solving special thirdorder differential equations 119910

101584010158401015840= 119891(119909 119910)rdquo Applied Mathemat-

ics and Computation vol 240 pp 281ndash293 2014[39] J H Verner ldquoExplicit Runge-Kutta pairs with lower stage-

orderrdquo Numerical Algorithms vol 65 no 3 pp 555ndash577 2014

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