solving oscillatory delay differential equations using block hybrid … · 2019. 7. 30. ·...

Research ArticleSolving Oscillatory Delay Differential EquationsUsing Block Hybrid Methods

Sufia Zulfa Ahmad 1 Fudziah Ismail 12 and Norazak Senu 12

1Department of Mathematics Faculty of Science Universiti Putra Malaysia 43400 Serdang Selangor Malaysia2Institute for Mathematical Research Universiti Putra Malaysia 43400 Serdang Selangor Malaysia

Correspondence should be addressed to Fudziah Ismail fudziah iyahoocommy

Received 17 April 2018 Accepted 3 September 2018 Published 1 October 2018

Guest Editor Muhammad Ozair

Copyright copy 2018 Sufia Zulfa Ahmad et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A set of order condition for block explicit hybrid method up to order five is presented and based on the order conditions two-pointblock explicit hybrid method of order five for the approximation of special second order delay differential equations is derived Themethod is then trigonometrically fitted and used to integrate second-order delay differential equations with oscillatory solutionsThe efficiency curves based on the log of maximum errors versus the CPU time taken to do the integration are plotted which clearlydemonstrated the superiority of the trigonometrically fitted block hybrid method

1 Introduction

Differential equations with a time delay are used tomodel theprocess which does not only depend on the current state of asystem but also the past states This type of equation is calleddelay differential equations (DDEs) in which the derivative atany time depends on the solution at prior times The specialsecond order DDE can be written in the form of

11991010158401015840 (119905) = 119891 (119905 119910 (119905) 119910 (119905 minus 120591)) 119886 le 119905 le 119887119910 (1199050) = 11991001199101015840 (1199050) = 11991010158400

119905 isin [minus120591 119886](1)

where 120591 is the delay term and the first derivative does notappear explicitly It is a more realistic model which includessome of the past history of the system to determine thefuture behavior DDEs have become an important criteriato investigate the complexities of the real-world problemsconcerning infectious diseases biotic population neuronalnetworks and population dynamics

Methods such as Runge-Kutta (RK) Runge-KuttaNystrom (RKN) hybrid and multistep are widely used forsolving DDEs Ismail et al [1] used RK method and Hermiteinterpolation to solve first-order DDEs Taiwo and Odetunde[2] worked on decomposition method as an integrator fordelay differential equations Some authors also derivedblock linear multistep method (LMM) to solve DDEs andsuch work can be seen in [3ndash6] Hoo et al [7] constructedAdams-Moulton Method for directly solving second-orderDDEs Mechee et al [8] in their paper has adapted RKN fordirectly solving second-order DDEs

However all the studies previously mentioned have notbeen applied for solving DDE problems with oscillatoryproperties Hence in this paper we derived order conditionfor block explicit hybrid method up to order five usingcomputer algebra system as proposed in Gander and Gruntz[9] The reason to derive block hybrid method is that afaster numerical solutions can be obtained since the methodapproximates the solution at more than one point per stepFrom the order conditions we constructed a two-point three-stage fifth-order block explicit hybrid method which is thentrigonometrically fitted so that it is suitable for solving

HindawiJournal of MathematicsVolume 2018 Article ID 2960237 7 pageshttpsdoiorg10115520182960237

2 Journal of Mathematics

second-order DDEs which are oscillatory in nature Finallywe tested the new methods using DDEs test problems toindicate that it is superior and more efficient for solvingoscillatory second order DDEs

2 Derivation of Order Condition for BlockExplicit Hybrid Method

The general formula of two-step explicit hybrid methodfor solving the special second-order ordinary differentialequations is given as

119884119894 = (1 + 119888119894) 119910119899 minus 119888119894119910119899minus1 + ℎ2 119904sum119895=1

119886119894119895119891 (119909119899 + 119888119895ℎ 119884119895) (2)

119910119899+1 = 2119910119899 minus 119910119899minus1 + ℎ2 119904sum119894=1

119887119894119891 (119909119899 + 119888119894ℎ 119884119894) (3)

where 119894 = 1 119904 and 119894 gt 119895 The method coefficients of 119887119894 119888119894and 119886119894119895 can be represented in Butcher tableau as follows

The explicit hybrid methods

119888 119860

119887119879

(4)

where119860 = [ 11988611 d1198861199041 119886119904119904minus1

] 119888 = [ 1198881119888119904

] and 119887119879 = [ 1198871119887119904

]In this section we derived the order condition for block

explicit hybrid method (BEHM)TheTaylor series expansionfor 119910119899+119898 and 119910119899minus119898 is as follows

119910119899+119898 = 119910119899 + 119898ℎ1199101198991015840 + 11989822 ℎ211991011989910158401015840 + 1198983

3 ℎ3119910119899101584010158401015840

+ 11989844 ℎ4119910119899(4) + sdot sdot sdot(5)

119910119899minus119898 = 119910119899 minus 119898ℎ1199101198991015840 + 11989822 ℎ211991011989910158401015840 minus 1198983

3 ℎ3119910119899101584010158401015840

+ 11989844 ℎ4119910119899(4) + sdot sdot sdot(6)

where ℎ is step sizes and 119898 is the number of step forwardAdding (5) and (6) we obtain

119910119899+119898 minus 119910119899minus119898 = 2119910119899 + 1198982ℎ211991011989910158401015840 + 119898412 ℎ4119910119899(4) + sdot sdot sdot (7)

or

119910119899+119898 = 2119910119899 minus 119910119899minus119898 + 1198982ℎ211991011989910158401015840 + 119898412 ℎ4119910119899(4) + sdot sdot sdot (8)

which can be expressed as

119910119899+119898 = 2119910119899 minus 119910119899minus119898 + ℎ2Δ (119909119899 119910119899 ℎ) (9)

The increment function is

Δ (119909119899 119910119899 ℎ) = 119898211991011989910158401015840 + 119898412 ℎ2119910119899(4) + sdot sdot sdot (10)

Then consider the general formula for block explicit hybridmethod in the form of

119910119899+119898 = 2119910119899 minus 119910119899minus119898 + ℎ2 119904sum119894=1

(119898)119887119894119891 (119909119899 119910119899 ℎ) (11)

Equation (11) can be simplified as

119910119899+119898 = 2119910119899 minus 119910119899minus119898 + ℎ2120601 (119909119899 119910119899 ℎ) (12)such that the Taylor series increment may be written as

120601 (119909119899 119910119899 ℎ) = 119904sum119894=1

(119898)119887119894119891 (119909119899 + 119888119894ℎ 119884119894)

= 119904sum119894=1

(119898)119887119894119865(2)1 + ℎ 119904sum119894=1

(119898)119887119894119888119894119865(3)1+ ℎ22 (

119904sum119894=1

(119898)1198871198941198881198942119865(4)1 + 119904sum119894=1

(119898)119887119894119886119894119895119865(4)2 )

(13)

where the first few elementary differential are

119865(2)1 = 119891119865(3)1 = 119891119909 + 1198911199101199101015840119865(4)1 = 119891119909119909 + 21199101015840119891119909119910 + (1199101015840)2119891119910119910 + 119891119891119910

The local truncation errors of the solution is obtain bysubtracting (12) from (9) This gives us

119905119899+1 = ℎ2 [120601 minus Δ] = ℎ2 [ 119904sum119894=1

(119898)119887119894119891 (119909119899 + 119888119894ℎ 119884119894)

minus (119898211991011989910158401015840 + 119898412 ℎ2119910119899(4) + sdot sdot sdot)](14)

Therefore from (14) we obtain the order condition for blockexplicit hybrid method up to order 5 as follows

Order 2 119904sum119894=1

(119898)119887119894 = 1198982

Order 3 119904sum119894=1

(119898)119887119894119888119894 = 0

Order 4 119904sum119894=1

(119898)1198871198941198881198942 = 11989846119904sum119894=1

(119898)119887119894119886119894119895 = 119898412Order 5 119904sum

119894=1

(119898)1198871198941198881198943 = 0119904sum119894=1

(119898)119887119894119888119894119886119894119895 = 119898412119904sum119894=1

(119898)119887119894119886119894119895119888119895 = 0for 119898 = 1 2

(15)

Journal of Mathematics 3

For the first point (119898 = 1) the order condition is the sameas the order conditions for explicit hybrid method given in

Coleman [10] while for the second point (119898 = 2) the orderconditions are given as follows

Order 2 119904sum119894=1

(2)119887119894 = 4 (16)

Order 3 119904sum119894=1

(2)119887119894119888119894 = 0 (17)

Order 4 119904sum119894=1

(2)1198871198941198881198942 = 83 (18)

119904sum119894=1

(2)119887119894119886119894119895 = 43 (19)

Order 5 119904sum119894=1

(2)1198871198941198881198943 = 0 (20)

119904sum119894=1

(2)119887119894119888119894119886119894119895 = 43 (21)

119904sum119894=1

(2)119887119894119886119894119895119888119895 = 0(Note The notation (119898)119887119894 means the coefficient of 119887 for 119898ℎ step forward)

(22)

The general formula for block explicit hybrid method(BEHM) for119898 = 1 2 can be written as

119884119894 = (1 + 119888119894) 119910119899 minus 119888119894119910119899minus1 + ℎ2 119904sum119895=1

119886119894119895119891 (119909119899 + 119888119895ℎ 119884119895) (23)

119910119899+1 = 2119910119899 minus 119910119899minus1 + ℎ2 119904sum119894=1

(1)119887119894119891 (119909119899 + 119888119894ℎ 119884119894) (24)

119910119899+2 = 2119910119899 minus 119910119899minus2 + ℎ2 119904sum119894=1

(2)119887119894119891 (119909119899 + 119888119894ℎ 119884119894) (25)

3 Construction of Trigonometrically FittedBlock Explicit Hybrid Method

In order to derive the two-point BEHMwe used the algebraiccoefficients of the original three-stage explicit hybrid methodin Franco [11] as the first point (119898 = 1) The block methodcan be written as follows

The coefficient of the block explicit hybrid method

119888 119860

(1)119887119879

(2)119887119879

=minus101 0 1112 56 112(2)1198871 (2)1198872 (2)1198873

(26)

First and second point of the BEHM share the same valuesof 119888 and 119860 By solving (16)-(19) simultaneously we obtain aunique solution of (2)119887119894

(2)1198871 = 43 (2)1198872 = 43 (2)1198873 = 43

(27)

By checking the fifth-order conditions for the first point wenoticed that the method at the first point ((1)1198871 = 112 (1)1198872 =56 (1)1198873 = 112 1198881 = minus1 1198882 = 0 1198883 = 1 11988631 = 0 11988632 = 1)satisfies sum3119894=1 (1)1198871198941198881198943 = 0sum3119894=1 (1)119887119894119888119894119886119894119895 = 112sum3119894=1 (1)119887119894119886119894119895119888119895 =0

Hence the method in Franco [11] is order fiveAnd by checking fifth-order conditions for the second

point we noticed that themethod at the second point satisfies3sum119894=1

(2)1198871198941198881198943 = 03sum119894=1

(2)119887119894119888119894119886119894119895 = 43 3sum119894=1

(2)119887119894119886119894119895119888119895 = 0

(28)


Hence the block explicit hybrid method that we have derivedis three-stage and order five denoted as BEHM3(5)

To trigonometrically fit the method we require (23) (24)and (25) to integrate exactly the linear combination of thefunctions sin(120596119905) cos(120596119905) for isin R Hence the followingequations are obtained

cos (1198883119867)= 1 + 1198883 minus 1198883 cos (119867) minus 1198672 11988631 cos (119867) + 11988632 (29)

sin (1198883119867) = 1198883 sin (119867) + 1198672 11988631 sin (119867) (30)

2 cos (119867)= 2 minus 1198672 (1)1198871 cos (119867) + (1)1198872 + (1)1198873 cos (1198883119867) (31)

(1)1198871 sin (119867) = (1)1198873 sin (1198883119867) (32)

2 cos (2119867)= 2 minus 1198672 (2)1198871 cos (119867) + (2)1198872 + (2)1198873 cos (1198883119867) (33)

(2)1198871 sin (119867) = (2)1198873 sin (1198883119867) (34)

where119867 = ℎ120596 ℎ is step size and 120596 is the fitted frequency ofthe problem (120596 depends on the problems)

By solving (29) (30) with 1198881 = minus1 1198882 = 0 and 1198883 = 1 weobtain the remaining values in terms of119867 as follows

11988631 = 011988632 = minus2 (cos (119867) minus 1)1198672 (35)

To find (1)119887119894 values we solve (31) (32) with choice ofcoefficients 1198881 = minus1 1198882 = 0 and 1198883 = 1 we obtained thefollowing

(1)1198871 = minus12 2 cos (119867) minus 2 + 1198672

1198672 (cos (119867) minus 1) (1)1198872 = 2 cos (119867) minus 2 + 119867

2 cos (119867)1198672 (cos (119867) minus 1) (1)1198873 = minus12 2 cos (119867) minus 2 + 119867

2

1198672 (cos (119867) minus 1) (36)

Then for the (2)119887119894 values we solve (33) (34) with choice ofcoefficients 1198883 = 1 and letting (2)1198871 = 43 and (2)1198873 = 43 weobtained (2)1198872 in terms of119867 as

(2)b2 = minus23sdot 3 cos (2119867) sin (119867) minus 3 sin (119867) + 41198672 cos (119867) sin (119867)

H2 sin (119867) (37)

Hence we denote the new method as three-stage fifth-ordertrigonometrically fitted block explicit hybrid method (TF-BEHM3(4))The method is shown as follows

The coefficient of the trigonometrically fitted block explicit hybrid method

(TF-BEHM3(5))

minus101 0 11988632(1)1198871 (1)1198872 (1)119887343 (2)1198872 43

(38)

The coefficients can be written in Taylor expansion to avoidheavy cancellation in the implementation of the method

11988632 = 1 minus 121198672 + 13601198674 minus 1201601198676+ 118144001198678 + 119874 (11986710)

(1)1198871 = 112 + 12401198672 + 160481198674 + 11728001198676

+ 119874 (1198678) (1)1198872 = 56 minus 11201198672 minus 130241198674 minus 1864001198676

+ 119874 (1198678) (1)1198873 = 112 + 12401198672 + 160481198674 + 11728001198676

+ 119874 (1198678)


(2)1198872 = 43 + 1151198674 minus 1718901198676 + 1132268001198678minus 742768011986710 + 119874 (11986711)

(39)

4 Problems Tested and Numerical Results

In this section the new methods BEHM3(5) and TF-BEHM3(5) are used to solve oscillatory delay differentialequations problems The delay terms are evaluated usingNewton divided different interpolation A measure of theaccuracy is examined using absolute error which is definedby

Absolute error = max 1003816100381610038161003816119910 (119905119899) minus 1199101198991003816100381610038161003816 (40)

where 119910(119905119899) is the exact solution and 119910119899 is the computedsolution

The test problems are listed as follows

Problem 1 (source Schmidt [12])

11991010158401015840 (119905) = minus12119910 (119905) + 12119910 (119905 minus 120587) 0 le 119905 le 8120587 119910119900 = 0The fitted frequency is V = 1 Exact solution is 119910 (119905)= sin (119905)

(41)


11991010158401015840 (119905) minus 119910 (119905) + 120578 (119905) 119910 ( 1199052) = 0 0 le 119905 le 2120587where 120578 (119905) = 4 sin (119905)

(2 minus 2 cos (119905))12 120578 (0) = 4Fitted frequency is V = 2 Exact solution is 119910 (119905)= sin (119905)

(42)

Problem 3 (source Ladas and Stavroulakis [13])

11991010158401015840 (119905) = 119910 (119905 minus 120587) 0 le 119905 le 8120587 119910119900 = 0Fitted frequency is V = 1 Exact solution is 119910 (119905)= sin (119905)

(43)

Problem 4 (source Bhagat Singh[14])

11991010158401015840 (119905) = minus sin (119905)2 minus sin (119905)119910 (119905 minus 120587) 0 le 119905 le 8120587 119910119900 = 2Fitted frequency is V = 1 Exact solution is 119910 (119905)= 2 + sin (119905)

(44)

The following notations are used in Figures 1ndash4

TF-BEHM3(5) A three-stage fifth-order trigonomet-rically fitted block explicit hybrid method derived inthis paper

Time (seconds)

Log(

max

- er

ror)

1 2 3 4 5 6

minus1

0

minus2

minus3

minus4

minus5

minus6

TF-BEHM3(5)

MPAFRKN4(4)SIHM4(5)PFRKN4(4)EHM4(5)

BEHM3(5)DIRKN4(4)

Figure 1 The efficiency curve of Problem 1 for ℎ = 1205874119894 (119894 =1 5)

BEHM3(5) A three-stage fifth-order new blockexplicit hybrid method derived in this paperSIHM4(5) A four-stage fifth-order semi-implicithybrid method in Ahmad et al[15]MPAFRKN4(4) A modified phase-fitted andamplification-fitted RKN method of four-stagefourth-order by Papadopoulos et al [16]PFRKN4(4) A phase-fitted RKN method of four-stage fourth-order by Papadopoulos et al [17]DIRKN4(4) A four-stage fourth-order diagonallyimplicit RKNmethod by Senu et al [18]EHM4(5) A four-stage fifth-order phase-fittedhybrid method by Franco [11]

5 Discussion and Conclusion

Figures 1ndash4 show the efficiency curves the logarithm of themaximum global error versus the CPU time taken in secondFrom our observation for all the problems TF-BEHM3(5)required lesser time to do the computation In Problems 1 3and 4 TF-BEHM3(5) has better accuracy compared to all themethods in comparison However for Problem 2 efficiencyof TF-BEHM3(5) is comparable to EHM4(5) and has betterperformance compared to the other methods For Problems2 3 and 4 too BEHM3(4) is comparable to the other existingmethods in comparison

The existing methods MPAFRKN4(4) and PFRKN4(4)were derived using specific fitting techniques SIHM4(5)EHM4(5) and DIRKN4(4) were derived with higher orderof dispersion and dissipation and all the methods purposelyderived for solving oscillatory problems

In this paper we derived the order conditions of theblock hybrid method up to order five based on the orderconditions we derived a two-point three-stage fifth-order

6 Journal of MathematicsLo

g(m

ax -

erro

r)

2 4 6 8 10

minus3

minus4

minus5

minus6

minus7

Time (seconds)TF-BEHM3(5)


BEHM3(5)DIRKN4(4)


2 3 4 5

minus1

0

minus2

1

2

minus3

minus4

Log(

max

- er

ror)



BEHM3(5)DIRKN4(4)


block explicit hybrid method (BEHM3(5)) and then themethod is trigonometrically fitted Both the fitted and non-fitted methods evaluate the solution of the problem at twopoints for each time step hence lesser time is needed to dothe computation making them faster and cheaper comparedto the other methods From the numerical results we canconclude that BEHM3(4) is a good method for directlysolving second-order DDEs however trigonometrically fit-ting the block method does improve the efficiency of themethod tremendously It can be said that TF-BEHM3(5) isa promising tool for solving special second-order DDEs withoscillatory solutions

Log(

max

- er

ror)

minus1

0

minus2

minus3

minus4

minus5

minus6

Time (seconds)1 2 3 4 5

TF-BEHM3(5)


BEHM3(5)DIRKN4(4)


Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to acknowledge Universiti PutraMalaysia for funding the research through Putra ResearchGrant vote number 9543500

References

[1] F Ismail R A Al-Khasawneh A S Lwin and M SuleimanldquoNumerical Treatment of Delay Differential Equations ByRunge-Kutta Method Using Hermite Interpolationrdquo Matem-atika vol 18 no 2 pp 79ndash90 2002

[2] OA Taiwo andO S Odetunde ldquoOn the numerical approxima-tion of delay differential equations by a decompositionmethodrdquoAsian Journal of Mathematics amp Statistics vol 3 no 4 pp 237ndash243 2010

[3] F Ishak M B Suleiman and Z Omar ldquoTwo-point predictor-correctorblockmethod for solving delay differential equationsrdquoMatematika vol 24 no 2 pp 131ndash140 2008

[4] H C San Z A Majid and M Othman ldquoSolving delay differ-ential equations using coupled block methodrdquo in Proceedings ofthe 4th International Conference on Modeling Simulation andApplied Optimization (ICMSAO rsquo11) pp 1ndash4 Kuala LumpurMalaysia April 2011

[5] H M Radzi Z A Majid F Ismail andM Suleiman ldquoTwo andthree point one-step blockmethods for solving delay differential


equationsrdquo Journal of Quality Measurement and Analysis vol82 no 1 pp 29ndash41 1823

[6] L K Yap and F Ismail ldquoBlock Hybrid-like Method for SolvingDelay Differential Equationsrdquo in Proceedings of the AIP Con-ference Proceeding of International Conference on MathematicsEngineering Industrial Applications (IComedia vol 1660 pp28ndash30 2014

[7] Hoo Yann Seong Zanariah Abdul Majid and Fudziah IsmailldquoSolving Second-Order Delay Differential Equations by DirectAdams-Moulton MethodrdquoMathematical Problems in Engineer-ing vol 2013 Article ID 261240 7 pages 2013

[8] MMechee F Ismail N Senu andZ Siri ldquoDirectly Solving Spe-cial Second Order Delay Differential Equations Using Runge-Kutta-Nystrom Methodrdquo Mathematical Problems in Engineer-ing vol 2013 Article ID 830317 7 pages 2013

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

[10] J P Coleman ldquoOrder conditions for a class of two-stepmethodsfor y10158401015840 = 119891(119909 119910)rdquo IMA Journal of Numerical Analysis (IMA-JNA) vol 23 no 2 pp 197ndash220 2003

[11] J M Franco ldquoA class of explicit two-step hybrid methodsfor second-order IVPsrdquo Journal of Computational and AppliedMathematics vol 187 no 1 pp 41ndash57 2006

[12] K Schmitt ldquoComparison theorems for second order delaydifferential equationsrdquo RockyMountain Journal of Mathematicsvol 1 no 3 pp 459ndash467 1971

[13] G Ladas and I P Stavroulakis ldquoOn delay differential inequali-ties of first orderrdquo Fako de lrsquoFunkcialaj Ekvacioj Japana Matem-atika Societo Funkcialaj Ekvaciog Serio Internacia vol 25 no1 pp 105ndash113 1982

[14] B Singh ldquoAsymptotic nature on non-oscillatory solutions ofnth order retarded differential equationsrdquo SIAM Journal onMathematical Analysis vol 6 no 5 pp 784ndash795 1975

[15] S Z Ahmad F Ismail N Senu and M Suleiman ldquoSemiImplicit Hybrid Methods with Higher Order Dispersion forSolving Oscillatory Problemsrdquo Abstract and Applied Analysisvol 2013 Article ID 136961 10 pages 2013

[16] D F Papadopoulos Z A Anastassi and T E Simos ldquoAmodified phase-fitted and amplification-fitted Runge-Kutta-Nystrom method for the numerical solution of the radialSchrodinger equationrdquo Journal of Molecular Modeling vol 16no 8 pp 1339ndash1346 2010

[17] D F Papadopoulos Z A Anastassi and T E Simos ldquoA phase-fitted Runge-Kutta Nystrommethod for the numerical solutionof initial value problems with oscillating solutionsrdquo ComputerPhysics Communications vol 180 no 10 pp 1839ndash1846 2009

[18] N Senu M Suleiman F Ismail and M Othman ldquoA SinglyDiagonally Implicit Runge-Kutta Nystrom Method for SolvingOscillatory Problemsrdquo IAENG International Journal of AppliedMathematics vol 41 no 2 pp 155ndash161 2011

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second-order DDEs which are oscillatory in nature Finallywe tested the new methods using DDEs test problems toindicate that it is superior and more efficient for solvingoscillatory second order DDEs

2 Derivation of Order Condition for BlockExplicit Hybrid Method

The general formula of two-step explicit hybrid methodfor solving the special second-order ordinary differentialequations is given as

119884119894 = (1 + 119888119894) 119910119899 minus 119888119894119910119899minus1 + ℎ2 119904sum119895=1

119886119894119895119891 (119909119899 + 119888119895ℎ 119884119895) (2)

119910119899+1 = 2119910119899 minus 119910119899minus1 + ℎ2 119904sum119894=1

119887119894119891 (119909119899 + 119888119894ℎ 119884119894) (3)

where 119894 = 1 119904 and 119894 gt 119895 The method coefficients of 119887119894 119888119894and 119886119894119895 can be represented in Butcher tableau as follows

The explicit hybrid methods

119888 119860

119887119879

(4)

where119860 = [ 11988611 d1198861199041 119886119904119904minus1

] 119888 = [ 1198881119888119904

] and 119887119879 = [ 1198871119887119904

]In this section we derived the order condition for block

explicit hybrid method (BEHM)TheTaylor series expansionfor 119910119899+119898 and 119910119899minus119898 is as follows

119910119899+119898 = 119910119899 + 119898ℎ1199101198991015840 + 11989822 ℎ211991011989910158401015840 + 1198983

3 ℎ3119910119899101584010158401015840

+ 11989844 ℎ4119910119899(4) + sdot sdot sdot(5)

119910119899minus119898 = 119910119899 minus 119898ℎ1199101198991015840 + 11989822 ℎ211991011989910158401015840 minus 1198983

3 ℎ3119910119899101584010158401015840

+ 11989844 ℎ4119910119899(4) + sdot sdot sdot(6)

where ℎ is step sizes and 119898 is the number of step forwardAdding (5) and (6) we obtain

119910119899+119898 minus 119910119899minus119898 = 2119910119899 + 1198982ℎ211991011989910158401015840 + 119898412 ℎ4119910119899(4) + sdot sdot sdot (7)

or

119910119899+119898 = 2119910119899 minus 119910119899minus119898 + 1198982ℎ211991011989910158401015840 + 119898412 ℎ4119910119899(4) + sdot sdot sdot (8)

which can be expressed as

119910119899+119898 = 2119910119899 minus 119910119899minus119898 + ℎ2Δ (119909119899 119910119899 ℎ) (9)

The increment function is

Δ (119909119899 119910119899 ℎ) = 119898211991011989910158401015840 + 119898412 ℎ2119910119899(4) + sdot sdot sdot (10)

Then consider the general formula for block explicit hybridmethod in the form of

119910119899+119898 = 2119910119899 minus 119910119899minus119898 + ℎ2 119904sum119894=1

(119898)119887119894119891 (119909119899 119910119899 ℎ) (11)

Equation (11) can be simplified as

119910119899+119898 = 2119910119899 minus 119910119899minus119898 + ℎ2120601 (119909119899 119910119899 ℎ) (12)such that the Taylor series increment may be written as

120601 (119909119899 119910119899 ℎ) = 119904sum119894=1

(119898)119887119894119891 (119909119899 + 119888119894ℎ 119884119894)

= 119904sum119894=1

(119898)119887119894119865(2)1 + ℎ 119904sum119894=1

(119898)119887119894119888119894119865(3)1+ ℎ22 (

119904sum119894=1

(119898)1198871198941198881198942119865(4)1 + 119904sum119894=1

(119898)119887119894119886119894119895119865(4)2 )

(13)

where the first few elementary differential are

119865(2)1 = 119891119865(3)1 = 119891119909 + 1198911199101199101015840119865(4)1 = 119891119909119909 + 21199101015840119891119909119910 + (1199101015840)2119891119910119910 + 119891119891119910

The local truncation errors of the solution is obtain bysubtracting (12) from (9) This gives us

119905119899+1 = ℎ2 [120601 minus Δ] = ℎ2 [ 119904sum119894=1

(119898)119887119894119891 (119909119899 + 119888119894ℎ 119884119894)

minus (119898211991011989910158401015840 + 119898412 ℎ2119910119899(4) + sdot sdot sdot)](14)

Therefore from (14) we obtain the order condition for blockexplicit hybrid method up to order 5 as follows

Order 2 119904sum119894=1

(119898)119887119894 = 1198982

Order 3 119904sum119894=1

(119898)119887119894119888119894 = 0

Order 4 119904sum119894=1

(119898)1198871198941198881198942 = 11989846119904sum119894=1

(119898)119887119894119886119894119895 = 119898412Order 5 119904sum

119894=1

(119898)1198871198941198881198943 = 0119904sum119894=1

(119898)119887119894119888119894119886119894119895 = 119898412119904sum119894=1

(119898)119887119894119886119894119895119888119895 = 0for 119898 = 1 2

(15)




Order 2 119904sum119894=1

(2)119887119894 = 4 (16)

Order 3 119904sum119894=1

(2)119887119894119888119894 = 0 (17)

Order 4 119904sum119894=1

(2)1198871198941198881198942 = 83 (18)

119904sum119894=1

(2)119887119894119886119894119895 = 43 (19)

Order 5 119904sum119894=1

(2)1198871198941198881198943 = 0 (20)

119904sum119894=1

(2)119887119894119888119894119886119894119895 = 43 (21)

119904sum119894=1


(22)


119884119894 = (1 + 119888119894) 119910119899 minus 119888119894119910119899minus1 + ℎ2 119904sum119895=1

119886119894119895119891 (119909119899 + 119888119895ℎ 119884119895) (23)

119910119899+1 = 2119910119899 minus 119910119899minus1 + ℎ2 119904sum119894=1

(1)119887119894119891 (119909119899 + 119888119894ℎ 119884119894) (24)

119910119899+2 = 2119910119899 minus 119910119899minus2 + ℎ2 119904sum119894=1

(2)119887119894119891 (119909119899 + 119888119894ℎ 119884119894) (25)




119888 119860

(1)119887119879

(2)119887119879

=minus101 0 1112 56 112(2)1198871 (2)1198872 (2)1198873

(26)


(2)1198871 = 43 (2)1198872 = 43 (2)1198873 = 43

(27)




(2)1198871198941198881198943 = 03sum119894=1

(2)119887119894119888119894119886119894119895 = 43 3sum119894=1

(2)119887119894119886119894119895119888119895 = 0

(28)





sin (1198883119867) = 1198883 sin (119867) + 1198672 11988631 sin (119867) (30)

2 cos (119867)= 2 minus 1198672 (1)1198871 cos (119867) + (1)1198872 + (1)1198873 cos (1198883119867) (31)

(1)1198871 sin (119867) = (1)1198873 sin (1198883119867) (32)

2 cos (2119867)= 2 minus 1198672 (2)1198871 cos (119867) + (2)1198872 + (2)1198873 cos (1198883119867) (33)

(2)1198871 sin (119867) = (2)1198873 sin (1198883119867) (34)



11988631 = 011988632 = minus2 (cos (119867) minus 1)1198672 (35)


(1)1198871 = minus12 2 cos (119867) minus 2 + 1198672



2

1198672 (cos (119867) minus 1) (36)



H2 sin (119867) (37)



(TF-BEHM3(5))

minus101 0 11988632(1)1198871 (1)1198872 (1)119887343 (2)1198872 43

(38)


11988632 = 1 minus 121198672 + 13601198674 minus 1201601198676+ 118144001198678 + 119874 (11986710)

(1)1198871 = 112 + 12401198672 + 160481198674 + 11728001198676


+ 119874 (1198678) (1)1198873 = 112 + 12401198672 + 160481198674 + 11728001198676

+ 119874 (1198678)


(2)1198872 = 43 + 1151198674 minus 1718901198676 + 1132268001198678minus 742768011986710 + 119874 (11986711)

(39)








(41)




(42)



(43)



(44)



Time (seconds)

Log(

max

- er

ror)

1 2 3 4 5 6

minus1

0

minus2

minus3

minus4

minus5

minus6

TF-BEHM3(5)


BEHM3(5)DIRKN4(4)








g(m

ax -

erro

r)

2 4 6 8 10

minus3

minus4

minus5

minus6

minus7



BEHM3(5)DIRKN4(4)


2 3 4 5

minus1

0

minus2

1

2

minus3

minus4

Log(

max

- er

ror)



BEHM3(5)DIRKN4(4)



Log(

max

- er

ror)

minus1

0

minus2

minus3

minus4

minus5

minus6


TF-BEHM3(5)


BEHM3(5)DIRKN4(4)


Data Availability




Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018











Order 2 119904sum119894=1

(2)119887119894 = 4 (16)

Order 3 119904sum119894=1

(2)119887119894119888119894 = 0 (17)

Order 4 119904sum119894=1

(2)1198871198941198881198942 = 83 (18)

119904sum119894=1

(2)119887119894119886119894119895 = 43 (19)

Order 5 119904sum119894=1

(2)1198871198941198881198943 = 0 (20)

119904sum119894=1

(2)119887119894119888119894119886119894119895 = 43 (21)

119904sum119894=1


(22)


119884119894 = (1 + 119888119894) 119910119899 minus 119888119894119910119899minus1 + ℎ2 119904sum119895=1

119886119894119895119891 (119909119899 + 119888119895ℎ 119884119895) (23)

119910119899+1 = 2119910119899 minus 119910119899minus1 + ℎ2 119904sum119894=1

(1)119887119894119891 (119909119899 + 119888119894ℎ 119884119894) (24)

119910119899+2 = 2119910119899 minus 119910119899minus2 + ℎ2 119904sum119894=1

(2)119887119894119891 (119909119899 + 119888119894ℎ 119884119894) (25)




119888 119860

(1)119887119879

(2)119887119879

=minus101 0 1112 56 112(2)1198871 (2)1198872 (2)1198873

(26)


(2)1198871 = 43 (2)1198872 = 43 (2)1198873 = 43

(27)




(2)1198871198941198881198943 = 03sum119894=1

(2)119887119894119888119894119886119894119895 = 43 3sum119894=1

(2)119887119894119886119894119895119888119895 = 0

(28)





sin (1198883119867) = 1198883 sin (119867) + 1198672 11988631 sin (119867) (30)

2 cos (119867)= 2 minus 1198672 (1)1198871 cos (119867) + (1)1198872 + (1)1198873 cos (1198883119867) (31)

(1)1198871 sin (119867) = (1)1198873 sin (1198883119867) (32)

2 cos (2119867)= 2 minus 1198672 (2)1198871 cos (119867) + (2)1198872 + (2)1198873 cos (1198883119867) (33)

(2)1198871 sin (119867) = (2)1198873 sin (1198883119867) (34)



11988631 = 011988632 = minus2 (cos (119867) minus 1)1198672 (35)


(1)1198871 = minus12 2 cos (119867) minus 2 + 1198672



2

1198672 (cos (119867) minus 1) (36)



H2 sin (119867) (37)



(TF-BEHM3(5))

minus101 0 11988632(1)1198871 (1)1198872 (1)119887343 (2)1198872 43

(38)


11988632 = 1 minus 121198672 + 13601198674 minus 1201601198676+ 118144001198678 + 119874 (11986710)

(1)1198871 = 112 + 12401198672 + 160481198674 + 11728001198676


+ 119874 (1198678) (1)1198873 = 112 + 12401198672 + 160481198674 + 11728001198676

+ 119874 (1198678)


(2)1198872 = 43 + 1151198674 minus 1718901198676 + 1132268001198678minus 742768011986710 + 119874 (11986711)

(39)








(41)




(42)



(43)



(44)



Time (seconds)

Log(

max

- er

ror)

1 2 3 4 5 6

minus1

0

minus2

minus3

minus4

minus5

minus6

TF-BEHM3(5)


BEHM3(5)DIRKN4(4)








g(m

ax -

erro

r)

2 4 6 8 10

minus3

minus4

minus5

minus6

minus7



BEHM3(5)DIRKN4(4)


2 3 4 5

minus1

0

minus2

1

2

minus3

minus4

Log(

max

- er

ror)



BEHM3(5)DIRKN4(4)



Log(

max

- er

ror)

minus1

0

minus2

minus3

minus4

minus5

minus6


TF-BEHM3(5)


BEHM3(5)DIRKN4(4)


Data Availability




Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018












sin (1198883119867) = 1198883 sin (119867) + 1198672 11988631 sin (119867) (30)

2 cos (119867)= 2 minus 1198672 (1)1198871 cos (119867) + (1)1198872 + (1)1198873 cos (1198883119867) (31)

(1)1198871 sin (119867) = (1)1198873 sin (1198883119867) (32)

2 cos (2119867)= 2 minus 1198672 (2)1198871 cos (119867) + (2)1198872 + (2)1198873 cos (1198883119867) (33)

(2)1198871 sin (119867) = (2)1198873 sin (1198883119867) (34)



11988631 = 011988632 = minus2 (cos (119867) minus 1)1198672 (35)


(1)1198871 = minus12 2 cos (119867) minus 2 + 1198672



2

1198672 (cos (119867) minus 1) (36)



H2 sin (119867) (37)



(TF-BEHM3(5))

minus101 0 11988632(1)1198871 (1)1198872 (1)119887343 (2)1198872 43

(38)


11988632 = 1 minus 121198672 + 13601198674 minus 1201601198676+ 118144001198678 + 119874 (11986710)

(1)1198871 = 112 + 12401198672 + 160481198674 + 11728001198676


+ 119874 (1198678) (1)1198873 = 112 + 12401198672 + 160481198674 + 11728001198676

+ 119874 (1198678)


(2)1198872 = 43 + 1151198674 minus 1718901198676 + 1132268001198678minus 742768011986710 + 119874 (11986711)

(39)








(41)




(42)



(43)



(44)



Time (seconds)

Log(

max

- er

ror)

1 2 3 4 5 6

minus1

0

minus2

minus3

minus4

minus5

minus6

TF-BEHM3(5)


BEHM3(5)DIRKN4(4)








g(m

ax -

erro

r)

2 4 6 8 10

minus3

minus4

minus5

minus6

minus7



BEHM3(5)DIRKN4(4)


2 3 4 5

minus1

0

minus2

1

2

minus3

minus4

Log(

max

- er

ror)



BEHM3(5)DIRKN4(4)



Log(

max

- er

ror)

minus1

0

minus2

minus3

minus4

minus5

minus6


TF-BEHM3(5)


BEHM3(5)DIRKN4(4)


Data Availability




Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018









(2)1198872 = 43 + 1151198674 minus 1718901198676 + 1132268001198678minus 742768011986710 + 119874 (11986711)

(39)








(41)




(42)



(43)



(44)



Time (seconds)

Log(

max

- er

ror)

1 2 3 4 5 6

minus1

0

minus2

minus3

minus4

minus5

minus6

TF-BEHM3(5)


BEHM3(5)DIRKN4(4)








g(m

ax -

erro

r)

2 4 6 8 10

minus3

minus4

minus5

minus6

minus7



BEHM3(5)DIRKN4(4)


2 3 4 5

minus1

0

minus2

1

2

minus3

minus4

Log(

max

- er

ror)



BEHM3(5)DIRKN4(4)



Log(

max

- er

ror)

minus1

0

minus2

minus3

minus4

minus5

minus6


TF-BEHM3(5)


BEHM3(5)DIRKN4(4)


Data Availability




Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018









g(m

ax -

erro

r)

2 4 6 8 10

minus3

minus4

minus5

minus6

minus7



BEHM3(5)DIRKN4(4)


2 3 4 5

minus1

0

minus2

1

2

minus3

minus4

Log(

max

- er

ror)



BEHM3(5)DIRKN4(4)



Log(

max

- er

ror)

minus1

0

minus2

minus3

minus4

minus5

minus6


TF-BEHM3(5)


BEHM3(5)DIRKN4(4)


Data Availability




Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018






























Journal of












Journal of







Volume 2018






Volume 2018








solving oscillatory delay differential equations using block hybrid … · 2019. 7. 30. ·...

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