Research ArticleApproximate Solutions of Delay Differential Equationswith Constant and Variable Coefficients by the EnhancedMultistage Homotopy Perturbation Method

D Olvera1 A Eliacuteas-Zuacutentildeiga1 L N Loacutepez de Lacalle2 and C A Rodriacuteguez1

1Center for Innovation in Design and Technology Tecnologico de Monterrey Campus Monterrey E Garza Sada 250164849 Monterrey NL Mexico2Department of Mechanical Engineering University of the Basque Country Alameda de Urquijo sn Bilbao 48013 Bizkaia Spain

Correspondence should be addressed to A Elıas-Zuniga aeliasitesmmx

Received 26 December 2013 Revised 21 July 2014 Accepted 16 August 2014

Academic Editor Zhichun Yang

Copyright copy 2015 D Olvera et alThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We expand the application of the enhanced multistage homotopy perturbation method (EMHPM) to solve delay differentialequations (DDEs) with constant and variable coefficients This EMHPM is based on a sequence of subintervals that provideapproximate solutions that require less CPU time than those computed from the dde23 MATLAB numerical integration algorithmsolutions To address the accuracy of our proposed approachwe examine the solutions of severalDDEs having constant and variablecoefficients finding predictions with a good match relative to the corresponding numerical integration solutions

1 Introduction

Delayed differential equations (DDEs) are used to describemany physical phenomena of interest in biology medicinechemistry physics engineering and economics among oth-ers Since the introduction of the first delayed models manypublications have appeared as summarizing theorems andhomotopy methods of solution that deal with the stabilityproperties of delayed systems (see [1ndash3] and references citedthere in)

For instance Shakeri andDehghan introduced an approachto find the solution of delay differential equations by meansof the homotopy perturbation technique (HPM) with resultsthat agree well with exact solutions [1] Wu in [2] usedthe homotopy analysis method to obtain the approximatesolution of a strong nonlinear ENSOdelayed oscillatormodelthat provides good agreement when compared to its exactsolution under the condition of 119861 = 0 Alomari andcoworkers in [3] developed an algorithm to obtain approx-imate analytical solutions for DDEs by using the homotopyanalysismethod (HAM) and themodified homotopy analysismethod (MHAM)They used their derived method to obtain

the approximate solution of various linear and nonlinearDDEs with numerical predictions that agree well with thenumerical integration solutions and they also proved thattheir derived solutions converge to the exact ones By apply-ing the homotopy perturbation method (HPM) Biazar andBehzad found approximate solutions of neutral differentialequations with proportional delays which describe well theircorresponding numerical integration solutions [4] RecentlyAnakira and co-workers in [5] extended the applicability ofthe so called optimal homotopy asymptoticmethod (OHAM)that does not depend on small or large parameters to findthe approximate analytic solution of DDEs They used theirproposed approach to compare the derived approximatesolutions of several DDEswith their exact analytical solutionswith predictions that compare well with the exact ones

On the other hand Insperger and Stepan in [6] used thesemidiscretization method to determine the stability lobes ofDDEs that model the dynamics of cutting machine opera-tions Based on the properties of the Chebyshev polynomialsButcher and coworkers in [7] developed a methodology toobtain the stability lobes of milling machine operations and

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2015 Article ID 382475 12 pageshttpdxdoiorg1011552015382475

2 Abstract and Applied Analysis

they proved that this technique is faster than that of thefull and the semidiscretization methods since these solutiontechniques approximate the original DDEs by a series ofODEs [8]

Here in this paper we develop a generalized procedure tosolve linear and nonlinear DDEs by introducing some mod-ifications to the multistage homotopy perturbation method(MHPM) derived by Hashim and Chowdhury to obtainapproximate solutions of ordinary differential equations [9]The proposed enhanced multistage homotopy perturbationmethod (EMHPM) is based on a sequence of subintervalsthat allow us to find more accurate approximated solutionsunder a numerical-analytical procedure that requires lessCPU time when compared to the numerical integrationsolutions provided by the MATLAB dde23 algorithm writtenby Shampine and Thompson in [10] The EMHPM is basedon a homotopy function that could be divided into a linearoperator and a nonlinear operator to satisfy its assumed initialsolution This split of the homotopy function allows us tomodify the nonlinear operator to guarantee by using theenhanced homotopy perturbation method the stability ofthe proposed approximate solutions of nonlinear differentialequations [11]

To clarify our proposed method we briefly review inSection 2 some basic concepts of the homotopy perturbationmethod and then in Section 3 we introduce the EMHPMto solve DDEs The difference between the HPM and theEMHPM is discussed in Section 4 by addressing the approx-imate solutions of a nonlinear delayed differential equationwith variable coefficients Finally the general solution oftwo DDEs that describe the dynamics of two engineeringproblems by using the EMHPM is discussed in Section 5

2 Homotopy Perturbation Method

The homotopy perturbation method (HPM) is a couplingof the traditional perturbation method and homotopy intopology which eliminates the limitation of the small param-eter assumed in the perturbation methods [12] Under thisapproach a nonlinear problem can be transformed into aninfinite number of simple problems without the restrictionof having small nonlinear parameter values This homotopyperturbationmethod takes themain advantages of traditionalperturbation methods together with homotopy analysis [13ndash15]

To illustrate the basic ideas of the HPM let us considerthe following nonlinear differential equation

119860 (119906) minus 119891 (119903) = 0 119903 isin Ω (1)

with boundary conditions

119861(119906120597119906

120597119899) = 0 119903 isin Γ (2)

where 119860 is a general differential operator 119861 is a boundaryoperator 119891(119903) is a known analytic function and Γ is theboundary of the domainΩ

The operator 119860 can generally be divided into two parts 119871and119873 where 119871 involves the linear terms and119873 the nonlinearones Equation (1) therefore can be rewritten as follows

119871 (119906) + 119873 (119906) minus 119891 (119903) = 0 (3)

By the homotopy perturbation technique we construct a homo-topy V(119903 119901) Ω times [0 1] rarr R that satisfies

119867(V 119901) = 119871 (V) minus 119871 (1199060) + 119901119871 (1199060) + 119901 [119873 (V) minus 119891 (119903)] = 0(4)

where 119901 isin [0 1] is an embedding parameter and 1199060is

an initial approximation of (1) which satisfies the boundaryconditions (2) Thus from (4) we have

119867(V 0) = 119871 (V) minus 119871 (1199060) = 0

119867 (V 1) = 119860 (V) minus 119891 (119903) = 0(5)

The changing process of 119901 from zero to unity is just thatof V(119903 119901) from 119906

0(119903) to 119906(119903) In topology this is called

deformation and 119871(V) minus 119871(1199060) and 119860(V) minus 119891(119903) are called

homotopicHe in [12] uses the embedding parameter 119901 as the small

parameter and assumed that the solution of (4) can be writtenas a power series of 119901 in the form

V = V0+ 119901V1+ 1199012V2+ sdot sdot sdot (6)

By setting 119901 = 1 He obtained the approximate solution of (1)as

119906 = lim119901rarr1

= V0+ V1+ V2+ sdot sdot sdot (7)

Then this method was applied to obtain the approximatesolution of some nonlinear ordinary differential equationsvalid not only for small but also for large nonlinear parametervalues

We next will introduce an approach based on homotopymethods to obtain the solution of DDEs with constant andvariable coefficients

3 The EMHPM Methodology to Solve DDEs

TheHPM is an asymptotic method that depends on the aux-iliary linear operator form and the initial guess of the initialconditions Therefore the convergence of the approximatesolution cannot be guaranteed in some cases [16] Hashimand Chowdhury showed in [9] that the solutions obtained bythe standard HPM were not valid for large time span unlessmore terms are calculated Thus they proposed a multistagehomotopy perturbation method (MHPM) which treated theHPMalgorithm in a sequence of subintervals in an attempt toimprove the accuracy of the approximate solutions of linearand nonlinear ordinary differential equations (ODEs)

However when the MHPM is applied to obtain theapproximate solutions of ODEs which contain coefficientsas a function of time this method cannot provide accuratesolutions when Δ119905 rarr 0 In this work we introduce some

Abstract and Applied Analysis 3

modifications to the MHPM and focus on the derivation ofapproximate solutions of DDEs equations with variable coef-ficient terms This new approach is based on the enhancedmultistage homotopy perturbationmethod (EMHPM) intro-duced in [17] to obtain the solution of nonlinear ordinarydifferential equations

The EMHPM is an algorithm which approximates theHPM solution by subintervals utilizing the following trans-formation rule 119906(119905) rarr 119906

119894(119879) where 119906

119894satisfies the initial

condition 119906119894(0) = 119906

119894minus1(119905119894minus1) 119879 is a shifted time scale used

to determine the approximate solution in each subintervaland 119906

119894(119879) represents the approximate solution in the 119894th

subinterval In this case the initial suggested solution in the119894th subinterval is given by 119906

1198940(119879) = 119906

119894minus1(119905119894minus1) where 119905

119894minus1

represents the time at the end of the previous subinterval(ie the value of the approximate solution at the end of theprevious subinterval represents the initial conditions of thenext subinterval under consideration)

To apply the homotopy technique to solve delay differen-tial equations we also assume the following

(1) The linear operator119871(119906119894) represents119871(119906

119894) = (119889119889119879)119906

119894

where the assumed approximate solution 1199061198940(119879) is

set equal to the initial condition 119906119894minus1(119905119894minus1) that is

1199061198940= 119906119894minus1(119905119894minus1) To simplify the notation we let 119906

119894minus1equiv

119906119894minus1(119905119894minus1)

(2) The transformation 119879 = 119905 minus 119905119894minus1

on 0 lt 119879 le 119905119894minus 119905119894minus1

holds in the homotopy 119894-subinterval Thus higherorder equations are integratedwith respect to119879 whilethe terms related to the independent variable 119905 areassumed to remain constant

Therefore we may conclude that the 119898 order approximatesolution by applying the EMHPM can be written as

119906119894(119879 119906119894minus1) =

119898

sum

119896=0

119880119894119896(119879 119906119894minus1) (8)

where the solution 119906119894(119879 119906119894minus1) is valid only in the 119894th subin-

terval [119905119894minus1 119905119894] Hence the solution 119906(119905) on the 119894th subinterval

(119905119894minus1 119905119894] can be written as

119906 (119905) asymp 119906119894(119905 minus 119905119894minus1) (9)

with initial condition 119906119894minus1(119905119894minus1) and 119894 = 1 2 119895 Thus the

approximate solution of 119906 at the time 119905119894is given by

119906119894(119905 minus 119905119894minus1)1003816100381610038161003816119905=119905119894= 119906119894+1(119905 minus 119905119894)1003816100381610038161003816119905=119905119894= 119906119894+1(0) = 119906

119894 (10)

In summary the solution 119906(119905) for an open-closed interval(1199050 1199051] is divided into 119895 subintervals that in general are

not equally spaced [1199050 1199051] [1199051 1199052] [119905

119895minus1 119905119895] Thus the

approximated solution of 119906(119905) for the span time interval isobtained by coupling the 119906

119894(119905) solutions

4 Approximate Solutions of Some DDEs byApplying the EMHPM

In this sectionwe focus on the solution ofDDEswith constantand variable coefficients and examine the applicability of theEMHPM to find the corresponding approximate solutions

41 Delay Differential Equations with Constant CoefficientsFirst let us consider the simplest DDE of the form

(119905) + 119909 (119905 minus 120591) = 0 (11)

with initial condition 119909(0) = 119888 Here the independent vari-able 119909 is a scalar 119909(119905) isin R the dot stands for differentiationwith respect to time 119905 and 120591 is the time delay To evaluate(11) on 119886 le 119905 le 119887 the term 119909(119905 minus 120591) must represent a knownfunction 119909(119905) on [119886 minus 120591 le 119905 le 119886] For instance if 119886 = 0the solution of (11) can be obtained in the interval (0 120591] byassuming an initial function that satisfies the initial conditionBy using this solution it becomes possible to obtain thesolution of (11) in the next 119894th interval [(119894 minus 1)120591 119894120591] 119894 =2 3 119895 where 119895 is an integer number that can be chosenas 2 le 119895 le infin With this approach we can apply the HPM tofind the solution of (11) by assuming that the previous delayedfunction is 1199091205910(119879) = 119888 thus the solution for the first intervalis given by 1199091205911(119879) valid on [0 120591] In terms of (4) we nowconstruct the homotopy of (11)

119867(1198831205911 119901) =

119889

1198891198791198831205911 + 119901119909

1205910 = 0 (12)

We next substitute the first order expansion1198831205911 = 11988312059110+1199011198831205911

1

in (12) and balance the terms with identical power of 119901 toobtain the following set of linear differential equations

1199010 1198891198891198791198831205911

0= 0 119883

1205911

0(0) = 119888 = 119883

1205910(120591)

1199011 1198891198891198791198831205911

1= minus1198831205910 119883

1205911

1(0) = 0

(13)

Integration of (13) yields

1198831205911

0= 119888

1198831205911

1= minus 119888119879

(14)

Hence the first order solution of (12) is given by

1199091205911(119879) = 119888 minus 119888119879 (15)

Notice that (15) represents the exact solution of (11) on thefirst interval By following the same procedure it is easy toshow that the exact solution of (11) for the second and thirdintervals is given respectively as

1199091205912(119879) = 119888 minus 119888120591 minus 119888119879 +

1

21198881198792

1199091205913(119879) = 119888 minus 2119888120591 +

1

21198881205912minus (119888 minus 119888120591) 119879 +

1

21198881198792minus1

61198881198793

(16)

Figure 1 shows the exact solution of (11) obtained by couplingat each interval the solution obtained by following HPMprocedure for 119905 = 10120591

It is easy to show that the solution of (11) by the EMHPMcoincides with the solution obtained by using the HPM since(11) is a delay differential equation with constant coefficients

4 Abstract and Applied Analysis

0 1 2 3 4 5 6 7 8 9 10

00

0505

11

Time t

HPM by intervals

minus05minus05

Am

plitu

dex

Figure 1 Exact solution of (11) obtained by using theHPMand 120591 = 1

42 Delay Differential Equations with Variable CoefficientsWe next show how the EMHPM approach can be applied toobtain the approximate solution of nonlinear delay differen-tial equation with variable coefficients In this case we obtainthe approximate solutions of a DDE of the form

+ 119909 (119905 minus 120591) minus cos (120587119905) 1199092 = 0 120591 = 1 119909 (0) = 119888 = 1199091205910 (120591)(17)

in which the solution 1199091205910(119879) = 1198881holds on (minus120591 0] In order to

find the solution 1199091205911 in the interval [0 120591] we assume that thehomotopy representation of (17) can be given as

119867(1198831205911 119901) =

119889

1198891198791198831205911 + 119901 [119909

1205910 minus cos (120587119905) (1198831205911)2] = 0(18)

Notice that the variable 119883 depends on the time 119879 for which0 le 119879 le 120591 If we now substitute the second order expansion1198831205911 = 119883

1205911

0+ 1199011198831205911

1+ 11990121198831205911

2in (18) and after balancing the 119901

terms we get that

1199010 1198891198891198791198831205911

0= 0 119883

0 (119879 = 0) = 1198881 = 1198831205910(119879 = 120591)

1199011 1198891198891198791198831205911

1= minus1199091205910 + cos (120587119905) (1198831205911

0)2

= 0 1198831(0) = 0

1199012 1198891198891198791198831205911

2= 2 cos (120587119905)1198831205911

01198831205911

1 1198832(0) = 0

1199013 1198891198891198791198831205911

3= cos (120587119905) (21198831205911

01198831205911

2+ (1198831205911

1)2

) = 0 1198833 (0) = 0

(19)

Equations (19) have the following solutions

1198831205911

0= 1198881

1198831205911

1= minus119879 (119909

1205910 minus 1198882

1cos120587119905)

1198831205911

2= minus11988811198792(cos120587119905) (1199091205910 minus 1198882

1cos120587119905)

1198831205911

3=1

31198793(cos120587119905) [31198884

1cos2120587119905 minus 41198882

11199091205910 + (119909

1205910)2]

(20)

Thus the approximate solution of (17) by using the EMHPMis given by

1199091205911(119879) asymp 119883

1205911

0+ 1198831205911

1+ 1198831205911

2+ 1198831205911

3 (21)

In this case the exact solution of 1199091205911(119879) is unknown Toobtain 1199091205912 we compute again the approximate solution of1199091205911(119879) by applying our EMHPM and the value of the delayed

time is assumed to remain constant in each subinterval Todetermine 1199091205912 we next use the homotopy representation of(17) for the interval (120591 2120591]

119867(1198831205912 119901) =

119889

1198891198791198831205912 + 119901 [119909

1205911 minus cos (120587119905) (1198831205912)2] = 0(22)

Substituting the second order expansion in (22) we get

1198831205912

0= 1198882

1198831205912

1= minus119879 (119909

1205911 minus 1198882

2cos120587119905)

1198831205912

2= minus11988821198792(cos120587119905) (1199091205911 minus 1198882

2cos120587119905)

1198831205912

3=1

31198793(cos120587119905) (31198884

2cos2120587119905 minus 41198882

21199091205911 + (119909

1205911)2)

(23)

Note that (20) and (23) provide approximate solutions to(17) but evaluated at different interval time delays To findthe third order approximate solution of (17) we can use ahomotopy of the form

119867(119883120591119894 119901) =

119889

119889119879119883120591119894 + 119901 [119883

120591119894minus1 minus cos (120587119905) (119883120591119894)2] = 0(24)

Then by using our EMPHM approach we have that

119883120591119894

0= 119888

119883120591119894

1= minus119879 (119909

120591119894minus1 minus 1198882 cos120587119905)

119883120591119894

2= minus119888119879

2(cos120587119905) (119909120591119894minus1 minus 1198882 cos120587119905)

119883120591119894

3=1

31198793(cos120587119905) (31198884cos2120587119905 minus 41198882119909120591119894minus1 + (119909120591119894minus1)2)

(25)

Notice from (25) that the 119896th order approximate solution of(17) can be written as

119883120591119894

0= 119888

119883120591119894

119896=119879

119896(minus119909120591119894minus1119892 (119896) + cos120587119905

119896minus1

sum

1198991=0

119883120591119894

1198991119883120591119894

119896minus1minus1198991)

(26)

where 119896 gt 0 119892(119896) = 1 when 119896 = 1 and zero otherwiseFigure 2 shows the approximate solution of (17) obtained

by using the EMHPM approach compared to its numericalintegration solution by using the dde23MATLAB subroutineprogram This case assumes two different initial solutions ofthe form 1199091205910(119879) = cos(120587(119879 + 1)) 1199091205910(119879) = 119890119879+1 and a timesubintervals Δ119905 = 001 We can see from Figure 2 that bothsimulations agree well for the time span showed

To further assess the applicability of our proposedEMHPM approach to high order delay differential equationswe will next describe a methodology to obtain the approx-imate solutions of well-known high order delay differentialequations by generalizing our EMHPM approach

Abstract and Applied Analysis 5

0 5 10 15

0

1

2

EMHPMdde23

minus1minus1

Am

plitu

dex

Time t

x1205910(T) = cos(120587(T + 1))

x1205910(T) = eT+1

Figure 2 EMHPM and dde23 solution of (17)

x x iminusNx iminusN+1

timinusN+1t titiminus1timinusN

Δt

x i(t)x i+1(t)

120591 = (N minus 1)Δt

Figure 3 Schematic of the zeroth order polynomial used to fit theapproximate EMHPM solution

5 Generalized Solution of Linear DDEsby the EMHPM Approach

Let us consider an 119899-dimensional delay differential equationof the form

x (119905) = A (119905) x + B (119905) x (119905 minus 120591) (27)

whereA(119905 + 120591) = A(119905) B(119905 + 120591) = B(119905) x(119905) is the state vectorand 120591 is the time delay By following our EMHPM procedurewe can write (27) in equivalent form as

x119894 (119879) minus A119905x119894 (119879) asymp B119905x

120591

119894(119879) (28)

where x119894(119879) denotes the 119898 order solution of (27) in the 119894th

subinterval that satisfies the initial conditions x119894(0) = x

119894minus1and

A119905and B

119905represent the values of the periodic coefficients at

the time 119905 In order to approximate the delayed term x120591119894(119879)

in (28) the period [1199050minus 120591 1199050] is discretized in 119873 points

equally spaced as shown in Figure 3 Here we assume thatthe function x120591

119894(119879) in the delay subinterval [119905

119894minus119873 119905119894minus119873+1

] isapproximated by a constant value

x120591119894(119879) = 119909

119894minus119873+1(119879) asymp x

119894minus119873 (29)

as shown in Figure 3 By following the homotopy perturba-tion technique we can write the homotopy representation of(28) as

119867(X119894 119901) = 119871 (X

119894) minus 119871 (x

1198940) + 119901119871 (x

1198940) = 119901 (A

119905X119894+ B119905x119894minus119873)

(30)

Substituting the 119898 order expansion X119894= X1198940+ 119901X1198941+ sdot sdot sdot +

119901119898X119894119898

in (30) and by assuming an initial approximation ofthe form x

1198940= x119894minus1

we get after applying the proposed

EMHPMapproach the following set of first order linear delaydifferential equations

1199010 119889119889119879

X1198940+119889

119889119879x119894minus1= 0 X

119894(0) = x

119894minus1

1199011 119889119889119879

X1198941= A119905X1198940+ B119905x119894minus119873 X1198941(0) = 0

1199012 119889119889119879

X1198942= A119905X1198941 X1198942(0) = 0

119901119898 119889119889119879

X119894119898= A119905X119894(119898minus1)

X119894119898(0) = 0

(31)

By solving (31) we get

X1198940= x119894minus1

X1198941= A119905x119894minus1119879 + B

119905x119894minus119873119879

X1198942=1

2A2119905x119894minus11198792+1

2A119905B119905x119894minus1198731198792

X119894119898=1

119898A119898119905x119894minus1119879119898+1

119898A119898minus1119905

B119905x119894minus119873119879119898

(32)

Equations (32) can be written as

X119894119896=119879

119896(A119905X119894(119896minus1)

+ 119892 (119896)B119905x119894minus119873) 119896 = 1 2 3 (33)

where X1198940= x119894minus1

and 119892(119896) = 1 for 119896 = 1 and 119892(119896) = 0otherwise Thus the solution of (27) is obtained by addingthe X119894119896approximate solutions

x119894 (119879) asymp

119898

sum

119896=0

X119894119896 (119879) (34)

Notice however that solution (34) may be further improvedby using a first order polynomial representation of x120591

119894(119879) as

shown in Figure 4 Then the function x120591119894(119879) in the delay

subinterval [119905119894minus119873 119905119894minus119873+1

] takes the form

x120591119894(119879) = x119894minus119873+1 (119879) asymp x119894minus119873 +

(119873 minus 1)

120591(x119894minus119873+1

minus x119894minus119873) 119879

(35)

Substituting (35) into (28) gives

x119894(119879) minus A

119905x119894(119879)

asymp B119905x119894minus119873minus(119873 minus 1)

120591B119905x119894minus119873119879 +(119873 minus 1)

120591B119905x119894minus119873+1

119879

(36)

6 Abstract and Applied Analysis

We next assume that the homotopy representation of (36) isgiven as

119867(X119894 119901) = 119871 (X

119894) minus 119871 (x

1198940) + 119901119871 (x

1198940)

minus 119901 (AX119894+ Bx119894minus119873minus(119873 minus 1)

120591Bx119894minus119873119879

+(119873 minus 1)

120591Bx119894minus119873+1

119879) = 0

(37)

Substituting the 119898 order expansion X119894(119879) = X

1198940(119879) +

119901X1198941(119879) + sdot sdot sdot 119901

119898X119894119898(119879) in (37) and assuming that the initial

approximation is given by x1198940= x119894minus1

we get

1199010 119889119889119879

X1198940+119889

119889119879x119894minus1= 0 X

119894 (0) = x119894minus1

1199011 119889119889119879

X1198941= A119905X1198940+ B119905x119894minus119873minus119873 minus 1

120591B119905x119894minus119873119879

+119873 minus 1

120591B119905x119894minus119873+1

119879 X1198941(0) = 0

1199012 119889119889119879

X1198942= AX

1198941 X1198942(0) = 0

119901119898 119889119889119879

X119894119898= AX

119894(119898minus1) X119894119898(0) = 0

(38)

By solving (38) and by following the EMHPM procedure weget

X1198940= x119894minus1

X1198941= A119905x119894minus1119879 + B

119905x119894minus119873119879 minus1

2

119873 minus 1

120591B119905x119894minus1198731198792

+1

2

119873 minus 1

120591B119905x119894minus119873+1

1198792

X1198942=1

2A2119905x119894minus11198792+1

2A119905B119905x119894minus1198731198792

minus1

6

119873 minus 1

120591A119905B119905x119894minus1198731198793+1

6

119873 minus 1

120591A119905B119905x119894minus119873+1

1198793

X119894119898=1

119898A119898119905x119894minus1119879119898+1

119898A119898minus1119905

B119905x119894minus119873119879119898

minus1

(119898 + 1)

119873 minus 1

120591A119898minus1119905

B119905x119894minus119873119879119898+1

+1

(119898 + 1)

119873 minus 1

120591A119898minus1119905

B119905x119894minus119873+1

119879119898+1

(39)

Here the recursive form of X119894119896(119879) is written as

X119894119896= Xa119894119896+ Xb119894119896119896 = 1 2 3 (40)

timinusN+1 ttitiminus1timinusN

Δt

x i(t)x i+1(t)

x x iminusNx iminusN+1

120591 = (N minus 1)Δt

Figure 4 Schematic EMHPM solution using first polynomial toapproximate delay subinterval

where Xa1198940= x119894minus1Xb1198940= 0 and

Xa119894119896=119879

119896(A119905Xa119894(119896minus1)

+ 119892 (119896)B119905x119894minus119873)

Xb119894119896=119879

119896 + 1(A119905Xb119894(119896minus1)

+119892 (119896) [119873 minus 1

120591119879 (minusB

119905x119894minus119873+ B119905x119894minus119873+1

)])

(41)

Thus the approximate solution of (27) by the EMHPM canbe obtained by substituting (40) into (34)

In the next section we will apply our EMHPM procedureto obtain the solution of two second order delay differentialequations (a) the dampedMathieu equation with time delayand (b) the well-known delay differential equation thatdescribes the dynamics in one degree-of-freedom millingmachine operations

51 Solution of theDampedMathieu EquationwithTimeDelayIn order to assess the accuracy of our EMHPM approach wefirst obtain the solution of the damped Mathieu differentialequation with time delay that combines the effect of paramet-ric excitation and dampingThis equation is described by thefollowing equation

+ 120581 + (120575 + 120576 cos(2120587119905119879))119909 = 119887119909 (119905 minus 120591) (42)

where 120581 120575 120576 120591 and 119879 are system parameters whose valuedepends on the physics of the system The approximatesolution of (42) obtained by using the semidiscretizationmethod is widely discussed in [18 19] Here we focus ourattention on applying the EMHPM to find the approximatesolution of (42) and we also assess the accuracy of the derivedsolution by comparing it with the corresponding numericalintegration solution of (42)

By following the EMHPM procedure we first write (42)in the following equivalent form

119894(119879) + 120581

119894(119879) + 120572

119905119909119894(119879) asymp 119887119909

119894minus119873+1(119879) (43)

where 119909119894(119905) denotes the 119898 order solution of (43) in the

119894th subinterval that satisfies the following initial conditions119909119894(0) = 119909

119894minus1and

119894(0) =

119894minus1 The space state form

representation of (43) is given by

x119894(119879) = A

119905x119894(119879) + B

119905x119894minus119873+1

(119879) (44)

Abstract and Applied Analysis 7

0 2 4 6 8 10 12

005

1

dde23Zeroth EMHPMFirst EMHPM

minus05minus05

minus15minus15

minus1minus1Am

plitu

dex

Time t

times10minus3

Figure 5 Numerical solutions of the damped Mathieu equationwith time delay by dde23 the zeroth EMHPM and the first EMHPMwith119873 = 50 and119898 = 4

Table 1 Computer time needed to solve the damped Mathieuequation with time delay The 119898 order solution of the EMPHMapproach is chosen to guarantee the convergence of its approximatesolution

dde23 [ms] EMHPM119873 119898 Zeroth [ms] First [ms]

19

15 5 4 520 4 6 640 3 12 1260 2 17 1860 5 18 1960 10 20 21

where

A119905= [0 1

minus120572119905minus120581] B

119905= [0 0

1198871199050] (45)

and 120572119905= (120575 + 120576 cos(119905)) is a time periodic term The EMHPM

approximate solution of (44) is illustrated in Figure 5 wherewe have assumed an unstable system behavior for which 120581 =02 120575 = 30 120576 = 1 119887 = minus1 and 119879 = 120591 = 2120587 See[20] As we can see from Figure 5 our approximate EMHPMsolution to (42) is compared with its numerical integrationsolution obtained from dde23 MATLAB algorithm for thetime interval of 2119879 by assuming that 119873 = 50 with thefollowing initial values 119909

minus50(119879) = 119909

minus49(119879) = sdot sdot sdot 119909

0(119879) =

0001 and minus50(119879) =

minus49(119879) = sdot sdot sdot

0(119879) = 0

It can be seen fromFigure 5 that in the interval [0119879] boththe zeroth and the first order solutions are the same since thedelay subintervals are constant See Figure 3 However in thenext interval [119879 2119879] it is clear that the first order EMPHMsolution provides a better approximation on the delay subin-terval The computation total time to calculate the solutionsin the MATLAB code is listed in Table 1 The order 119898 andthe discretized time intervals 119873 in the EMPHM approachare chosen to guarantee the convergence of our approximatesolution to the exact one To provide a full understanding ofhow the solution is computed by the EMHPM approach weattached in Algorithm 1 the corresponding MATLAB code

10 20 30 40 50 60 70 80 90 100

Abso

lute

erro

r

10minus4

10minus3

10minus2

Zeroth order m = 2

First order m = 2

Zeroth order m = 5

First order m = 5

Zeroth order m = 10

First order m = 10

Discretizations N

Figure 6 Estimated relative error values between the numericalsolution dde23 and the EMHPMapproximate solutionsHerewe usefor the EMHPM the values of119898 = 2 5 and 10

Figure 6 shows the relative error between our approxi-mate EMPHM and the dde23 solution and its relationshipwith the order119898 and the discretized time intervals119873 Noticethat the relative error values coincide at values of 119873 ge 45Also we can see from Figure 6 that the computed relativeerror values for approximate solutions of order119898 ge 5 remainunchanged

52 A Practical Application Cutting Operation on MillingMachine We next use our EMHPM procedure to obtainthe solution of the single degree-of-freedom milling opera-tion We use the simplified form based on [20ndash22]

(119905) + 2120577120596119899 (119905) + 1205962

119899119909 (119905) = minus

119886119901119870119904 (119905)

119898119898

(119909 (119905) minus 119909 (119905 minus 120591))

(46)

where 120596119899is the angular natural frequency of the system 120577 is

the damping ratio 119886119901is the depth of cut 119898

119898is the modal

mass of the tool 120591 represents the time delay which is equalto the tooth passing period and 119870

119904(119905) is the specific cutting

force coefficient which can be determined from119870119904 (119905)

=

119911119899

sum

119895=1

119892 (120601119895(119905)) sin (120601

119895(119905)) (119870

119905cos120601119895(119905) + 119870

119899sin120601119895(119905))

(47)

where 119911119899is the tool number of teeth 119870

119905and 119870

119899are the

tangential and the normal linear cutting force coefficientsrespectively 120601

119895(119905) is the angular position of the 119895-tooth

defined as

120601119895(119905) = (

2120587119899

60) 119905 +

2120587119895

119911119899

(48)

and 119899 is the spindle speed in rpm [20] The function 119892(120593119895(119905))

is a switching function which has a unity value when the 119895-tooth is cutting and zero otherwise

119892 (120601119895 (119905)) =

1 120601st lt 120601119895 (119905) lt 120601ex0 otherwise

(49)

8 Abstract and Applied Analysis

function dde23 ddeEM mathieu paper

disp(lsquoMathieu equation solutionrsquo) Solution by EMHPM with zeroth and first order solution

eps=1 kappa=02 T=2lowastpi tau=T b=minus1 delta=3 Mathieu Parameters

pntDelay=1 N=50 m ord=5 dt=tau(Nminus1) ktau=2 tspan=[0ktaulowastT] EMHPM Parameters

Solution

Solution by dde23

tdde=linspace(0ktaulowasttauktaulowast(Nminus1)+1)

dde=(tyz) mathieu dde(tyzkappadeltaepsbtau)

te dde=ticsol = dde23(ddetauhistorytspan) toc(te dde) xdde=deval(soltdde)

Solution by zeroth EMHPM

dde emhpm fun=(tc0ttzmzntauN) mathieu zeroth(tc0ttzmzntauNm ord

kappadeltaepsbT)

tic[t0z0]=ddeEMHPM(dde emhpm funtspanhistorytaudtpntDelay) toc

Solution by First EMHPM

dde emhpm fun=(tc0ttzmzntauN) mathieu first(tc0ttzmzntauNm ordkappadeltaepsbT)

tic[t1z1]=ddeEMHPM(dde emhpm funtspanhistorytaudtpntDelay) toc

Plot results

ind0=12ktaulowast(Nminus1) ind1=22ktaulowast(Nminus1)

Parent1=figure (1)

axes1 = axes(lsquoParentrsquoParent1lsquoFontSizersquo12lsquoFontNamersquolsquoTimes New Romanrsquo)box(axes1lsquoonrsquo) hold(axes1lsquoallrsquo)plot(tddexdde(1)lsquoParentrsquoaxes1lsquoLineWidthrsquo2lsquoColorrsquo[0502 0502 0502]lsquoDisplayNamersquolsquoNumericaldd23rsquo)plot(t0(ind0)z0(ind01)lsquoMarkerSizersquo5lsquoMarkerrsquolsquoorsquolsquoLineStylersquo lsquononersquolsquoDisplayNamersquolsquoZerothEMHPMrsquolsquoColorrsquo[0 0 0])

plot(t1(ind1)z1(ind11)lsquoMarkerSizersquo7lsquoMarkerrsquolsquoxrsquolsquoLineStylersquolsquononersquolsquoDisplayNamersquolsquoFirstEMHPMrsquolsquoColorrsquo[0 0 0])

xlabel(lsquoittrsquolsquoFontSizersquo12lsquoFontNamersquolsquoTimes New Romanrsquo)ylabel(lsquoitxrsquolsquoFontSizersquo12lsquoFontNamersquolsquoTimes New Romanrsquo)end

Mathieu definitions

function dydt = mathieu dde(tyzkapadltepsbT)

dydt = [y(2)-kapalowasty(2)minus(dlt+epslowastcos(2lowastpilowasttT))lowasty(1)+blowastz(1)]

end

function Z = mathieu zeroth(tc0ttzmzntauNmkpadltepsbT)

Z=[c0(1)c0(2)] alf=dlt+epslowastcos(2lowastpiTlowasttt)

for ik=1m

Z(ik+11)=Z(ik2)lowasttik

Z(ik+12)=minuskpalowastZ(ik2)minusalflowastZ(ik1)

if ik==1 Z(ik+12)=Z(ik+12)+blowastzm(1) end

Z(ik+12)=Z(ik+12)lowasttik

end

Z=sum(Z)

end

function Z = mathieu first(tc0ttzmzntauNmkpadltepsbT)

alf=dlt+epslowastcos(2lowastpiTlowasttt) Z=[c0(1)c0(2)] Z =[00]

for ik=1m

Z(ik+1)=[Z(ik2)lowasttik minuskpalowastZ(ik2)minusalflowastZ(ik1)]

Z (ik+1)=[Z (ik2)lowasttik minuskpalowastZ (ik2)minusalflowastZ (ik1)]

if ik==1

Z(ik+12)=Z(ik+12)+blowastzm(1)Z (ik+12)=Z (ik+12)+blowast(Nminus1)taulowast(zn(1)minuszm(1))lowastt

end

Z(ik+12)=Z(ik+12)lowasttik

Z (ik+12)=Z (ik+12)lowastt(ik+1)

Algorithm 1 Continued

Abstract and Applied Analysis 9

end

Z=sum(Z)+sum(Z )

end

function out=history(t)

out=[1Eminus3+0lowastt 0+0lowastt]

end

EMHPM algorithm for ODE solutions

function [tz]= odeEMHPM(ndetspanz0Deltatpnts)

ode solver by Enhanced Multistage Homotopy Perturbation Method

tini=tspan(1) tfin=tspan(end) tstart=tini

tini=tiniminuststart tfin=tfinminuststart shifted time set to zero

Handle errors

if tini == tfin

error(lsquoThe ending and starting time values must be differentrsquo)elseif abs(tini)gt abs(tfin)

tspan=flipud(fliplr(tspan)) tini=tspan(1) tfin=tspan(end)

end

tdir=sign(tfinminustini)

if any(tdirlowastdiff(tspan) lt= 0)

error(lsquotspan entries must be strictly sortedrsquo)end

incT=Deltatpnts

if incTlt=0

error (lsquoIncreasing must be greater than zerorsquo)end

z(1)=z01015840 t(1)=tini iteT=2 set initial values

t(iteT)=tini+incTlowasttdir

while tdirlowastt(iteT)lttdirlowast(tfin+tdirlowastincT)

P act=ceil(99999lowast(t(iteT)minustini)lowasttdirDeltat) count sub-intervals

c=z((DeltatincT)lowast(P actminus1)+1) set the corresponding initial condition

tsub(iteTminus1)=t(iteT)-(P actminus1)lowastDeltatlowasttdir evaluate the solution at the shifted time

temp=tsub(iteTminus1)

z(iteT)=nde(tempctstart+t(iteT))

if tdirlowastt(iteT)gt=tdirlowasttfinlowast099999 repeat for the next sub-interval

break

else

iteT=iteT+1 t(iteT)=tini+(iteTminus1)lowastincTlowasttdir

end

end

t=t 1015840 +tstart

end

EMHPM algorithm for DDE solutions

function [t z]=ddeEMHPM(ddetspanhistorytauDeltatpntDelta)

dde solver by Enhanced Multistage Homotopy Perturbation Method

tini=tspan(1) tfin=tspan(end) span where the solution is founded

if taultDeltat

error(lsquoSubtinterval must not be greater than taursquo)end

if mod(tauDeltat)sim=0

pastDlt=Deltat

Deltat=tauround(tauDeltat)

warning(lsquoSubtinterval was modified from 05g to 05grsquopastDltDeltat)end

pnt=round(tauDeltat) samplesminus1 [minustau 0]

incT=DeltatpntDelta step for set resolution

t=minustau+tini+(0pntlowastpntDelta)rsquolowastincT evaluation of the initial solution [minustau 0]

z=history(t) initial behavior in the inverval [minustau 0]

c=z(end) initial condition

Algorithm 1 Continued

10 Abstract and Applied Analysis

m=pntlowastpntDelta samples between tau

itDlt=0

while (t(end)+eps)lttfin

tspan=[tini+itDltlowastDeltattini+(itDlt+1)lowastDeltat] preparing the span for next Deltat

zm=z(length(z)minusm)rsquo zn=z(1+length(z)minusm)rsquo previous tau solution

emhpm fun=(timecT) dde(timecTzmzntaum) application of the odeEMHPM

[t aux z aux]=odeEMHPM(emhpm funtspanz(end)DeltatpntDelta)

z=[z(1endminus1)z aux] t=[t(1endminus1)t aux] joining the solutions

itDlt=itDlt+1

end

z=z(pntlowastpntDelta+1end)t=t(pntlowastpntDelta+1end)

end

Algorithm 1 MATLAB algorithm

Here 120601st and 120601ex are the angles where the teeth enter and exitthe workpiece For upmilling 120601st = 0 and 120601ex = arccos(1 minus2119886119889) for downmilling 120601st = arccos(2119886

119889minus 1) and 120601ex = 120587

where 119886119889is the radial depth of cut ratio

By following the EMHPM procedure we can write (46)in equivalent form as

119894(119879) + 2120577120596

119899119894(119879) + 120596

2

119899119909119894(119879)

asymp minus

119886119901119870st

119898119898

(119909119894 (119879) minus 119909119894minus119873+1 (119879))

(50)

where 119909119894(119879) denotes the 119898 order solution of (46) on the 119894th

subinterval that satisfies the initial conditions 119909119894(0) = 119909

119894minus1

119894(0) =

119894minus1 and ℎ

119905= ℎ(119905) and 119909

minus120591is given by (35)

Introducing the transformation x119894= [119909119894 119894]119879 (50) can be

written as a system of first order linear delay differentialequations of the form

x119894(119879) = A

119905x119894(119879) + B

119905x119894minus119873+1

(119879) (51)

where

A119905= [

[

0 1

minus1205962

119899minus119908

119898119898

119870st minus2120577120596119899]

]

B119905= [

[

0 0119908

119898119898

119870st 0]

]

(52)

We next apply the EMHPM procedure to solve (46) byconsidering a downmilling operation with the followingparameter values 119911

119899= 2 119886

119889= 01 120596

119899= 5793 rads

120577 = 0011 119898119898= 003993 kg 119870

119905= 6 times 10

8Nm2 and119870119899= 2 times 10

8Nm2 As we can see from Figures 7 and 8 andfor the depth of cut values of 119886

119901= 2mm (stable) and 119886

119901=

3mm (unstable) our EMHPM approximate solutions followclosely the numerical integration solutions of (46) obtainedby using the dde23 algorithm

Figure 9 shows the relative error between the EMPHMand the dde23 numerical solution while Table 2 shows theCPU time needed for each solution Here we use 119873 = 75since the average step size of the dde23 algorithm is aroundΔ119905 asymp 120591119873 Note that for 119898 = 7 the zeroth order EMHPMapproximate solution has the fastest CPU time We can seefrom Figure 9 that the value of the relative error becomesbasically the same for119898 = 2 7 and 10 and119873 ge 20

0 0005 001 0015 002

0

1

2

Time (s)

Disp

lace

men

t (m

)

dd23Zeroth EMHPMFirst EMHPM

minus2

minus1

times10minus3

Figure 7 EMHPM approximate solutions of (46) with parametervalues of 119886

119901= 2mm 119899 = 10000 rpm Stable machine operation

0 0005 001 0015 002

00005

0010015

Time (s)

Disp

lace

men

t (m

)

dd23Zeroth EMHPMFirst EMHPM

minus001

minus0005

Figure 8 EMHPM approximate solutions of (46) with parametervalues of 119886

119901= 3mmand 119899 = 10000 rpmUnstable (chatter)machine

operation

6 Conclusions

We have developed a new algorithm based on the homotopyperturbation method to solve delay differential equationsThe proposed EMHPM approach is based on a sequence ofsubintervals that approximate the solution of delayed differ-ential equations by using the transformation rule 119906(119905) rarr119906119894(119879) where 119906

119894satisfies the initial conditions We have

Abstract and Applied Analysis 11

10 20 30 40 50 60 70 80 90 100

Abso

lute

erro

r

Zeroth order m = 2

First order m = 2

Zeroth order m = 7

First order m = 7

Zeroth order m = 10

First order m = 10

10minus3

10minus2

10minus1

Discretizations N

Figure 9 Estimated relative error values between dde23 and theEMHPM approximate solutions Here we have used the systemparameter values of 119886

119901= 2mm and 119899 = 1000 rpm and values of

119898 = 2 7 and 10

Table 2 CPU time comparison among the approximate solutions of(46) by using dde23 the zeroth order and the first order EMHPMsolutions

Solution method Time [ms]dde23 139Zeroth order EMHPM (119898 = 7) 77First order EMHPM (119898 = 7) 87

shown that our proposed EMHPM approach can be appliedto obtain the approximate solution of a delay differentialequation not only with constant but also with variablecoefficients with theoretical predictions that follow well thenumerical integration solutions To further assess the validityof this new approach we have compared the approximatesolutions of two delayed differential equations with respect totheir corresponding numerical integration solutions obtainedfrom the MATLAB dde23 algorithm The test cases were (a)the damped Mathieu differential equation with time delayand (b) the governing equation of motion of downmillingoperations We have found that the EMHPM closely followsthe numerical integration solutions of the correspondingequations and that these require less CPU time and havesmaller relative errors

Conflict of Interests

Theauthors declare that they have no conflict of interests withany mentioned entities in the paper

Acknowledgments

This work was funded by Tecnologico deMonterrey CampusMonterrey through the ResearchGroup inNanomaterials forMedical Devices and the Research Group in Advanced Man-ufacturing Additional support was provided from ConsejoNacional de Ciencia y Tecnologıa (Conacyt) Mexico

References

[1] F Shakeri and M Dehghan ldquoSolution of delay differentialequations via a homotopy perturbation methodrdquoMathematicaland Computer Modelling vol 48 no 3-4 pp 486ndash498 2008

[2] Z-K Wu ldquoSolution of the enso delayed oscillator with homo-topy analysis methodrdquo Journal of Hydrodynamics vol 21 no 1pp 131ndash135 2009

[3] A K Alomari M S Noorani and R Nazar ldquoSolution of delaydifferential equation by means of homotopy analysis methodrdquoActa Applicandae Mathematicae vol 108 no 2 pp 395ndash4122009

[4] J Biazar and G Behzad ldquoThe homotopy perturbation methodfor solving neutral functional-differential equations with pro-portional delaysrdquo Journal of King Saud UniversitymdashScience vol24 no 1 pp 33ndash37 2012

[5] NRAnakiraAKAlomari and IHashim ldquoOptimal homotopyasymptotic method for solving delay differential equationsrdquoMathematical Problems in Engineering vol 2013 Article ID498902 11 pages 2013

[6] T Insperger and G Stepan Semi-Discretization for Time-DelaySystems Stability and Engineering Applications Springer NewYork NY USA 2011

[7] E A Butcher P Nindujarla and E Bueler ldquoStability of up-and down-milling using chebyshev collocation methodrdquo inProceedings of the ASME International Design Engineering Tech-nical Conferences amp Computers and Information in EngineeringConference vol 6 part AndashC pp 841ndash850 2005

[8] T Insperger ldquoFull-discretization and semi-discretization formilling stability prediction some commentsrdquo InternationalJournal of Machine Tools and Manufacture vol 50 no 7 pp658ndash662 2010

[9] I Hashim and M S Chowdhury ldquoAdaptation of homotopy-perturbation method for numeric-analytic solution of systemof ODEsrdquo Physics Letters A vol 372 no 4 pp 470ndash481 2008

[10] L F Shampine and S Thompson ldquoSolving DDEs in MatlabrdquoAppliedNumericalMathematics vol 37 no 4 pp 441ndash458 2001

[11] SHHNiaANRanjbarDDGanjiH Soltani and JGhasemildquoMaintaining the stability of nonlinear differential equations bythe enhancement of HPMrdquo Physics Letters A vol 372 no 16 pp2855ndash2861 2008

[12] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[13] J-H He ldquoHomotopy perturbation method a new nonlinearanalytical techniquerdquo Applied Mathematics and Computationvol 135 no 1 pp 73ndash79 2003

[14] J H He ldquoComparison of homotopy perturbation method andhomotopy analysis methodrdquo Applied Mathematics and Compu-tation vol 156 no 2 pp 527ndash539 2004

[15] JHHe ldquoNew interpretation of homotopy perturbationmethodrdquoInternational Journal of Modern Physics B vol 20 no 18 pp2561ndash2568 2006

[16] S Liao ldquoComparison between the homotopy analysis methodand homotopy perturbationmethodrdquoAppliedMathematics andComputation vol 169 no 2 pp 1186ndash1194 2005

[17] D Olvera and A Elıas-Zuniga ldquoEnhanced multistage homo-topy perturbation method approximate solutions of nonlineardynamic systemsrdquo Abstract and Applied Analysis vol 2014Article ID 486509 12 pages 2014

12 Abstract and Applied Analysis

[18] T Insperger and G Stepan ldquoStability of the damped Mathieuequationwith timedelayrdquo Journal ofDynamic SystemsMeasure-ment and Control vol 125 no 2 pp 166ndash171 2003

[19] T Insperger and G Stepan ldquoStability chart for the delayedMathieu equationrdquo The Royal Society of London ProceedingsSeries A Mathematical Physical and Engineering Sciences vol458 no 2024 pp 1989ndash1998 2002

[20] T Insperger andGStepan ldquoUpdated semi-discretizationmethodfor periodic delay-differential equations with discrete delayrdquoInternational Journal for Numerical Methods in Engineering vol61 no 1 pp 117ndash141 2004

[21] T Insperger B P Mann G Stepan and P V Bayly ldquoStabilityof up-milling and down-milling part 1 alternative analyticalmethodsrdquo International Journal of Machine Tools and Manufac-ture vol 43 no 1 pp 25ndash34 2003

[22] F ICompeanDOlvera F J Campa LN L de Lacalle A Elıas-Zuniga and C A Rodrıguez ldquoCharacterization and stabilityanalysis of a multivariable milling tool by the enhanced mul-tistage homotopy perturbation methodrdquo International Journalof Machine Tools and Manufacture vol 57 pp 27ndash33 2012

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

they proved that this technique is faster than that of thefull and the semidiscretization methods since these solutiontechniques approximate the original DDEs by a series ofODEs [8]

Here in this paper we develop a generalized procedure tosolve linear and nonlinear DDEs by introducing some mod-ifications to the multistage homotopy perturbation method(MHPM) derived by Hashim and Chowdhury to obtainapproximate solutions of ordinary differential equations [9]The proposed enhanced multistage homotopy perturbationmethod (EMHPM) is based on a sequence of subintervalsthat allow us to find more accurate approximated solutionsunder a numerical-analytical procedure that requires lessCPU time when compared to the numerical integrationsolutions provided by the MATLAB dde23 algorithm writtenby Shampine and Thompson in [10] The EMHPM is basedon a homotopy function that could be divided into a linearoperator and a nonlinear operator to satisfy its assumed initialsolution This split of the homotopy function allows us tomodify the nonlinear operator to guarantee by using theenhanced homotopy perturbation method the stability ofthe proposed approximate solutions of nonlinear differentialequations [11]

To clarify our proposed method we briefly review inSection 2 some basic concepts of the homotopy perturbationmethod and then in Section 3 we introduce the EMHPMto solve DDEs The difference between the HPM and theEMHPM is discussed in Section 4 by addressing the approx-imate solutions of a nonlinear delayed differential equationwith variable coefficients Finally the general solution oftwo DDEs that describe the dynamics of two engineeringproblems by using the EMHPM is discussed in Section 5

2 Homotopy Perturbation Method

The homotopy perturbation method (HPM) is a couplingof the traditional perturbation method and homotopy intopology which eliminates the limitation of the small param-eter assumed in the perturbation methods [12] Under thisapproach a nonlinear problem can be transformed into aninfinite number of simple problems without the restrictionof having small nonlinear parameter values This homotopyperturbationmethod takes themain advantages of traditionalperturbation methods together with homotopy analysis [13ndash15]

To illustrate the basic ideas of the HPM let us considerthe following nonlinear differential equation

119860 (119906) minus 119891 (119903) = 0 119903 isin Ω (1)

with boundary conditions

119861(119906120597119906

120597119899) = 0 119903 isin Γ (2)

where 119860 is a general differential operator 119861 is a boundaryoperator 119891(119903) is a known analytic function and Γ is theboundary of the domainΩ

The operator 119860 can generally be divided into two parts 119871and119873 where 119871 involves the linear terms and119873 the nonlinearones Equation (1) therefore can be rewritten as follows

119871 (119906) + 119873 (119906) minus 119891 (119903) = 0 (3)

By the homotopy perturbation technique we construct a homo-topy V(119903 119901) Ω times [0 1] rarr R that satisfies

119867(V 119901) = 119871 (V) minus 119871 (1199060) + 119901119871 (1199060) + 119901 [119873 (V) minus 119891 (119903)] = 0(4)

where 119901 isin [0 1] is an embedding parameter and 1199060is

an initial approximation of (1) which satisfies the boundaryconditions (2) Thus from (4) we have

119867(V 0) = 119871 (V) minus 119871 (1199060) = 0

119867 (V 1) = 119860 (V) minus 119891 (119903) = 0(5)

The changing process of 119901 from zero to unity is just thatof V(119903 119901) from 119906

0(119903) to 119906(119903) In topology this is called

deformation and 119871(V) minus 119871(1199060) and 119860(V) minus 119891(119903) are called

homotopicHe in [12] uses the embedding parameter 119901 as the small

parameter and assumed that the solution of (4) can be writtenas a power series of 119901 in the form

V = V0+ 119901V1+ 1199012V2+ sdot sdot sdot (6)

By setting 119901 = 1 He obtained the approximate solution of (1)as

119906 = lim119901rarr1

= V0+ V1+ V2+ sdot sdot sdot (7)

Then this method was applied to obtain the approximatesolution of some nonlinear ordinary differential equationsvalid not only for small but also for large nonlinear parametervalues

We next will introduce an approach based on homotopymethods to obtain the solution of DDEs with constant andvariable coefficients

3 The EMHPM Methodology to Solve DDEs

TheHPM is an asymptotic method that depends on the aux-iliary linear operator form and the initial guess of the initialconditions Therefore the convergence of the approximatesolution cannot be guaranteed in some cases [16] Hashimand Chowdhury showed in [9] that the solutions obtained bythe standard HPM were not valid for large time span unlessmore terms are calculated Thus they proposed a multistagehomotopy perturbation method (MHPM) which treated theHPMalgorithm in a sequence of subintervals in an attempt toimprove the accuracy of the approximate solutions of linearand nonlinear ordinary differential equations (ODEs)

However when the MHPM is applied to obtain theapproximate solutions of ODEs which contain coefficientsas a function of time this method cannot provide accuratesolutions when Δ119905 rarr 0 In this work we introduce some

Abstract and Applied Analysis 3

modifications to the MHPM and focus on the derivation ofapproximate solutions of DDEs equations with variable coef-ficient terms This new approach is based on the enhancedmultistage homotopy perturbationmethod (EMHPM) intro-duced in [17] to obtain the solution of nonlinear ordinarydifferential equations

The EMHPM is an algorithm which approximates theHPM solution by subintervals utilizing the following trans-formation rule 119906(119905) rarr 119906

119894(119879) where 119906

119894satisfies the initial

condition 119906119894(0) = 119906

119894minus1(119905119894minus1) 119879 is a shifted time scale used

to determine the approximate solution in each subintervaland 119906

119894(119879) represents the approximate solution in the 119894th

subinterval In this case the initial suggested solution in the119894th subinterval is given by 119906

1198940(119879) = 119906

119894minus1(119905119894minus1) where 119905

119894minus1

represents the time at the end of the previous subinterval(ie the value of the approximate solution at the end of theprevious subinterval represents the initial conditions of thenext subinterval under consideration)

To apply the homotopy technique to solve delay differen-tial equations we also assume the following

(1) The linear operator119871(119906119894) represents119871(119906

119894) = (119889119889119879)119906

119894

where the assumed approximate solution 1199061198940(119879) is

set equal to the initial condition 119906119894minus1(119905119894minus1) that is

1199061198940= 119906119894minus1(119905119894minus1) To simplify the notation we let 119906

119894minus1equiv

119906119894minus1(119905119894minus1)

(2) The transformation 119879 = 119905 minus 119905119894minus1

on 0 lt 119879 le 119905119894minus 119905119894minus1

holds in the homotopy 119894-subinterval Thus higherorder equations are integratedwith respect to119879 whilethe terms related to the independent variable 119905 areassumed to remain constant

Therefore we may conclude that the 119898 order approximatesolution by applying the EMHPM can be written as

119906119894(119879 119906119894minus1) =

119898

sum

119896=0

119880119894119896(119879 119906119894minus1) (8)

where the solution 119906119894(119879 119906119894minus1) is valid only in the 119894th subin-

terval [119905119894minus1 119905119894] Hence the solution 119906(119905) on the 119894th subinterval

(119905119894minus1 119905119894] can be written as

119906 (119905) asymp 119906119894(119905 minus 119905119894minus1) (9)

with initial condition 119906119894minus1(119905119894minus1) and 119894 = 1 2 119895 Thus the

approximate solution of 119906 at the time 119905119894is given by

119906119894(119905 minus 119905119894minus1)1003816100381610038161003816119905=119905119894= 119906119894+1(119905 minus 119905119894)1003816100381610038161003816119905=119905119894= 119906119894+1(0) = 119906

119894 (10)

In summary the solution 119906(119905) for an open-closed interval(1199050 1199051] is divided into 119895 subintervals that in general are

not equally spaced [1199050 1199051] [1199051 1199052] [119905

119895minus1 119905119895] Thus the

approximated solution of 119906(119905) for the span time interval isobtained by coupling the 119906

119894(119905) solutions

4 Approximate Solutions of Some DDEs byApplying the EMHPM

In this sectionwe focus on the solution ofDDEswith constantand variable coefficients and examine the applicability of theEMHPM to find the corresponding approximate solutions

41 Delay Differential Equations with Constant CoefficientsFirst let us consider the simplest DDE of the form

(119905) + 119909 (119905 minus 120591) = 0 (11)

with initial condition 119909(0) = 119888 Here the independent vari-able 119909 is a scalar 119909(119905) isin R the dot stands for differentiationwith respect to time 119905 and 120591 is the time delay To evaluate(11) on 119886 le 119905 le 119887 the term 119909(119905 minus 120591) must represent a knownfunction 119909(119905) on [119886 minus 120591 le 119905 le 119886] For instance if 119886 = 0the solution of (11) can be obtained in the interval (0 120591] byassuming an initial function that satisfies the initial conditionBy using this solution it becomes possible to obtain thesolution of (11) in the next 119894th interval [(119894 minus 1)120591 119894120591] 119894 =2 3 119895 where 119895 is an integer number that can be chosenas 2 le 119895 le infin With this approach we can apply the HPM tofind the solution of (11) by assuming that the previous delayedfunction is 1199091205910(119879) = 119888 thus the solution for the first intervalis given by 1199091205911(119879) valid on [0 120591] In terms of (4) we nowconstruct the homotopy of (11)

119867(1198831205911 119901) =

119889

1198891198791198831205911 + 119901119909

1205910 = 0 (12)

We next substitute the first order expansion1198831205911 = 11988312059110+1199011198831205911

1

in (12) and balance the terms with identical power of 119901 toobtain the following set of linear differential equations

1199010 1198891198891198791198831205911

0= 0 119883

1205911

0(0) = 119888 = 119883

1205910(120591)

1199011 1198891198891198791198831205911

1= minus1198831205910 119883

1205911

1(0) = 0

(13)

Integration of (13) yields

1198831205911

0= 119888

1198831205911

1= minus 119888119879

(14)

Hence the first order solution of (12) is given by

1199091205911(119879) = 119888 minus 119888119879 (15)

Notice that (15) represents the exact solution of (11) on thefirst interval By following the same procedure it is easy toshow that the exact solution of (11) for the second and thirdintervals is given respectively as

1199091205912(119879) = 119888 minus 119888120591 minus 119888119879 +

1

21198881198792

1199091205913(119879) = 119888 minus 2119888120591 +

1

21198881205912minus (119888 minus 119888120591) 119879 +

1

21198881198792minus1

61198881198793

(16)

Figure 1 shows the exact solution of (11) obtained by couplingat each interval the solution obtained by following HPMprocedure for 119905 = 10120591

It is easy to show that the solution of (11) by the EMHPMcoincides with the solution obtained by using the HPM since(11) is a delay differential equation with constant coefficients

4 Abstract and Applied Analysis

0 1 2 3 4 5 6 7 8 9 10

00

0505

11

Time t

HPM by intervals

minus05minus05

Am

plitu

dex

Figure 1 Exact solution of (11) obtained by using theHPMand 120591 = 1

42 Delay Differential Equations with Variable CoefficientsWe next show how the EMHPM approach can be applied toobtain the approximate solution of nonlinear delay differen-tial equation with variable coefficients In this case we obtainthe approximate solutions of a DDE of the form

+ 119909 (119905 minus 120591) minus cos (120587119905) 1199092 = 0 120591 = 1 119909 (0) = 119888 = 1199091205910 (120591)(17)

in which the solution 1199091205910(119879) = 1198881holds on (minus120591 0] In order to

find the solution 1199091205911 in the interval [0 120591] we assume that thehomotopy representation of (17) can be given as

119867(1198831205911 119901) =

119889

1198891198791198831205911 + 119901 [119909

1205910 minus cos (120587119905) (1198831205911)2] = 0(18)

Notice that the variable 119883 depends on the time 119879 for which0 le 119879 le 120591 If we now substitute the second order expansion1198831205911 = 119883

1205911

0+ 1199011198831205911

1+ 11990121198831205911

2in (18) and after balancing the 119901

terms we get that

1199010 1198891198891198791198831205911

0= 0 119883

0 (119879 = 0) = 1198881 = 1198831205910(119879 = 120591)

1199011 1198891198891198791198831205911

1= minus1199091205910 + cos (120587119905) (1198831205911

0)2

= 0 1198831(0) = 0

1199012 1198891198891198791198831205911

2= 2 cos (120587119905)1198831205911

01198831205911

1 1198832(0) = 0

1199013 1198891198891198791198831205911

3= cos (120587119905) (21198831205911

01198831205911

2+ (1198831205911

1)2

) = 0 1198833 (0) = 0

(19)

Equations (19) have the following solutions

1198831205911

0= 1198881

1198831205911

1= minus119879 (119909

1205910 minus 1198882

1cos120587119905)

1198831205911

2= minus11988811198792(cos120587119905) (1199091205910 minus 1198882

1cos120587119905)

1198831205911

3=1

31198793(cos120587119905) [31198884

1cos2120587119905 minus 41198882

11199091205910 + (119909

1205910)2]

(20)

Thus the approximate solution of (17) by using the EMHPMis given by

1199091205911(119879) asymp 119883

1205911

0+ 1198831205911

1+ 1198831205911

2+ 1198831205911

3 (21)

In this case the exact solution of 1199091205911(119879) is unknown Toobtain 1199091205912 we compute again the approximate solution of1199091205911(119879) by applying our EMHPM and the value of the delayed

time is assumed to remain constant in each subinterval Todetermine 1199091205912 we next use the homotopy representation of(17) for the interval (120591 2120591]

119867(1198831205912 119901) =

119889

1198891198791198831205912 + 119901 [119909

1205911 minus cos (120587119905) (1198831205912)2] = 0(22)

Substituting the second order expansion in (22) we get

1198831205912

0= 1198882

1198831205912

1= minus119879 (119909

1205911 minus 1198882

2cos120587119905)

1198831205912

2= minus11988821198792(cos120587119905) (1199091205911 minus 1198882

2cos120587119905)

1198831205912

3=1

31198793(cos120587119905) (31198884

2cos2120587119905 minus 41198882

21199091205911 + (119909

1205911)2)

(23)

Note that (20) and (23) provide approximate solutions to(17) but evaluated at different interval time delays To findthe third order approximate solution of (17) we can use ahomotopy of the form

119867(119883120591119894 119901) =

119889

119889119879119883120591119894 + 119901 [119883

120591119894minus1 minus cos (120587119905) (119883120591119894)2] = 0(24)

Then by using our EMPHM approach we have that

119883120591119894

0= 119888

119883120591119894

1= minus119879 (119909

120591119894minus1 minus 1198882 cos120587119905)

119883120591119894

2= minus119888119879

2(cos120587119905) (119909120591119894minus1 minus 1198882 cos120587119905)

119883120591119894

3=1

31198793(cos120587119905) (31198884cos2120587119905 minus 41198882119909120591119894minus1 + (119909120591119894minus1)2)

(25)

Notice from (25) that the 119896th order approximate solution of(17) can be written as

119883120591119894

0= 119888

119883120591119894

119896=119879

119896(minus119909120591119894minus1119892 (119896) + cos120587119905

119896minus1

sum

1198991=0

119883120591119894

1198991119883120591119894

119896minus1minus1198991)

(26)

where 119896 gt 0 119892(119896) = 1 when 119896 = 1 and zero otherwiseFigure 2 shows the approximate solution of (17) obtained

by using the EMHPM approach compared to its numericalintegration solution by using the dde23MATLAB subroutineprogram This case assumes two different initial solutions ofthe form 1199091205910(119879) = cos(120587(119879 + 1)) 1199091205910(119879) = 119890119879+1 and a timesubintervals Δ119905 = 001 We can see from Figure 2 that bothsimulations agree well for the time span showed

To further assess the applicability of our proposedEMHPM approach to high order delay differential equationswe will next describe a methodology to obtain the approx-imate solutions of well-known high order delay differentialequations by generalizing our EMHPM approach

Abstract and Applied Analysis 5

0 5 10 15

0

1

2

EMHPMdde23

minus1minus1

Am

plitu

dex

Time t

x1205910(T) = cos(120587(T + 1))

x1205910(T) = eT+1

Figure 2 EMHPM and dde23 solution of (17)

x x iminusNx iminusN+1

timinusN+1t titiminus1timinusN

Δt

x i(t)x i+1(t)

120591 = (N minus 1)Δt

Figure 3 Schematic of the zeroth order polynomial used to fit theapproximate EMHPM solution

5 Generalized Solution of Linear DDEsby the EMHPM Approach

Let us consider an 119899-dimensional delay differential equationof the form

x (119905) = A (119905) x + B (119905) x (119905 minus 120591) (27)

whereA(119905 + 120591) = A(119905) B(119905 + 120591) = B(119905) x(119905) is the state vectorand 120591 is the time delay By following our EMHPM procedurewe can write (27) in equivalent form as

x119894 (119879) minus A119905x119894 (119879) asymp B119905x

120591

119894(119879) (28)

where x119894(119879) denotes the 119898 order solution of (27) in the 119894th

subinterval that satisfies the initial conditions x119894(0) = x

119894minus1and

A119905and B

119905represent the values of the periodic coefficients at

the time 119905 In order to approximate the delayed term x120591119894(119879)

in (28) the period [1199050minus 120591 1199050] is discretized in 119873 points

equally spaced as shown in Figure 3 Here we assume thatthe function x120591

119894(119879) in the delay subinterval [119905

119894minus119873 119905119894minus119873+1

] isapproximated by a constant value

x120591119894(119879) = 119909

119894minus119873+1(119879) asymp x

119894minus119873 (29)

as shown in Figure 3 By following the homotopy perturba-tion technique we can write the homotopy representation of(28) as

119867(X119894 119901) = 119871 (X

119894) minus 119871 (x

1198940) + 119901119871 (x

1198940) = 119901 (A

119905X119894+ B119905x119894minus119873)

(30)

Substituting the 119898 order expansion X119894= X1198940+ 119901X1198941+ sdot sdot sdot +

119901119898X119894119898

in (30) and by assuming an initial approximation ofthe form x

1198940= x119894minus1

we get after applying the proposed

EMHPMapproach the following set of first order linear delaydifferential equations

1199010 119889119889119879

X1198940+119889

119889119879x119894minus1= 0 X

119894(0) = x

119894minus1

1199011 119889119889119879

X1198941= A119905X1198940+ B119905x119894minus119873 X1198941(0) = 0

1199012 119889119889119879

X1198942= A119905X1198941 X1198942(0) = 0

119901119898 119889119889119879

X119894119898= A119905X119894(119898minus1)

X119894119898(0) = 0

(31)

By solving (31) we get

X1198940= x119894minus1

X1198941= A119905x119894minus1119879 + B

119905x119894minus119873119879

X1198942=1

2A2119905x119894minus11198792+1

2A119905B119905x119894minus1198731198792

X119894119898=1

119898A119898119905x119894minus1119879119898+1

119898A119898minus1119905

B119905x119894minus119873119879119898

(32)

Equations (32) can be written as

X119894119896=119879

119896(A119905X119894(119896minus1)

+ 119892 (119896)B119905x119894minus119873) 119896 = 1 2 3 (33)

where X1198940= x119894minus1

and 119892(119896) = 1 for 119896 = 1 and 119892(119896) = 0otherwise Thus the solution of (27) is obtained by addingthe X119894119896approximate solutions

x119894 (119879) asymp

119898

sum

119896=0

X119894119896 (119879) (34)

Notice however that solution (34) may be further improvedby using a first order polynomial representation of x120591

119894(119879) as

shown in Figure 4 Then the function x120591119894(119879) in the delay

subinterval [119905119894minus119873 119905119894minus119873+1

] takes the form

x120591119894(119879) = x119894minus119873+1 (119879) asymp x119894minus119873 +

(119873 minus 1)

120591(x119894minus119873+1

minus x119894minus119873) 119879

(35)

Substituting (35) into (28) gives

x119894(119879) minus A

119905x119894(119879)

asymp B119905x119894minus119873minus(119873 minus 1)

120591B119905x119894minus119873119879 +(119873 minus 1)

120591B119905x119894minus119873+1

119879

(36)

6 Abstract and Applied Analysis

We next assume that the homotopy representation of (36) isgiven as

119867(X119894 119901) = 119871 (X

119894) minus 119871 (x

1198940) + 119901119871 (x

1198940)

minus 119901 (AX119894+ Bx119894minus119873minus(119873 minus 1)

120591Bx119894minus119873119879

+(119873 minus 1)

120591Bx119894minus119873+1

119879) = 0

(37)

Substituting the 119898 order expansion X119894(119879) = X

1198940(119879) +

119901X1198941(119879) + sdot sdot sdot 119901

119898X119894119898(119879) in (37) and assuming that the initial

approximation is given by x1198940= x119894minus1

we get

1199010 119889119889119879

X1198940+119889

119889119879x119894minus1= 0 X

119894 (0) = x119894minus1

1199011 119889119889119879

X1198941= A119905X1198940+ B119905x119894minus119873minus119873 minus 1

120591B119905x119894minus119873119879

+119873 minus 1

120591B119905x119894minus119873+1

119879 X1198941(0) = 0

1199012 119889119889119879

X1198942= AX

1198941 X1198942(0) = 0

119901119898 119889119889119879

X119894119898= AX

119894(119898minus1) X119894119898(0) = 0

(38)

By solving (38) and by following the EMHPM procedure weget

X1198940= x119894minus1

X1198941= A119905x119894minus1119879 + B

119905x119894minus119873119879 minus1

2

119873 minus 1

120591B119905x119894minus1198731198792

+1

2

119873 minus 1

120591B119905x119894minus119873+1

1198792

X1198942=1

2A2119905x119894minus11198792+1

2A119905B119905x119894minus1198731198792

minus1

6

119873 minus 1

120591A119905B119905x119894minus1198731198793+1

6

119873 minus 1

120591A119905B119905x119894minus119873+1

1198793

X119894119898=1

119898A119898119905x119894minus1119879119898+1

119898A119898minus1119905

B119905x119894minus119873119879119898

minus1

(119898 + 1)

119873 minus 1

120591A119898minus1119905

B119905x119894minus119873119879119898+1

+1

(119898 + 1)

119873 minus 1

120591A119898minus1119905

B119905x119894minus119873+1

119879119898+1

(39)

Here the recursive form of X119894119896(119879) is written as

X119894119896= Xa119894119896+ Xb119894119896119896 = 1 2 3 (40)

timinusN+1 ttitiminus1timinusN

Δt

x i(t)x i+1(t)

x x iminusNx iminusN+1

120591 = (N minus 1)Δt

Figure 4 Schematic EMHPM solution using first polynomial toapproximate delay subinterval

where Xa1198940= x119894minus1Xb1198940= 0 and

Xa119894119896=119879

119896(A119905Xa119894(119896minus1)

+ 119892 (119896)B119905x119894minus119873)

Xb119894119896=119879

119896 + 1(A119905Xb119894(119896minus1)

+119892 (119896) [119873 minus 1

120591119879 (minusB

119905x119894minus119873+ B119905x119894minus119873+1

)])

(41)

Thus the approximate solution of (27) by the EMHPM canbe obtained by substituting (40) into (34)

In the next section we will apply our EMHPM procedureto obtain the solution of two second order delay differentialequations (a) the dampedMathieu equation with time delayand (b) the well-known delay differential equation thatdescribes the dynamics in one degree-of-freedom millingmachine operations

51 Solution of theDampedMathieu EquationwithTimeDelayIn order to assess the accuracy of our EMHPM approach wefirst obtain the solution of the damped Mathieu differentialequation with time delay that combines the effect of paramet-ric excitation and dampingThis equation is described by thefollowing equation

+ 120581 + (120575 + 120576 cos(2120587119905119879))119909 = 119887119909 (119905 minus 120591) (42)

where 120581 120575 120576 120591 and 119879 are system parameters whose valuedepends on the physics of the system The approximatesolution of (42) obtained by using the semidiscretizationmethod is widely discussed in [18 19] Here we focus ourattention on applying the EMHPM to find the approximatesolution of (42) and we also assess the accuracy of the derivedsolution by comparing it with the corresponding numericalintegration solution of (42)

By following the EMHPM procedure we first write (42)in the following equivalent form

119894(119879) + 120581

119894(119879) + 120572

119905119909119894(119879) asymp 119887119909

119894minus119873+1(119879) (43)

where 119909119894(119905) denotes the 119898 order solution of (43) in the

119894th subinterval that satisfies the following initial conditions119909119894(0) = 119909

119894minus1and

119894(0) =

119894minus1 The space state form

representation of (43) is given by

x119894(119879) = A

119905x119894(119879) + B

119905x119894minus119873+1

(119879) (44)

Abstract and Applied Analysis 7

0 2 4 6 8 10 12

005

1

dde23Zeroth EMHPMFirst EMHPM

minus05minus05

minus15minus15

minus1minus1Am

plitu

dex

Time t

times10minus3

Figure 5 Numerical solutions of the damped Mathieu equationwith time delay by dde23 the zeroth EMHPM and the first EMHPMwith119873 = 50 and119898 = 4

Table 1 Computer time needed to solve the damped Mathieuequation with time delay The 119898 order solution of the EMPHMapproach is chosen to guarantee the convergence of its approximatesolution

dde23 [ms] EMHPM119873 119898 Zeroth [ms] First [ms]

19

15 5 4 520 4 6 640 3 12 1260 2 17 1860 5 18 1960 10 20 21

where

A119905= [0 1

minus120572119905minus120581] B

119905= [0 0

1198871199050] (45)

and 120572119905= (120575 + 120576 cos(119905)) is a time periodic term The EMHPM

approximate solution of (44) is illustrated in Figure 5 wherewe have assumed an unstable system behavior for which 120581 =02 120575 = 30 120576 = 1 119887 = minus1 and 119879 = 120591 = 2120587 See[20] As we can see from Figure 5 our approximate EMHPMsolution to (42) is compared with its numerical integrationsolution obtained from dde23 MATLAB algorithm for thetime interval of 2119879 by assuming that 119873 = 50 with thefollowing initial values 119909

minus50(119879) = 119909

minus49(119879) = sdot sdot sdot 119909

0(119879) =

0001 and minus50(119879) =

minus49(119879) = sdot sdot sdot

0(119879) = 0

It can be seen fromFigure 5 that in the interval [0119879] boththe zeroth and the first order solutions are the same since thedelay subintervals are constant See Figure 3 However in thenext interval [119879 2119879] it is clear that the first order EMPHMsolution provides a better approximation on the delay subin-terval The computation total time to calculate the solutionsin the MATLAB code is listed in Table 1 The order 119898 andthe discretized time intervals 119873 in the EMPHM approachare chosen to guarantee the convergence of our approximatesolution to the exact one To provide a full understanding ofhow the solution is computed by the EMHPM approach weattached in Algorithm 1 the corresponding MATLAB code

10 20 30 40 50 60 70 80 90 100

Abso

lute

erro

r

10minus4

10minus3

10minus2

Zeroth order m = 2

First order m = 2

Zeroth order m = 5

First order m = 5

Zeroth order m = 10

First order m = 10

Discretizations N

Figure 6 Estimated relative error values between the numericalsolution dde23 and the EMHPMapproximate solutionsHerewe usefor the EMHPM the values of119898 = 2 5 and 10

Figure 6 shows the relative error between our approxi-mate EMPHM and the dde23 solution and its relationshipwith the order119898 and the discretized time intervals119873 Noticethat the relative error values coincide at values of 119873 ge 45Also we can see from Figure 6 that the computed relativeerror values for approximate solutions of order119898 ge 5 remainunchanged

52 A Practical Application Cutting Operation on MillingMachine We next use our EMHPM procedure to obtainthe solution of the single degree-of-freedom milling opera-tion We use the simplified form based on [20ndash22]

(119905) + 2120577120596119899 (119905) + 1205962

119899119909 (119905) = minus

119886119901119870119904 (119905)

119898119898

(119909 (119905) minus 119909 (119905 minus 120591))

(46)

where 120596119899is the angular natural frequency of the system 120577 is

the damping ratio 119886119901is the depth of cut 119898

119898is the modal

mass of the tool 120591 represents the time delay which is equalto the tooth passing period and 119870

119904(119905) is the specific cutting

force coefficient which can be determined from119870119904 (119905)

=

119911119899

sum

119895=1

119892 (120601119895(119905)) sin (120601

119895(119905)) (119870

119905cos120601119895(119905) + 119870

119899sin120601119895(119905))

(47)

where 119911119899is the tool number of teeth 119870

119905and 119870

119899are the

tangential and the normal linear cutting force coefficientsrespectively 120601

119895(119905) is the angular position of the 119895-tooth

defined as

120601119895(119905) = (

2120587119899

60) 119905 +

2120587119895

119911119899

(48)

and 119899 is the spindle speed in rpm [20] The function 119892(120593119895(119905))

is a switching function which has a unity value when the 119895-tooth is cutting and zero otherwise

119892 (120601119895 (119905)) =

1 120601st lt 120601119895 (119905) lt 120601ex0 otherwise

(49)

8 Abstract and Applied Analysis

function dde23 ddeEM mathieu paper

disp(lsquoMathieu equation solutionrsquo) Solution by EMHPM with zeroth and first order solution

eps=1 kappa=02 T=2lowastpi tau=T b=minus1 delta=3 Mathieu Parameters

pntDelay=1 N=50 m ord=5 dt=tau(Nminus1) ktau=2 tspan=[0ktaulowastT] EMHPM Parameters

Solution

Solution by dde23

tdde=linspace(0ktaulowasttauktaulowast(Nminus1)+1)

dde=(tyz) mathieu dde(tyzkappadeltaepsbtau)

te dde=ticsol = dde23(ddetauhistorytspan) toc(te dde) xdde=deval(soltdde)

Solution by zeroth EMHPM

dde emhpm fun=(tc0ttzmzntauN) mathieu zeroth(tc0ttzmzntauNm ord

kappadeltaepsbT)

tic[t0z0]=ddeEMHPM(dde emhpm funtspanhistorytaudtpntDelay) toc

Solution by First EMHPM

dde emhpm fun=(tc0ttzmzntauN) mathieu first(tc0ttzmzntauNm ordkappadeltaepsbT)

tic[t1z1]=ddeEMHPM(dde emhpm funtspanhistorytaudtpntDelay) toc

Plot results

ind0=12ktaulowast(Nminus1) ind1=22ktaulowast(Nminus1)

Parent1=figure (1)

axes1 = axes(lsquoParentrsquoParent1lsquoFontSizersquo12lsquoFontNamersquolsquoTimes New Romanrsquo)box(axes1lsquoonrsquo) hold(axes1lsquoallrsquo)plot(tddexdde(1)lsquoParentrsquoaxes1lsquoLineWidthrsquo2lsquoColorrsquo[0502 0502 0502]lsquoDisplayNamersquolsquoNumericaldd23rsquo)plot(t0(ind0)z0(ind01)lsquoMarkerSizersquo5lsquoMarkerrsquolsquoorsquolsquoLineStylersquo lsquononersquolsquoDisplayNamersquolsquoZerothEMHPMrsquolsquoColorrsquo[0 0 0])

plot(t1(ind1)z1(ind11)lsquoMarkerSizersquo7lsquoMarkerrsquolsquoxrsquolsquoLineStylersquolsquononersquolsquoDisplayNamersquolsquoFirstEMHPMrsquolsquoColorrsquo[0 0 0])

xlabel(lsquoittrsquolsquoFontSizersquo12lsquoFontNamersquolsquoTimes New Romanrsquo)ylabel(lsquoitxrsquolsquoFontSizersquo12lsquoFontNamersquolsquoTimes New Romanrsquo)end

Mathieu definitions

function dydt = mathieu dde(tyzkapadltepsbT)

dydt = [y(2)-kapalowasty(2)minus(dlt+epslowastcos(2lowastpilowasttT))lowasty(1)+blowastz(1)]

end

function Z = mathieu zeroth(tc0ttzmzntauNmkpadltepsbT)

Z=[c0(1)c0(2)] alf=dlt+epslowastcos(2lowastpiTlowasttt)

for ik=1m

Z(ik+11)=Z(ik2)lowasttik

Z(ik+12)=minuskpalowastZ(ik2)minusalflowastZ(ik1)

if ik==1 Z(ik+12)=Z(ik+12)+blowastzm(1) end

Z(ik+12)=Z(ik+12)lowasttik

end

Z=sum(Z)

end

function Z = mathieu first(tc0ttzmzntauNmkpadltepsbT)

alf=dlt+epslowastcos(2lowastpiTlowasttt) Z=[c0(1)c0(2)] Z =[00]

for ik=1m

Z(ik+1)=[Z(ik2)lowasttik minuskpalowastZ(ik2)minusalflowastZ(ik1)]

Z (ik+1)=[Z (ik2)lowasttik minuskpalowastZ (ik2)minusalflowastZ (ik1)]

if ik==1

Z(ik+12)=Z(ik+12)+blowastzm(1)Z (ik+12)=Z (ik+12)+blowast(Nminus1)taulowast(zn(1)minuszm(1))lowastt

end

Z(ik+12)=Z(ik+12)lowasttik

Z (ik+12)=Z (ik+12)lowastt(ik+1)

Algorithm 1 Continued

Abstract and Applied Analysis 9

end

Z=sum(Z)+sum(Z )

end

function out=history(t)

out=[1Eminus3+0lowastt 0+0lowastt]

end

EMHPM algorithm for ODE solutions

function [tz]= odeEMHPM(ndetspanz0Deltatpnts)

ode solver by Enhanced Multistage Homotopy Perturbation Method

tini=tspan(1) tfin=tspan(end) tstart=tini

tini=tiniminuststart tfin=tfinminuststart shifted time set to zero

Handle errors

if tini == tfin

error(lsquoThe ending and starting time values must be differentrsquo)elseif abs(tini)gt abs(tfin)

tspan=flipud(fliplr(tspan)) tini=tspan(1) tfin=tspan(end)

end

tdir=sign(tfinminustini)

if any(tdirlowastdiff(tspan) lt= 0)

error(lsquotspan entries must be strictly sortedrsquo)end

incT=Deltatpnts

if incTlt=0

error (lsquoIncreasing must be greater than zerorsquo)end

z(1)=z01015840 t(1)=tini iteT=2 set initial values

t(iteT)=tini+incTlowasttdir

while tdirlowastt(iteT)lttdirlowast(tfin+tdirlowastincT)

P act=ceil(99999lowast(t(iteT)minustini)lowasttdirDeltat) count sub-intervals

c=z((DeltatincT)lowast(P actminus1)+1) set the corresponding initial condition

tsub(iteTminus1)=t(iteT)-(P actminus1)lowastDeltatlowasttdir evaluate the solution at the shifted time

temp=tsub(iteTminus1)

z(iteT)=nde(tempctstart+t(iteT))

if tdirlowastt(iteT)gt=tdirlowasttfinlowast099999 repeat for the next sub-interval

break

else

iteT=iteT+1 t(iteT)=tini+(iteTminus1)lowastincTlowasttdir

end

end

t=t 1015840 +tstart

end

EMHPM algorithm for DDE solutions

function [t z]=ddeEMHPM(ddetspanhistorytauDeltatpntDelta)

dde solver by Enhanced Multistage Homotopy Perturbation Method

tini=tspan(1) tfin=tspan(end) span where the solution is founded

if taultDeltat

error(lsquoSubtinterval must not be greater than taursquo)end

if mod(tauDeltat)sim=0

pastDlt=Deltat

Deltat=tauround(tauDeltat)

warning(lsquoSubtinterval was modified from 05g to 05grsquopastDltDeltat)end

pnt=round(tauDeltat) samplesminus1 [minustau 0]

incT=DeltatpntDelta step for set resolution

t=minustau+tini+(0pntlowastpntDelta)rsquolowastincT evaluation of the initial solution [minustau 0]

z=history(t) initial behavior in the inverval [minustau 0]

c=z(end) initial condition

Algorithm 1 Continued

10 Abstract and Applied Analysis

m=pntlowastpntDelta samples between tau

itDlt=0

while (t(end)+eps)lttfin

tspan=[tini+itDltlowastDeltattini+(itDlt+1)lowastDeltat] preparing the span for next Deltat

zm=z(length(z)minusm)rsquo zn=z(1+length(z)minusm)rsquo previous tau solution

emhpm fun=(timecT) dde(timecTzmzntaum) application of the odeEMHPM

[t aux z aux]=odeEMHPM(emhpm funtspanz(end)DeltatpntDelta)

z=[z(1endminus1)z aux] t=[t(1endminus1)t aux] joining the solutions

itDlt=itDlt+1

end

z=z(pntlowastpntDelta+1end)t=t(pntlowastpntDelta+1end)

end

Algorithm 1 MATLAB algorithm

Here 120601st and 120601ex are the angles where the teeth enter and exitthe workpiece For upmilling 120601st = 0 and 120601ex = arccos(1 minus2119886119889) for downmilling 120601st = arccos(2119886

119889minus 1) and 120601ex = 120587

where 119886119889is the radial depth of cut ratio

By following the EMHPM procedure we can write (46)in equivalent form as

119894(119879) + 2120577120596

119899119894(119879) + 120596

2

119899119909119894(119879)

asymp minus

119886119901119870st

119898119898

(119909119894 (119879) minus 119909119894minus119873+1 (119879))

(50)

where 119909119894(119879) denotes the 119898 order solution of (46) on the 119894th

subinterval that satisfies the initial conditions 119909119894(0) = 119909

119894minus1

119894(0) =

119894minus1 and ℎ

119905= ℎ(119905) and 119909

minus120591is given by (35)

Introducing the transformation x119894= [119909119894 119894]119879 (50) can be

written as a system of first order linear delay differentialequations of the form

x119894(119879) = A

119905x119894(119879) + B

119905x119894minus119873+1

(119879) (51)

where

A119905= [

[

0 1

minus1205962

119899minus119908

119898119898

119870st minus2120577120596119899]

]

B119905= [

[

0 0119908

119898119898

119870st 0]

]

(52)

We next apply the EMHPM procedure to solve (46) byconsidering a downmilling operation with the followingparameter values 119911

119899= 2 119886

119889= 01 120596

119899= 5793 rads

120577 = 0011 119898119898= 003993 kg 119870

119905= 6 times 10

8Nm2 and119870119899= 2 times 10

8Nm2 As we can see from Figures 7 and 8 andfor the depth of cut values of 119886

119901= 2mm (stable) and 119886

119901=

3mm (unstable) our EMHPM approximate solutions followclosely the numerical integration solutions of (46) obtainedby using the dde23 algorithm

Figure 9 shows the relative error between the EMPHMand the dde23 numerical solution while Table 2 shows theCPU time needed for each solution Here we use 119873 = 75since the average step size of the dde23 algorithm is aroundΔ119905 asymp 120591119873 Note that for 119898 = 7 the zeroth order EMHPMapproximate solution has the fastest CPU time We can seefrom Figure 9 that the value of the relative error becomesbasically the same for119898 = 2 7 and 10 and119873 ge 20

0 0005 001 0015 002

0

1

2

Time (s)

Disp

lace

men

t (m

)

dd23Zeroth EMHPMFirst EMHPM

minus2

minus1

times10minus3

Figure 7 EMHPM approximate solutions of (46) with parametervalues of 119886

119901= 2mm 119899 = 10000 rpm Stable machine operation

0 0005 001 0015 002

00005

0010015

Time (s)

Disp

lace

men

t (m

)

dd23Zeroth EMHPMFirst EMHPM

minus001

minus0005

Figure 8 EMHPM approximate solutions of (46) with parametervalues of 119886

119901= 3mmand 119899 = 10000 rpmUnstable (chatter)machine

operation

6 Conclusions

We have developed a new algorithm based on the homotopyperturbation method to solve delay differential equationsThe proposed EMHPM approach is based on a sequence ofsubintervals that approximate the solution of delayed differ-ential equations by using the transformation rule 119906(119905) rarr119906119894(119879) where 119906

119894satisfies the initial conditions We have

Abstract and Applied Analysis 11

10 20 30 40 50 60 70 80 90 100

Abso

lute

erro

r

Zeroth order m = 2

First order m = 2

Zeroth order m = 7

First order m = 7

Zeroth order m = 10

First order m = 10

10minus3

10minus2

10minus1

Discretizations N

Figure 9 Estimated relative error values between dde23 and theEMHPM approximate solutions Here we have used the systemparameter values of 119886

119901= 2mm and 119899 = 1000 rpm and values of

119898 = 2 7 and 10

Table 2 CPU time comparison among the approximate solutions of(46) by using dde23 the zeroth order and the first order EMHPMsolutions

Solution method Time [ms]dde23 139Zeroth order EMHPM (119898 = 7) 77First order EMHPM (119898 = 7) 87

shown that our proposed EMHPM approach can be appliedto obtain the approximate solution of a delay differentialequation not only with constant but also with variablecoefficients with theoretical predictions that follow well thenumerical integration solutions To further assess the validityof this new approach we have compared the approximatesolutions of two delayed differential equations with respect totheir corresponding numerical integration solutions obtainedfrom the MATLAB dde23 algorithm The test cases were (a)the damped Mathieu differential equation with time delayand (b) the governing equation of motion of downmillingoperations We have found that the EMHPM closely followsthe numerical integration solutions of the correspondingequations and that these require less CPU time and havesmaller relative errors

Conflict of Interests

Theauthors declare that they have no conflict of interests withany mentioned entities in the paper

Acknowledgments

This work was funded by Tecnologico deMonterrey CampusMonterrey through the ResearchGroup inNanomaterials forMedical Devices and the Research Group in Advanced Man-ufacturing Additional support was provided from ConsejoNacional de Ciencia y Tecnologıa (Conacyt) Mexico

References

[1] F Shakeri and M Dehghan ldquoSolution of delay differentialequations via a homotopy perturbation methodrdquoMathematicaland Computer Modelling vol 48 no 3-4 pp 486ndash498 2008

[2] Z-K Wu ldquoSolution of the enso delayed oscillator with homo-topy analysis methodrdquo Journal of Hydrodynamics vol 21 no 1pp 131ndash135 2009

[3] A K Alomari M S Noorani and R Nazar ldquoSolution of delaydifferential equation by means of homotopy analysis methodrdquoActa Applicandae Mathematicae vol 108 no 2 pp 395ndash4122009

[4] J Biazar and G Behzad ldquoThe homotopy perturbation methodfor solving neutral functional-differential equations with pro-portional delaysrdquo Journal of King Saud UniversitymdashScience vol24 no 1 pp 33ndash37 2012

[5] NRAnakiraAKAlomari and IHashim ldquoOptimal homotopyasymptotic method for solving delay differential equationsrdquoMathematical Problems in Engineering vol 2013 Article ID498902 11 pages 2013

[6] T Insperger and G Stepan Semi-Discretization for Time-DelaySystems Stability and Engineering Applications Springer NewYork NY USA 2011

[7] E A Butcher P Nindujarla and E Bueler ldquoStability of up-and down-milling using chebyshev collocation methodrdquo inProceedings of the ASME International Design Engineering Tech-nical Conferences amp Computers and Information in EngineeringConference vol 6 part AndashC pp 841ndash850 2005

[8] T Insperger ldquoFull-discretization and semi-discretization formilling stability prediction some commentsrdquo InternationalJournal of Machine Tools and Manufacture vol 50 no 7 pp658ndash662 2010

[9] I Hashim and M S Chowdhury ldquoAdaptation of homotopy-perturbation method for numeric-analytic solution of systemof ODEsrdquo Physics Letters A vol 372 no 4 pp 470ndash481 2008

[10] L F Shampine and S Thompson ldquoSolving DDEs in MatlabrdquoAppliedNumericalMathematics vol 37 no 4 pp 441ndash458 2001

[11] SHHNiaANRanjbarDDGanjiH Soltani and JGhasemildquoMaintaining the stability of nonlinear differential equations bythe enhancement of HPMrdquo Physics Letters A vol 372 no 16 pp2855ndash2861 2008

[12] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[13] J-H He ldquoHomotopy perturbation method a new nonlinearanalytical techniquerdquo Applied Mathematics and Computationvol 135 no 1 pp 73ndash79 2003

[14] J H He ldquoComparison of homotopy perturbation method andhomotopy analysis methodrdquo Applied Mathematics and Compu-tation vol 156 no 2 pp 527ndash539 2004

[15] JHHe ldquoNew interpretation of homotopy perturbationmethodrdquoInternational Journal of Modern Physics B vol 20 no 18 pp2561ndash2568 2006

[16] S Liao ldquoComparison between the homotopy analysis methodand homotopy perturbationmethodrdquoAppliedMathematics andComputation vol 169 no 2 pp 1186ndash1194 2005

[17] D Olvera and A Elıas-Zuniga ldquoEnhanced multistage homo-topy perturbation method approximate solutions of nonlineardynamic systemsrdquo Abstract and Applied Analysis vol 2014Article ID 486509 12 pages 2014

12 Abstract and Applied Analysis

[18] T Insperger and G Stepan ldquoStability of the damped Mathieuequationwith timedelayrdquo Journal ofDynamic SystemsMeasure-ment and Control vol 125 no 2 pp 166ndash171 2003

[19] T Insperger and G Stepan ldquoStability chart for the delayedMathieu equationrdquo The Royal Society of London ProceedingsSeries A Mathematical Physical and Engineering Sciences vol458 no 2024 pp 1989ndash1998 2002

[20] T Insperger andGStepan ldquoUpdated semi-discretizationmethodfor periodic delay-differential equations with discrete delayrdquoInternational Journal for Numerical Methods in Engineering vol61 no 1 pp 117ndash141 2004

[21] T Insperger B P Mann G Stepan and P V Bayly ldquoStabilityof up-milling and down-milling part 1 alternative analyticalmethodsrdquo International Journal of Machine Tools and Manufac-ture vol 43 no 1 pp 25ndash34 2003

[22] F ICompeanDOlvera F J Campa LN L de Lacalle A Elıas-Zuniga and C A Rodrıguez ldquoCharacterization and stabilityanalysis of a multivariable milling tool by the enhanced mul-tistage homotopy perturbation methodrdquo International Journalof Machine Tools and Manufacture vol 57 pp 27ndash33 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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

Abstract and Applied Analysis 3

modifications to the MHPM and focus on the derivation ofapproximate solutions of DDEs equations with variable coef-ficient terms This new approach is based on the enhancedmultistage homotopy perturbationmethod (EMHPM) intro-duced in [17] to obtain the solution of nonlinear ordinarydifferential equations

The EMHPM is an algorithm which approximates theHPM solution by subintervals utilizing the following trans-formation rule 119906(119905) rarr 119906

119894(119879) where 119906

119894satisfies the initial

condition 119906119894(0) = 119906

119894minus1(119905119894minus1) 119879 is a shifted time scale used

to determine the approximate solution in each subintervaland 119906

119894(119879) represents the approximate solution in the 119894th

subinterval In this case the initial suggested solution in the119894th subinterval is given by 119906

1198940(119879) = 119906

119894minus1(119905119894minus1) where 119905

119894minus1

represents the time at the end of the previous subinterval(ie the value of the approximate solution at the end of theprevious subinterval represents the initial conditions of thenext subinterval under consideration)

To apply the homotopy technique to solve delay differen-tial equations we also assume the following

(1) The linear operator119871(119906119894) represents119871(119906

119894) = (119889119889119879)119906

119894

where the assumed approximate solution 1199061198940(119879) is

set equal to the initial condition 119906119894minus1(119905119894minus1) that is

1199061198940= 119906119894minus1(119905119894minus1) To simplify the notation we let 119906

119894minus1equiv

119906119894minus1(119905119894minus1)

(2) The transformation 119879 = 119905 minus 119905119894minus1

on 0 lt 119879 le 119905119894minus 119905119894minus1

holds in the homotopy 119894-subinterval Thus higherorder equations are integratedwith respect to119879 whilethe terms related to the independent variable 119905 areassumed to remain constant

Therefore we may conclude that the 119898 order approximatesolution by applying the EMHPM can be written as

119906119894(119879 119906119894minus1) =

119898

sum

119896=0

119880119894119896(119879 119906119894minus1) (8)

where the solution 119906119894(119879 119906119894minus1) is valid only in the 119894th subin-

terval [119905119894minus1 119905119894] Hence the solution 119906(119905) on the 119894th subinterval

(119905119894minus1 119905119894] can be written as

119906 (119905) asymp 119906119894(119905 minus 119905119894minus1) (9)

with initial condition 119906119894minus1(119905119894minus1) and 119894 = 1 2 119895 Thus the

approximate solution of 119906 at the time 119905119894is given by

119906119894(119905 minus 119905119894minus1)1003816100381610038161003816119905=119905119894= 119906119894+1(119905 minus 119905119894)1003816100381610038161003816119905=119905119894= 119906119894+1(0) = 119906

119894 (10)

In summary the solution 119906(119905) for an open-closed interval(1199050 1199051] is divided into 119895 subintervals that in general are

not equally spaced [1199050 1199051] [1199051 1199052] [119905

119895minus1 119905119895] Thus the

approximated solution of 119906(119905) for the span time interval isobtained by coupling the 119906

119894(119905) solutions

4 Approximate Solutions of Some DDEs byApplying the EMHPM

In this sectionwe focus on the solution ofDDEswith constantand variable coefficients and examine the applicability of theEMHPM to find the corresponding approximate solutions

41 Delay Differential Equations with Constant CoefficientsFirst let us consider the simplest DDE of the form

(119905) + 119909 (119905 minus 120591) = 0 (11)

with initial condition 119909(0) = 119888 Here the independent vari-able 119909 is a scalar 119909(119905) isin R the dot stands for differentiationwith respect to time 119905 and 120591 is the time delay To evaluate(11) on 119886 le 119905 le 119887 the term 119909(119905 minus 120591) must represent a knownfunction 119909(119905) on [119886 minus 120591 le 119905 le 119886] For instance if 119886 = 0the solution of (11) can be obtained in the interval (0 120591] byassuming an initial function that satisfies the initial conditionBy using this solution it becomes possible to obtain thesolution of (11) in the next 119894th interval [(119894 minus 1)120591 119894120591] 119894 =2 3 119895 where 119895 is an integer number that can be chosenas 2 le 119895 le infin With this approach we can apply the HPM tofind the solution of (11) by assuming that the previous delayedfunction is 1199091205910(119879) = 119888 thus the solution for the first intervalis given by 1199091205911(119879) valid on [0 120591] In terms of (4) we nowconstruct the homotopy of (11)

119867(1198831205911 119901) =

119889

1198891198791198831205911 + 119901119909

1205910 = 0 (12)

We next substitute the first order expansion1198831205911 = 11988312059110+1199011198831205911

1

in (12) and balance the terms with identical power of 119901 toobtain the following set of linear differential equations

1199010 1198891198891198791198831205911

0= 0 119883

1205911

0(0) = 119888 = 119883

1205910(120591)

1199011 1198891198891198791198831205911

1= minus1198831205910 119883

1205911

1(0) = 0

(13)

Integration of (13) yields

1198831205911

0= 119888

1198831205911

1= minus 119888119879

(14)

Hence the first order solution of (12) is given by

1199091205911(119879) = 119888 minus 119888119879 (15)

Notice that (15) represents the exact solution of (11) on thefirst interval By following the same procedure it is easy toshow that the exact solution of (11) for the second and thirdintervals is given respectively as

1199091205912(119879) = 119888 minus 119888120591 minus 119888119879 +

1

21198881198792

1199091205913(119879) = 119888 minus 2119888120591 +

1

21198881205912minus (119888 minus 119888120591) 119879 +

1

21198881198792minus1

61198881198793

(16)

Figure 1 shows the exact solution of (11) obtained by couplingat each interval the solution obtained by following HPMprocedure for 119905 = 10120591

It is easy to show that the solution of (11) by the EMHPMcoincides with the solution obtained by using the HPM since(11) is a delay differential equation with constant coefficients

4 Abstract and Applied Analysis

0 1 2 3 4 5 6 7 8 9 10

00

0505

11

Time t

HPM by intervals

minus05minus05

Am

plitu

dex

Figure 1 Exact solution of (11) obtained by using theHPMand 120591 = 1

42 Delay Differential Equations with Variable CoefficientsWe next show how the EMHPM approach can be applied toobtain the approximate solution of nonlinear delay differen-tial equation with variable coefficients In this case we obtainthe approximate solutions of a DDE of the form

+ 119909 (119905 minus 120591) minus cos (120587119905) 1199092 = 0 120591 = 1 119909 (0) = 119888 = 1199091205910 (120591)(17)

in which the solution 1199091205910(119879) = 1198881holds on (minus120591 0] In order to

find the solution 1199091205911 in the interval [0 120591] we assume that thehomotopy representation of (17) can be given as

119867(1198831205911 119901) =

119889

1198891198791198831205911 + 119901 [119909

1205910 minus cos (120587119905) (1198831205911)2] = 0(18)

Notice that the variable 119883 depends on the time 119879 for which0 le 119879 le 120591 If we now substitute the second order expansion1198831205911 = 119883

1205911

0+ 1199011198831205911

1+ 11990121198831205911

2in (18) and after balancing the 119901

terms we get that

1199010 1198891198891198791198831205911

0= 0 119883

0 (119879 = 0) = 1198881 = 1198831205910(119879 = 120591)

1199011 1198891198891198791198831205911

1= minus1199091205910 + cos (120587119905) (1198831205911

0)2

= 0 1198831(0) = 0

1199012 1198891198891198791198831205911

2= 2 cos (120587119905)1198831205911

01198831205911

1 1198832(0) = 0

1199013 1198891198891198791198831205911

3= cos (120587119905) (21198831205911

01198831205911

2+ (1198831205911

1)2

) = 0 1198833 (0) = 0

(19)

Equations (19) have the following solutions

1198831205911

0= 1198881

1198831205911

1= minus119879 (119909

1205910 minus 1198882

1cos120587119905)

1198831205911

2= minus11988811198792(cos120587119905) (1199091205910 minus 1198882

1cos120587119905)

1198831205911

3=1

31198793(cos120587119905) [31198884

1cos2120587119905 minus 41198882

11199091205910 + (119909

1205910)2]

(20)

Thus the approximate solution of (17) by using the EMHPMis given by

1199091205911(119879) asymp 119883

1205911

0+ 1198831205911

1+ 1198831205911

2+ 1198831205911

3 (21)

In this case the exact solution of 1199091205911(119879) is unknown Toobtain 1199091205912 we compute again the approximate solution of1199091205911(119879) by applying our EMHPM and the value of the delayed

time is assumed to remain constant in each subinterval Todetermine 1199091205912 we next use the homotopy representation of(17) for the interval (120591 2120591]

119867(1198831205912 119901) =

119889

1198891198791198831205912 + 119901 [119909

1205911 minus cos (120587119905) (1198831205912)2] = 0(22)

Substituting the second order expansion in (22) we get

1198831205912

0= 1198882

1198831205912

1= minus119879 (119909

1205911 minus 1198882

2cos120587119905)

1198831205912

2= minus11988821198792(cos120587119905) (1199091205911 minus 1198882

2cos120587119905)

1198831205912

3=1

31198793(cos120587119905) (31198884

2cos2120587119905 minus 41198882

21199091205911 + (119909

1205911)2)

(23)

Note that (20) and (23) provide approximate solutions to(17) but evaluated at different interval time delays To findthe third order approximate solution of (17) we can use ahomotopy of the form

119867(119883120591119894 119901) =

119889

119889119879119883120591119894 + 119901 [119883

120591119894minus1 minus cos (120587119905) (119883120591119894)2] = 0(24)

Then by using our EMPHM approach we have that

119883120591119894

0= 119888

119883120591119894

1= minus119879 (119909

120591119894minus1 minus 1198882 cos120587119905)

119883120591119894

2= minus119888119879

2(cos120587119905) (119909120591119894minus1 minus 1198882 cos120587119905)

119883120591119894

3=1

31198793(cos120587119905) (31198884cos2120587119905 minus 41198882119909120591119894minus1 + (119909120591119894minus1)2)

(25)

Notice from (25) that the 119896th order approximate solution of(17) can be written as

119883120591119894

0= 119888

119883120591119894

119896=119879

119896(minus119909120591119894minus1119892 (119896) + cos120587119905

119896minus1

sum

1198991=0

119883120591119894

1198991119883120591119894

119896minus1minus1198991)

(26)

where 119896 gt 0 119892(119896) = 1 when 119896 = 1 and zero otherwiseFigure 2 shows the approximate solution of (17) obtained

by using the EMHPM approach compared to its numericalintegration solution by using the dde23MATLAB subroutineprogram This case assumes two different initial solutions ofthe form 1199091205910(119879) = cos(120587(119879 + 1)) 1199091205910(119879) = 119890119879+1 and a timesubintervals Δ119905 = 001 We can see from Figure 2 that bothsimulations agree well for the time span showed

To further assess the applicability of our proposedEMHPM approach to high order delay differential equationswe will next describe a methodology to obtain the approx-imate solutions of well-known high order delay differentialequations by generalizing our EMHPM approach

Abstract and Applied Analysis 5

0 5 10 15

0

1

2

EMHPMdde23

minus1minus1

Am

plitu

dex

Time t

x1205910(T) = cos(120587(T + 1))

x1205910(T) = eT+1

Figure 2 EMHPM and dde23 solution of (17)

x x iminusNx iminusN+1

timinusN+1t titiminus1timinusN

Δt

x i(t)x i+1(t)

120591 = (N minus 1)Δt

Figure 3 Schematic of the zeroth order polynomial used to fit theapproximate EMHPM solution

5 Generalized Solution of Linear DDEsby the EMHPM Approach

Let us consider an 119899-dimensional delay differential equationof the form

x (119905) = A (119905) x + B (119905) x (119905 minus 120591) (27)

whereA(119905 + 120591) = A(119905) B(119905 + 120591) = B(119905) x(119905) is the state vectorand 120591 is the time delay By following our EMHPM procedurewe can write (27) in equivalent form as

x119894 (119879) minus A119905x119894 (119879) asymp B119905x

120591

119894(119879) (28)

where x119894(119879) denotes the 119898 order solution of (27) in the 119894th

subinterval that satisfies the initial conditions x119894(0) = x

119894minus1and

A119905and B

119905represent the values of the periodic coefficients at

the time 119905 In order to approximate the delayed term x120591119894(119879)

in (28) the period [1199050minus 120591 1199050] is discretized in 119873 points

equally spaced as shown in Figure 3 Here we assume thatthe function x120591

119894(119879) in the delay subinterval [119905

119894minus119873 119905119894minus119873+1

] isapproximated by a constant value

x120591119894(119879) = 119909

119894minus119873+1(119879) asymp x

119894minus119873 (29)

as shown in Figure 3 By following the homotopy perturba-tion technique we can write the homotopy representation of(28) as

119867(X119894 119901) = 119871 (X

119894) minus 119871 (x

1198940) + 119901119871 (x

1198940) = 119901 (A

119905X119894+ B119905x119894minus119873)

(30)

Substituting the 119898 order expansion X119894= X1198940+ 119901X1198941+ sdot sdot sdot +

119901119898X119894119898

in (30) and by assuming an initial approximation ofthe form x

1198940= x119894minus1

we get after applying the proposed

EMHPMapproach the following set of first order linear delaydifferential equations

1199010 119889119889119879

X1198940+119889

119889119879x119894minus1= 0 X

119894(0) = x

119894minus1

1199011 119889119889119879

X1198941= A119905X1198940+ B119905x119894minus119873 X1198941(0) = 0

1199012 119889119889119879

X1198942= A119905X1198941 X1198942(0) = 0

119901119898 119889119889119879

X119894119898= A119905X119894(119898minus1)

X119894119898(0) = 0

(31)

By solving (31) we get

X1198940= x119894minus1

X1198941= A119905x119894minus1119879 + B

119905x119894minus119873119879

X1198942=1

2A2119905x119894minus11198792+1

2A119905B119905x119894minus1198731198792

X119894119898=1

119898A119898119905x119894minus1119879119898+1

119898A119898minus1119905

B119905x119894minus119873119879119898

(32)

Equations (32) can be written as

X119894119896=119879

119896(A119905X119894(119896minus1)

+ 119892 (119896)B119905x119894minus119873) 119896 = 1 2 3 (33)

where X1198940= x119894minus1

and 119892(119896) = 1 for 119896 = 1 and 119892(119896) = 0otherwise Thus the solution of (27) is obtained by addingthe X119894119896approximate solutions

x119894 (119879) asymp

119898

sum

119896=0

X119894119896 (119879) (34)

Notice however that solution (34) may be further improvedby using a first order polynomial representation of x120591

119894(119879) as

shown in Figure 4 Then the function x120591119894(119879) in the delay

subinterval [119905119894minus119873 119905119894minus119873+1

] takes the form

x120591119894(119879) = x119894minus119873+1 (119879) asymp x119894minus119873 +

(119873 minus 1)

120591(x119894minus119873+1

minus x119894minus119873) 119879

(35)

Substituting (35) into (28) gives

x119894(119879) minus A

119905x119894(119879)

asymp B119905x119894minus119873minus(119873 minus 1)

120591B119905x119894minus119873119879 +(119873 minus 1)

120591B119905x119894minus119873+1

119879

(36)

6 Abstract and Applied Analysis

We next assume that the homotopy representation of (36) isgiven as

119867(X119894 119901) = 119871 (X

119894) minus 119871 (x

1198940) + 119901119871 (x

1198940)

minus 119901 (AX119894+ Bx119894minus119873minus(119873 minus 1)

120591Bx119894minus119873119879

+(119873 minus 1)

120591Bx119894minus119873+1

119879) = 0

(37)

Substituting the 119898 order expansion X119894(119879) = X

1198940(119879) +

119901X1198941(119879) + sdot sdot sdot 119901

119898X119894119898(119879) in (37) and assuming that the initial

approximation is given by x1198940= x119894minus1

we get

1199010 119889119889119879

X1198940+119889

119889119879x119894minus1= 0 X

119894 (0) = x119894minus1

1199011 119889119889119879

X1198941= A119905X1198940+ B119905x119894minus119873minus119873 minus 1

120591B119905x119894minus119873119879

+119873 minus 1

120591B119905x119894minus119873+1

119879 X1198941(0) = 0

1199012 119889119889119879

X1198942= AX

1198941 X1198942(0) = 0

119901119898 119889119889119879

X119894119898= AX

119894(119898minus1) X119894119898(0) = 0

(38)

By solving (38) and by following the EMHPM procedure weget

X1198940= x119894minus1

X1198941= A119905x119894minus1119879 + B

119905x119894minus119873119879 minus1

2

119873 minus 1

120591B119905x119894minus1198731198792

+1

2

119873 minus 1

120591B119905x119894minus119873+1

1198792

X1198942=1

2A2119905x119894minus11198792+1

2A119905B119905x119894minus1198731198792

minus1

6

119873 minus 1

120591A119905B119905x119894minus1198731198793+1

6

119873 minus 1

120591A119905B119905x119894minus119873+1

1198793

X119894119898=1

119898A119898119905x119894minus1119879119898+1

119898A119898minus1119905

B119905x119894minus119873119879119898

minus1

(119898 + 1)

119873 minus 1

120591A119898minus1119905

B119905x119894minus119873119879119898+1

+1

(119898 + 1)

119873 minus 1

120591A119898minus1119905

B119905x119894minus119873+1

119879119898+1

(39)

Here the recursive form of X119894119896(119879) is written as

X119894119896= Xa119894119896+ Xb119894119896119896 = 1 2 3 (40)

timinusN+1 ttitiminus1timinusN

Δt

x i(t)x i+1(t)

x x iminusNx iminusN+1

120591 = (N minus 1)Δt

Figure 4 Schematic EMHPM solution using first polynomial toapproximate delay subinterval

where Xa1198940= x119894minus1Xb1198940= 0 and

Xa119894119896=119879

119896(A119905Xa119894(119896minus1)

+ 119892 (119896)B119905x119894minus119873)

Xb119894119896=119879

119896 + 1(A119905Xb119894(119896minus1)

+119892 (119896) [119873 minus 1

120591119879 (minusB

119905x119894minus119873+ B119905x119894minus119873+1

)])

(41)

Thus the approximate solution of (27) by the EMHPM canbe obtained by substituting (40) into (34)

In the next section we will apply our EMHPM procedureto obtain the solution of two second order delay differentialequations (a) the dampedMathieu equation with time delayand (b) the well-known delay differential equation thatdescribes the dynamics in one degree-of-freedom millingmachine operations

51 Solution of theDampedMathieu EquationwithTimeDelayIn order to assess the accuracy of our EMHPM approach wefirst obtain the solution of the damped Mathieu differentialequation with time delay that combines the effect of paramet-ric excitation and dampingThis equation is described by thefollowing equation

+ 120581 + (120575 + 120576 cos(2120587119905119879))119909 = 119887119909 (119905 minus 120591) (42)

where 120581 120575 120576 120591 and 119879 are system parameters whose valuedepends on the physics of the system The approximatesolution of (42) obtained by using the semidiscretizationmethod is widely discussed in [18 19] Here we focus ourattention on applying the EMHPM to find the approximatesolution of (42) and we also assess the accuracy of the derivedsolution by comparing it with the corresponding numericalintegration solution of (42)

By following the EMHPM procedure we first write (42)in the following equivalent form

119894(119879) + 120581

119894(119879) + 120572

119905119909119894(119879) asymp 119887119909

119894minus119873+1(119879) (43)

where 119909119894(119905) denotes the 119898 order solution of (43) in the

119894th subinterval that satisfies the following initial conditions119909119894(0) = 119909

119894minus1and

119894(0) =

119894minus1 The space state form

representation of (43) is given by

x119894(119879) = A

119905x119894(119879) + B

119905x119894minus119873+1

(119879) (44)

Abstract and Applied Analysis 7

0 2 4 6 8 10 12

005

1

dde23Zeroth EMHPMFirst EMHPM

minus05minus05

minus15minus15

minus1minus1Am

plitu

dex

Time t

times10minus3

Figure 5 Numerical solutions of the damped Mathieu equationwith time delay by dde23 the zeroth EMHPM and the first EMHPMwith119873 = 50 and119898 = 4

Table 1 Computer time needed to solve the damped Mathieuequation with time delay The 119898 order solution of the EMPHMapproach is chosen to guarantee the convergence of its approximatesolution

dde23 [ms] EMHPM119873 119898 Zeroth [ms] First [ms]

19

15 5 4 520 4 6 640 3 12 1260 2 17 1860 5 18 1960 10 20 21

where

A119905= [0 1

minus120572119905minus120581] B

119905= [0 0

1198871199050] (45)

and 120572119905= (120575 + 120576 cos(119905)) is a time periodic term The EMHPM

approximate solution of (44) is illustrated in Figure 5 wherewe have assumed an unstable system behavior for which 120581 =02 120575 = 30 120576 = 1 119887 = minus1 and 119879 = 120591 = 2120587 See[20] As we can see from Figure 5 our approximate EMHPMsolution to (42) is compared with its numerical integrationsolution obtained from dde23 MATLAB algorithm for thetime interval of 2119879 by assuming that 119873 = 50 with thefollowing initial values 119909

minus50(119879) = 119909

minus49(119879) = sdot sdot sdot 119909

0(119879) =

0001 and minus50(119879) =

minus49(119879) = sdot sdot sdot

0(119879) = 0

It can be seen fromFigure 5 that in the interval [0119879] boththe zeroth and the first order solutions are the same since thedelay subintervals are constant See Figure 3 However in thenext interval [119879 2119879] it is clear that the first order EMPHMsolution provides a better approximation on the delay subin-terval The computation total time to calculate the solutionsin the MATLAB code is listed in Table 1 The order 119898 andthe discretized time intervals 119873 in the EMPHM approachare chosen to guarantee the convergence of our approximatesolution to the exact one To provide a full understanding ofhow the solution is computed by the EMHPM approach weattached in Algorithm 1 the corresponding MATLAB code

10 20 30 40 50 60 70 80 90 100

Abso

lute

erro

r

10minus4

10minus3

10minus2

Zeroth order m = 2

First order m = 2

Zeroth order m = 5

First order m = 5

Zeroth order m = 10

First order m = 10

Discretizations N

Figure 6 Estimated relative error values between the numericalsolution dde23 and the EMHPMapproximate solutionsHerewe usefor the EMHPM the values of119898 = 2 5 and 10

Figure 6 shows the relative error between our approxi-mate EMPHM and the dde23 solution and its relationshipwith the order119898 and the discretized time intervals119873 Noticethat the relative error values coincide at values of 119873 ge 45Also we can see from Figure 6 that the computed relativeerror values for approximate solutions of order119898 ge 5 remainunchanged

52 A Practical Application Cutting Operation on MillingMachine We next use our EMHPM procedure to obtainthe solution of the single degree-of-freedom milling opera-tion We use the simplified form based on [20ndash22]

(119905) + 2120577120596119899 (119905) + 1205962

119899119909 (119905) = minus

119886119901119870119904 (119905)

119898119898

(119909 (119905) minus 119909 (119905 minus 120591))

(46)

where 120596119899is the angular natural frequency of the system 120577 is

the damping ratio 119886119901is the depth of cut 119898

119898is the modal

mass of the tool 120591 represents the time delay which is equalto the tooth passing period and 119870

119904(119905) is the specific cutting

force coefficient which can be determined from119870119904 (119905)

=

119911119899

sum

119895=1

119892 (120601119895(119905)) sin (120601

119895(119905)) (119870

119905cos120601119895(119905) + 119870

119899sin120601119895(119905))

(47)

where 119911119899is the tool number of teeth 119870

119905and 119870

119899are the

tangential and the normal linear cutting force coefficientsrespectively 120601

119895(119905) is the angular position of the 119895-tooth

defined as

120601119895(119905) = (

2120587119899

60) 119905 +

2120587119895

119911119899

(48)

and 119899 is the spindle speed in rpm [20] The function 119892(120593119895(119905))

is a switching function which has a unity value when the 119895-tooth is cutting and zero otherwise

119892 (120601119895 (119905)) =

1 120601st lt 120601119895 (119905) lt 120601ex0 otherwise

(49)

8 Abstract and Applied Analysis

function dde23 ddeEM mathieu paper

disp(lsquoMathieu equation solutionrsquo) Solution by EMHPM with zeroth and first order solution

eps=1 kappa=02 T=2lowastpi tau=T b=minus1 delta=3 Mathieu Parameters

pntDelay=1 N=50 m ord=5 dt=tau(Nminus1) ktau=2 tspan=[0ktaulowastT] EMHPM Parameters

Solution

Solution by dde23

tdde=linspace(0ktaulowasttauktaulowast(Nminus1)+1)

dde=(tyz) mathieu dde(tyzkappadeltaepsbtau)

te dde=ticsol = dde23(ddetauhistorytspan) toc(te dde) xdde=deval(soltdde)

Solution by zeroth EMHPM

dde emhpm fun=(tc0ttzmzntauN) mathieu zeroth(tc0ttzmzntauNm ord

kappadeltaepsbT)

tic[t0z0]=ddeEMHPM(dde emhpm funtspanhistorytaudtpntDelay) toc

Solution by First EMHPM

dde emhpm fun=(tc0ttzmzntauN) mathieu first(tc0ttzmzntauNm ordkappadeltaepsbT)

tic[t1z1]=ddeEMHPM(dde emhpm funtspanhistorytaudtpntDelay) toc

Plot results

ind0=12ktaulowast(Nminus1) ind1=22ktaulowast(Nminus1)

Parent1=figure (1)

axes1 = axes(lsquoParentrsquoParent1lsquoFontSizersquo12lsquoFontNamersquolsquoTimes New Romanrsquo)box(axes1lsquoonrsquo) hold(axes1lsquoallrsquo)plot(tddexdde(1)lsquoParentrsquoaxes1lsquoLineWidthrsquo2lsquoColorrsquo[0502 0502 0502]lsquoDisplayNamersquolsquoNumericaldd23rsquo)plot(t0(ind0)z0(ind01)lsquoMarkerSizersquo5lsquoMarkerrsquolsquoorsquolsquoLineStylersquo lsquononersquolsquoDisplayNamersquolsquoZerothEMHPMrsquolsquoColorrsquo[0 0 0])

plot(t1(ind1)z1(ind11)lsquoMarkerSizersquo7lsquoMarkerrsquolsquoxrsquolsquoLineStylersquolsquononersquolsquoDisplayNamersquolsquoFirstEMHPMrsquolsquoColorrsquo[0 0 0])

xlabel(lsquoittrsquolsquoFontSizersquo12lsquoFontNamersquolsquoTimes New Romanrsquo)ylabel(lsquoitxrsquolsquoFontSizersquo12lsquoFontNamersquolsquoTimes New Romanrsquo)end

Mathieu definitions

function dydt = mathieu dde(tyzkapadltepsbT)

dydt = [y(2)-kapalowasty(2)minus(dlt+epslowastcos(2lowastpilowasttT))lowasty(1)+blowastz(1)]

end

function Z = mathieu zeroth(tc0ttzmzntauNmkpadltepsbT)

Z=[c0(1)c0(2)] alf=dlt+epslowastcos(2lowastpiTlowasttt)

for ik=1m

Z(ik+11)=Z(ik2)lowasttik

Z(ik+12)=minuskpalowastZ(ik2)minusalflowastZ(ik1)

if ik==1 Z(ik+12)=Z(ik+12)+blowastzm(1) end

Z(ik+12)=Z(ik+12)lowasttik

end

Z=sum(Z)

end

function Z = mathieu first(tc0ttzmzntauNmkpadltepsbT)

alf=dlt+epslowastcos(2lowastpiTlowasttt) Z=[c0(1)c0(2)] Z =[00]

for ik=1m

Z(ik+1)=[Z(ik2)lowasttik minuskpalowastZ(ik2)minusalflowastZ(ik1)]

Z (ik+1)=[Z (ik2)lowasttik minuskpalowastZ (ik2)minusalflowastZ (ik1)]

if ik==1

Z(ik+12)=Z(ik+12)+blowastzm(1)Z (ik+12)=Z (ik+12)+blowast(Nminus1)taulowast(zn(1)minuszm(1))lowastt

end

Z(ik+12)=Z(ik+12)lowasttik

Z (ik+12)=Z (ik+12)lowastt(ik+1)

Algorithm 1 Continued

Abstract and Applied Analysis 9

end

Z=sum(Z)+sum(Z )

end

function out=history(t)

out=[1Eminus3+0lowastt 0+0lowastt]

end

EMHPM algorithm for ODE solutions

function [tz]= odeEMHPM(ndetspanz0Deltatpnts)

ode solver by Enhanced Multistage Homotopy Perturbation Method

tini=tspan(1) tfin=tspan(end) tstart=tini

tini=tiniminuststart tfin=tfinminuststart shifted time set to zero

Handle errors

if tini == tfin

error(lsquoThe ending and starting time values must be differentrsquo)elseif abs(tini)gt abs(tfin)

tspan=flipud(fliplr(tspan)) tini=tspan(1) tfin=tspan(end)

end

tdir=sign(tfinminustini)

if any(tdirlowastdiff(tspan) lt= 0)

error(lsquotspan entries must be strictly sortedrsquo)end

incT=Deltatpnts

if incTlt=0

error (lsquoIncreasing must be greater than zerorsquo)end

z(1)=z01015840 t(1)=tini iteT=2 set initial values

t(iteT)=tini+incTlowasttdir

while tdirlowastt(iteT)lttdirlowast(tfin+tdirlowastincT)

P act=ceil(99999lowast(t(iteT)minustini)lowasttdirDeltat) count sub-intervals

c=z((DeltatincT)lowast(P actminus1)+1) set the corresponding initial condition

tsub(iteTminus1)=t(iteT)-(P actminus1)lowastDeltatlowasttdir evaluate the solution at the shifted time

temp=tsub(iteTminus1)

z(iteT)=nde(tempctstart+t(iteT))

if tdirlowastt(iteT)gt=tdirlowasttfinlowast099999 repeat for the next sub-interval

break

else

iteT=iteT+1 t(iteT)=tini+(iteTminus1)lowastincTlowasttdir

end

end

t=t 1015840 +tstart

end

EMHPM algorithm for DDE solutions

function [t z]=ddeEMHPM(ddetspanhistorytauDeltatpntDelta)

dde solver by Enhanced Multistage Homotopy Perturbation Method

tini=tspan(1) tfin=tspan(end) span where the solution is founded

if taultDeltat

error(lsquoSubtinterval must not be greater than taursquo)end

if mod(tauDeltat)sim=0

pastDlt=Deltat

Deltat=tauround(tauDeltat)

warning(lsquoSubtinterval was modified from 05g to 05grsquopastDltDeltat)end

pnt=round(tauDeltat) samplesminus1 [minustau 0]

incT=DeltatpntDelta step for set resolution

t=minustau+tini+(0pntlowastpntDelta)rsquolowastincT evaluation of the initial solution [minustau 0]

z=history(t) initial behavior in the inverval [minustau 0]

c=z(end) initial condition

Algorithm 1 Continued

10 Abstract and Applied Analysis

m=pntlowastpntDelta samples between tau

itDlt=0

while (t(end)+eps)lttfin

tspan=[tini+itDltlowastDeltattini+(itDlt+1)lowastDeltat] preparing the span for next Deltat

zm=z(length(z)minusm)rsquo zn=z(1+length(z)minusm)rsquo previous tau solution

emhpm fun=(timecT) dde(timecTzmzntaum) application of the odeEMHPM

[t aux z aux]=odeEMHPM(emhpm funtspanz(end)DeltatpntDelta)

z=[z(1endminus1)z aux] t=[t(1endminus1)t aux] joining the solutions

itDlt=itDlt+1

end

z=z(pntlowastpntDelta+1end)t=t(pntlowastpntDelta+1end)

end

Algorithm 1 MATLAB algorithm

Here 120601st and 120601ex are the angles where the teeth enter and exitthe workpiece For upmilling 120601st = 0 and 120601ex = arccos(1 minus2119886119889) for downmilling 120601st = arccos(2119886

119889minus 1) and 120601ex = 120587

where 119886119889is the radial depth of cut ratio

By following the EMHPM procedure we can write (46)in equivalent form as

119894(119879) + 2120577120596

119899119894(119879) + 120596

2

119899119909119894(119879)

asymp minus

119886119901119870st

119898119898

(119909119894 (119879) minus 119909119894minus119873+1 (119879))

(50)

where 119909119894(119879) denotes the 119898 order solution of (46) on the 119894th

subinterval that satisfies the initial conditions 119909119894(0) = 119909

119894minus1

119894(0) =

119894minus1 and ℎ

119905= ℎ(119905) and 119909

minus120591is given by (35)

Introducing the transformation x119894= [119909119894 119894]119879 (50) can be

written as a system of first order linear delay differentialequations of the form

x119894(119879) = A

119905x119894(119879) + B

119905x119894minus119873+1

(119879) (51)

where

A119905= [

[

0 1

minus1205962

119899minus119908

119898119898

119870st minus2120577120596119899]

]

B119905= [

[

0 0119908

119898119898

119870st 0]

]

(52)

We next apply the EMHPM procedure to solve (46) byconsidering a downmilling operation with the followingparameter values 119911

119899= 2 119886

119889= 01 120596

119899= 5793 rads

120577 = 0011 119898119898= 003993 kg 119870

119905= 6 times 10

8Nm2 and119870119899= 2 times 10

8Nm2 As we can see from Figures 7 and 8 andfor the depth of cut values of 119886

119901= 2mm (stable) and 119886

119901=

3mm (unstable) our EMHPM approximate solutions followclosely the numerical integration solutions of (46) obtainedby using the dde23 algorithm

Figure 9 shows the relative error between the EMPHMand the dde23 numerical solution while Table 2 shows theCPU time needed for each solution Here we use 119873 = 75since the average step size of the dde23 algorithm is aroundΔ119905 asymp 120591119873 Note that for 119898 = 7 the zeroth order EMHPMapproximate solution has the fastest CPU time We can seefrom Figure 9 that the value of the relative error becomesbasically the same for119898 = 2 7 and 10 and119873 ge 20

0 0005 001 0015 002

0

1

2

Time (s)

Disp

lace

men

t (m

)

dd23Zeroth EMHPMFirst EMHPM

minus2

minus1

times10minus3

Figure 7 EMHPM approximate solutions of (46) with parametervalues of 119886

119901= 2mm 119899 = 10000 rpm Stable machine operation

0 0005 001 0015 002

00005

0010015

Time (s)

Disp

lace

men

t (m

)

dd23Zeroth EMHPMFirst EMHPM

minus001

minus0005

Figure 8 EMHPM approximate solutions of (46) with parametervalues of 119886

119901= 3mmand 119899 = 10000 rpmUnstable (chatter)machine

operation

6 Conclusions

We have developed a new algorithm based on the homotopyperturbation method to solve delay differential equationsThe proposed EMHPM approach is based on a sequence ofsubintervals that approximate the solution of delayed differ-ential equations by using the transformation rule 119906(119905) rarr119906119894(119879) where 119906

119894satisfies the initial conditions We have

Abstract and Applied Analysis 11

10 20 30 40 50 60 70 80 90 100

Abso

lute

erro

r

Zeroth order m = 2

First order m = 2

Zeroth order m = 7

First order m = 7

Zeroth order m = 10

First order m = 10

10minus3

10minus2

10minus1

Discretizations N

Figure 9 Estimated relative error values between dde23 and theEMHPM approximate solutions Here we have used the systemparameter values of 119886

119901= 2mm and 119899 = 1000 rpm and values of

119898 = 2 7 and 10

Table 2 CPU time comparison among the approximate solutions of(46) by using dde23 the zeroth order and the first order EMHPMsolutions

Solution method Time [ms]dde23 139Zeroth order EMHPM (119898 = 7) 77First order EMHPM (119898 = 7) 87

shown that our proposed EMHPM approach can be appliedto obtain the approximate solution of a delay differentialequation not only with constant but also with variablecoefficients with theoretical predictions that follow well thenumerical integration solutions To further assess the validityof this new approach we have compared the approximatesolutions of two delayed differential equations with respect totheir corresponding numerical integration solutions obtainedfrom the MATLAB dde23 algorithm The test cases were (a)the damped Mathieu differential equation with time delayand (b) the governing equation of motion of downmillingoperations We have found that the EMHPM closely followsthe numerical integration solutions of the correspondingequations and that these require less CPU time and havesmaller relative errors

Conflict of Interests

Theauthors declare that they have no conflict of interests withany mentioned entities in the paper

Acknowledgments

This work was funded by Tecnologico deMonterrey CampusMonterrey through the ResearchGroup inNanomaterials forMedical Devices and the Research Group in Advanced Man-ufacturing Additional support was provided from ConsejoNacional de Ciencia y Tecnologıa (Conacyt) Mexico

References

[1] F Shakeri and M Dehghan ldquoSolution of delay differentialequations via a homotopy perturbation methodrdquoMathematicaland Computer Modelling vol 48 no 3-4 pp 486ndash498 2008

[2] Z-K Wu ldquoSolution of the enso delayed oscillator with homo-topy analysis methodrdquo Journal of Hydrodynamics vol 21 no 1pp 131ndash135 2009

[3] A K Alomari M S Noorani and R Nazar ldquoSolution of delaydifferential equation by means of homotopy analysis methodrdquoActa Applicandae Mathematicae vol 108 no 2 pp 395ndash4122009

[4] J Biazar and G Behzad ldquoThe homotopy perturbation methodfor solving neutral functional-differential equations with pro-portional delaysrdquo Journal of King Saud UniversitymdashScience vol24 no 1 pp 33ndash37 2012

[5] NRAnakiraAKAlomari and IHashim ldquoOptimal homotopyasymptotic method for solving delay differential equationsrdquoMathematical Problems in Engineering vol 2013 Article ID498902 11 pages 2013

[6] T Insperger and G Stepan Semi-Discretization for Time-DelaySystems Stability and Engineering Applications Springer NewYork NY USA 2011

[7] E A Butcher P Nindujarla and E Bueler ldquoStability of up-and down-milling using chebyshev collocation methodrdquo inProceedings of the ASME International Design Engineering Tech-nical Conferences amp Computers and Information in EngineeringConference vol 6 part AndashC pp 841ndash850 2005

[8] T Insperger ldquoFull-discretization and semi-discretization formilling stability prediction some commentsrdquo InternationalJournal of Machine Tools and Manufacture vol 50 no 7 pp658ndash662 2010

[9] I Hashim and M S Chowdhury ldquoAdaptation of homotopy-perturbation method for numeric-analytic solution of systemof ODEsrdquo Physics Letters A vol 372 no 4 pp 470ndash481 2008

[10] L F Shampine and S Thompson ldquoSolving DDEs in MatlabrdquoAppliedNumericalMathematics vol 37 no 4 pp 441ndash458 2001

[11] SHHNiaANRanjbarDDGanjiH Soltani and JGhasemildquoMaintaining the stability of nonlinear differential equations bythe enhancement of HPMrdquo Physics Letters A vol 372 no 16 pp2855ndash2861 2008

[12] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[13] J-H He ldquoHomotopy perturbation method a new nonlinearanalytical techniquerdquo Applied Mathematics and Computationvol 135 no 1 pp 73ndash79 2003

[14] J H He ldquoComparison of homotopy perturbation method andhomotopy analysis methodrdquo Applied Mathematics and Compu-tation vol 156 no 2 pp 527ndash539 2004

[15] JHHe ldquoNew interpretation of homotopy perturbationmethodrdquoInternational Journal of Modern Physics B vol 20 no 18 pp2561ndash2568 2006

[16] S Liao ldquoComparison between the homotopy analysis methodand homotopy perturbationmethodrdquoAppliedMathematics andComputation vol 169 no 2 pp 1186ndash1194 2005

[17] D Olvera and A Elıas-Zuniga ldquoEnhanced multistage homo-topy perturbation method approximate solutions of nonlineardynamic systemsrdquo Abstract and Applied Analysis vol 2014Article ID 486509 12 pages 2014

12 Abstract and Applied Analysis

[18] T Insperger and G Stepan ldquoStability of the damped Mathieuequationwith timedelayrdquo Journal ofDynamic SystemsMeasure-ment and Control vol 125 no 2 pp 166ndash171 2003

[19] T Insperger and G Stepan ldquoStability chart for the delayedMathieu equationrdquo The Royal Society of London ProceedingsSeries A Mathematical Physical and Engineering Sciences vol458 no 2024 pp 1989ndash1998 2002

[20] T Insperger andGStepan ldquoUpdated semi-discretizationmethodfor periodic delay-differential equations with discrete delayrdquoInternational Journal for Numerical Methods in Engineering vol61 no 1 pp 117ndash141 2004

[21] T Insperger B P Mann G Stepan and P V Bayly ldquoStabilityof up-milling and down-milling part 1 alternative analyticalmethodsrdquo International Journal of Machine Tools and Manufac-ture vol 43 no 1 pp 25ndash34 2003

[22] F ICompeanDOlvera F J Campa LN L de Lacalle A Elıas-Zuniga and C A Rodrıguez ldquoCharacterization and stabilityanalysis of a multivariable milling tool by the enhanced mul-tistage homotopy perturbation methodrdquo International Journalof Machine Tools and Manufacture vol 57 pp 27ndash33 2012

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 Abstract and Applied Analysis

0 1 2 3 4 5 6 7 8 9 10

00

0505

11

Time t

HPM by intervals

minus05minus05

Am

plitu

dex

Figure 1 Exact solution of (11) obtained by using theHPMand 120591 = 1

42 Delay Differential Equations with Variable CoefficientsWe next show how the EMHPM approach can be applied toobtain the approximate solution of nonlinear delay differen-tial equation with variable coefficients In this case we obtainthe approximate solutions of a DDE of the form

+ 119909 (119905 minus 120591) minus cos (120587119905) 1199092 = 0 120591 = 1 119909 (0) = 119888 = 1199091205910 (120591)(17)

in which the solution 1199091205910(119879) = 1198881holds on (minus120591 0] In order to

find the solution 1199091205911 in the interval [0 120591] we assume that thehomotopy representation of (17) can be given as

119867(1198831205911 119901) =

119889

1198891198791198831205911 + 119901 [119909

1205910 minus cos (120587119905) (1198831205911)2] = 0(18)

Notice that the variable 119883 depends on the time 119879 for which0 le 119879 le 120591 If we now substitute the second order expansion1198831205911 = 119883

1205911

0+ 1199011198831205911

1+ 11990121198831205911

2in (18) and after balancing the 119901

terms we get that

1199010 1198891198891198791198831205911

0= 0 119883

0 (119879 = 0) = 1198881 = 1198831205910(119879 = 120591)

1199011 1198891198891198791198831205911

1= minus1199091205910 + cos (120587119905) (1198831205911

0)2

= 0 1198831(0) = 0

1199012 1198891198891198791198831205911

2= 2 cos (120587119905)1198831205911

01198831205911

1 1198832(0) = 0

1199013 1198891198891198791198831205911

3= cos (120587119905) (21198831205911

01198831205911

2+ (1198831205911

1)2

) = 0 1198833 (0) = 0

(19)

Equations (19) have the following solutions

1198831205911

0= 1198881

1198831205911

1= minus119879 (119909

1205910 minus 1198882

1cos120587119905)

1198831205911

2= minus11988811198792(cos120587119905) (1199091205910 minus 1198882

1cos120587119905)

1198831205911

3=1

31198793(cos120587119905) [31198884

1cos2120587119905 minus 41198882

11199091205910 + (119909

1205910)2]

(20)

Thus the approximate solution of (17) by using the EMHPMis given by

1199091205911(119879) asymp 119883

1205911

0+ 1198831205911

1+ 1198831205911

2+ 1198831205911

3 (21)

In this case the exact solution of 1199091205911(119879) is unknown Toobtain 1199091205912 we compute again the approximate solution of1199091205911(119879) by applying our EMHPM and the value of the delayed

time is assumed to remain constant in each subinterval Todetermine 1199091205912 we next use the homotopy representation of(17) for the interval (120591 2120591]

119867(1198831205912 119901) =

119889

1198891198791198831205912 + 119901 [119909

1205911 minus cos (120587119905) (1198831205912)2] = 0(22)

Substituting the second order expansion in (22) we get

1198831205912

0= 1198882

1198831205912

1= minus119879 (119909

1205911 minus 1198882

2cos120587119905)

1198831205912

2= minus11988821198792(cos120587119905) (1199091205911 minus 1198882

2cos120587119905)

1198831205912

3=1

31198793(cos120587119905) (31198884

2cos2120587119905 minus 41198882

21199091205911 + (119909

1205911)2)

(23)

Note that (20) and (23) provide approximate solutions to(17) but evaluated at different interval time delays To findthe third order approximate solution of (17) we can use ahomotopy of the form

119867(119883120591119894 119901) =

119889

119889119879119883120591119894 + 119901 [119883

120591119894minus1 minus cos (120587119905) (119883120591119894)2] = 0(24)

Then by using our EMPHM approach we have that

119883120591119894

0= 119888

119883120591119894

1= minus119879 (119909

120591119894minus1 minus 1198882 cos120587119905)

119883120591119894

2= minus119888119879

2(cos120587119905) (119909120591119894minus1 minus 1198882 cos120587119905)

119883120591119894

3=1

31198793(cos120587119905) (31198884cos2120587119905 minus 41198882119909120591119894minus1 + (119909120591119894minus1)2)

(25)

Notice from (25) that the 119896th order approximate solution of(17) can be written as

119883120591119894

0= 119888

119883120591119894

119896=119879

119896(minus119909120591119894minus1119892 (119896) + cos120587119905

119896minus1

sum

1198991=0

119883120591119894

1198991119883120591119894

119896minus1minus1198991)

(26)

where 119896 gt 0 119892(119896) = 1 when 119896 = 1 and zero otherwiseFigure 2 shows the approximate solution of (17) obtained

by using the EMHPM approach compared to its numericalintegration solution by using the dde23MATLAB subroutineprogram This case assumes two different initial solutions ofthe form 1199091205910(119879) = cos(120587(119879 + 1)) 1199091205910(119879) = 119890119879+1 and a timesubintervals Δ119905 = 001 We can see from Figure 2 that bothsimulations agree well for the time span showed

To further assess the applicability of our proposedEMHPM approach to high order delay differential equationswe will next describe a methodology to obtain the approx-imate solutions of well-known high order delay differentialequations by generalizing our EMHPM approach

Abstract and Applied Analysis 5

0 5 10 15

0

1

2

EMHPMdde23

minus1minus1

Am

plitu

dex

Time t

x1205910(T) = cos(120587(T + 1))

x1205910(T) = eT+1

Figure 2 EMHPM and dde23 solution of (17)

x x iminusNx iminusN+1

timinusN+1t titiminus1timinusN

Δt

x i(t)x i+1(t)

120591 = (N minus 1)Δt

Figure 3 Schematic of the zeroth order polynomial used to fit theapproximate EMHPM solution

5 Generalized Solution of Linear DDEsby the EMHPM Approach

Let us consider an 119899-dimensional delay differential equationof the form

x (119905) = A (119905) x + B (119905) x (119905 minus 120591) (27)

whereA(119905 + 120591) = A(119905) B(119905 + 120591) = B(119905) x(119905) is the state vectorand 120591 is the time delay By following our EMHPM procedurewe can write (27) in equivalent form as

x119894 (119879) minus A119905x119894 (119879) asymp B119905x

120591

119894(119879) (28)

where x119894(119879) denotes the 119898 order solution of (27) in the 119894th

subinterval that satisfies the initial conditions x119894(0) = x

119894minus1and

A119905and B

119905represent the values of the periodic coefficients at

the time 119905 In order to approximate the delayed term x120591119894(119879)

in (28) the period [1199050minus 120591 1199050] is discretized in 119873 points

equally spaced as shown in Figure 3 Here we assume thatthe function x120591

119894(119879) in the delay subinterval [119905

119894minus119873 119905119894minus119873+1

] isapproximated by a constant value

x120591119894(119879) = 119909

119894minus119873+1(119879) asymp x

119894minus119873 (29)

as shown in Figure 3 By following the homotopy perturba-tion technique we can write the homotopy representation of(28) as

119867(X119894 119901) = 119871 (X

119894) minus 119871 (x

1198940) + 119901119871 (x

1198940) = 119901 (A

119905X119894+ B119905x119894minus119873)

(30)

Substituting the 119898 order expansion X119894= X1198940+ 119901X1198941+ sdot sdot sdot +

119901119898X119894119898

in (30) and by assuming an initial approximation ofthe form x

1198940= x119894minus1

we get after applying the proposed

EMHPMapproach the following set of first order linear delaydifferential equations

1199010 119889119889119879

X1198940+119889

119889119879x119894minus1= 0 X

119894(0) = x

119894minus1

1199011 119889119889119879

X1198941= A119905X1198940+ B119905x119894minus119873 X1198941(0) = 0

1199012 119889119889119879

X1198942= A119905X1198941 X1198942(0) = 0

119901119898 119889119889119879

X119894119898= A119905X119894(119898minus1)

X119894119898(0) = 0

(31)

By solving (31) we get

X1198940= x119894minus1

X1198941= A119905x119894minus1119879 + B

119905x119894minus119873119879

X1198942=1

2A2119905x119894minus11198792+1

2A119905B119905x119894minus1198731198792

X119894119898=1

119898A119898119905x119894minus1119879119898+1

119898A119898minus1119905

B119905x119894minus119873119879119898

(32)

Equations (32) can be written as

X119894119896=119879

119896(A119905X119894(119896minus1)

+ 119892 (119896)B119905x119894minus119873) 119896 = 1 2 3 (33)

where X1198940= x119894minus1

and 119892(119896) = 1 for 119896 = 1 and 119892(119896) = 0otherwise Thus the solution of (27) is obtained by addingthe X119894119896approximate solutions

x119894 (119879) asymp

119898

sum

119896=0

X119894119896 (119879) (34)

Notice however that solution (34) may be further improvedby using a first order polynomial representation of x120591

119894(119879) as

shown in Figure 4 Then the function x120591119894(119879) in the delay

subinterval [119905119894minus119873 119905119894minus119873+1

] takes the form

x120591119894(119879) = x119894minus119873+1 (119879) asymp x119894minus119873 +

(119873 minus 1)

120591(x119894minus119873+1

minus x119894minus119873) 119879

(35)

Substituting (35) into (28) gives

x119894(119879) minus A

119905x119894(119879)

asymp B119905x119894minus119873minus(119873 minus 1)

120591B119905x119894minus119873119879 +(119873 minus 1)

120591B119905x119894minus119873+1

119879

(36)

6 Abstract and Applied Analysis

We next assume that the homotopy representation of (36) isgiven as

119867(X119894 119901) = 119871 (X

119894) minus 119871 (x

1198940) + 119901119871 (x

1198940)

minus 119901 (AX119894+ Bx119894minus119873minus(119873 minus 1)

120591Bx119894minus119873119879

+(119873 minus 1)

120591Bx119894minus119873+1

119879) = 0

(37)

Substituting the 119898 order expansion X119894(119879) = X

1198940(119879) +

119901X1198941(119879) + sdot sdot sdot 119901

119898X119894119898(119879) in (37) and assuming that the initial

approximation is given by x1198940= x119894minus1

we get

1199010 119889119889119879

X1198940+119889

119889119879x119894minus1= 0 X

119894 (0) = x119894minus1

1199011 119889119889119879

X1198941= A119905X1198940+ B119905x119894minus119873minus119873 minus 1

120591B119905x119894minus119873119879

+119873 minus 1

120591B119905x119894minus119873+1

119879 X1198941(0) = 0

1199012 119889119889119879

X1198942= AX

1198941 X1198942(0) = 0

119901119898 119889119889119879

X119894119898= AX

119894(119898minus1) X119894119898(0) = 0

(38)

By solving (38) and by following the EMHPM procedure weget

X1198940= x119894minus1

X1198941= A119905x119894minus1119879 + B

119905x119894minus119873119879 minus1

2

119873 minus 1

120591B119905x119894minus1198731198792

+1

2

119873 minus 1

120591B119905x119894minus119873+1

1198792

X1198942=1

2A2119905x119894minus11198792+1

2A119905B119905x119894minus1198731198792

minus1

6

119873 minus 1

120591A119905B119905x119894minus1198731198793+1

6

119873 minus 1

120591A119905B119905x119894minus119873+1

1198793

X119894119898=1

119898A119898119905x119894minus1119879119898+1

119898A119898minus1119905

B119905x119894minus119873119879119898

minus1

(119898 + 1)

119873 minus 1

120591A119898minus1119905

B119905x119894minus119873119879119898+1

+1

(119898 + 1)

119873 minus 1

120591A119898minus1119905

B119905x119894minus119873+1

119879119898+1

(39)

Here the recursive form of X119894119896(119879) is written as

X119894119896= Xa119894119896+ Xb119894119896119896 = 1 2 3 (40)

timinusN+1 ttitiminus1timinusN

Δt

x i(t)x i+1(t)

x x iminusNx iminusN+1

120591 = (N minus 1)Δt

Figure 4 Schematic EMHPM solution using first polynomial toapproximate delay subinterval

where Xa1198940= x119894minus1Xb1198940= 0 and

Xa119894119896=119879

119896(A119905Xa119894(119896minus1)

+ 119892 (119896)B119905x119894minus119873)

Xb119894119896=119879

119896 + 1(A119905Xb119894(119896minus1)

+119892 (119896) [119873 minus 1

120591119879 (minusB

119905x119894minus119873+ B119905x119894minus119873+1

)])

(41)

Thus the approximate solution of (27) by the EMHPM canbe obtained by substituting (40) into (34)

In the next section we will apply our EMHPM procedureto obtain the solution of two second order delay differentialequations (a) the dampedMathieu equation with time delayand (b) the well-known delay differential equation thatdescribes the dynamics in one degree-of-freedom millingmachine operations

51 Solution of theDampedMathieu EquationwithTimeDelayIn order to assess the accuracy of our EMHPM approach wefirst obtain the solution of the damped Mathieu differentialequation with time delay that combines the effect of paramet-ric excitation and dampingThis equation is described by thefollowing equation

+ 120581 + (120575 + 120576 cos(2120587119905119879))119909 = 119887119909 (119905 minus 120591) (42)

where 120581 120575 120576 120591 and 119879 are system parameters whose valuedepends on the physics of the system The approximatesolution of (42) obtained by using the semidiscretizationmethod is widely discussed in [18 19] Here we focus ourattention on applying the EMHPM to find the approximatesolution of (42) and we also assess the accuracy of the derivedsolution by comparing it with the corresponding numericalintegration solution of (42)

By following the EMHPM procedure we first write (42)in the following equivalent form

119894(119879) + 120581

119894(119879) + 120572

119905119909119894(119879) asymp 119887119909

119894minus119873+1(119879) (43)

where 119909119894(119905) denotes the 119898 order solution of (43) in the

119894th subinterval that satisfies the following initial conditions119909119894(0) = 119909

119894minus1and

119894(0) =

119894minus1 The space state form

representation of (43) is given by

x119894(119879) = A

119905x119894(119879) + B

119905x119894minus119873+1

(119879) (44)

Abstract and Applied Analysis 7

0 2 4 6 8 10 12

005

1

dde23Zeroth EMHPMFirst EMHPM

minus05minus05

minus15minus15

minus1minus1Am

plitu

dex

Time t

times10minus3

Figure 5 Numerical solutions of the damped Mathieu equationwith time delay by dde23 the zeroth EMHPM and the first EMHPMwith119873 = 50 and119898 = 4

Table 1 Computer time needed to solve the damped Mathieuequation with time delay The 119898 order solution of the EMPHMapproach is chosen to guarantee the convergence of its approximatesolution

dde23 [ms] EMHPM119873 119898 Zeroth [ms] First [ms]

19

15 5 4 520 4 6 640 3 12 1260 2 17 1860 5 18 1960 10 20 21

where

A119905= [0 1

minus120572119905minus120581] B

119905= [0 0

1198871199050] (45)

and 120572119905= (120575 + 120576 cos(119905)) is a time periodic term The EMHPM

approximate solution of (44) is illustrated in Figure 5 wherewe have assumed an unstable system behavior for which 120581 =02 120575 = 30 120576 = 1 119887 = minus1 and 119879 = 120591 = 2120587 See[20] As we can see from Figure 5 our approximate EMHPMsolution to (42) is compared with its numerical integrationsolution obtained from dde23 MATLAB algorithm for thetime interval of 2119879 by assuming that 119873 = 50 with thefollowing initial values 119909

minus50(119879) = 119909

minus49(119879) = sdot sdot sdot 119909

0(119879) =

0001 and minus50(119879) =

minus49(119879) = sdot sdot sdot

0(119879) = 0

It can be seen fromFigure 5 that in the interval [0119879] boththe zeroth and the first order solutions are the same since thedelay subintervals are constant See Figure 3 However in thenext interval [119879 2119879] it is clear that the first order EMPHMsolution provides a better approximation on the delay subin-terval The computation total time to calculate the solutionsin the MATLAB code is listed in Table 1 The order 119898 andthe discretized time intervals 119873 in the EMPHM approachare chosen to guarantee the convergence of our approximatesolution to the exact one To provide a full understanding ofhow the solution is computed by the EMHPM approach weattached in Algorithm 1 the corresponding MATLAB code

10 20 30 40 50 60 70 80 90 100

Abso

lute

erro

r

10minus4

10minus3

10minus2

Zeroth order m = 2

First order m = 2

Zeroth order m = 5

First order m = 5

Zeroth order m = 10

First order m = 10

Discretizations N

Figure 6 Estimated relative error values between the numericalsolution dde23 and the EMHPMapproximate solutionsHerewe usefor the EMHPM the values of119898 = 2 5 and 10

Figure 6 shows the relative error between our approxi-mate EMPHM and the dde23 solution and its relationshipwith the order119898 and the discretized time intervals119873 Noticethat the relative error values coincide at values of 119873 ge 45Also we can see from Figure 6 that the computed relativeerror values for approximate solutions of order119898 ge 5 remainunchanged

52 A Practical Application Cutting Operation on MillingMachine We next use our EMHPM procedure to obtainthe solution of the single degree-of-freedom milling opera-tion We use the simplified form based on [20ndash22]

(119905) + 2120577120596119899 (119905) + 1205962

119899119909 (119905) = minus

119886119901119870119904 (119905)

119898119898

(119909 (119905) minus 119909 (119905 minus 120591))

(46)

where 120596119899is the angular natural frequency of the system 120577 is

the damping ratio 119886119901is the depth of cut 119898

119898is the modal

mass of the tool 120591 represents the time delay which is equalto the tooth passing period and 119870

119904(119905) is the specific cutting

force coefficient which can be determined from119870119904 (119905)

=

119911119899

sum

119895=1

119892 (120601119895(119905)) sin (120601

119895(119905)) (119870

119905cos120601119895(119905) + 119870

119899sin120601119895(119905))

(47)

where 119911119899is the tool number of teeth 119870

119905and 119870

119899are the

tangential and the normal linear cutting force coefficientsrespectively 120601

119895(119905) is the angular position of the 119895-tooth

defined as

120601119895(119905) = (

2120587119899

60) 119905 +

2120587119895

119911119899

(48)

and 119899 is the spindle speed in rpm [20] The function 119892(120593119895(119905))

is a switching function which has a unity value when the 119895-tooth is cutting and zero otherwise

119892 (120601119895 (119905)) =

1 120601st lt 120601119895 (119905) lt 120601ex0 otherwise

(49)

8 Abstract and Applied Analysis

function dde23 ddeEM mathieu paper

disp(lsquoMathieu equation solutionrsquo) Solution by EMHPM with zeroth and first order solution

eps=1 kappa=02 T=2lowastpi tau=T b=minus1 delta=3 Mathieu Parameters

pntDelay=1 N=50 m ord=5 dt=tau(Nminus1) ktau=2 tspan=[0ktaulowastT] EMHPM Parameters

Solution

Solution by dde23

tdde=linspace(0ktaulowasttauktaulowast(Nminus1)+1)

dde=(tyz) mathieu dde(tyzkappadeltaepsbtau)

te dde=ticsol = dde23(ddetauhistorytspan) toc(te dde) xdde=deval(soltdde)

Solution by zeroth EMHPM

dde emhpm fun=(tc0ttzmzntauN) mathieu zeroth(tc0ttzmzntauNm ord

kappadeltaepsbT)

tic[t0z0]=ddeEMHPM(dde emhpm funtspanhistorytaudtpntDelay) toc

Solution by First EMHPM

dde emhpm fun=(tc0ttzmzntauN) mathieu first(tc0ttzmzntauNm ordkappadeltaepsbT)

tic[t1z1]=ddeEMHPM(dde emhpm funtspanhistorytaudtpntDelay) toc

Plot results

ind0=12ktaulowast(Nminus1) ind1=22ktaulowast(Nminus1)

Parent1=figure (1)

axes1 = axes(lsquoParentrsquoParent1lsquoFontSizersquo12lsquoFontNamersquolsquoTimes New Romanrsquo)box(axes1lsquoonrsquo) hold(axes1lsquoallrsquo)plot(tddexdde(1)lsquoParentrsquoaxes1lsquoLineWidthrsquo2lsquoColorrsquo[0502 0502 0502]lsquoDisplayNamersquolsquoNumericaldd23rsquo)plot(t0(ind0)z0(ind01)lsquoMarkerSizersquo5lsquoMarkerrsquolsquoorsquolsquoLineStylersquo lsquononersquolsquoDisplayNamersquolsquoZerothEMHPMrsquolsquoColorrsquo[0 0 0])

plot(t1(ind1)z1(ind11)lsquoMarkerSizersquo7lsquoMarkerrsquolsquoxrsquolsquoLineStylersquolsquononersquolsquoDisplayNamersquolsquoFirstEMHPMrsquolsquoColorrsquo[0 0 0])

xlabel(lsquoittrsquolsquoFontSizersquo12lsquoFontNamersquolsquoTimes New Romanrsquo)ylabel(lsquoitxrsquolsquoFontSizersquo12lsquoFontNamersquolsquoTimes New Romanrsquo)end

Mathieu definitions

function dydt = mathieu dde(tyzkapadltepsbT)

dydt = [y(2)-kapalowasty(2)minus(dlt+epslowastcos(2lowastpilowasttT))lowasty(1)+blowastz(1)]

end

function Z = mathieu zeroth(tc0ttzmzntauNmkpadltepsbT)

Z=[c0(1)c0(2)] alf=dlt+epslowastcos(2lowastpiTlowasttt)

for ik=1m

Z(ik+11)=Z(ik2)lowasttik

Z(ik+12)=minuskpalowastZ(ik2)minusalflowastZ(ik1)

if ik==1 Z(ik+12)=Z(ik+12)+blowastzm(1) end

Z(ik+12)=Z(ik+12)lowasttik

end

Z=sum(Z)

end

function Z = mathieu first(tc0ttzmzntauNmkpadltepsbT)

alf=dlt+epslowastcos(2lowastpiTlowasttt) Z=[c0(1)c0(2)] Z =[00]

for ik=1m

Z(ik+1)=[Z(ik2)lowasttik minuskpalowastZ(ik2)minusalflowastZ(ik1)]

Z (ik+1)=[Z (ik2)lowasttik minuskpalowastZ (ik2)minusalflowastZ (ik1)]

if ik==1

Z(ik+12)=Z(ik+12)+blowastzm(1)Z (ik+12)=Z (ik+12)+blowast(Nminus1)taulowast(zn(1)minuszm(1))lowastt

end

Z(ik+12)=Z(ik+12)lowasttik

Z (ik+12)=Z (ik+12)lowastt(ik+1)

Algorithm 1 Continued

Abstract and Applied Analysis 9

end

Z=sum(Z)+sum(Z )

end

function out=history(t)

out=[1Eminus3+0lowastt 0+0lowastt]

end

EMHPM algorithm for ODE solutions

function [tz]= odeEMHPM(ndetspanz0Deltatpnts)

ode solver by Enhanced Multistage Homotopy Perturbation Method

tini=tspan(1) tfin=tspan(end) tstart=tini

tini=tiniminuststart tfin=tfinminuststart shifted time set to zero

Handle errors

if tini == tfin

error(lsquoThe ending and starting time values must be differentrsquo)elseif abs(tini)gt abs(tfin)

tspan=flipud(fliplr(tspan)) tini=tspan(1) tfin=tspan(end)

end

tdir=sign(tfinminustini)

if any(tdirlowastdiff(tspan) lt= 0)

error(lsquotspan entries must be strictly sortedrsquo)end

incT=Deltatpnts

if incTlt=0

error (lsquoIncreasing must be greater than zerorsquo)end

z(1)=z01015840 t(1)=tini iteT=2 set initial values

t(iteT)=tini+incTlowasttdir

while tdirlowastt(iteT)lttdirlowast(tfin+tdirlowastincT)

P act=ceil(99999lowast(t(iteT)minustini)lowasttdirDeltat) count sub-intervals

c=z((DeltatincT)lowast(P actminus1)+1) set the corresponding initial condition

tsub(iteTminus1)=t(iteT)-(P actminus1)lowastDeltatlowasttdir evaluate the solution at the shifted time

temp=tsub(iteTminus1)

z(iteT)=nde(tempctstart+t(iteT))

if tdirlowastt(iteT)gt=tdirlowasttfinlowast099999 repeat for the next sub-interval

break

else

iteT=iteT+1 t(iteT)=tini+(iteTminus1)lowastincTlowasttdir

end

end

t=t 1015840 +tstart

end

EMHPM algorithm for DDE solutions

function [t z]=ddeEMHPM(ddetspanhistorytauDeltatpntDelta)

dde solver by Enhanced Multistage Homotopy Perturbation Method

tini=tspan(1) tfin=tspan(end) span where the solution is founded

if taultDeltat

error(lsquoSubtinterval must not be greater than taursquo)end

if mod(tauDeltat)sim=0

pastDlt=Deltat

Deltat=tauround(tauDeltat)

warning(lsquoSubtinterval was modified from 05g to 05grsquopastDltDeltat)end

pnt=round(tauDeltat) samplesminus1 [minustau 0]

incT=DeltatpntDelta step for set resolution

t=minustau+tini+(0pntlowastpntDelta)rsquolowastincT evaluation of the initial solution [minustau 0]

z=history(t) initial behavior in the inverval [minustau 0]

c=z(end) initial condition

Algorithm 1 Continued

10 Abstract and Applied Analysis

m=pntlowastpntDelta samples between tau

itDlt=0

while (t(end)+eps)lttfin

tspan=[tini+itDltlowastDeltattini+(itDlt+1)lowastDeltat] preparing the span for next Deltat

zm=z(length(z)minusm)rsquo zn=z(1+length(z)minusm)rsquo previous tau solution

emhpm fun=(timecT) dde(timecTzmzntaum) application of the odeEMHPM

[t aux z aux]=odeEMHPM(emhpm funtspanz(end)DeltatpntDelta)

z=[z(1endminus1)z aux] t=[t(1endminus1)t aux] joining the solutions

itDlt=itDlt+1

end

z=z(pntlowastpntDelta+1end)t=t(pntlowastpntDelta+1end)

end

Algorithm 1 MATLAB algorithm

Here 120601st and 120601ex are the angles where the teeth enter and exitthe workpiece For upmilling 120601st = 0 and 120601ex = arccos(1 minus2119886119889) for downmilling 120601st = arccos(2119886

119889minus 1) and 120601ex = 120587

where 119886119889is the radial depth of cut ratio

By following the EMHPM procedure we can write (46)in equivalent form as

119894(119879) + 2120577120596

119899119894(119879) + 120596

2

119899119909119894(119879)

asymp minus

119886119901119870st

119898119898

(119909119894 (119879) minus 119909119894minus119873+1 (119879))

(50)

where 119909119894(119879) denotes the 119898 order solution of (46) on the 119894th

subinterval that satisfies the initial conditions 119909119894(0) = 119909

119894minus1

119894(0) =

119894minus1 and ℎ

119905= ℎ(119905) and 119909

minus120591is given by (35)

Introducing the transformation x119894= [119909119894 119894]119879 (50) can be

written as a system of first order linear delay differentialequations of the form

x119894(119879) = A

119905x119894(119879) + B

119905x119894minus119873+1

(119879) (51)

where

A119905= [

[

0 1

minus1205962

119899minus119908

119898119898

119870st minus2120577120596119899]

]

B119905= [

[

0 0119908

119898119898

119870st 0]

]

(52)

We next apply the EMHPM procedure to solve (46) byconsidering a downmilling operation with the followingparameter values 119911

119899= 2 119886

119889= 01 120596

119899= 5793 rads

120577 = 0011 119898119898= 003993 kg 119870

119905= 6 times 10

8Nm2 and119870119899= 2 times 10

8Nm2 As we can see from Figures 7 and 8 andfor the depth of cut values of 119886

119901= 2mm (stable) and 119886

119901=

3mm (unstable) our EMHPM approximate solutions followclosely the numerical integration solutions of (46) obtainedby using the dde23 algorithm

Figure 9 shows the relative error between the EMPHMand the dde23 numerical solution while Table 2 shows theCPU time needed for each solution Here we use 119873 = 75since the average step size of the dde23 algorithm is aroundΔ119905 asymp 120591119873 Note that for 119898 = 7 the zeroth order EMHPMapproximate solution has the fastest CPU time We can seefrom Figure 9 that the value of the relative error becomesbasically the same for119898 = 2 7 and 10 and119873 ge 20

0 0005 001 0015 002

0

1

2

Time (s)

Disp

lace

men

t (m

)

dd23Zeroth EMHPMFirst EMHPM

minus2

minus1

times10minus3

Figure 7 EMHPM approximate solutions of (46) with parametervalues of 119886

119901= 2mm 119899 = 10000 rpm Stable machine operation

0 0005 001 0015 002

00005

0010015

Time (s)

Disp

lace

men

t (m

)

dd23Zeroth EMHPMFirst EMHPM

minus001

minus0005

Figure 8 EMHPM approximate solutions of (46) with parametervalues of 119886

119901= 3mmand 119899 = 10000 rpmUnstable (chatter)machine

operation

6 Conclusions

We have developed a new algorithm based on the homotopyperturbation method to solve delay differential equationsThe proposed EMHPM approach is based on a sequence ofsubintervals that approximate the solution of delayed differ-ential equations by using the transformation rule 119906(119905) rarr119906119894(119879) where 119906

119894satisfies the initial conditions We have

Abstract and Applied Analysis 11

10 20 30 40 50 60 70 80 90 100

Abso

lute

erro

r

Zeroth order m = 2

First order m = 2

Zeroth order m = 7

First order m = 7

Zeroth order m = 10

First order m = 10

10minus3

10minus2

10minus1

Discretizations N

Figure 9 Estimated relative error values between dde23 and theEMHPM approximate solutions Here we have used the systemparameter values of 119886

119901= 2mm and 119899 = 1000 rpm and values of

119898 = 2 7 and 10

Table 2 CPU time comparison among the approximate solutions of(46) by using dde23 the zeroth order and the first order EMHPMsolutions

Solution method Time [ms]dde23 139Zeroth order EMHPM (119898 = 7) 77First order EMHPM (119898 = 7) 87

shown that our proposed EMHPM approach can be appliedto obtain the approximate solution of a delay differentialequation not only with constant but also with variablecoefficients with theoretical predictions that follow well thenumerical integration solutions To further assess the validityof this new approach we have compared the approximatesolutions of two delayed differential equations with respect totheir corresponding numerical integration solutions obtainedfrom the MATLAB dde23 algorithm The test cases were (a)the damped Mathieu differential equation with time delayand (b) the governing equation of motion of downmillingoperations We have found that the EMHPM closely followsthe numerical integration solutions of the correspondingequations and that these require less CPU time and havesmaller relative errors

Conflict of Interests

Theauthors declare that they have no conflict of interests withany mentioned entities in the paper

Acknowledgments

This work was funded by Tecnologico deMonterrey CampusMonterrey through the ResearchGroup inNanomaterials forMedical Devices and the Research Group in Advanced Man-ufacturing Additional support was provided from ConsejoNacional de Ciencia y Tecnologıa (Conacyt) Mexico

References

[1] F Shakeri and M Dehghan ldquoSolution of delay differentialequations via a homotopy perturbation methodrdquoMathematicaland Computer Modelling vol 48 no 3-4 pp 486ndash498 2008

[2] Z-K Wu ldquoSolution of the enso delayed oscillator with homo-topy analysis methodrdquo Journal of Hydrodynamics vol 21 no 1pp 131ndash135 2009

[3] A K Alomari M S Noorani and R Nazar ldquoSolution of delaydifferential equation by means of homotopy analysis methodrdquoActa Applicandae Mathematicae vol 108 no 2 pp 395ndash4122009

[4] J Biazar and G Behzad ldquoThe homotopy perturbation methodfor solving neutral functional-differential equations with pro-portional delaysrdquo Journal of King Saud UniversitymdashScience vol24 no 1 pp 33ndash37 2012

[5] NRAnakiraAKAlomari and IHashim ldquoOptimal homotopyasymptotic method for solving delay differential equationsrdquoMathematical Problems in Engineering vol 2013 Article ID498902 11 pages 2013

[6] T Insperger and G Stepan Semi-Discretization for Time-DelaySystems Stability and Engineering Applications Springer NewYork NY USA 2011

[7] E A Butcher P Nindujarla and E Bueler ldquoStability of up-and down-milling using chebyshev collocation methodrdquo inProceedings of the ASME International Design Engineering Tech-nical Conferences amp Computers and Information in EngineeringConference vol 6 part AndashC pp 841ndash850 2005

[8] T Insperger ldquoFull-discretization and semi-discretization formilling stability prediction some commentsrdquo InternationalJournal of Machine Tools and Manufacture vol 50 no 7 pp658ndash662 2010

[9] I Hashim and M S Chowdhury ldquoAdaptation of homotopy-perturbation method for numeric-analytic solution of systemof ODEsrdquo Physics Letters A vol 372 no 4 pp 470ndash481 2008

[10] L F Shampine and S Thompson ldquoSolving DDEs in MatlabrdquoAppliedNumericalMathematics vol 37 no 4 pp 441ndash458 2001

[11] SHHNiaANRanjbarDDGanjiH Soltani and JGhasemildquoMaintaining the stability of nonlinear differential equations bythe enhancement of HPMrdquo Physics Letters A vol 372 no 16 pp2855ndash2861 2008

[12] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[13] J-H He ldquoHomotopy perturbation method a new nonlinearanalytical techniquerdquo Applied Mathematics and Computationvol 135 no 1 pp 73ndash79 2003

[14] J H He ldquoComparison of homotopy perturbation method andhomotopy analysis methodrdquo Applied Mathematics and Compu-tation vol 156 no 2 pp 527ndash539 2004

[15] JHHe ldquoNew interpretation of homotopy perturbationmethodrdquoInternational Journal of Modern Physics B vol 20 no 18 pp2561ndash2568 2006

[16] S Liao ldquoComparison between the homotopy analysis methodand homotopy perturbationmethodrdquoAppliedMathematics andComputation vol 169 no 2 pp 1186ndash1194 2005

[17] D Olvera and A Elıas-Zuniga ldquoEnhanced multistage homo-topy perturbation method approximate solutions of nonlineardynamic systemsrdquo Abstract and Applied Analysis vol 2014Article ID 486509 12 pages 2014

12 Abstract and Applied Analysis

[18] T Insperger and G Stepan ldquoStability of the damped Mathieuequationwith timedelayrdquo Journal ofDynamic SystemsMeasure-ment and Control vol 125 no 2 pp 166ndash171 2003

[19] T Insperger and G Stepan ldquoStability chart for the delayedMathieu equationrdquo The Royal Society of London ProceedingsSeries A Mathematical Physical and Engineering Sciences vol458 no 2024 pp 1989ndash1998 2002

[20] T Insperger andGStepan ldquoUpdated semi-discretizationmethodfor periodic delay-differential equations with discrete delayrdquoInternational Journal for Numerical Methods in Engineering vol61 no 1 pp 117ndash141 2004

[21] T Insperger B P Mann G Stepan and P V Bayly ldquoStabilityof up-milling and down-milling part 1 alternative analyticalmethodsrdquo International Journal of Machine Tools and Manufac-ture vol 43 no 1 pp 25ndash34 2003

[22] F ICompeanDOlvera F J Campa LN L de Lacalle A Elıas-Zuniga and C A Rodrıguez ldquoCharacterization and stabilityanalysis of a multivariable milling tool by the enhanced mul-tistage homotopy perturbation methodrdquo International Journalof Machine Tools and Manufacture vol 57 pp 27ndash33 2012

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

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Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

0 5 10 15

0

1

2

EMHPMdde23

minus1minus1

Am

plitu

dex

Time t

x1205910(T) = cos(120587(T + 1))

x1205910(T) = eT+1

Figure 2 EMHPM and dde23 solution of (17)

x x iminusNx iminusN+1

timinusN+1t titiminus1timinusN

Δt

x i(t)x i+1(t)

120591 = (N minus 1)Δt

Figure 3 Schematic of the zeroth order polynomial used to fit theapproximate EMHPM solution

5 Generalized Solution of Linear DDEsby the EMHPM Approach

Let us consider an 119899-dimensional delay differential equationof the form

x (119905) = A (119905) x + B (119905) x (119905 minus 120591) (27)

whereA(119905 + 120591) = A(119905) B(119905 + 120591) = B(119905) x(119905) is the state vectorand 120591 is the time delay By following our EMHPM procedurewe can write (27) in equivalent form as

x119894 (119879) minus A119905x119894 (119879) asymp B119905x

120591

119894(119879) (28)

where x119894(119879) denotes the 119898 order solution of (27) in the 119894th

subinterval that satisfies the initial conditions x119894(0) = x

119894minus1and

A119905and B

119905represent the values of the periodic coefficients at

the time 119905 In order to approximate the delayed term x120591119894(119879)

in (28) the period [1199050minus 120591 1199050] is discretized in 119873 points

equally spaced as shown in Figure 3 Here we assume thatthe function x120591

119894(119879) in the delay subinterval [119905

119894minus119873 119905119894minus119873+1

] isapproximated by a constant value

x120591119894(119879) = 119909

119894minus119873+1(119879) asymp x

119894minus119873 (29)

as shown in Figure 3 By following the homotopy perturba-tion technique we can write the homotopy representation of(28) as

119867(X119894 119901) = 119871 (X

119894) minus 119871 (x

1198940) + 119901119871 (x

1198940) = 119901 (A

119905X119894+ B119905x119894minus119873)

(30)

Substituting the 119898 order expansion X119894= X1198940+ 119901X1198941+ sdot sdot sdot +

119901119898X119894119898

in (30) and by assuming an initial approximation ofthe form x

1198940= x119894minus1

we get after applying the proposed

EMHPMapproach the following set of first order linear delaydifferential equations

1199010 119889119889119879

X1198940+119889

119889119879x119894minus1= 0 X

119894(0) = x

119894minus1

1199011 119889119889119879

X1198941= A119905X1198940+ B119905x119894minus119873 X1198941(0) = 0

1199012 119889119889119879

X1198942= A119905X1198941 X1198942(0) = 0

119901119898 119889119889119879

X119894119898= A119905X119894(119898minus1)

X119894119898(0) = 0

(31)

By solving (31) we get

X1198940= x119894minus1

X1198941= A119905x119894minus1119879 + B

119905x119894minus119873119879

X1198942=1

2A2119905x119894minus11198792+1

2A119905B119905x119894minus1198731198792

X119894119898=1

119898A119898119905x119894minus1119879119898+1

119898A119898minus1119905

B119905x119894minus119873119879119898

(32)

Equations (32) can be written as

X119894119896=119879

119896(A119905X119894(119896minus1)

+ 119892 (119896)B119905x119894minus119873) 119896 = 1 2 3 (33)

where X1198940= x119894minus1

and 119892(119896) = 1 for 119896 = 1 and 119892(119896) = 0otherwise Thus the solution of (27) is obtained by addingthe X119894119896approximate solutions

x119894 (119879) asymp

119898

sum

119896=0

X119894119896 (119879) (34)

Notice however that solution (34) may be further improvedby using a first order polynomial representation of x120591

119894(119879) as

shown in Figure 4 Then the function x120591119894(119879) in the delay

subinterval [119905119894minus119873 119905119894minus119873+1

] takes the form

x120591119894(119879) = x119894minus119873+1 (119879) asymp x119894minus119873 +

(119873 minus 1)

120591(x119894minus119873+1

minus x119894minus119873) 119879

(35)

Substituting (35) into (28) gives

x119894(119879) minus A

119905x119894(119879)

asymp B119905x119894minus119873minus(119873 minus 1)

120591B119905x119894minus119873119879 +(119873 minus 1)

120591B119905x119894minus119873+1

119879

(36)

6 Abstract and Applied Analysis

We next assume that the homotopy representation of (36) isgiven as

119867(X119894 119901) = 119871 (X

119894) minus 119871 (x

1198940) + 119901119871 (x

1198940)

minus 119901 (AX119894+ Bx119894minus119873minus(119873 minus 1)

120591Bx119894minus119873119879

+(119873 minus 1)

120591Bx119894minus119873+1

119879) = 0

(37)

Substituting the 119898 order expansion X119894(119879) = X

1198940(119879) +

119901X1198941(119879) + sdot sdot sdot 119901

119898X119894119898(119879) in (37) and assuming that the initial

approximation is given by x1198940= x119894minus1

we get

1199010 119889119889119879

X1198940+119889

119889119879x119894minus1= 0 X

119894 (0) = x119894minus1

1199011 119889119889119879

X1198941= A119905X1198940+ B119905x119894minus119873minus119873 minus 1

120591B119905x119894minus119873119879

+119873 minus 1

120591B119905x119894minus119873+1

119879 X1198941(0) = 0

1199012 119889119889119879

X1198942= AX

1198941 X1198942(0) = 0

119901119898 119889119889119879

X119894119898= AX

119894(119898minus1) X119894119898(0) = 0

(38)

By solving (38) and by following the EMHPM procedure weget

X1198940= x119894minus1

X1198941= A119905x119894minus1119879 + B

119905x119894minus119873119879 minus1

2

119873 minus 1

120591B119905x119894minus1198731198792

+1

2

119873 minus 1

120591B119905x119894minus119873+1

1198792

X1198942=1

2A2119905x119894minus11198792+1

2A119905B119905x119894minus1198731198792

minus1

6

119873 minus 1

120591A119905B119905x119894minus1198731198793+1

6

119873 minus 1

120591A119905B119905x119894minus119873+1

1198793

X119894119898=1

119898A119898119905x119894minus1119879119898+1

119898A119898minus1119905

B119905x119894minus119873119879119898

minus1

(119898 + 1)

119873 minus 1

120591A119898minus1119905

B119905x119894minus119873119879119898+1

+1

(119898 + 1)

119873 minus 1

120591A119898minus1119905

B119905x119894minus119873+1

119879119898+1

(39)

Here the recursive form of X119894119896(119879) is written as

X119894119896= Xa119894119896+ Xb119894119896119896 = 1 2 3 (40)

timinusN+1 ttitiminus1timinusN

Δt

x i(t)x i+1(t)

x x iminusNx iminusN+1

120591 = (N minus 1)Δt

Figure 4 Schematic EMHPM solution using first polynomial toapproximate delay subinterval

where Xa1198940= x119894minus1Xb1198940= 0 and

Xa119894119896=119879

119896(A119905Xa119894(119896minus1)

+ 119892 (119896)B119905x119894minus119873)

Xb119894119896=119879

119896 + 1(A119905Xb119894(119896minus1)

+119892 (119896) [119873 minus 1

120591119879 (minusB

119905x119894minus119873+ B119905x119894minus119873+1

)])

(41)

Thus the approximate solution of (27) by the EMHPM canbe obtained by substituting (40) into (34)

In the next section we will apply our EMHPM procedureto obtain the solution of two second order delay differentialequations (a) the dampedMathieu equation with time delayand (b) the well-known delay differential equation thatdescribes the dynamics in one degree-of-freedom millingmachine operations

51 Solution of theDampedMathieu EquationwithTimeDelayIn order to assess the accuracy of our EMHPM approach wefirst obtain the solution of the damped Mathieu differentialequation with time delay that combines the effect of paramet-ric excitation and dampingThis equation is described by thefollowing equation

+ 120581 + (120575 + 120576 cos(2120587119905119879))119909 = 119887119909 (119905 minus 120591) (42)

where 120581 120575 120576 120591 and 119879 are system parameters whose valuedepends on the physics of the system The approximatesolution of (42) obtained by using the semidiscretizationmethod is widely discussed in [18 19] Here we focus ourattention on applying the EMHPM to find the approximatesolution of (42) and we also assess the accuracy of the derivedsolution by comparing it with the corresponding numericalintegration solution of (42)

By following the EMHPM procedure we first write (42)in the following equivalent form

119894(119879) + 120581

119894(119879) + 120572

119905119909119894(119879) asymp 119887119909

119894minus119873+1(119879) (43)

where 119909119894(119905) denotes the 119898 order solution of (43) in the

119894th subinterval that satisfies the following initial conditions119909119894(0) = 119909

119894minus1and

119894(0) =

119894minus1 The space state form

representation of (43) is given by

x119894(119879) = A

119905x119894(119879) + B

119905x119894minus119873+1

(119879) (44)

Abstract and Applied Analysis 7

0 2 4 6 8 10 12

005

1

dde23Zeroth EMHPMFirst EMHPM

minus05minus05

minus15minus15

minus1minus1Am

plitu

dex

Time t

times10minus3

Figure 5 Numerical solutions of the damped Mathieu equationwith time delay by dde23 the zeroth EMHPM and the first EMHPMwith119873 = 50 and119898 = 4

Table 1 Computer time needed to solve the damped Mathieuequation with time delay The 119898 order solution of the EMPHMapproach is chosen to guarantee the convergence of its approximatesolution

dde23 [ms] EMHPM119873 119898 Zeroth [ms] First [ms]

19

15 5 4 520 4 6 640 3 12 1260 2 17 1860 5 18 1960 10 20 21

where

A119905= [0 1

minus120572119905minus120581] B

119905= [0 0

1198871199050] (45)

and 120572119905= (120575 + 120576 cos(119905)) is a time periodic term The EMHPM

approximate solution of (44) is illustrated in Figure 5 wherewe have assumed an unstable system behavior for which 120581 =02 120575 = 30 120576 = 1 119887 = minus1 and 119879 = 120591 = 2120587 See[20] As we can see from Figure 5 our approximate EMHPMsolution to (42) is compared with its numerical integrationsolution obtained from dde23 MATLAB algorithm for thetime interval of 2119879 by assuming that 119873 = 50 with thefollowing initial values 119909

minus50(119879) = 119909

minus49(119879) = sdot sdot sdot 119909

0(119879) =

0001 and minus50(119879) =

minus49(119879) = sdot sdot sdot

0(119879) = 0

It can be seen fromFigure 5 that in the interval [0119879] boththe zeroth and the first order solutions are the same since thedelay subintervals are constant See Figure 3 However in thenext interval [119879 2119879] it is clear that the first order EMPHMsolution provides a better approximation on the delay subin-terval The computation total time to calculate the solutionsin the MATLAB code is listed in Table 1 The order 119898 andthe discretized time intervals 119873 in the EMPHM approachare chosen to guarantee the convergence of our approximatesolution to the exact one To provide a full understanding ofhow the solution is computed by the EMHPM approach weattached in Algorithm 1 the corresponding MATLAB code

10 20 30 40 50 60 70 80 90 100

Abso

lute

erro

r

10minus4

10minus3

10minus2

Zeroth order m = 2

First order m = 2

Zeroth order m = 5

First order m = 5

Zeroth order m = 10

First order m = 10

Discretizations N

Figure 6 Estimated relative error values between the numericalsolution dde23 and the EMHPMapproximate solutionsHerewe usefor the EMHPM the values of119898 = 2 5 and 10

Figure 6 shows the relative error between our approxi-mate EMPHM and the dde23 solution and its relationshipwith the order119898 and the discretized time intervals119873 Noticethat the relative error values coincide at values of 119873 ge 45Also we can see from Figure 6 that the computed relativeerror values for approximate solutions of order119898 ge 5 remainunchanged

52 A Practical Application Cutting Operation on MillingMachine We next use our EMHPM procedure to obtainthe solution of the single degree-of-freedom milling opera-tion We use the simplified form based on [20ndash22]

(119905) + 2120577120596119899 (119905) + 1205962

119899119909 (119905) = minus

119886119901119870119904 (119905)

119898119898

(119909 (119905) minus 119909 (119905 minus 120591))

(46)

where 120596119899is the angular natural frequency of the system 120577 is

the damping ratio 119886119901is the depth of cut 119898

119898is the modal

mass of the tool 120591 represents the time delay which is equalto the tooth passing period and 119870

119904(119905) is the specific cutting

force coefficient which can be determined from119870119904 (119905)

=

119911119899

sum

119895=1

119892 (120601119895(119905)) sin (120601

119895(119905)) (119870

119905cos120601119895(119905) + 119870

119899sin120601119895(119905))

(47)

where 119911119899is the tool number of teeth 119870

119905and 119870

119899are the

tangential and the normal linear cutting force coefficientsrespectively 120601

119895(119905) is the angular position of the 119895-tooth

defined as

120601119895(119905) = (

2120587119899

60) 119905 +

2120587119895

119911119899

(48)

and 119899 is the spindle speed in rpm [20] The function 119892(120593119895(119905))

is a switching function which has a unity value when the 119895-tooth is cutting and zero otherwise

119892 (120601119895 (119905)) =

1 120601st lt 120601119895 (119905) lt 120601ex0 otherwise

(49)

8 Abstract and Applied Analysis

function dde23 ddeEM mathieu paper

disp(lsquoMathieu equation solutionrsquo) Solution by EMHPM with zeroth and first order solution

eps=1 kappa=02 T=2lowastpi tau=T b=minus1 delta=3 Mathieu Parameters

pntDelay=1 N=50 m ord=5 dt=tau(Nminus1) ktau=2 tspan=[0ktaulowastT] EMHPM Parameters

Solution

Solution by dde23

tdde=linspace(0ktaulowasttauktaulowast(Nminus1)+1)

dde=(tyz) mathieu dde(tyzkappadeltaepsbtau)

te dde=ticsol = dde23(ddetauhistorytspan) toc(te dde) xdde=deval(soltdde)

Solution by zeroth EMHPM

dde emhpm fun=(tc0ttzmzntauN) mathieu zeroth(tc0ttzmzntauNm ord

kappadeltaepsbT)

tic[t0z0]=ddeEMHPM(dde emhpm funtspanhistorytaudtpntDelay) toc

Solution by First EMHPM

dde emhpm fun=(tc0ttzmzntauN) mathieu first(tc0ttzmzntauNm ordkappadeltaepsbT)

tic[t1z1]=ddeEMHPM(dde emhpm funtspanhistorytaudtpntDelay) toc

Plot results

ind0=12ktaulowast(Nminus1) ind1=22ktaulowast(Nminus1)

Parent1=figure (1)

axes1 = axes(lsquoParentrsquoParent1lsquoFontSizersquo12lsquoFontNamersquolsquoTimes New Romanrsquo)box(axes1lsquoonrsquo) hold(axes1lsquoallrsquo)plot(tddexdde(1)lsquoParentrsquoaxes1lsquoLineWidthrsquo2lsquoColorrsquo[0502 0502 0502]lsquoDisplayNamersquolsquoNumericaldd23rsquo)plot(t0(ind0)z0(ind01)lsquoMarkerSizersquo5lsquoMarkerrsquolsquoorsquolsquoLineStylersquo lsquononersquolsquoDisplayNamersquolsquoZerothEMHPMrsquolsquoColorrsquo[0 0 0])

plot(t1(ind1)z1(ind11)lsquoMarkerSizersquo7lsquoMarkerrsquolsquoxrsquolsquoLineStylersquolsquononersquolsquoDisplayNamersquolsquoFirstEMHPMrsquolsquoColorrsquo[0 0 0])

xlabel(lsquoittrsquolsquoFontSizersquo12lsquoFontNamersquolsquoTimes New Romanrsquo)ylabel(lsquoitxrsquolsquoFontSizersquo12lsquoFontNamersquolsquoTimes New Romanrsquo)end

Mathieu definitions

function dydt = mathieu dde(tyzkapadltepsbT)

dydt = [y(2)-kapalowasty(2)minus(dlt+epslowastcos(2lowastpilowasttT))lowasty(1)+blowastz(1)]

end

function Z = mathieu zeroth(tc0ttzmzntauNmkpadltepsbT)

Z=[c0(1)c0(2)] alf=dlt+epslowastcos(2lowastpiTlowasttt)

for ik=1m

Z(ik+11)=Z(ik2)lowasttik

Z(ik+12)=minuskpalowastZ(ik2)minusalflowastZ(ik1)

if ik==1 Z(ik+12)=Z(ik+12)+blowastzm(1) end

Z(ik+12)=Z(ik+12)lowasttik

end

Z=sum(Z)

end

function Z = mathieu first(tc0ttzmzntauNmkpadltepsbT)

alf=dlt+epslowastcos(2lowastpiTlowasttt) Z=[c0(1)c0(2)] Z =[00]

for ik=1m

Z(ik+1)=[Z(ik2)lowasttik minuskpalowastZ(ik2)minusalflowastZ(ik1)]

Z (ik+1)=[Z (ik2)lowasttik minuskpalowastZ (ik2)minusalflowastZ (ik1)]

if ik==1

Z(ik+12)=Z(ik+12)+blowastzm(1)Z (ik+12)=Z (ik+12)+blowast(Nminus1)taulowast(zn(1)minuszm(1))lowastt

end

Z(ik+12)=Z(ik+12)lowasttik

Z (ik+12)=Z (ik+12)lowastt(ik+1)

Algorithm 1 Continued

Abstract and Applied Analysis 9

end

Z=sum(Z)+sum(Z )

end

function out=history(t)

out=[1Eminus3+0lowastt 0+0lowastt]

end

EMHPM algorithm for ODE solutions

function [tz]= odeEMHPM(ndetspanz0Deltatpnts)

ode solver by Enhanced Multistage Homotopy Perturbation Method

tini=tspan(1) tfin=tspan(end) tstart=tini

tini=tiniminuststart tfin=tfinminuststart shifted time set to zero

Handle errors

if tini == tfin

error(lsquoThe ending and starting time values must be differentrsquo)elseif abs(tini)gt abs(tfin)

tspan=flipud(fliplr(tspan)) tini=tspan(1) tfin=tspan(end)

end

tdir=sign(tfinminustini)

if any(tdirlowastdiff(tspan) lt= 0)

error(lsquotspan entries must be strictly sortedrsquo)end

incT=Deltatpnts

if incTlt=0

error (lsquoIncreasing must be greater than zerorsquo)end

z(1)=z01015840 t(1)=tini iteT=2 set initial values

t(iteT)=tini+incTlowasttdir

while tdirlowastt(iteT)lttdirlowast(tfin+tdirlowastincT)

P act=ceil(99999lowast(t(iteT)minustini)lowasttdirDeltat) count sub-intervals

c=z((DeltatincT)lowast(P actminus1)+1) set the corresponding initial condition

tsub(iteTminus1)=t(iteT)-(P actminus1)lowastDeltatlowasttdir evaluate the solution at the shifted time

temp=tsub(iteTminus1)

z(iteT)=nde(tempctstart+t(iteT))

if tdirlowastt(iteT)gt=tdirlowasttfinlowast099999 repeat for the next sub-interval

break

else

iteT=iteT+1 t(iteT)=tini+(iteTminus1)lowastincTlowasttdir

end

end

t=t 1015840 +tstart

end

EMHPM algorithm for DDE solutions

function [t z]=ddeEMHPM(ddetspanhistorytauDeltatpntDelta)

dde solver by Enhanced Multistage Homotopy Perturbation Method

tini=tspan(1) tfin=tspan(end) span where the solution is founded

if taultDeltat

error(lsquoSubtinterval must not be greater than taursquo)end

if mod(tauDeltat)sim=0

pastDlt=Deltat

Deltat=tauround(tauDeltat)

warning(lsquoSubtinterval was modified from 05g to 05grsquopastDltDeltat)end

pnt=round(tauDeltat) samplesminus1 [minustau 0]

incT=DeltatpntDelta step for set resolution

t=minustau+tini+(0pntlowastpntDelta)rsquolowastincT evaluation of the initial solution [minustau 0]

z=history(t) initial behavior in the inverval [minustau 0]

c=z(end) initial condition

Algorithm 1 Continued

10 Abstract and Applied Analysis

m=pntlowastpntDelta samples between tau

itDlt=0

while (t(end)+eps)lttfin

tspan=[tini+itDltlowastDeltattini+(itDlt+1)lowastDeltat] preparing the span for next Deltat

zm=z(length(z)minusm)rsquo zn=z(1+length(z)minusm)rsquo previous tau solution

emhpm fun=(timecT) dde(timecTzmzntaum) application of the odeEMHPM

[t aux z aux]=odeEMHPM(emhpm funtspanz(end)DeltatpntDelta)

z=[z(1endminus1)z aux] t=[t(1endminus1)t aux] joining the solutions

itDlt=itDlt+1

end

z=z(pntlowastpntDelta+1end)t=t(pntlowastpntDelta+1end)

end

Algorithm 1 MATLAB algorithm

Here 120601st and 120601ex are the angles where the teeth enter and exitthe workpiece For upmilling 120601st = 0 and 120601ex = arccos(1 minus2119886119889) for downmilling 120601st = arccos(2119886

119889minus 1) and 120601ex = 120587

where 119886119889is the radial depth of cut ratio

By following the EMHPM procedure we can write (46)in equivalent form as

119894(119879) + 2120577120596

119899119894(119879) + 120596

2

119899119909119894(119879)

asymp minus

119886119901119870st

119898119898

(119909119894 (119879) minus 119909119894minus119873+1 (119879))

(50)

where 119909119894(119879) denotes the 119898 order solution of (46) on the 119894th

subinterval that satisfies the initial conditions 119909119894(0) = 119909

119894minus1

119894(0) =

119894minus1 and ℎ

119905= ℎ(119905) and 119909

minus120591is given by (35)

Introducing the transformation x119894= [119909119894 119894]119879 (50) can be

written as a system of first order linear delay differentialequations of the form

x119894(119879) = A

119905x119894(119879) + B

119905x119894minus119873+1

(119879) (51)

where

A119905= [

[

0 1

minus1205962

119899minus119908

119898119898

119870st minus2120577120596119899]

]

B119905= [

[

0 0119908

119898119898

119870st 0]

]

(52)

We next apply the EMHPM procedure to solve (46) byconsidering a downmilling operation with the followingparameter values 119911

119899= 2 119886

119889= 01 120596

119899= 5793 rads

120577 = 0011 119898119898= 003993 kg 119870

119905= 6 times 10

8Nm2 and119870119899= 2 times 10

8Nm2 As we can see from Figures 7 and 8 andfor the depth of cut values of 119886

119901= 2mm (stable) and 119886

119901=

3mm (unstable) our EMHPM approximate solutions followclosely the numerical integration solutions of (46) obtainedby using the dde23 algorithm

Figure 9 shows the relative error between the EMPHMand the dde23 numerical solution while Table 2 shows theCPU time needed for each solution Here we use 119873 = 75since the average step size of the dde23 algorithm is aroundΔ119905 asymp 120591119873 Note that for 119898 = 7 the zeroth order EMHPMapproximate solution has the fastest CPU time We can seefrom Figure 9 that the value of the relative error becomesbasically the same for119898 = 2 7 and 10 and119873 ge 20

0 0005 001 0015 002

0

1

2

Time (s)

Disp

lace

men

t (m

)

dd23Zeroth EMHPMFirst EMHPM

minus2

minus1

times10minus3

Figure 7 EMHPM approximate solutions of (46) with parametervalues of 119886

119901= 2mm 119899 = 10000 rpm Stable machine operation

0 0005 001 0015 002

00005

0010015

Time (s)

Disp

lace

men

t (m

)

dd23Zeroth EMHPMFirst EMHPM

minus001

minus0005

Figure 8 EMHPM approximate solutions of (46) with parametervalues of 119886

119901= 3mmand 119899 = 10000 rpmUnstable (chatter)machine

operation

6 Conclusions

We have developed a new algorithm based on the homotopyperturbation method to solve delay differential equationsThe proposed EMHPM approach is based on a sequence ofsubintervals that approximate the solution of delayed differ-ential equations by using the transformation rule 119906(119905) rarr119906119894(119879) where 119906

119894satisfies the initial conditions We have

Abstract and Applied Analysis 11

10 20 30 40 50 60 70 80 90 100

Abso

lute

erro

r

Zeroth order m = 2

First order m = 2

Zeroth order m = 7

First order m = 7

Zeroth order m = 10

First order m = 10

10minus3

10minus2

10minus1

Discretizations N

Figure 9 Estimated relative error values between dde23 and theEMHPM approximate solutions Here we have used the systemparameter values of 119886

119901= 2mm and 119899 = 1000 rpm and values of

119898 = 2 7 and 10

Table 2 CPU time comparison among the approximate solutions of(46) by using dde23 the zeroth order and the first order EMHPMsolutions

Solution method Time [ms]dde23 139Zeroth order EMHPM (119898 = 7) 77First order EMHPM (119898 = 7) 87

shown that our proposed EMHPM approach can be appliedto obtain the approximate solution of a delay differentialequation not only with constant but also with variablecoefficients with theoretical predictions that follow well thenumerical integration solutions To further assess the validityof this new approach we have compared the approximatesolutions of two delayed differential equations with respect totheir corresponding numerical integration solutions obtainedfrom the MATLAB dde23 algorithm The test cases were (a)the damped Mathieu differential equation with time delayand (b) the governing equation of motion of downmillingoperations We have found that the EMHPM closely followsthe numerical integration solutions of the correspondingequations and that these require less CPU time and havesmaller relative errors

Conflict of Interests

Theauthors declare that they have no conflict of interests withany mentioned entities in the paper

Acknowledgments

This work was funded by Tecnologico deMonterrey CampusMonterrey through the ResearchGroup inNanomaterials forMedical Devices and the Research Group in Advanced Man-ufacturing Additional support was provided from ConsejoNacional de Ciencia y Tecnologıa (Conacyt) Mexico

References

[1] F Shakeri and M Dehghan ldquoSolution of delay differentialequations via a homotopy perturbation methodrdquoMathematicaland Computer Modelling vol 48 no 3-4 pp 486ndash498 2008

[2] Z-K Wu ldquoSolution of the enso delayed oscillator with homo-topy analysis methodrdquo Journal of Hydrodynamics vol 21 no 1pp 131ndash135 2009

[3] A K Alomari M S Noorani and R Nazar ldquoSolution of delaydifferential equation by means of homotopy analysis methodrdquoActa Applicandae Mathematicae vol 108 no 2 pp 395ndash4122009

[4] J Biazar and G Behzad ldquoThe homotopy perturbation methodfor solving neutral functional-differential equations with pro-portional delaysrdquo Journal of King Saud UniversitymdashScience vol24 no 1 pp 33ndash37 2012

[5] NRAnakiraAKAlomari and IHashim ldquoOptimal homotopyasymptotic method for solving delay differential equationsrdquoMathematical Problems in Engineering vol 2013 Article ID498902 11 pages 2013

[6] T Insperger and G Stepan Semi-Discretization for Time-DelaySystems Stability and Engineering Applications Springer NewYork NY USA 2011

[7] E A Butcher P Nindujarla and E Bueler ldquoStability of up-and down-milling using chebyshev collocation methodrdquo inProceedings of the ASME International Design Engineering Tech-nical Conferences amp Computers and Information in EngineeringConference vol 6 part AndashC pp 841ndash850 2005

[8] T Insperger ldquoFull-discretization and semi-discretization formilling stability prediction some commentsrdquo InternationalJournal of Machine Tools and Manufacture vol 50 no 7 pp658ndash662 2010

[9] I Hashim and M S Chowdhury ldquoAdaptation of homotopy-perturbation method for numeric-analytic solution of systemof ODEsrdquo Physics Letters A vol 372 no 4 pp 470ndash481 2008

[10] L F Shampine and S Thompson ldquoSolving DDEs in MatlabrdquoAppliedNumericalMathematics vol 37 no 4 pp 441ndash458 2001

[11] SHHNiaANRanjbarDDGanjiH Soltani and JGhasemildquoMaintaining the stability of nonlinear differential equations bythe enhancement of HPMrdquo Physics Letters A vol 372 no 16 pp2855ndash2861 2008

[12] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[13] J-H He ldquoHomotopy perturbation method a new nonlinearanalytical techniquerdquo Applied Mathematics and Computationvol 135 no 1 pp 73ndash79 2003

[14] J H He ldquoComparison of homotopy perturbation method andhomotopy analysis methodrdquo Applied Mathematics and Compu-tation vol 156 no 2 pp 527ndash539 2004

[15] JHHe ldquoNew interpretation of homotopy perturbationmethodrdquoInternational Journal of Modern Physics B vol 20 no 18 pp2561ndash2568 2006

[16] S Liao ldquoComparison between the homotopy analysis methodand homotopy perturbationmethodrdquoAppliedMathematics andComputation vol 169 no 2 pp 1186ndash1194 2005

[17] D Olvera and A Elıas-Zuniga ldquoEnhanced multistage homo-topy perturbation method approximate solutions of nonlineardynamic systemsrdquo Abstract and Applied Analysis vol 2014Article ID 486509 12 pages 2014

12 Abstract and Applied Analysis

[18] T Insperger and G Stepan ldquoStability of the damped Mathieuequationwith timedelayrdquo Journal ofDynamic SystemsMeasure-ment and Control vol 125 no 2 pp 166ndash171 2003

[19] T Insperger and G Stepan ldquoStability chart for the delayedMathieu equationrdquo The Royal Society of London ProceedingsSeries A Mathematical Physical and Engineering Sciences vol458 no 2024 pp 1989ndash1998 2002

[20] T Insperger andGStepan ldquoUpdated semi-discretizationmethodfor periodic delay-differential equations with discrete delayrdquoInternational Journal for Numerical Methods in Engineering vol61 no 1 pp 117ndash141 2004

[21] T Insperger B P Mann G Stepan and P V Bayly ldquoStabilityof up-milling and down-milling part 1 alternative analyticalmethodsrdquo International Journal of Machine Tools and Manufac-ture vol 43 no 1 pp 25ndash34 2003

[22] F ICompeanDOlvera F J Campa LN L de Lacalle A Elıas-Zuniga and C A Rodrıguez ldquoCharacterization and stabilityanalysis of a multivariable milling tool by the enhanced mul-tistage homotopy perturbation methodrdquo International Journalof Machine Tools and Manufacture vol 57 pp 27ndash33 2012

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 Abstract and Applied Analysis

We next assume that the homotopy representation of (36) isgiven as

119867(X119894 119901) = 119871 (X

119894) minus 119871 (x

1198940) + 119901119871 (x

1198940)

minus 119901 (AX119894+ Bx119894minus119873minus(119873 minus 1)

120591Bx119894minus119873119879

+(119873 minus 1)

120591Bx119894minus119873+1

119879) = 0

(37)

Substituting the 119898 order expansion X119894(119879) = X

1198940(119879) +

119901X1198941(119879) + sdot sdot sdot 119901

119898X119894119898(119879) in (37) and assuming that the initial

approximation is given by x1198940= x119894minus1

we get

1199010 119889119889119879

X1198940+119889

119889119879x119894minus1= 0 X

119894 (0) = x119894minus1

1199011 119889119889119879

X1198941= A119905X1198940+ B119905x119894minus119873minus119873 minus 1

120591B119905x119894minus119873119879

+119873 minus 1

120591B119905x119894minus119873+1

119879 X1198941(0) = 0

1199012 119889119889119879

X1198942= AX

1198941 X1198942(0) = 0

119901119898 119889119889119879

X119894119898= AX

119894(119898minus1) X119894119898(0) = 0

(38)

By solving (38) and by following the EMHPM procedure weget

X1198940= x119894minus1

X1198941= A119905x119894minus1119879 + B

119905x119894minus119873119879 minus1

2

119873 minus 1

120591B119905x119894minus1198731198792

+1

2

119873 minus 1

120591B119905x119894minus119873+1

1198792

X1198942=1

2A2119905x119894minus11198792+1

2A119905B119905x119894minus1198731198792

minus1

6

119873 minus 1

120591A119905B119905x119894minus1198731198793+1

6

119873 minus 1

120591A119905B119905x119894minus119873+1

1198793

X119894119898=1

119898A119898119905x119894minus1119879119898+1

119898A119898minus1119905

B119905x119894minus119873119879119898

minus1

(119898 + 1)

119873 minus 1

120591A119898minus1119905

B119905x119894minus119873119879119898+1

+1

(119898 + 1)

119873 minus 1

120591A119898minus1119905

B119905x119894minus119873+1

119879119898+1

(39)

Here the recursive form of X119894119896(119879) is written as

X119894119896= Xa119894119896+ Xb119894119896119896 = 1 2 3 (40)

timinusN+1 ttitiminus1timinusN

Δt

x i(t)x i+1(t)

x x iminusNx iminusN+1

120591 = (N minus 1)Δt

Figure 4 Schematic EMHPM solution using first polynomial toapproximate delay subinterval

where Xa1198940= x119894minus1Xb1198940= 0 and

Xa119894119896=119879

119896(A119905Xa119894(119896minus1)

+ 119892 (119896)B119905x119894minus119873)

Xb119894119896=119879

119896 + 1(A119905Xb119894(119896minus1)

+119892 (119896) [119873 minus 1

120591119879 (minusB

119905x119894minus119873+ B119905x119894minus119873+1

)])

(41)

Thus the approximate solution of (27) by the EMHPM canbe obtained by substituting (40) into (34)

In the next section we will apply our EMHPM procedureto obtain the solution of two second order delay differentialequations (a) the dampedMathieu equation with time delayand (b) the well-known delay differential equation thatdescribes the dynamics in one degree-of-freedom millingmachine operations

51 Solution of theDampedMathieu EquationwithTimeDelayIn order to assess the accuracy of our EMHPM approach wefirst obtain the solution of the damped Mathieu differentialequation with time delay that combines the effect of paramet-ric excitation and dampingThis equation is described by thefollowing equation

+ 120581 + (120575 + 120576 cos(2120587119905119879))119909 = 119887119909 (119905 minus 120591) (42)

where 120581 120575 120576 120591 and 119879 are system parameters whose valuedepends on the physics of the system The approximatesolution of (42) obtained by using the semidiscretizationmethod is widely discussed in [18 19] Here we focus ourattention on applying the EMHPM to find the approximatesolution of (42) and we also assess the accuracy of the derivedsolution by comparing it with the corresponding numericalintegration solution of (42)

By following the EMHPM procedure we first write (42)in the following equivalent form

119894(119879) + 120581

119894(119879) + 120572

119905119909119894(119879) asymp 119887119909

119894minus119873+1(119879) (43)

where 119909119894(119905) denotes the 119898 order solution of (43) in the

119894th subinterval that satisfies the following initial conditions119909119894(0) = 119909

119894minus1and

119894(0) =

119894minus1 The space state form

representation of (43) is given by

x119894(119879) = A

119905x119894(119879) + B

119905x119894minus119873+1

(119879) (44)

Abstract and Applied Analysis 7

0 2 4 6 8 10 12

005

1

dde23Zeroth EMHPMFirst EMHPM

minus05minus05

minus15minus15

minus1minus1Am

plitu

dex

Time t

times10minus3

Figure 5 Numerical solutions of the damped Mathieu equationwith time delay by dde23 the zeroth EMHPM and the first EMHPMwith119873 = 50 and119898 = 4

Table 1 Computer time needed to solve the damped Mathieuequation with time delay The 119898 order solution of the EMPHMapproach is chosen to guarantee the convergence of its approximatesolution

dde23 [ms] EMHPM119873 119898 Zeroth [ms] First [ms]

19

15 5 4 520 4 6 640 3 12 1260 2 17 1860 5 18 1960 10 20 21

where

A119905= [0 1

minus120572119905minus120581] B

119905= [0 0

1198871199050] (45)

and 120572119905= (120575 + 120576 cos(119905)) is a time periodic term The EMHPM

approximate solution of (44) is illustrated in Figure 5 wherewe have assumed an unstable system behavior for which 120581 =02 120575 = 30 120576 = 1 119887 = minus1 and 119879 = 120591 = 2120587 See[20] As we can see from Figure 5 our approximate EMHPMsolution to (42) is compared with its numerical integrationsolution obtained from dde23 MATLAB algorithm for thetime interval of 2119879 by assuming that 119873 = 50 with thefollowing initial values 119909

minus50(119879) = 119909

minus49(119879) = sdot sdot sdot 119909

0(119879) =

0001 and minus50(119879) =

minus49(119879) = sdot sdot sdot

0(119879) = 0

It can be seen fromFigure 5 that in the interval [0119879] boththe zeroth and the first order solutions are the same since thedelay subintervals are constant See Figure 3 However in thenext interval [119879 2119879] it is clear that the first order EMPHMsolution provides a better approximation on the delay subin-terval The computation total time to calculate the solutionsin the MATLAB code is listed in Table 1 The order 119898 andthe discretized time intervals 119873 in the EMPHM approachare chosen to guarantee the convergence of our approximatesolution to the exact one To provide a full understanding ofhow the solution is computed by the EMHPM approach weattached in Algorithm 1 the corresponding MATLAB code

10 20 30 40 50 60 70 80 90 100

Abso

lute

erro

r

10minus4

10minus3

10minus2

Zeroth order m = 2

First order m = 2

Zeroth order m = 5

First order m = 5

Zeroth order m = 10

First order m = 10

Discretizations N

Figure 6 Estimated relative error values between the numericalsolution dde23 and the EMHPMapproximate solutionsHerewe usefor the EMHPM the values of119898 = 2 5 and 10

Figure 6 shows the relative error between our approxi-mate EMPHM and the dde23 solution and its relationshipwith the order119898 and the discretized time intervals119873 Noticethat the relative error values coincide at values of 119873 ge 45Also we can see from Figure 6 that the computed relativeerror values for approximate solutions of order119898 ge 5 remainunchanged

52 A Practical Application Cutting Operation on MillingMachine We next use our EMHPM procedure to obtainthe solution of the single degree-of-freedom milling opera-tion We use the simplified form based on [20ndash22]

(119905) + 2120577120596119899 (119905) + 1205962

119899119909 (119905) = minus

119886119901119870119904 (119905)

119898119898

(119909 (119905) minus 119909 (119905 minus 120591))

(46)

where 120596119899is the angular natural frequency of the system 120577 is

the damping ratio 119886119901is the depth of cut 119898

119898is the modal

mass of the tool 120591 represents the time delay which is equalto the tooth passing period and 119870

119904(119905) is the specific cutting

force coefficient which can be determined from119870119904 (119905)

=

119911119899

sum

119895=1

119892 (120601119895(119905)) sin (120601

119895(119905)) (119870

119905cos120601119895(119905) + 119870

119899sin120601119895(119905))

(47)

where 119911119899is the tool number of teeth 119870

119905and 119870

119899are the

tangential and the normal linear cutting force coefficientsrespectively 120601

119895(119905) is the angular position of the 119895-tooth

defined as

120601119895(119905) = (

2120587119899

60) 119905 +

2120587119895

119911119899

(48)

and 119899 is the spindle speed in rpm [20] The function 119892(120593119895(119905))

is a switching function which has a unity value when the 119895-tooth is cutting and zero otherwise

119892 (120601119895 (119905)) =

1 120601st lt 120601119895 (119905) lt 120601ex0 otherwise

(49)

8 Abstract and Applied Analysis

function dde23 ddeEM mathieu paper

disp(lsquoMathieu equation solutionrsquo) Solution by EMHPM with zeroth and first order solution

eps=1 kappa=02 T=2lowastpi tau=T b=minus1 delta=3 Mathieu Parameters

pntDelay=1 N=50 m ord=5 dt=tau(Nminus1) ktau=2 tspan=[0ktaulowastT] EMHPM Parameters

Solution

Solution by dde23

tdde=linspace(0ktaulowasttauktaulowast(Nminus1)+1)

dde=(tyz) mathieu dde(tyzkappadeltaepsbtau)

te dde=ticsol = dde23(ddetauhistorytspan) toc(te dde) xdde=deval(soltdde)

Solution by zeroth EMHPM

dde emhpm fun=(tc0ttzmzntauN) mathieu zeroth(tc0ttzmzntauNm ord

kappadeltaepsbT)

tic[t0z0]=ddeEMHPM(dde emhpm funtspanhistorytaudtpntDelay) toc

Solution by First EMHPM

dde emhpm fun=(tc0ttzmzntauN) mathieu first(tc0ttzmzntauNm ordkappadeltaepsbT)

tic[t1z1]=ddeEMHPM(dde emhpm funtspanhistorytaudtpntDelay) toc

Plot results

ind0=12ktaulowast(Nminus1) ind1=22ktaulowast(Nminus1)

Parent1=figure (1)

axes1 = axes(lsquoParentrsquoParent1lsquoFontSizersquo12lsquoFontNamersquolsquoTimes New Romanrsquo)box(axes1lsquoonrsquo) hold(axes1lsquoallrsquo)plot(tddexdde(1)lsquoParentrsquoaxes1lsquoLineWidthrsquo2lsquoColorrsquo[0502 0502 0502]lsquoDisplayNamersquolsquoNumericaldd23rsquo)plot(t0(ind0)z0(ind01)lsquoMarkerSizersquo5lsquoMarkerrsquolsquoorsquolsquoLineStylersquo lsquononersquolsquoDisplayNamersquolsquoZerothEMHPMrsquolsquoColorrsquo[0 0 0])

plot(t1(ind1)z1(ind11)lsquoMarkerSizersquo7lsquoMarkerrsquolsquoxrsquolsquoLineStylersquolsquononersquolsquoDisplayNamersquolsquoFirstEMHPMrsquolsquoColorrsquo[0 0 0])

xlabel(lsquoittrsquolsquoFontSizersquo12lsquoFontNamersquolsquoTimes New Romanrsquo)ylabel(lsquoitxrsquolsquoFontSizersquo12lsquoFontNamersquolsquoTimes New Romanrsquo)end

Mathieu definitions

function dydt = mathieu dde(tyzkapadltepsbT)

dydt = [y(2)-kapalowasty(2)minus(dlt+epslowastcos(2lowastpilowasttT))lowasty(1)+blowastz(1)]

end

function Z = mathieu zeroth(tc0ttzmzntauNmkpadltepsbT)

Z=[c0(1)c0(2)] alf=dlt+epslowastcos(2lowastpiTlowasttt)

for ik=1m

Z(ik+11)=Z(ik2)lowasttik

Z(ik+12)=minuskpalowastZ(ik2)minusalflowastZ(ik1)

if ik==1 Z(ik+12)=Z(ik+12)+blowastzm(1) end

Z(ik+12)=Z(ik+12)lowasttik

end

Z=sum(Z)

end

function Z = mathieu first(tc0ttzmzntauNmkpadltepsbT)

alf=dlt+epslowastcos(2lowastpiTlowasttt) Z=[c0(1)c0(2)] Z =[00]

for ik=1m

Z(ik+1)=[Z(ik2)lowasttik minuskpalowastZ(ik2)minusalflowastZ(ik1)]

Z (ik+1)=[Z (ik2)lowasttik minuskpalowastZ (ik2)minusalflowastZ (ik1)]

if ik==1

Z(ik+12)=Z(ik+12)+blowastzm(1)Z (ik+12)=Z (ik+12)+blowast(Nminus1)taulowast(zn(1)minuszm(1))lowastt

end

Z(ik+12)=Z(ik+12)lowasttik

Z (ik+12)=Z (ik+12)lowastt(ik+1)

Algorithm 1 Continued

Abstract and Applied Analysis 9

end

Z=sum(Z)+sum(Z )

end

function out=history(t)

out=[1Eminus3+0lowastt 0+0lowastt]

end

EMHPM algorithm for ODE solutions

function [tz]= odeEMHPM(ndetspanz0Deltatpnts)

ode solver by Enhanced Multistage Homotopy Perturbation Method

tini=tspan(1) tfin=tspan(end) tstart=tini

tini=tiniminuststart tfin=tfinminuststart shifted time set to zero

Handle errors

if tini == tfin

error(lsquoThe ending and starting time values must be differentrsquo)elseif abs(tini)gt abs(tfin)

tspan=flipud(fliplr(tspan)) tini=tspan(1) tfin=tspan(end)

end

tdir=sign(tfinminustini)

if any(tdirlowastdiff(tspan) lt= 0)

error(lsquotspan entries must be strictly sortedrsquo)end

incT=Deltatpnts

if incTlt=0

error (lsquoIncreasing must be greater than zerorsquo)end

z(1)=z01015840 t(1)=tini iteT=2 set initial values

t(iteT)=tini+incTlowasttdir

while tdirlowastt(iteT)lttdirlowast(tfin+tdirlowastincT)

P act=ceil(99999lowast(t(iteT)minustini)lowasttdirDeltat) count sub-intervals

c=z((DeltatincT)lowast(P actminus1)+1) set the corresponding initial condition

tsub(iteTminus1)=t(iteT)-(P actminus1)lowastDeltatlowasttdir evaluate the solution at the shifted time

temp=tsub(iteTminus1)

z(iteT)=nde(tempctstart+t(iteT))

if tdirlowastt(iteT)gt=tdirlowasttfinlowast099999 repeat for the next sub-interval

break

else

iteT=iteT+1 t(iteT)=tini+(iteTminus1)lowastincTlowasttdir

end

end

t=t 1015840 +tstart

end

EMHPM algorithm for DDE solutions

function [t z]=ddeEMHPM(ddetspanhistorytauDeltatpntDelta)

dde solver by Enhanced Multistage Homotopy Perturbation Method

tini=tspan(1) tfin=tspan(end) span where the solution is founded

if taultDeltat

error(lsquoSubtinterval must not be greater than taursquo)end

if mod(tauDeltat)sim=0

pastDlt=Deltat

Deltat=tauround(tauDeltat)

warning(lsquoSubtinterval was modified from 05g to 05grsquopastDltDeltat)end

pnt=round(tauDeltat) samplesminus1 [minustau 0]

incT=DeltatpntDelta step for set resolution

t=minustau+tini+(0pntlowastpntDelta)rsquolowastincT evaluation of the initial solution [minustau 0]

z=history(t) initial behavior in the inverval [minustau 0]

c=z(end) initial condition

Algorithm 1 Continued

10 Abstract and Applied Analysis

m=pntlowastpntDelta samples between tau

itDlt=0

while (t(end)+eps)lttfin

tspan=[tini+itDltlowastDeltattini+(itDlt+1)lowastDeltat] preparing the span for next Deltat

zm=z(length(z)minusm)rsquo zn=z(1+length(z)minusm)rsquo previous tau solution

emhpm fun=(timecT) dde(timecTzmzntaum) application of the odeEMHPM

[t aux z aux]=odeEMHPM(emhpm funtspanz(end)DeltatpntDelta)

z=[z(1endminus1)z aux] t=[t(1endminus1)t aux] joining the solutions

itDlt=itDlt+1

end

z=z(pntlowastpntDelta+1end)t=t(pntlowastpntDelta+1end)

end

Algorithm 1 MATLAB algorithm

Here 120601st and 120601ex are the angles where the teeth enter and exitthe workpiece For upmilling 120601st = 0 and 120601ex = arccos(1 minus2119886119889) for downmilling 120601st = arccos(2119886

119889minus 1) and 120601ex = 120587

where 119886119889is the radial depth of cut ratio

By following the EMHPM procedure we can write (46)in equivalent form as

119894(119879) + 2120577120596

119899119894(119879) + 120596

2

119899119909119894(119879)

asymp minus

119886119901119870st

119898119898

(119909119894 (119879) minus 119909119894minus119873+1 (119879))

(50)

where 119909119894(119879) denotes the 119898 order solution of (46) on the 119894th

subinterval that satisfies the initial conditions 119909119894(0) = 119909

119894minus1

119894(0) =

119894minus1 and ℎ

119905= ℎ(119905) and 119909

minus120591is given by (35)

Introducing the transformation x119894= [119909119894 119894]119879 (50) can be

written as a system of first order linear delay differentialequations of the form

x119894(119879) = A

119905x119894(119879) + B

119905x119894minus119873+1

(119879) (51)

where

A119905= [

[

0 1

minus1205962

119899minus119908

119898119898

119870st minus2120577120596119899]

]

B119905= [

[

0 0119908

119898119898

119870st 0]

]

(52)

We next apply the EMHPM procedure to solve (46) byconsidering a downmilling operation with the followingparameter values 119911

119899= 2 119886

119889= 01 120596

119899= 5793 rads

120577 = 0011 119898119898= 003993 kg 119870

119905= 6 times 10

8Nm2 and119870119899= 2 times 10

8Nm2 As we can see from Figures 7 and 8 andfor the depth of cut values of 119886

119901= 2mm (stable) and 119886

119901=

3mm (unstable) our EMHPM approximate solutions followclosely the numerical integration solutions of (46) obtainedby using the dde23 algorithm

Figure 9 shows the relative error between the EMPHMand the dde23 numerical solution while Table 2 shows theCPU time needed for each solution Here we use 119873 = 75since the average step size of the dde23 algorithm is aroundΔ119905 asymp 120591119873 Note that for 119898 = 7 the zeroth order EMHPMapproximate solution has the fastest CPU time We can seefrom Figure 9 that the value of the relative error becomesbasically the same for119898 = 2 7 and 10 and119873 ge 20

0 0005 001 0015 002

0

1

2

Time (s)

Disp

lace

men

t (m

)

dd23Zeroth EMHPMFirst EMHPM

minus2

minus1

times10minus3

Figure 7 EMHPM approximate solutions of (46) with parametervalues of 119886

119901= 2mm 119899 = 10000 rpm Stable machine operation

0 0005 001 0015 002

00005

0010015

Time (s)

Disp

lace

men

t (m

)

dd23Zeroth EMHPMFirst EMHPM

minus001

minus0005

Figure 8 EMHPM approximate solutions of (46) with parametervalues of 119886

119901= 3mmand 119899 = 10000 rpmUnstable (chatter)machine

operation

6 Conclusions

We have developed a new algorithm based on the homotopyperturbation method to solve delay differential equationsThe proposed EMHPM approach is based on a sequence ofsubintervals that approximate the solution of delayed differ-ential equations by using the transformation rule 119906(119905) rarr119906119894(119879) where 119906

119894satisfies the initial conditions We have

Abstract and Applied Analysis 11

10 20 30 40 50 60 70 80 90 100

Abso

lute

erro

r

Zeroth order m = 2

First order m = 2

Zeroth order m = 7

First order m = 7

Zeroth order m = 10

First order m = 10

10minus3

10minus2

10minus1

Discretizations N

Figure 9 Estimated relative error values between dde23 and theEMHPM approximate solutions Here we have used the systemparameter values of 119886

119901= 2mm and 119899 = 1000 rpm and values of

119898 = 2 7 and 10

Table 2 CPU time comparison among the approximate solutions of(46) by using dde23 the zeroth order and the first order EMHPMsolutions

Solution method Time [ms]dde23 139Zeroth order EMHPM (119898 = 7) 77First order EMHPM (119898 = 7) 87

shown that our proposed EMHPM approach can be appliedto obtain the approximate solution of a delay differentialequation not only with constant but also with variablecoefficients with theoretical predictions that follow well thenumerical integration solutions To further assess the validityof this new approach we have compared the approximatesolutions of two delayed differential equations with respect totheir corresponding numerical integration solutions obtainedfrom the MATLAB dde23 algorithm The test cases were (a)the damped Mathieu differential equation with time delayand (b) the governing equation of motion of downmillingoperations We have found that the EMHPM closely followsthe numerical integration solutions of the correspondingequations and that these require less CPU time and havesmaller relative errors

Conflict of Interests

Theauthors declare that they have no conflict of interests withany mentioned entities in the paper

Acknowledgments

This work was funded by Tecnologico deMonterrey CampusMonterrey through the ResearchGroup inNanomaterials forMedical Devices and the Research Group in Advanced Man-ufacturing Additional support was provided from ConsejoNacional de Ciencia y Tecnologıa (Conacyt) Mexico

References

[1] F Shakeri and M Dehghan ldquoSolution of delay differentialequations via a homotopy perturbation methodrdquoMathematicaland Computer Modelling vol 48 no 3-4 pp 486ndash498 2008

[2] Z-K Wu ldquoSolution of the enso delayed oscillator with homo-topy analysis methodrdquo Journal of Hydrodynamics vol 21 no 1pp 131ndash135 2009

[3] A K Alomari M S Noorani and R Nazar ldquoSolution of delaydifferential equation by means of homotopy analysis methodrdquoActa Applicandae Mathematicae vol 108 no 2 pp 395ndash4122009

[4] J Biazar and G Behzad ldquoThe homotopy perturbation methodfor solving neutral functional-differential equations with pro-portional delaysrdquo Journal of King Saud UniversitymdashScience vol24 no 1 pp 33ndash37 2012

[5] NRAnakiraAKAlomari and IHashim ldquoOptimal homotopyasymptotic method for solving delay differential equationsrdquoMathematical Problems in Engineering vol 2013 Article ID498902 11 pages 2013

[6] T Insperger and G Stepan Semi-Discretization for Time-DelaySystems Stability and Engineering Applications Springer NewYork NY USA 2011

[7] E A Butcher P Nindujarla and E Bueler ldquoStability of up-and down-milling using chebyshev collocation methodrdquo inProceedings of the ASME International Design Engineering Tech-nical Conferences amp Computers and Information in EngineeringConference vol 6 part AndashC pp 841ndash850 2005

[8] T Insperger ldquoFull-discretization and semi-discretization formilling stability prediction some commentsrdquo InternationalJournal of Machine Tools and Manufacture vol 50 no 7 pp658ndash662 2010

[9] I Hashim and M S Chowdhury ldquoAdaptation of homotopy-perturbation method for numeric-analytic solution of systemof ODEsrdquo Physics Letters A vol 372 no 4 pp 470ndash481 2008

[10] L F Shampine and S Thompson ldquoSolving DDEs in MatlabrdquoAppliedNumericalMathematics vol 37 no 4 pp 441ndash458 2001

[11] SHHNiaANRanjbarDDGanjiH Soltani and JGhasemildquoMaintaining the stability of nonlinear differential equations bythe enhancement of HPMrdquo Physics Letters A vol 372 no 16 pp2855ndash2861 2008

[12] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[13] J-H He ldquoHomotopy perturbation method a new nonlinearanalytical techniquerdquo Applied Mathematics and Computationvol 135 no 1 pp 73ndash79 2003

[14] J H He ldquoComparison of homotopy perturbation method andhomotopy analysis methodrdquo Applied Mathematics and Compu-tation vol 156 no 2 pp 527ndash539 2004

[15] JHHe ldquoNew interpretation of homotopy perturbationmethodrdquoInternational Journal of Modern Physics B vol 20 no 18 pp2561ndash2568 2006

[16] S Liao ldquoComparison between the homotopy analysis methodand homotopy perturbationmethodrdquoAppliedMathematics andComputation vol 169 no 2 pp 1186ndash1194 2005

[17] D Olvera and A Elıas-Zuniga ldquoEnhanced multistage homo-topy perturbation method approximate solutions of nonlineardynamic systemsrdquo Abstract and Applied Analysis vol 2014Article ID 486509 12 pages 2014

12 Abstract and Applied Analysis

[18] T Insperger and G Stepan ldquoStability of the damped Mathieuequationwith timedelayrdquo Journal ofDynamic SystemsMeasure-ment and Control vol 125 no 2 pp 166ndash171 2003

[19] T Insperger and G Stepan ldquoStability chart for the delayedMathieu equationrdquo The Royal Society of London ProceedingsSeries A Mathematical Physical and Engineering Sciences vol458 no 2024 pp 1989ndash1998 2002

[20] T Insperger andGStepan ldquoUpdated semi-discretizationmethodfor periodic delay-differential equations with discrete delayrdquoInternational Journal for Numerical Methods in Engineering vol61 no 1 pp 117ndash141 2004

[21] T Insperger B P Mann G Stepan and P V Bayly ldquoStabilityof up-milling and down-milling part 1 alternative analyticalmethodsrdquo International Journal of Machine Tools and Manufac-ture vol 43 no 1 pp 25ndash34 2003

[22] F ICompeanDOlvera F J Campa LN L de Lacalle A Elıas-Zuniga and C A Rodrıguez ldquoCharacterization and stabilityanalysis of a multivariable milling tool by the enhanced mul-tistage homotopy perturbation methodrdquo International Journalof Machine Tools and Manufacture vol 57 pp 27ndash33 2012

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

Abstract and Applied Analysis 7

0 2 4 6 8 10 12

005

1

dde23Zeroth EMHPMFirst EMHPM

minus05minus05

minus15minus15

minus1minus1Am

plitu

dex

Time t

times10minus3

Figure 5 Numerical solutions of the damped Mathieu equationwith time delay by dde23 the zeroth EMHPM and the first EMHPMwith119873 = 50 and119898 = 4

Table 1 Computer time needed to solve the damped Mathieuequation with time delay The 119898 order solution of the EMPHMapproach is chosen to guarantee the convergence of its approximatesolution

dde23 [ms] EMHPM119873 119898 Zeroth [ms] First [ms]

19

15 5 4 520 4 6 640 3 12 1260 2 17 1860 5 18 1960 10 20 21

where

A119905= [0 1

minus120572119905minus120581] B

119905= [0 0

1198871199050] (45)

and 120572119905= (120575 + 120576 cos(119905)) is a time periodic term The EMHPM

approximate solution of (44) is illustrated in Figure 5 wherewe have assumed an unstable system behavior for which 120581 =02 120575 = 30 120576 = 1 119887 = minus1 and 119879 = 120591 = 2120587 See[20] As we can see from Figure 5 our approximate EMHPMsolution to (42) is compared with its numerical integrationsolution obtained from dde23 MATLAB algorithm for thetime interval of 2119879 by assuming that 119873 = 50 with thefollowing initial values 119909

minus50(119879) = 119909

minus49(119879) = sdot sdot sdot 119909

0(119879) =

0001 and minus50(119879) =

minus49(119879) = sdot sdot sdot

0(119879) = 0

It can be seen fromFigure 5 that in the interval [0119879] boththe zeroth and the first order solutions are the same since thedelay subintervals are constant See Figure 3 However in thenext interval [119879 2119879] it is clear that the first order EMPHMsolution provides a better approximation on the delay subin-terval The computation total time to calculate the solutionsin the MATLAB code is listed in Table 1 The order 119898 andthe discretized time intervals 119873 in the EMPHM approachare chosen to guarantee the convergence of our approximatesolution to the exact one To provide a full understanding ofhow the solution is computed by the EMHPM approach weattached in Algorithm 1 the corresponding MATLAB code

10 20 30 40 50 60 70 80 90 100

Abso

lute

erro

r

10minus4

10minus3

10minus2

Zeroth order m = 2

First order m = 2

Zeroth order m = 5

First order m = 5

Zeroth order m = 10

First order m = 10

Discretizations N

Figure 6 Estimated relative error values between the numericalsolution dde23 and the EMHPMapproximate solutionsHerewe usefor the EMHPM the values of119898 = 2 5 and 10

Figure 6 shows the relative error between our approxi-mate EMPHM and the dde23 solution and its relationshipwith the order119898 and the discretized time intervals119873 Noticethat the relative error values coincide at values of 119873 ge 45Also we can see from Figure 6 that the computed relativeerror values for approximate solutions of order119898 ge 5 remainunchanged

52 A Practical Application Cutting Operation on MillingMachine We next use our EMHPM procedure to obtainthe solution of the single degree-of-freedom milling opera-tion We use the simplified form based on [20ndash22]

(119905) + 2120577120596119899 (119905) + 1205962

119899119909 (119905) = minus

119886119901119870119904 (119905)

119898119898

(119909 (119905) minus 119909 (119905 minus 120591))

(46)

where 120596119899is the angular natural frequency of the system 120577 is

the damping ratio 119886119901is the depth of cut 119898

119898is the modal

mass of the tool 120591 represents the time delay which is equalto the tooth passing period and 119870

119904(119905) is the specific cutting

force coefficient which can be determined from119870119904 (119905)

=

119911119899

sum

119895=1

119892 (120601119895(119905)) sin (120601

119895(119905)) (119870

119905cos120601119895(119905) + 119870

119899sin120601119895(119905))

(47)

where 119911119899is the tool number of teeth 119870

119905and 119870

119899are the

tangential and the normal linear cutting force coefficientsrespectively 120601

119895(119905) is the angular position of the 119895-tooth

defined as

120601119895(119905) = (

2120587119899

60) 119905 +

2120587119895

119911119899

(48)

and 119899 is the spindle speed in rpm [20] The function 119892(120593119895(119905))

is a switching function which has a unity value when the 119895-tooth is cutting and zero otherwise

119892 (120601119895 (119905)) =

1 120601st lt 120601119895 (119905) lt 120601ex0 otherwise

(49)

8 Abstract and Applied Analysis

function dde23 ddeEM mathieu paper

disp(lsquoMathieu equation solutionrsquo) Solution by EMHPM with zeroth and first order solution

eps=1 kappa=02 T=2lowastpi tau=T b=minus1 delta=3 Mathieu Parameters

pntDelay=1 N=50 m ord=5 dt=tau(Nminus1) ktau=2 tspan=[0ktaulowastT] EMHPM Parameters

Solution

Solution by dde23

tdde=linspace(0ktaulowasttauktaulowast(Nminus1)+1)

dde=(tyz) mathieu dde(tyzkappadeltaepsbtau)

te dde=ticsol = dde23(ddetauhistorytspan) toc(te dde) xdde=deval(soltdde)

Solution by zeroth EMHPM

dde emhpm fun=(tc0ttzmzntauN) mathieu zeroth(tc0ttzmzntauNm ord

kappadeltaepsbT)

tic[t0z0]=ddeEMHPM(dde emhpm funtspanhistorytaudtpntDelay) toc

Solution by First EMHPM

dde emhpm fun=(tc0ttzmzntauN) mathieu first(tc0ttzmzntauNm ordkappadeltaepsbT)

tic[t1z1]=ddeEMHPM(dde emhpm funtspanhistorytaudtpntDelay) toc

Plot results

ind0=12ktaulowast(Nminus1) ind1=22ktaulowast(Nminus1)

Parent1=figure (1)

axes1 = axes(lsquoParentrsquoParent1lsquoFontSizersquo12lsquoFontNamersquolsquoTimes New Romanrsquo)box(axes1lsquoonrsquo) hold(axes1lsquoallrsquo)plot(tddexdde(1)lsquoParentrsquoaxes1lsquoLineWidthrsquo2lsquoColorrsquo[0502 0502 0502]lsquoDisplayNamersquolsquoNumericaldd23rsquo)plot(t0(ind0)z0(ind01)lsquoMarkerSizersquo5lsquoMarkerrsquolsquoorsquolsquoLineStylersquo lsquononersquolsquoDisplayNamersquolsquoZerothEMHPMrsquolsquoColorrsquo[0 0 0])

plot(t1(ind1)z1(ind11)lsquoMarkerSizersquo7lsquoMarkerrsquolsquoxrsquolsquoLineStylersquolsquononersquolsquoDisplayNamersquolsquoFirstEMHPMrsquolsquoColorrsquo[0 0 0])

xlabel(lsquoittrsquolsquoFontSizersquo12lsquoFontNamersquolsquoTimes New Romanrsquo)ylabel(lsquoitxrsquolsquoFontSizersquo12lsquoFontNamersquolsquoTimes New Romanrsquo)end

Mathieu definitions

function dydt = mathieu dde(tyzkapadltepsbT)

dydt = [y(2)-kapalowasty(2)minus(dlt+epslowastcos(2lowastpilowasttT))lowasty(1)+blowastz(1)]

end

function Z = mathieu zeroth(tc0ttzmzntauNmkpadltepsbT)

Z=[c0(1)c0(2)] alf=dlt+epslowastcos(2lowastpiTlowasttt)

for ik=1m

Z(ik+11)=Z(ik2)lowasttik

Z(ik+12)=minuskpalowastZ(ik2)minusalflowastZ(ik1)

if ik==1 Z(ik+12)=Z(ik+12)+blowastzm(1) end

Z(ik+12)=Z(ik+12)lowasttik

end

Z=sum(Z)

end

function Z = mathieu first(tc0ttzmzntauNmkpadltepsbT)

alf=dlt+epslowastcos(2lowastpiTlowasttt) Z=[c0(1)c0(2)] Z =[00]

for ik=1m

Z(ik+1)=[Z(ik2)lowasttik minuskpalowastZ(ik2)minusalflowastZ(ik1)]

Z (ik+1)=[Z (ik2)lowasttik minuskpalowastZ (ik2)minusalflowastZ (ik1)]

if ik==1

Z(ik+12)=Z(ik+12)+blowastzm(1)Z (ik+12)=Z (ik+12)+blowast(Nminus1)taulowast(zn(1)minuszm(1))lowastt

end

Z(ik+12)=Z(ik+12)lowasttik

Z (ik+12)=Z (ik+12)lowastt(ik+1)

Algorithm 1 Continued

Abstract and Applied Analysis 9

end

Z=sum(Z)+sum(Z )

end

function out=history(t)

out=[1Eminus3+0lowastt 0+0lowastt]

end

EMHPM algorithm for ODE solutions

function [tz]= odeEMHPM(ndetspanz0Deltatpnts)

ode solver by Enhanced Multistage Homotopy Perturbation Method

tini=tspan(1) tfin=tspan(end) tstart=tini

tini=tiniminuststart tfin=tfinminuststart shifted time set to zero

Handle errors

if tini == tfin

error(lsquoThe ending and starting time values must be differentrsquo)elseif abs(tini)gt abs(tfin)

tspan=flipud(fliplr(tspan)) tini=tspan(1) tfin=tspan(end)

end

tdir=sign(tfinminustini)

if any(tdirlowastdiff(tspan) lt= 0)

error(lsquotspan entries must be strictly sortedrsquo)end

incT=Deltatpnts

if incTlt=0

error (lsquoIncreasing must be greater than zerorsquo)end

z(1)=z01015840 t(1)=tini iteT=2 set initial values

t(iteT)=tini+incTlowasttdir

while tdirlowastt(iteT)lttdirlowast(tfin+tdirlowastincT)

P act=ceil(99999lowast(t(iteT)minustini)lowasttdirDeltat) count sub-intervals

c=z((DeltatincT)lowast(P actminus1)+1) set the corresponding initial condition

tsub(iteTminus1)=t(iteT)-(P actminus1)lowastDeltatlowasttdir evaluate the solution at the shifted time

temp=tsub(iteTminus1)

z(iteT)=nde(tempctstart+t(iteT))

if tdirlowastt(iteT)gt=tdirlowasttfinlowast099999 repeat for the next sub-interval

break

else

iteT=iteT+1 t(iteT)=tini+(iteTminus1)lowastincTlowasttdir

end

end

t=t 1015840 +tstart

end

EMHPM algorithm for DDE solutions

function [t z]=ddeEMHPM(ddetspanhistorytauDeltatpntDelta)

dde solver by Enhanced Multistage Homotopy Perturbation Method

tini=tspan(1) tfin=tspan(end) span where the solution is founded

if taultDeltat

error(lsquoSubtinterval must not be greater than taursquo)end

if mod(tauDeltat)sim=0

pastDlt=Deltat

Deltat=tauround(tauDeltat)

warning(lsquoSubtinterval was modified from 05g to 05grsquopastDltDeltat)end

pnt=round(tauDeltat) samplesminus1 [minustau 0]

incT=DeltatpntDelta step for set resolution

t=minustau+tini+(0pntlowastpntDelta)rsquolowastincT evaluation of the initial solution [minustau 0]

z=history(t) initial behavior in the inverval [minustau 0]

c=z(end) initial condition

Algorithm 1 Continued

10 Abstract and Applied Analysis

m=pntlowastpntDelta samples between tau

itDlt=0

while (t(end)+eps)lttfin

tspan=[tini+itDltlowastDeltattini+(itDlt+1)lowastDeltat] preparing the span for next Deltat

zm=z(length(z)minusm)rsquo zn=z(1+length(z)minusm)rsquo previous tau solution

emhpm fun=(timecT) dde(timecTzmzntaum) application of the odeEMHPM

[t aux z aux]=odeEMHPM(emhpm funtspanz(end)DeltatpntDelta)

z=[z(1endminus1)z aux] t=[t(1endminus1)t aux] joining the solutions

itDlt=itDlt+1

end

z=z(pntlowastpntDelta+1end)t=t(pntlowastpntDelta+1end)

end

Algorithm 1 MATLAB algorithm

Here 120601st and 120601ex are the angles where the teeth enter and exitthe workpiece For upmilling 120601st = 0 and 120601ex = arccos(1 minus2119886119889) for downmilling 120601st = arccos(2119886

119889minus 1) and 120601ex = 120587

where 119886119889is the radial depth of cut ratio

By following the EMHPM procedure we can write (46)in equivalent form as

119894(119879) + 2120577120596

119899119894(119879) + 120596

2

119899119909119894(119879)

asymp minus

119886119901119870st

119898119898

(119909119894 (119879) minus 119909119894minus119873+1 (119879))

(50)

where 119909119894(119879) denotes the 119898 order solution of (46) on the 119894th

subinterval that satisfies the initial conditions 119909119894(0) = 119909

119894minus1

119894(0) =

119894minus1 and ℎ

119905= ℎ(119905) and 119909

minus120591is given by (35)

Introducing the transformation x119894= [119909119894 119894]119879 (50) can be

written as a system of first order linear delay differentialequations of the form

x119894(119879) = A

119905x119894(119879) + B

119905x119894minus119873+1

(119879) (51)

where

A119905= [

[

0 1

minus1205962

119899minus119908

119898119898

119870st minus2120577120596119899]

]

B119905= [

[

0 0119908

119898119898

119870st 0]

]

(52)

We next apply the EMHPM procedure to solve (46) byconsidering a downmilling operation with the followingparameter values 119911

119899= 2 119886

119889= 01 120596

119899= 5793 rads

120577 = 0011 119898119898= 003993 kg 119870

119905= 6 times 10

8Nm2 and119870119899= 2 times 10

8Nm2 As we can see from Figures 7 and 8 andfor the depth of cut values of 119886

119901= 2mm (stable) and 119886

119901=

3mm (unstable) our EMHPM approximate solutions followclosely the numerical integration solutions of (46) obtainedby using the dde23 algorithm

Figure 9 shows the relative error between the EMPHMand the dde23 numerical solution while Table 2 shows theCPU time needed for each solution Here we use 119873 = 75since the average step size of the dde23 algorithm is aroundΔ119905 asymp 120591119873 Note that for 119898 = 7 the zeroth order EMHPMapproximate solution has the fastest CPU time We can seefrom Figure 9 that the value of the relative error becomesbasically the same for119898 = 2 7 and 10 and119873 ge 20

0 0005 001 0015 002

0

1

2

Time (s)

Disp

lace

men

t (m

)

dd23Zeroth EMHPMFirst EMHPM

minus2

minus1

times10minus3

Figure 7 EMHPM approximate solutions of (46) with parametervalues of 119886

119901= 2mm 119899 = 10000 rpm Stable machine operation

0 0005 001 0015 002

00005

0010015

Time (s)

Disp

lace

men

t (m

)

dd23Zeroth EMHPMFirst EMHPM

minus001

minus0005

Figure 8 EMHPM approximate solutions of (46) with parametervalues of 119886

119901= 3mmand 119899 = 10000 rpmUnstable (chatter)machine

operation

6 Conclusions

We have developed a new algorithm based on the homotopyperturbation method to solve delay differential equationsThe proposed EMHPM approach is based on a sequence ofsubintervals that approximate the solution of delayed differ-ential equations by using the transformation rule 119906(119905) rarr119906119894(119879) where 119906

119894satisfies the initial conditions We have

Abstract and Applied Analysis 11

10 20 30 40 50 60 70 80 90 100

Abso

lute

erro

r

Zeroth order m = 2

First order m = 2

Zeroth order m = 7

First order m = 7

Zeroth order m = 10

First order m = 10

10minus3

10minus2

10minus1

Discretizations N

Figure 9 Estimated relative error values between dde23 and theEMHPM approximate solutions Here we have used the systemparameter values of 119886

119901= 2mm and 119899 = 1000 rpm and values of

119898 = 2 7 and 10

Table 2 CPU time comparison among the approximate solutions of(46) by using dde23 the zeroth order and the first order EMHPMsolutions

Solution method Time [ms]dde23 139Zeroth order EMHPM (119898 = 7) 77First order EMHPM (119898 = 7) 87

shown that our proposed EMHPM approach can be appliedto obtain the approximate solution of a delay differentialequation not only with constant but also with variablecoefficients with theoretical predictions that follow well thenumerical integration solutions To further assess the validityof this new approach we have compared the approximatesolutions of two delayed differential equations with respect totheir corresponding numerical integration solutions obtainedfrom the MATLAB dde23 algorithm The test cases were (a)the damped Mathieu differential equation with time delayand (b) the governing equation of motion of downmillingoperations We have found that the EMHPM closely followsthe numerical integration solutions of the correspondingequations and that these require less CPU time and havesmaller relative errors

Conflict of Interests

Theauthors declare that they have no conflict of interests withany mentioned entities in the paper

Acknowledgments

This work was funded by Tecnologico deMonterrey CampusMonterrey through the ResearchGroup inNanomaterials forMedical Devices and the Research Group in Advanced Man-ufacturing Additional support was provided from ConsejoNacional de Ciencia y Tecnologıa (Conacyt) Mexico

References

[1] F Shakeri and M Dehghan ldquoSolution of delay differentialequations via a homotopy perturbation methodrdquoMathematicaland Computer Modelling vol 48 no 3-4 pp 486ndash498 2008

[2] Z-K Wu ldquoSolution of the enso delayed oscillator with homo-topy analysis methodrdquo Journal of Hydrodynamics vol 21 no 1pp 131ndash135 2009

[3] A K Alomari M S Noorani and R Nazar ldquoSolution of delaydifferential equation by means of homotopy analysis methodrdquoActa Applicandae Mathematicae vol 108 no 2 pp 395ndash4122009

[4] J Biazar and G Behzad ldquoThe homotopy perturbation methodfor solving neutral functional-differential equations with pro-portional delaysrdquo Journal of King Saud UniversitymdashScience vol24 no 1 pp 33ndash37 2012

[5] NRAnakiraAKAlomari and IHashim ldquoOptimal homotopyasymptotic method for solving delay differential equationsrdquoMathematical Problems in Engineering vol 2013 Article ID498902 11 pages 2013

[6] T Insperger and G Stepan Semi-Discretization for Time-DelaySystems Stability and Engineering Applications Springer NewYork NY USA 2011

[7] E A Butcher P Nindujarla and E Bueler ldquoStability of up-and down-milling using chebyshev collocation methodrdquo inProceedings of the ASME International Design Engineering Tech-nical Conferences amp Computers and Information in EngineeringConference vol 6 part AndashC pp 841ndash850 2005

[8] T Insperger ldquoFull-discretization and semi-discretization formilling stability prediction some commentsrdquo InternationalJournal of Machine Tools and Manufacture vol 50 no 7 pp658ndash662 2010

[9] I Hashim and M S Chowdhury ldquoAdaptation of homotopy-perturbation method for numeric-analytic solution of systemof ODEsrdquo Physics Letters A vol 372 no 4 pp 470ndash481 2008

[10] L F Shampine and S Thompson ldquoSolving DDEs in MatlabrdquoAppliedNumericalMathematics vol 37 no 4 pp 441ndash458 2001

[11] SHHNiaANRanjbarDDGanjiH Soltani and JGhasemildquoMaintaining the stability of nonlinear differential equations bythe enhancement of HPMrdquo Physics Letters A vol 372 no 16 pp2855ndash2861 2008

[12] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[13] J-H He ldquoHomotopy perturbation method a new nonlinearanalytical techniquerdquo Applied Mathematics and Computationvol 135 no 1 pp 73ndash79 2003

[14] J H He ldquoComparison of homotopy perturbation method andhomotopy analysis methodrdquo Applied Mathematics and Compu-tation vol 156 no 2 pp 527ndash539 2004

[15] JHHe ldquoNew interpretation of homotopy perturbationmethodrdquoInternational Journal of Modern Physics B vol 20 no 18 pp2561ndash2568 2006

[16] S Liao ldquoComparison between the homotopy analysis methodand homotopy perturbationmethodrdquoAppliedMathematics andComputation vol 169 no 2 pp 1186ndash1194 2005

[17] D Olvera and A Elıas-Zuniga ldquoEnhanced multistage homo-topy perturbation method approximate solutions of nonlineardynamic systemsrdquo Abstract and Applied Analysis vol 2014Article ID 486509 12 pages 2014

12 Abstract and Applied Analysis

[18] T Insperger and G Stepan ldquoStability of the damped Mathieuequationwith timedelayrdquo Journal ofDynamic SystemsMeasure-ment and Control vol 125 no 2 pp 166ndash171 2003

[19] T Insperger and G Stepan ldquoStability chart for the delayedMathieu equationrdquo The Royal Society of London ProceedingsSeries A Mathematical Physical and Engineering Sciences vol458 no 2024 pp 1989ndash1998 2002

[20] T Insperger andGStepan ldquoUpdated semi-discretizationmethodfor periodic delay-differential equations with discrete delayrdquoInternational Journal for Numerical Methods in Engineering vol61 no 1 pp 117ndash141 2004

[21] T Insperger B P Mann G Stepan and P V Bayly ldquoStabilityof up-milling and down-milling part 1 alternative analyticalmethodsrdquo International Journal of Machine Tools and Manufac-ture vol 43 no 1 pp 25ndash34 2003

[22] F ICompeanDOlvera F J Campa LN L de Lacalle A Elıas-Zuniga and C A Rodrıguez ldquoCharacterization and stabilityanalysis of a multivariable milling tool by the enhanced mul-tistage homotopy perturbation methodrdquo International Journalof Machine Tools and Manufacture vol 57 pp 27ndash33 2012

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 Abstract and Applied Analysis

function dde23 ddeEM mathieu paper

disp(lsquoMathieu equation solutionrsquo) Solution by EMHPM with zeroth and first order solution

eps=1 kappa=02 T=2lowastpi tau=T b=minus1 delta=3 Mathieu Parameters

pntDelay=1 N=50 m ord=5 dt=tau(Nminus1) ktau=2 tspan=[0ktaulowastT] EMHPM Parameters

Solution

Solution by dde23

tdde=linspace(0ktaulowasttauktaulowast(Nminus1)+1)

dde=(tyz) mathieu dde(tyzkappadeltaepsbtau)

te dde=ticsol = dde23(ddetauhistorytspan) toc(te dde) xdde=deval(soltdde)

Solution by zeroth EMHPM

dde emhpm fun=(tc0ttzmzntauN) mathieu zeroth(tc0ttzmzntauNm ord

kappadeltaepsbT)

tic[t0z0]=ddeEMHPM(dde emhpm funtspanhistorytaudtpntDelay) toc

Solution by First EMHPM

dde emhpm fun=(tc0ttzmzntauN) mathieu first(tc0ttzmzntauNm ordkappadeltaepsbT)

tic[t1z1]=ddeEMHPM(dde emhpm funtspanhistorytaudtpntDelay) toc

Plot results

ind0=12ktaulowast(Nminus1) ind1=22ktaulowast(Nminus1)

Parent1=figure (1)

axes1 = axes(lsquoParentrsquoParent1lsquoFontSizersquo12lsquoFontNamersquolsquoTimes New Romanrsquo)box(axes1lsquoonrsquo) hold(axes1lsquoallrsquo)plot(tddexdde(1)lsquoParentrsquoaxes1lsquoLineWidthrsquo2lsquoColorrsquo[0502 0502 0502]lsquoDisplayNamersquolsquoNumericaldd23rsquo)plot(t0(ind0)z0(ind01)lsquoMarkerSizersquo5lsquoMarkerrsquolsquoorsquolsquoLineStylersquo lsquononersquolsquoDisplayNamersquolsquoZerothEMHPMrsquolsquoColorrsquo[0 0 0])

plot(t1(ind1)z1(ind11)lsquoMarkerSizersquo7lsquoMarkerrsquolsquoxrsquolsquoLineStylersquolsquononersquolsquoDisplayNamersquolsquoFirstEMHPMrsquolsquoColorrsquo[0 0 0])

xlabel(lsquoittrsquolsquoFontSizersquo12lsquoFontNamersquolsquoTimes New Romanrsquo)ylabel(lsquoitxrsquolsquoFontSizersquo12lsquoFontNamersquolsquoTimes New Romanrsquo)end

Mathieu definitions

function dydt = mathieu dde(tyzkapadltepsbT)

dydt = [y(2)-kapalowasty(2)minus(dlt+epslowastcos(2lowastpilowasttT))lowasty(1)+blowastz(1)]

end

function Z = mathieu zeroth(tc0ttzmzntauNmkpadltepsbT)

Z=[c0(1)c0(2)] alf=dlt+epslowastcos(2lowastpiTlowasttt)

for ik=1m

Z(ik+11)=Z(ik2)lowasttik

Z(ik+12)=minuskpalowastZ(ik2)minusalflowastZ(ik1)

if ik==1 Z(ik+12)=Z(ik+12)+blowastzm(1) end

Z(ik+12)=Z(ik+12)lowasttik

end

Z=sum(Z)

end

function Z = mathieu first(tc0ttzmzntauNmkpadltepsbT)

alf=dlt+epslowastcos(2lowastpiTlowasttt) Z=[c0(1)c0(2)] Z =[00]

for ik=1m

Z(ik+1)=[Z(ik2)lowasttik minuskpalowastZ(ik2)minusalflowastZ(ik1)]

Z (ik+1)=[Z (ik2)lowasttik minuskpalowastZ (ik2)minusalflowastZ (ik1)]

if ik==1

Z(ik+12)=Z(ik+12)+blowastzm(1)Z (ik+12)=Z (ik+12)+blowast(Nminus1)taulowast(zn(1)minuszm(1))lowastt

end

Z(ik+12)=Z(ik+12)lowasttik

Z (ik+12)=Z (ik+12)lowastt(ik+1)

Algorithm 1 Continued

Abstract and Applied Analysis 9

end

Z=sum(Z)+sum(Z )

end

function out=history(t)

out=[1Eminus3+0lowastt 0+0lowastt]

end

EMHPM algorithm for ODE solutions

function [tz]= odeEMHPM(ndetspanz0Deltatpnts)

ode solver by Enhanced Multistage Homotopy Perturbation Method

tini=tspan(1) tfin=tspan(end) tstart=tini

tini=tiniminuststart tfin=tfinminuststart shifted time set to zero

Handle errors

if tini == tfin

error(lsquoThe ending and starting time values must be differentrsquo)elseif abs(tini)gt abs(tfin)

tspan=flipud(fliplr(tspan)) tini=tspan(1) tfin=tspan(end)

end

tdir=sign(tfinminustini)

if any(tdirlowastdiff(tspan) lt= 0)

error(lsquotspan entries must be strictly sortedrsquo)end

incT=Deltatpnts

if incTlt=0

error (lsquoIncreasing must be greater than zerorsquo)end

z(1)=z01015840 t(1)=tini iteT=2 set initial values

t(iteT)=tini+incTlowasttdir

while tdirlowastt(iteT)lttdirlowast(tfin+tdirlowastincT)

P act=ceil(99999lowast(t(iteT)minustini)lowasttdirDeltat) count sub-intervals

c=z((DeltatincT)lowast(P actminus1)+1) set the corresponding initial condition

tsub(iteTminus1)=t(iteT)-(P actminus1)lowastDeltatlowasttdir evaluate the solution at the shifted time

temp=tsub(iteTminus1)

z(iteT)=nde(tempctstart+t(iteT))

if tdirlowastt(iteT)gt=tdirlowasttfinlowast099999 repeat for the next sub-interval

break

else

iteT=iteT+1 t(iteT)=tini+(iteTminus1)lowastincTlowasttdir

end

end

t=t 1015840 +tstart

end

EMHPM algorithm for DDE solutions

function [t z]=ddeEMHPM(ddetspanhistorytauDeltatpntDelta)

dde solver by Enhanced Multistage Homotopy Perturbation Method

tini=tspan(1) tfin=tspan(end) span where the solution is founded

if taultDeltat

error(lsquoSubtinterval must not be greater than taursquo)end

if mod(tauDeltat)sim=0

pastDlt=Deltat

Deltat=tauround(tauDeltat)

warning(lsquoSubtinterval was modified from 05g to 05grsquopastDltDeltat)end

pnt=round(tauDeltat) samplesminus1 [minustau 0]

incT=DeltatpntDelta step for set resolution

t=minustau+tini+(0pntlowastpntDelta)rsquolowastincT evaluation of the initial solution [minustau 0]

z=history(t) initial behavior in the inverval [minustau 0]

c=z(end) initial condition

Algorithm 1 Continued

10 Abstract and Applied Analysis

m=pntlowastpntDelta samples between tau

itDlt=0

while (t(end)+eps)lttfin

tspan=[tini+itDltlowastDeltattini+(itDlt+1)lowastDeltat] preparing the span for next Deltat

zm=z(length(z)minusm)rsquo zn=z(1+length(z)minusm)rsquo previous tau solution

emhpm fun=(timecT) dde(timecTzmzntaum) application of the odeEMHPM

[t aux z aux]=odeEMHPM(emhpm funtspanz(end)DeltatpntDelta)

z=[z(1endminus1)z aux] t=[t(1endminus1)t aux] joining the solutions

itDlt=itDlt+1

end

z=z(pntlowastpntDelta+1end)t=t(pntlowastpntDelta+1end)

end

Algorithm 1 MATLAB algorithm

Here 120601st and 120601ex are the angles where the teeth enter and exitthe workpiece For upmilling 120601st = 0 and 120601ex = arccos(1 minus2119886119889) for downmilling 120601st = arccos(2119886

119889minus 1) and 120601ex = 120587

where 119886119889is the radial depth of cut ratio

By following the EMHPM procedure we can write (46)in equivalent form as

119894(119879) + 2120577120596

119899119894(119879) + 120596

2

119899119909119894(119879)

asymp minus

119886119901119870st

119898119898

(119909119894 (119879) minus 119909119894minus119873+1 (119879))

(50)

where 119909119894(119879) denotes the 119898 order solution of (46) on the 119894th

subinterval that satisfies the initial conditions 119909119894(0) = 119909

119894minus1

119894(0) =

119894minus1 and ℎ

119905= ℎ(119905) and 119909

minus120591is given by (35)

Introducing the transformation x119894= [119909119894 119894]119879 (50) can be

written as a system of first order linear delay differentialequations of the form

x119894(119879) = A

119905x119894(119879) + B

119905x119894minus119873+1

(119879) (51)

where

A119905= [

[

0 1

minus1205962

119899minus119908

119898119898

119870st minus2120577120596119899]

]

B119905= [

[

0 0119908

119898119898

119870st 0]

]

(52)

We next apply the EMHPM procedure to solve (46) byconsidering a downmilling operation with the followingparameter values 119911

119899= 2 119886

119889= 01 120596

119899= 5793 rads

120577 = 0011 119898119898= 003993 kg 119870

119905= 6 times 10

8Nm2 and119870119899= 2 times 10

8Nm2 As we can see from Figures 7 and 8 andfor the depth of cut values of 119886

119901= 2mm (stable) and 119886

119901=

3mm (unstable) our EMHPM approximate solutions followclosely the numerical integration solutions of (46) obtainedby using the dde23 algorithm

Figure 9 shows the relative error between the EMPHMand the dde23 numerical solution while Table 2 shows theCPU time needed for each solution Here we use 119873 = 75since the average step size of the dde23 algorithm is aroundΔ119905 asymp 120591119873 Note that for 119898 = 7 the zeroth order EMHPMapproximate solution has the fastest CPU time We can seefrom Figure 9 that the value of the relative error becomesbasically the same for119898 = 2 7 and 10 and119873 ge 20

0 0005 001 0015 002

0

1

2

Time (s)

Disp

lace

men

t (m

)

dd23Zeroth EMHPMFirst EMHPM

minus2

minus1

times10minus3

Figure 7 EMHPM approximate solutions of (46) with parametervalues of 119886

119901= 2mm 119899 = 10000 rpm Stable machine operation

0 0005 001 0015 002

00005

0010015

Time (s)

Disp

lace

men

t (m

)

dd23Zeroth EMHPMFirst EMHPM

minus001

minus0005

Figure 8 EMHPM approximate solutions of (46) with parametervalues of 119886

119901= 3mmand 119899 = 10000 rpmUnstable (chatter)machine

operation

6 Conclusions

We have developed a new algorithm based on the homotopyperturbation method to solve delay differential equationsThe proposed EMHPM approach is based on a sequence ofsubintervals that approximate the solution of delayed differ-ential equations by using the transformation rule 119906(119905) rarr119906119894(119879) where 119906

119894satisfies the initial conditions We have

Abstract and Applied Analysis 11

10 20 30 40 50 60 70 80 90 100

Abso

lute

erro

r

Zeroth order m = 2

First order m = 2

Zeroth order m = 7

First order m = 7

Zeroth order m = 10

First order m = 10

10minus3

10minus2

10minus1

Discretizations N

Figure 9 Estimated relative error values between dde23 and theEMHPM approximate solutions Here we have used the systemparameter values of 119886

119901= 2mm and 119899 = 1000 rpm and values of

119898 = 2 7 and 10

Table 2 CPU time comparison among the approximate solutions of(46) by using dde23 the zeroth order and the first order EMHPMsolutions

Solution method Time [ms]dde23 139Zeroth order EMHPM (119898 = 7) 77First order EMHPM (119898 = 7) 87

shown that our proposed EMHPM approach can be appliedto obtain the approximate solution of a delay differentialequation not only with constant but also with variablecoefficients with theoretical predictions that follow well thenumerical integration solutions To further assess the validityof this new approach we have compared the approximatesolutions of two delayed differential equations with respect totheir corresponding numerical integration solutions obtainedfrom the MATLAB dde23 algorithm The test cases were (a)the damped Mathieu differential equation with time delayand (b) the governing equation of motion of downmillingoperations We have found that the EMHPM closely followsthe numerical integration solutions of the correspondingequations and that these require less CPU time and havesmaller relative errors

Conflict of Interests

Theauthors declare that they have no conflict of interests withany mentioned entities in the paper

Acknowledgments

This work was funded by Tecnologico deMonterrey CampusMonterrey through the ResearchGroup inNanomaterials forMedical Devices and the Research Group in Advanced Man-ufacturing Additional support was provided from ConsejoNacional de Ciencia y Tecnologıa (Conacyt) Mexico

References

[1] F Shakeri and M Dehghan ldquoSolution of delay differentialequations via a homotopy perturbation methodrdquoMathematicaland Computer Modelling vol 48 no 3-4 pp 486ndash498 2008

[2] Z-K Wu ldquoSolution of the enso delayed oscillator with homo-topy analysis methodrdquo Journal of Hydrodynamics vol 21 no 1pp 131ndash135 2009

[3] A K Alomari M S Noorani and R Nazar ldquoSolution of delaydifferential equation by means of homotopy analysis methodrdquoActa Applicandae Mathematicae vol 108 no 2 pp 395ndash4122009

[4] J Biazar and G Behzad ldquoThe homotopy perturbation methodfor solving neutral functional-differential equations with pro-portional delaysrdquo Journal of King Saud UniversitymdashScience vol24 no 1 pp 33ndash37 2012

[5] NRAnakiraAKAlomari and IHashim ldquoOptimal homotopyasymptotic method for solving delay differential equationsrdquoMathematical Problems in Engineering vol 2013 Article ID498902 11 pages 2013

[6] T Insperger and G Stepan Semi-Discretization for Time-DelaySystems Stability and Engineering Applications Springer NewYork NY USA 2011

[7] E A Butcher P Nindujarla and E Bueler ldquoStability of up-and down-milling using chebyshev collocation methodrdquo inProceedings of the ASME International Design Engineering Tech-nical Conferences amp Computers and Information in EngineeringConference vol 6 part AndashC pp 841ndash850 2005

[8] T Insperger ldquoFull-discretization and semi-discretization formilling stability prediction some commentsrdquo InternationalJournal of Machine Tools and Manufacture vol 50 no 7 pp658ndash662 2010

[9] I Hashim and M S Chowdhury ldquoAdaptation of homotopy-perturbation method for numeric-analytic solution of systemof ODEsrdquo Physics Letters A vol 372 no 4 pp 470ndash481 2008

[10] L F Shampine and S Thompson ldquoSolving DDEs in MatlabrdquoAppliedNumericalMathematics vol 37 no 4 pp 441ndash458 2001

[11] SHHNiaANRanjbarDDGanjiH Soltani and JGhasemildquoMaintaining the stability of nonlinear differential equations bythe enhancement of HPMrdquo Physics Letters A vol 372 no 16 pp2855ndash2861 2008

[12] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[13] J-H He ldquoHomotopy perturbation method a new nonlinearanalytical techniquerdquo Applied Mathematics and Computationvol 135 no 1 pp 73ndash79 2003

[14] J H He ldquoComparison of homotopy perturbation method andhomotopy analysis methodrdquo Applied Mathematics and Compu-tation vol 156 no 2 pp 527ndash539 2004

[15] JHHe ldquoNew interpretation of homotopy perturbationmethodrdquoInternational Journal of Modern Physics B vol 20 no 18 pp2561ndash2568 2006

[16] S Liao ldquoComparison between the homotopy analysis methodand homotopy perturbationmethodrdquoAppliedMathematics andComputation vol 169 no 2 pp 1186ndash1194 2005

[17] D Olvera and A Elıas-Zuniga ldquoEnhanced multistage homo-topy perturbation method approximate solutions of nonlineardynamic systemsrdquo Abstract and Applied Analysis vol 2014Article ID 486509 12 pages 2014

12 Abstract and Applied Analysis

[18] T Insperger and G Stepan ldquoStability of the damped Mathieuequationwith timedelayrdquo Journal ofDynamic SystemsMeasure-ment and Control vol 125 no 2 pp 166ndash171 2003

[19] T Insperger and G Stepan ldquoStability chart for the delayedMathieu equationrdquo The Royal Society of London ProceedingsSeries A Mathematical Physical and Engineering Sciences vol458 no 2024 pp 1989ndash1998 2002

[20] T Insperger andGStepan ldquoUpdated semi-discretizationmethodfor periodic delay-differential equations with discrete delayrdquoInternational Journal for Numerical Methods in Engineering vol61 no 1 pp 117ndash141 2004

[21] T Insperger B P Mann G Stepan and P V Bayly ldquoStabilityof up-milling and down-milling part 1 alternative analyticalmethodsrdquo International Journal of Machine Tools and Manufac-ture vol 43 no 1 pp 25ndash34 2003

[22] F ICompeanDOlvera F J Campa LN L de Lacalle A Elıas-Zuniga and C A Rodrıguez ldquoCharacterization and stabilityanalysis of a multivariable milling tool by the enhanced mul-tistage homotopy perturbation methodrdquo International Journalof Machine Tools and Manufacture vol 57 pp 27ndash33 2012

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

Abstract and Applied Analysis 9

end

Z=sum(Z)+sum(Z )

end

function out=history(t)

out=[1Eminus3+0lowastt 0+0lowastt]

end

EMHPM algorithm for ODE solutions

function [tz]= odeEMHPM(ndetspanz0Deltatpnts)

ode solver by Enhanced Multistage Homotopy Perturbation Method

tini=tspan(1) tfin=tspan(end) tstart=tini

tini=tiniminuststart tfin=tfinminuststart shifted time set to zero

Handle errors

if tini == tfin

error(lsquoThe ending and starting time values must be differentrsquo)elseif abs(tini)gt abs(tfin)

tspan=flipud(fliplr(tspan)) tini=tspan(1) tfin=tspan(end)

end

tdir=sign(tfinminustini)

if any(tdirlowastdiff(tspan) lt= 0)

error(lsquotspan entries must be strictly sortedrsquo)end

incT=Deltatpnts

if incTlt=0

error (lsquoIncreasing must be greater than zerorsquo)end

z(1)=z01015840 t(1)=tini iteT=2 set initial values

t(iteT)=tini+incTlowasttdir

while tdirlowastt(iteT)lttdirlowast(tfin+tdirlowastincT)

P act=ceil(99999lowast(t(iteT)minustini)lowasttdirDeltat) count sub-intervals

c=z((DeltatincT)lowast(P actminus1)+1) set the corresponding initial condition

tsub(iteTminus1)=t(iteT)-(P actminus1)lowastDeltatlowasttdir evaluate the solution at the shifted time

temp=tsub(iteTminus1)

z(iteT)=nde(tempctstart+t(iteT))

if tdirlowastt(iteT)gt=tdirlowasttfinlowast099999 repeat for the next sub-interval

break

else

iteT=iteT+1 t(iteT)=tini+(iteTminus1)lowastincTlowasttdir

end

end

t=t 1015840 +tstart

end

EMHPM algorithm for DDE solutions

function [t z]=ddeEMHPM(ddetspanhistorytauDeltatpntDelta)

dde solver by Enhanced Multistage Homotopy Perturbation Method

tini=tspan(1) tfin=tspan(end) span where the solution is founded

if taultDeltat

error(lsquoSubtinterval must not be greater than taursquo)end

if mod(tauDeltat)sim=0

pastDlt=Deltat

Deltat=tauround(tauDeltat)

warning(lsquoSubtinterval was modified from 05g to 05grsquopastDltDeltat)end

pnt=round(tauDeltat) samplesminus1 [minustau 0]

incT=DeltatpntDelta step for set resolution

t=minustau+tini+(0pntlowastpntDelta)rsquolowastincT evaluation of the initial solution [minustau 0]

z=history(t) initial behavior in the inverval [minustau 0]

c=z(end) initial condition

Algorithm 1 Continued

10 Abstract and Applied Analysis

m=pntlowastpntDelta samples between tau

itDlt=0

while (t(end)+eps)lttfin

tspan=[tini+itDltlowastDeltattini+(itDlt+1)lowastDeltat] preparing the span for next Deltat

zm=z(length(z)minusm)rsquo zn=z(1+length(z)minusm)rsquo previous tau solution

emhpm fun=(timecT) dde(timecTzmzntaum) application of the odeEMHPM

[t aux z aux]=odeEMHPM(emhpm funtspanz(end)DeltatpntDelta)

z=[z(1endminus1)z aux] t=[t(1endminus1)t aux] joining the solutions

itDlt=itDlt+1

end

z=z(pntlowastpntDelta+1end)t=t(pntlowastpntDelta+1end)

end

Algorithm 1 MATLAB algorithm

Here 120601st and 120601ex are the angles where the teeth enter and exitthe workpiece For upmilling 120601st = 0 and 120601ex = arccos(1 minus2119886119889) for downmilling 120601st = arccos(2119886

119889minus 1) and 120601ex = 120587

where 119886119889is the radial depth of cut ratio

By following the EMHPM procedure we can write (46)in equivalent form as

119894(119879) + 2120577120596

119899119894(119879) + 120596

2

119899119909119894(119879)

asymp minus

119886119901119870st

119898119898

(119909119894 (119879) minus 119909119894minus119873+1 (119879))

(50)

where 119909119894(119879) denotes the 119898 order solution of (46) on the 119894th

subinterval that satisfies the initial conditions 119909119894(0) = 119909

119894minus1

119894(0) =

119894minus1 and ℎ

119905= ℎ(119905) and 119909

minus120591is given by (35)

Introducing the transformation x119894= [119909119894 119894]119879 (50) can be

written as a system of first order linear delay differentialequations of the form

x119894(119879) = A

119905x119894(119879) + B

119905x119894minus119873+1

(119879) (51)

where

A119905= [

[

0 1

minus1205962

119899minus119908

119898119898

119870st minus2120577120596119899]

]

B119905= [

[

0 0119908

119898119898

119870st 0]

]

(52)

We next apply the EMHPM procedure to solve (46) byconsidering a downmilling operation with the followingparameter values 119911

119899= 2 119886

119889= 01 120596

119899= 5793 rads

120577 = 0011 119898119898= 003993 kg 119870

119905= 6 times 10

8Nm2 and119870119899= 2 times 10

8Nm2 As we can see from Figures 7 and 8 andfor the depth of cut values of 119886

119901= 2mm (stable) and 119886

119901=

3mm (unstable) our EMHPM approximate solutions followclosely the numerical integration solutions of (46) obtainedby using the dde23 algorithm

Figure 9 shows the relative error between the EMPHMand the dde23 numerical solution while Table 2 shows theCPU time needed for each solution Here we use 119873 = 75since the average step size of the dde23 algorithm is aroundΔ119905 asymp 120591119873 Note that for 119898 = 7 the zeroth order EMHPMapproximate solution has the fastest CPU time We can seefrom Figure 9 that the value of the relative error becomesbasically the same for119898 = 2 7 and 10 and119873 ge 20

0 0005 001 0015 002

0

1

2

Time (s)

Disp

lace

men

t (m

)

dd23Zeroth EMHPMFirst EMHPM

minus2

minus1

times10minus3

Figure 7 EMHPM approximate solutions of (46) with parametervalues of 119886

119901= 2mm 119899 = 10000 rpm Stable machine operation

0 0005 001 0015 002

00005

0010015

Time (s)

Disp

lace

men

t (m

)

dd23Zeroth EMHPMFirst EMHPM

minus001

minus0005

Figure 8 EMHPM approximate solutions of (46) with parametervalues of 119886

119901= 3mmand 119899 = 10000 rpmUnstable (chatter)machine

operation

6 Conclusions

We have developed a new algorithm based on the homotopyperturbation method to solve delay differential equationsThe proposed EMHPM approach is based on a sequence ofsubintervals that approximate the solution of delayed differ-ential equations by using the transformation rule 119906(119905) rarr119906119894(119879) where 119906

119894satisfies the initial conditions We have

Abstract and Applied Analysis 11

10 20 30 40 50 60 70 80 90 100

Abso

lute

erro

r

Zeroth order m = 2

First order m = 2

Zeroth order m = 7

First order m = 7

Zeroth order m = 10

First order m = 10

10minus3

10minus2

10minus1

Discretizations N

Figure 9 Estimated relative error values between dde23 and theEMHPM approximate solutions Here we have used the systemparameter values of 119886

119901= 2mm and 119899 = 1000 rpm and values of

119898 = 2 7 and 10

Table 2 CPU time comparison among the approximate solutions of(46) by using dde23 the zeroth order and the first order EMHPMsolutions

Solution method Time [ms]dde23 139Zeroth order EMHPM (119898 = 7) 77First order EMHPM (119898 = 7) 87

shown that our proposed EMHPM approach can be appliedto obtain the approximate solution of a delay differentialequation not only with constant but also with variablecoefficients with theoretical predictions that follow well thenumerical integration solutions To further assess the validityof this new approach we have compared the approximatesolutions of two delayed differential equations with respect totheir corresponding numerical integration solutions obtainedfrom the MATLAB dde23 algorithm The test cases were (a)the damped Mathieu differential equation with time delayand (b) the governing equation of motion of downmillingoperations We have found that the EMHPM closely followsthe numerical integration solutions of the correspondingequations and that these require less CPU time and havesmaller relative errors

Conflict of Interests

Theauthors declare that they have no conflict of interests withany mentioned entities in the paper

Acknowledgments

This work was funded by Tecnologico deMonterrey CampusMonterrey through the ResearchGroup inNanomaterials forMedical Devices and the Research Group in Advanced Man-ufacturing Additional support was provided from ConsejoNacional de Ciencia y Tecnologıa (Conacyt) Mexico

References

[1] F Shakeri and M Dehghan ldquoSolution of delay differentialequations via a homotopy perturbation methodrdquoMathematicaland Computer Modelling vol 48 no 3-4 pp 486ndash498 2008

[2] Z-K Wu ldquoSolution of the enso delayed oscillator with homo-topy analysis methodrdquo Journal of Hydrodynamics vol 21 no 1pp 131ndash135 2009

[3] A K Alomari M S Noorani and R Nazar ldquoSolution of delaydifferential equation by means of homotopy analysis methodrdquoActa Applicandae Mathematicae vol 108 no 2 pp 395ndash4122009

[4] J Biazar and G Behzad ldquoThe homotopy perturbation methodfor solving neutral functional-differential equations with pro-portional delaysrdquo Journal of King Saud UniversitymdashScience vol24 no 1 pp 33ndash37 2012

[5] NRAnakiraAKAlomari and IHashim ldquoOptimal homotopyasymptotic method for solving delay differential equationsrdquoMathematical Problems in Engineering vol 2013 Article ID498902 11 pages 2013

[6] T Insperger and G Stepan Semi-Discretization for Time-DelaySystems Stability and Engineering Applications Springer NewYork NY USA 2011

[7] E A Butcher P Nindujarla and E Bueler ldquoStability of up-and down-milling using chebyshev collocation methodrdquo inProceedings of the ASME International Design Engineering Tech-nical Conferences amp Computers and Information in EngineeringConference vol 6 part AndashC pp 841ndash850 2005

[8] T Insperger ldquoFull-discretization and semi-discretization formilling stability prediction some commentsrdquo InternationalJournal of Machine Tools and Manufacture vol 50 no 7 pp658ndash662 2010

[9] I Hashim and M S Chowdhury ldquoAdaptation of homotopy-perturbation method for numeric-analytic solution of systemof ODEsrdquo Physics Letters A vol 372 no 4 pp 470ndash481 2008

[10] L F Shampine and S Thompson ldquoSolving DDEs in MatlabrdquoAppliedNumericalMathematics vol 37 no 4 pp 441ndash458 2001

[11] SHHNiaANRanjbarDDGanjiH Soltani and JGhasemildquoMaintaining the stability of nonlinear differential equations bythe enhancement of HPMrdquo Physics Letters A vol 372 no 16 pp2855ndash2861 2008

[12] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[13] J-H He ldquoHomotopy perturbation method a new nonlinearanalytical techniquerdquo Applied Mathematics and Computationvol 135 no 1 pp 73ndash79 2003

[14] J H He ldquoComparison of homotopy perturbation method andhomotopy analysis methodrdquo Applied Mathematics and Compu-tation vol 156 no 2 pp 527ndash539 2004

[15] JHHe ldquoNew interpretation of homotopy perturbationmethodrdquoInternational Journal of Modern Physics B vol 20 no 18 pp2561ndash2568 2006

[16] S Liao ldquoComparison between the homotopy analysis methodand homotopy perturbationmethodrdquoAppliedMathematics andComputation vol 169 no 2 pp 1186ndash1194 2005

[17] D Olvera and A Elıas-Zuniga ldquoEnhanced multistage homo-topy perturbation method approximate solutions of nonlineardynamic systemsrdquo Abstract and Applied Analysis vol 2014Article ID 486509 12 pages 2014

12 Abstract and Applied Analysis

[18] T Insperger and G Stepan ldquoStability of the damped Mathieuequationwith timedelayrdquo Journal ofDynamic SystemsMeasure-ment and Control vol 125 no 2 pp 166ndash171 2003

[19] T Insperger and G Stepan ldquoStability chart for the delayedMathieu equationrdquo The Royal Society of London ProceedingsSeries A Mathematical Physical and Engineering Sciences vol458 no 2024 pp 1989ndash1998 2002

[20] T Insperger andGStepan ldquoUpdated semi-discretizationmethodfor periodic delay-differential equations with discrete delayrdquoInternational Journal for Numerical Methods in Engineering vol61 no 1 pp 117ndash141 2004

[21] T Insperger B P Mann G Stepan and P V Bayly ldquoStabilityof up-milling and down-milling part 1 alternative analyticalmethodsrdquo International Journal of Machine Tools and Manufac-ture vol 43 no 1 pp 25ndash34 2003

[22] F ICompeanDOlvera F J Campa LN L de Lacalle A Elıas-Zuniga and C A Rodrıguez ldquoCharacterization and stabilityanalysis of a multivariable milling tool by the enhanced mul-tistage homotopy perturbation methodrdquo International Journalof Machine Tools and Manufacture vol 57 pp 27ndash33 2012

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 Abstract and Applied Analysis

m=pntlowastpntDelta samples between tau

itDlt=0

while (t(end)+eps)lttfin

tspan=[tini+itDltlowastDeltattini+(itDlt+1)lowastDeltat] preparing the span for next Deltat

zm=z(length(z)minusm)rsquo zn=z(1+length(z)minusm)rsquo previous tau solution

emhpm fun=(timecT) dde(timecTzmzntaum) application of the odeEMHPM

[t aux z aux]=odeEMHPM(emhpm funtspanz(end)DeltatpntDelta)

z=[z(1endminus1)z aux] t=[t(1endminus1)t aux] joining the solutions

itDlt=itDlt+1

end

z=z(pntlowastpntDelta+1end)t=t(pntlowastpntDelta+1end)

end

Algorithm 1 MATLAB algorithm

Here 120601st and 120601ex are the angles where the teeth enter and exitthe workpiece For upmilling 120601st = 0 and 120601ex = arccos(1 minus2119886119889) for downmilling 120601st = arccos(2119886

119889minus 1) and 120601ex = 120587

where 119886119889is the radial depth of cut ratio

By following the EMHPM procedure we can write (46)in equivalent form as

119894(119879) + 2120577120596

119899119894(119879) + 120596

2

119899119909119894(119879)

asymp minus

119886119901119870st

119898119898

(119909119894 (119879) minus 119909119894minus119873+1 (119879))

(50)

where 119909119894(119879) denotes the 119898 order solution of (46) on the 119894th

subinterval that satisfies the initial conditions 119909119894(0) = 119909

119894minus1

119894(0) =

119894minus1 and ℎ

119905= ℎ(119905) and 119909

minus120591is given by (35)

Introducing the transformation x119894= [119909119894 119894]119879 (50) can be

written as a system of first order linear delay differentialequations of the form

x119894(119879) = A

119905x119894(119879) + B

119905x119894minus119873+1

(119879) (51)

where

A119905= [

[

0 1

minus1205962

119899minus119908

119898119898

119870st minus2120577120596119899]

]

B119905= [

[

0 0119908

119898119898

119870st 0]

]

(52)

We next apply the EMHPM procedure to solve (46) byconsidering a downmilling operation with the followingparameter values 119911

119899= 2 119886

119889= 01 120596

119899= 5793 rads

120577 = 0011 119898119898= 003993 kg 119870

119905= 6 times 10

8Nm2 and119870119899= 2 times 10

8Nm2 As we can see from Figures 7 and 8 andfor the depth of cut values of 119886

119901= 2mm (stable) and 119886

119901=

3mm (unstable) our EMHPM approximate solutions followclosely the numerical integration solutions of (46) obtainedby using the dde23 algorithm

Figure 9 shows the relative error between the EMPHMand the dde23 numerical solution while Table 2 shows theCPU time needed for each solution Here we use 119873 = 75since the average step size of the dde23 algorithm is aroundΔ119905 asymp 120591119873 Note that for 119898 = 7 the zeroth order EMHPMapproximate solution has the fastest CPU time We can seefrom Figure 9 that the value of the relative error becomesbasically the same for119898 = 2 7 and 10 and119873 ge 20

0 0005 001 0015 002

0

1

2

Time (s)

Disp

lace

men

t (m

)

dd23Zeroth EMHPMFirst EMHPM

minus2

minus1

times10minus3

Figure 7 EMHPM approximate solutions of (46) with parametervalues of 119886

119901= 2mm 119899 = 10000 rpm Stable machine operation

0 0005 001 0015 002

00005

0010015

Time (s)

Disp

lace

men

t (m

)

dd23Zeroth EMHPMFirst EMHPM

minus001

minus0005

Figure 8 EMHPM approximate solutions of (46) with parametervalues of 119886

119901= 3mmand 119899 = 10000 rpmUnstable (chatter)machine

operation

6 Conclusions

We have developed a new algorithm based on the homotopyperturbation method to solve delay differential equationsThe proposed EMHPM approach is based on a sequence ofsubintervals that approximate the solution of delayed differ-ential equations by using the transformation rule 119906(119905) rarr119906119894(119879) where 119906

119894satisfies the initial conditions We have

Abstract and Applied Analysis 11

10 20 30 40 50 60 70 80 90 100

Abso

lute

erro

r

Zeroth order m = 2

First order m = 2

Zeroth order m = 7

First order m = 7

Zeroth order m = 10

First order m = 10

10minus3

10minus2

10minus1

Discretizations N

Figure 9 Estimated relative error values between dde23 and theEMHPM approximate solutions Here we have used the systemparameter values of 119886

119901= 2mm and 119899 = 1000 rpm and values of

119898 = 2 7 and 10

Table 2 CPU time comparison among the approximate solutions of(46) by using dde23 the zeroth order and the first order EMHPMsolutions

Solution method Time [ms]dde23 139Zeroth order EMHPM (119898 = 7) 77First order EMHPM (119898 = 7) 87

shown that our proposed EMHPM approach can be appliedto obtain the approximate solution of a delay differentialequation not only with constant but also with variablecoefficients with theoretical predictions that follow well thenumerical integration solutions To further assess the validityof this new approach we have compared the approximatesolutions of two delayed differential equations with respect totheir corresponding numerical integration solutions obtainedfrom the MATLAB dde23 algorithm The test cases were (a)the damped Mathieu differential equation with time delayand (b) the governing equation of motion of downmillingoperations We have found that the EMHPM closely followsthe numerical integration solutions of the correspondingequations and that these require less CPU time and havesmaller relative errors

Conflict of Interests

Theauthors declare that they have no conflict of interests withany mentioned entities in the paper

Acknowledgments

This work was funded by Tecnologico deMonterrey CampusMonterrey through the ResearchGroup inNanomaterials forMedical Devices and the Research Group in Advanced Man-ufacturing Additional support was provided from ConsejoNacional de Ciencia y Tecnologıa (Conacyt) Mexico

References

[1] F Shakeri and M Dehghan ldquoSolution of delay differentialequations via a homotopy perturbation methodrdquoMathematicaland Computer Modelling vol 48 no 3-4 pp 486ndash498 2008

[2] Z-K Wu ldquoSolution of the enso delayed oscillator with homo-topy analysis methodrdquo Journal of Hydrodynamics vol 21 no 1pp 131ndash135 2009

[3] A K Alomari M S Noorani and R Nazar ldquoSolution of delaydifferential equation by means of homotopy analysis methodrdquoActa Applicandae Mathematicae vol 108 no 2 pp 395ndash4122009

[4] J Biazar and G Behzad ldquoThe homotopy perturbation methodfor solving neutral functional-differential equations with pro-portional delaysrdquo Journal of King Saud UniversitymdashScience vol24 no 1 pp 33ndash37 2012

[5] NRAnakiraAKAlomari and IHashim ldquoOptimal homotopyasymptotic method for solving delay differential equationsrdquoMathematical Problems in Engineering vol 2013 Article ID498902 11 pages 2013

[6] T Insperger and G Stepan Semi-Discretization for Time-DelaySystems Stability and Engineering Applications Springer NewYork NY USA 2011

[7] E A Butcher P Nindujarla and E Bueler ldquoStability of up-and down-milling using chebyshev collocation methodrdquo inProceedings of the ASME International Design Engineering Tech-nical Conferences amp Computers and Information in EngineeringConference vol 6 part AndashC pp 841ndash850 2005

[8] T Insperger ldquoFull-discretization and semi-discretization formilling stability prediction some commentsrdquo InternationalJournal of Machine Tools and Manufacture vol 50 no 7 pp658ndash662 2010

[9] I Hashim and M S Chowdhury ldquoAdaptation of homotopy-perturbation method for numeric-analytic solution of systemof ODEsrdquo Physics Letters A vol 372 no 4 pp 470ndash481 2008

[10] L F Shampine and S Thompson ldquoSolving DDEs in MatlabrdquoAppliedNumericalMathematics vol 37 no 4 pp 441ndash458 2001

[11] SHHNiaANRanjbarDDGanjiH Soltani and JGhasemildquoMaintaining the stability of nonlinear differential equations bythe enhancement of HPMrdquo Physics Letters A vol 372 no 16 pp2855ndash2861 2008

[12] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[13] J-H He ldquoHomotopy perturbation method a new nonlinearanalytical techniquerdquo Applied Mathematics and Computationvol 135 no 1 pp 73ndash79 2003

[14] J H He ldquoComparison of homotopy perturbation method andhomotopy analysis methodrdquo Applied Mathematics and Compu-tation vol 156 no 2 pp 527ndash539 2004

[15] JHHe ldquoNew interpretation of homotopy perturbationmethodrdquoInternational Journal of Modern Physics B vol 20 no 18 pp2561ndash2568 2006

[16] S Liao ldquoComparison between the homotopy analysis methodand homotopy perturbationmethodrdquoAppliedMathematics andComputation vol 169 no 2 pp 1186ndash1194 2005

[17] D Olvera and A Elıas-Zuniga ldquoEnhanced multistage homo-topy perturbation method approximate solutions of nonlineardynamic systemsrdquo Abstract and Applied Analysis vol 2014Article ID 486509 12 pages 2014

12 Abstract and Applied Analysis

[18] T Insperger and G Stepan ldquoStability of the damped Mathieuequationwith timedelayrdquo Journal ofDynamic SystemsMeasure-ment and Control vol 125 no 2 pp 166ndash171 2003

[19] T Insperger and G Stepan ldquoStability chart for the delayedMathieu equationrdquo The Royal Society of London ProceedingsSeries A Mathematical Physical and Engineering Sciences vol458 no 2024 pp 1989ndash1998 2002

[20] T Insperger andGStepan ldquoUpdated semi-discretizationmethodfor periodic delay-differential equations with discrete delayrdquoInternational Journal for Numerical Methods in Engineering vol61 no 1 pp 117ndash141 2004

[21] T Insperger B P Mann G Stepan and P V Bayly ldquoStabilityof up-milling and down-milling part 1 alternative analyticalmethodsrdquo International Journal of Machine Tools and Manufac-ture vol 43 no 1 pp 25ndash34 2003

[22] F ICompeanDOlvera F J Campa LN L de Lacalle A Elıas-Zuniga and C A Rodrıguez ldquoCharacterization and stabilityanalysis of a multivariable milling tool by the enhanced mul-tistage homotopy perturbation methodrdquo International Journalof Machine Tools and Manufacture vol 57 pp 27ndash33 2012

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

Abstract and Applied Analysis 11

10 20 30 40 50 60 70 80 90 100

Abso

lute

erro

r

Zeroth order m = 2

First order m = 2

Zeroth order m = 7

First order m = 7

Zeroth order m = 10

First order m = 10

10minus3

10minus2

10minus1

Discretizations N

Figure 9 Estimated relative error values between dde23 and theEMHPM approximate solutions Here we have used the systemparameter values of 119886

119901= 2mm and 119899 = 1000 rpm and values of

119898 = 2 7 and 10

Table 2 CPU time comparison among the approximate solutions of(46) by using dde23 the zeroth order and the first order EMHPMsolutions

Solution method Time [ms]dde23 139Zeroth order EMHPM (119898 = 7) 77First order EMHPM (119898 = 7) 87

shown that our proposed EMHPM approach can be appliedto obtain the approximate solution of a delay differentialequation not only with constant but also with variablecoefficients with theoretical predictions that follow well thenumerical integration solutions To further assess the validityof this new approach we have compared the approximatesolutions of two delayed differential equations with respect totheir corresponding numerical integration solutions obtainedfrom the MATLAB dde23 algorithm The test cases were (a)the damped Mathieu differential equation with time delayand (b) the governing equation of motion of downmillingoperations We have found that the EMHPM closely followsthe numerical integration solutions of the correspondingequations and that these require less CPU time and havesmaller relative errors

Conflict of Interests

Theauthors declare that they have no conflict of interests withany mentioned entities in the paper

Acknowledgments

This work was funded by Tecnologico deMonterrey CampusMonterrey through the ResearchGroup inNanomaterials forMedical Devices and the Research Group in Advanced Man-ufacturing Additional support was provided from ConsejoNacional de Ciencia y Tecnologıa (Conacyt) Mexico

References

[1] F Shakeri and M Dehghan ldquoSolution of delay differentialequations via a homotopy perturbation methodrdquoMathematicaland Computer Modelling vol 48 no 3-4 pp 486ndash498 2008

[2] Z-K Wu ldquoSolution of the enso delayed oscillator with homo-topy analysis methodrdquo Journal of Hydrodynamics vol 21 no 1pp 131ndash135 2009

[3] A K Alomari M S Noorani and R Nazar ldquoSolution of delaydifferential equation by means of homotopy analysis methodrdquoActa Applicandae Mathematicae vol 108 no 2 pp 395ndash4122009

[4] J Biazar and G Behzad ldquoThe homotopy perturbation methodfor solving neutral functional-differential equations with pro-portional delaysrdquo Journal of King Saud UniversitymdashScience vol24 no 1 pp 33ndash37 2012

[5] NRAnakiraAKAlomari and IHashim ldquoOptimal homotopyasymptotic method for solving delay differential equationsrdquoMathematical Problems in Engineering vol 2013 Article ID498902 11 pages 2013

[6] T Insperger and G Stepan Semi-Discretization for Time-DelaySystems Stability and Engineering Applications Springer NewYork NY USA 2011

[7] E A Butcher P Nindujarla and E Bueler ldquoStability of up-and down-milling using chebyshev collocation methodrdquo inProceedings of the ASME International Design Engineering Tech-nical Conferences amp Computers and Information in EngineeringConference vol 6 part AndashC pp 841ndash850 2005

[8] T Insperger ldquoFull-discretization and semi-discretization formilling stability prediction some commentsrdquo InternationalJournal of Machine Tools and Manufacture vol 50 no 7 pp658ndash662 2010

[9] I Hashim and M S Chowdhury ldquoAdaptation of homotopy-perturbation method for numeric-analytic solution of systemof ODEsrdquo Physics Letters A vol 372 no 4 pp 470ndash481 2008

[10] L F Shampine and S Thompson ldquoSolving DDEs in MatlabrdquoAppliedNumericalMathematics vol 37 no 4 pp 441ndash458 2001

[11] SHHNiaANRanjbarDDGanjiH Soltani and JGhasemildquoMaintaining the stability of nonlinear differential equations bythe enhancement of HPMrdquo Physics Letters A vol 372 no 16 pp2855ndash2861 2008

[12] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[13] J-H He ldquoHomotopy perturbation method a new nonlinearanalytical techniquerdquo Applied Mathematics and Computationvol 135 no 1 pp 73ndash79 2003

[14] J H He ldquoComparison of homotopy perturbation method andhomotopy analysis methodrdquo Applied Mathematics and Compu-tation vol 156 no 2 pp 527ndash539 2004

[15] JHHe ldquoNew interpretation of homotopy perturbationmethodrdquoInternational Journal of Modern Physics B vol 20 no 18 pp2561ndash2568 2006

[16] S Liao ldquoComparison between the homotopy analysis methodand homotopy perturbationmethodrdquoAppliedMathematics andComputation vol 169 no 2 pp 1186ndash1194 2005

[17] D Olvera and A Elıas-Zuniga ldquoEnhanced multistage homo-topy perturbation method approximate solutions of nonlineardynamic systemsrdquo Abstract and Applied Analysis vol 2014Article ID 486509 12 pages 2014

12 Abstract and Applied Analysis

[18] T Insperger and G Stepan ldquoStability of the damped Mathieuequationwith timedelayrdquo Journal ofDynamic SystemsMeasure-ment and Control vol 125 no 2 pp 166ndash171 2003

[19] T Insperger and G Stepan ldquoStability chart for the delayedMathieu equationrdquo The Royal Society of London ProceedingsSeries A Mathematical Physical and Engineering Sciences vol458 no 2024 pp 1989ndash1998 2002

[20] T Insperger andGStepan ldquoUpdated semi-discretizationmethodfor periodic delay-differential equations with discrete delayrdquoInternational Journal for Numerical Methods in Engineering vol61 no 1 pp 117ndash141 2004

[21] T Insperger B P Mann G Stepan and P V Bayly ldquoStabilityof up-milling and down-milling part 1 alternative analyticalmethodsrdquo International Journal of Machine Tools and Manufac-ture vol 43 no 1 pp 25ndash34 2003

[22] F ICompeanDOlvera F J Campa LN L de Lacalle A Elıas-Zuniga and C A Rodrıguez ldquoCharacterization and stabilityanalysis of a multivariable milling tool by the enhanced mul-tistage homotopy perturbation methodrdquo International Journalof Machine Tools and Manufacture vol 57 pp 27ndash33 2012

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 Abstract and Applied Analysis

[18] T Insperger and G Stepan ldquoStability of the damped Mathieuequationwith timedelayrdquo Journal ofDynamic SystemsMeasure-ment and Control vol 125 no 2 pp 166ndash171 2003

[19] T Insperger and G Stepan ldquoStability chart for the delayedMathieu equationrdquo The Royal Society of London ProceedingsSeries A Mathematical Physical and Engineering Sciences vol458 no 2024 pp 1989ndash1998 2002

[20] T Insperger andGStepan ldquoUpdated semi-discretizationmethodfor periodic delay-differential equations with discrete delayrdquoInternational Journal for Numerical Methods in Engineering vol61 no 1 pp 117ndash141 2004

[21] T Insperger B P Mann G Stepan and P V Bayly ldquoStabilityof up-milling and down-milling part 1 alternative analyticalmethodsrdquo International Journal of Machine Tools and Manufac-ture vol 43 no 1 pp 25ndash34 2003

[22] F ICompeanDOlvera F J Campa LN L de Lacalle A Elıas-Zuniga and C A Rodrıguez ldquoCharacterization and stabilityanalysis of a multivariable milling tool by the enhanced mul-tistage homotopy perturbation methodrdquo International Journalof Machine Tools and Manufacture vol 57 pp 27ndash33 2012

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