Research ArticleNumerical Solution of Pantograph-Type Delay DifferentialEquations Using Perturbation-Iteration Algorithms

M Mustafa BahGi1 and Mehmet Ccedilevik2

1Department of Mechanical Engineering Celal Bayar University 45140 Manisa Turkey2Department of Mechanical Engineering Izmir Katip Celebi University 35620 Izmir Turkey

Correspondence should be addressed to Mehmet Cevik cevik2002gmailcom

Received 12 August 2015 Accepted 16 November 2015

Academic Editor Vasile Marinca

Copyright copy 2015 M M Bahsi and M Cevik This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Thepantograph equation is a special type of functional differential equationswith proportional delayThepresent study introduces acompound technique incorporating the perturbationmethod with an iteration algorithm to solve numerically the delay differentialequations of pantograph typeWe put forward two types of algorithms depending upon the order of derivatives in the Taylor seriesexpansion The crucial convenience of this method when compared with other perturbation methods is that this method does notrequire a small perturbation parameter Furthermore a relatively fast convergence of the iterations to the exact solutions and moreaccurate results can be achieved Several illustrative examples are given to demonstrate the efficiency and reliability of the techniqueeven for nonlinear cases

1 Introduction and Preliminaries

The pantograph equation is a special type of functionaldifferential equations with proportional delay It arises inrather different fields of pure and applied mathematicssuch as electrodynamics control systems number theoryprobability and quantum mechanics Many researchers havestudied the pantograph-type delay differential equation usinganalytical and numerical techniques [1ndash8] The second-orderpantograph-type delay differential equation is given as [7 8]

1198892119906 (119909)

1198891199092= 119891 (119909 119906 (119909) 119906 (120572119909)) 119909 isin (0 119879) (1)

with the boundary conditions

119906 (0) = 119887

119906 (119879) = 119888

(2)

where 119879 gt 0 and 0 lt 120572 lt 1

A brief review of recent literature on the methods ofsolution for pantograph-type delay differential equations canbe found in the paper by Trif [9]

On the other hand perturbation methods [10] have beenamong the most common approximate methods used forsolving algebraic equations differential equations integrod-ifferential equations and difference equations The primarylimitation of these methods is the necessity of a smallparameter This parameter might come out as an originalphysical parameter of the given equation or alternatively itmay be inserted as an artificial parameter Several differentmethods have been established in order to deal with thislimitation

In this study we present the application of a hybrid tech-nique combining the perturbation method with an iterationalgorithm to find a numerical solution for pantograph-typedelay differential equations Various researchers consideredthis new perturbation-iteration method [11ndash16] Pakdemirliand coworkers besides others have studied extensively thismethodThefirst study on algebraic equationswas carried out

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2015 Article ID 139821 10 pageshttpdxdoiorg1011552015139821

2 Journal of Applied Mathematics

by Pakdemirli and Boyacı [17] where they proposed new rootfinding algorithms using the new systematic approach Laterfourth-order [18] and fifth-order [19] derivative algorithmswere presented Aksoy and Pakdemirli [20] systematicallygenerated the present method for both linear and nonlinearsecond-order differential equations and applied to it Bratu-type equations Dolapci et al [21] used this method to solveFredholm and Volterra integral equations and argued that itcan be applied to both types of integral equations

The main purpose of the work presented in this paperis to apply the new perturbation-iteration solution for thepantograph-type delay differential equations To the authorsrsquoknowledge the application of perturbation-iteration algo-rithms to pantograph-type delay differential equations isnovel The two types of perturbation-iteration algorithmsPIA(1 1) and PIA(1 2) are introduced in the next sectionThen the method is applied to pantograph equations via sixexamples two of which are first-order linear one is first-ordernonlinear and the others are second-order nonlinear theyshowed excellent agreement with the published results andverified the accuracy and efficiency of the present methodThe results are discussed and commented on

The general form of pantograph-type delay differentialequations of both first- and second-order including (1) canbe stated as

119865 (11990610158401015840 1199061015840 119906 119906120572 120576 119909) = 0 (3)

where 119906 = 119906(119909) 119906120572

= 119906(120572119909) 0 lt 120572 lt 1 is a givenconstant and 120576 is the perturbation parameter In this studyweinvestigated a solution for (3) which is closed form Equa-tion (3) covers many different problems studied by variousresearchers

2 Perturbation-Iteration Algorithms

Perturbation-iteration algorithms are briefly called PIA(119899119898)Here 119899 denotes the number of correction terms in the per-turbation expansion and 119898 denotes the order of derivativesin the Taylor series expansion Generally 119899 is smaller thanor equal to 119898 otherwise the correction terms cannot becalculated In this study PIA(1 1) and PIA(1 2) algorithmsconstructed by taking correction term in the perturbationexpansion together with first- and second-order derivativesrespectively are employed

21 Perturbation-Iteration Algorithm PIA(1 1) In PIA(1 1)algorithm we propose a perturbation-iteration algorithm bytaking one correction term in the perturbation expansion andcorrection terms of only the first derivatives in the Taylorexpansion that is 119899 = 1 119898 = 1 Let us consider a second-order pantograph differential equation written in the formof (3) Taking only one correction term in the perturbationexpansion the straightforward expansion for the solution ofeach iteration can be written as follows

119906119899+1

= 119906119899+ 120576 (119906

119888)119899 (4)

where (119906119888)119899is the 119899th correction term of the perturbation-

iteration algorithm Then we substitute (4) into (3) andexpand it in a Taylor series with first derivatives to obtain

119865 (11990610158401015840

119899 1199061015840

119899 119906119899 (119906120572)119899 0)

+ 11986511990610158401015840 (11990610158401015840

119899 1199061015840

119899 119906119899 (119906120572)119899 0) 120576 (119906

10158401015840

119888)119899

+ 1198651199061015840 (11990610158401015840

119899 1199061015840

119899 119906119899 (119906120572)119899 0) 120576 (119906

1015840

119888)119899

+ 119865119906(11990610158401015840

119899 1199061015840

119899 119906119899 (119906120572)119899 0) 120576 (119906

119888)119899

+ 119865119906120572

(11990610158401015840

119899 1199061015840

119899 119906119899 (119906120572)119899 0) 120576 ((119906

120572)119888)119899

+ 119865120576(11990610158401015840

119899 1199061015840

119899 119906119899 (119906120572)119899 0) 120576 = 0

(5)

where ( )1015840 denotes differentiationwith respect to the indepen-dent variable and

11986511990610158401015840 =

120597119865

12059711990610158401015840

1198651199061015840 =

120597119865

1205971199061015840

119865119906=120597119865

120597119906

119865119906120572

=120597119865

120597119906120572

119865120576=120597119865

120597120576

(6)

Bearing in mind that all derivatives are evaluated at 120576 = 0 andrewriting the equation in the following more suitable form

11986511990610158401015840 (11990610158401015840

119888)119899+ 1198651199061015840 (1199061015840

119888)119899+ 119865119906(119906119888)119899+ 119865119906120572

((119906120572)119888)119899

= minus119865120576minus119865

120576

(7)

one may easily notice that (7) is a variable coefficientpantograph equation Starting with an initial guess 119906

0 first

we determine (119906119888)0from (7) and then substitute it into (4)

for calculating 1199061 This iteration procedure is repeated using

(4) and (7) until the approximation is sufficient within a user-defined threshold

22 Perturbation-Iteration Algorithm PIA(1 2) In PIA(1 2)algorithm we put forward a perturbation-iteration algorithmby taking one correction term in the perturbation expansionand correction terms of up to second derivatives in the Taylorexpansion that is 119899 = 1 119898 = 2 Only one correctionterm perturbation expansion was given before in (4) Now

Journal of Applied Mathematics 3

we substitute (4) into (3) and expand in a Taylor series of upto second-order derivatives to obtain

119865 + 11986511990610158401015840120576 (11990610158401015840

119888)119899+ 1198651199061015840120576 (1199061015840

119888)119899+ 119865119906120576 (119906119888)119899

+ 119865119906120572

120576 ((119906120572)119888)119899+ 119865120576120576 +

1

211986511990610158401015840119906101584010158401205762(11990610158401015840

119888)2

119899

+1

2119865119906101584011990610158401205762(1199061015840

119888)2

119899+1

21198651199061199061205762(119906119888)2

119899

+1

2119865119906120572119906120572

1205762((119906120572)119888)2

119899+1

21205762119865120576120576

+ 1198651199061015840101584011990610158401205762(11990610158401015840

119888)119899(1199061015840

119888)119899+ 119865119906101584010158401199061205762(11990610158401015840

119888)119899(119906119888)119899

+ 11986511990610158401015840119906120572

1205762(11990610158401015840

119888)119899((119906120572)119888)119899+ 119865119906101584010158401205761205762(11990610158401015840

119888)119899

+ 11986511990610158401199061205762(1199061015840

119888)119899(119906119888)119899+ 1198651199061015840119906120572

1205762(1199061015840

119888)119899((119906120572)119888)119899

+ 11986511990610158401205761205762(1199061015840

119888)119899+ 119865119906119906120572

1205762(119906119888)119899((119906120572)119888)119899

+ 1198651199061205761205762(119906119888)119899+ 1198651199061205721205761205762((119906120572)119888)119899= 0

(8)

Then bearing in mind that all derivatives are evaluated at120576 = 0 we rewrite this equation in the following more suitableform

(12057611986511990610158401015840 + 120576211986511990610158401015840120576) (11990610158401015840

119888)119899+ (1205761198651199061015840 + 12057621198651199061015840120576) (1199061015840

119888)119899

+ (120576119865119906+ 1205762119865119906120576) (119906119888)119899+ (120576119865119906120572

+1205762119865119906120572120576) ((119906120572)119888)119899

+1

212057621198651199061015840101584011990610158401015840 (11990610158401015840

119888)2

119899+1

2120576211986511990610158401199061015840 (1199061015840

119888)2

119899

+1

21205762119865119906119906(119906119888)2

119899+1

21205762119865119906120572119906120572

((119906120572)119888)2

119899

+ 1205762119865119906101584010158401199061015840 (11990610158401015840

119888)119899(1199061015840

119888)119899+ 120576211986511990610158401015840119906(11990610158401015840

119888)119899(119906119888)119899

+ 120576211986511990610158401015840119906120572

(11990610158401015840

119888)119899((119906120572)119888)119899+ 12057621198651199061015840119906(1199061015840

119888)119899(119906119888)119899

+ 12057621198651199061015840119906120572

(1199061015840

119888)119899((119906120572)119888)119899+ 1205762119865119906119906120572

(119906119888)119899((119906120572)119888)119899

= minus120576119865120576minus 119865 minus

1

21205762119865120576120576

(9)

Equation (9) is a variable coefficient pantograph equationof order two We repeat the previously mentioned iterationprocedure until the approximation is sufficient within a user-defined threshold

3 Application of theMethod to the Pantograph-TypeDelay Differential Equations

In order to illustrate the accuracy and applicability of thepresented method the new perturbation-iteration algorithmis applied to six pantograph-type delay differential equationsof different types For comparison purposes the solutionintervals of problems are chosen generally the same as thosein the references Yet the solution intervals are enlarged inillustrative examples 3 and 5 For a further wider intervalthe method should be modified such that the Taylor seriesexpansions are made around points other than zero

31 Illustrative Example 1 Consider the first-order linearpantograph differential equation [25]

1199061015840(119909) + 119906 (119909) minus

1

10119906 (

119909

5) = minus

1

10119890minus(15)119909

0 le 119909 le 1 (10)

with the initial condition

119906 (0) = 1 (11)

for which the exact solution is 119906(119909) = 119890minus119909 [25]

Equation (10) can be written in the following form

1199061015840(119909) + 119906 (119909) minus 120576119906 (

119909

5) +

1

10119890minus(15)119909

= 0 (12)

where 120576 = 01 is assumed to be the perturbation parameterThe nonzero terms of (7) in PIA(1 1) algorithm are

1198651199061015840 = 1

119865119906= 1

119865120576= minus119906119899(119909

5)

119865 = 1199061015840

119899(119909) + 119906

119899(119909) +

1

10119890minus(15)119909

(13)

Upon substituting the terms of (13) (7) reduces to

(1199061015840

119888)119899(119909) + (119906

119888)119899(119909)

= 119906119899(119909

5) minus

1199061015840

119899(119909) + 119906

119899(119909) + (110) 119890

minus(15)119909

01

(14)

When applying the iteration formula (4) we select an initiallyassumed function Here we start with the following trivialsolution which satisfies the given initial condition

1199060= 1 (15)

4 Journal of Applied Mathematics

Table 1 Comparison of the absolute errors of the PIA(1 1)method with other numerical methods for example 1

119909VIM [22] Taylor method [23] Continuous method [24]

JRC method120572 = 120573 = 05 [25] PIA(1 1)

119899 = 4 119873 = 64 119899 = 4

2minus1

240119890 minus 15 124119890 minus 10 333119890 minus 11 393119890 minus 14 143119890 minus 15

2minus2

791119890 minus 17 974119890 minus 11 413119890 minus 11 216119890 minus 14 791119890 minus 17

2minus3

253119890 minus 18 700119890 minus 11 462119890 minus 11 166119890 minus 15 253119890 minus 18

2minus4

803119890 minus 20 914119890 minus 11 489119890 minus 11 621119890 minus 15 803119890 minus 20

2minus5

252119890 minus 21 528119890 minus 11 505119890 minus 11 368119890 minus 14 252119890 minus 21

2minus6

700119890 minus 23 195119890 minus 11 474119890 minus 11 454119890 minus 14 792119890 minus 23

Using (4) and (14) the approximate solutions at each stepbecome

1199061=

1

10minus1

8119890minus(15)119909

+41

40119890minus119909

1199062=

1

102+

1

320119890minus(15)119909

minus5

384119890minus(125)119909

+9599

9600119890minus119909

1199063=

1

103minus

1

76800119890minus(15)119909

+1

3072119890minus(125)119909

minus125

95232119890minus(1125)119909

+11904001

11904000119890minus119909

1199064=

1

104+

1

95232000119890minus(15)119909

minus1

737280119890minus(125)119909

+25

761825119890minus(1125)119909

minus15625

118849536119890minus(1625)119909

+74280959999

74280960000119890minus119909

(16)

Table 1 gives a comparison of the absolute errors of thePIA(1 1) method with those of other numerical methods Itis apparent from the table that the absolute errors of PIA(1 1)method are smaller than those of the other methods for119899 = 4 and also considering the first four iteration solutionsgiven in (16) one may notice that the numerical solutionconverges to the exact solution Absolute errors obtained byPIA(1 1) method are almost the same as those obtained bythe Variational Iteration Method (VIM) [22] for 119899 = 4

32 Illustrative Example 2 Consider the first-order linearpantograph differential equation [3]

1199061015840(119909) + 119906 (119909) + 119906 (08119909) = 0 0 le 119909 le 1 (17)

with the initial condition

119906 (0) = 1 (18)

which does not have an exact solution Equation (17) can bewritten in the following form

1199061015840(119909) + 119906 (119909) + 120576119906 (08119909) = 0 (19)

where 120576 is the artificially introduced perturbation parameterThe nonzero terms of (7) in PIA(1 1) algorithm are

1198651199061015840 = 1

119865119906= 1

119865120576= 119906119899(08119909)

119865 = 1199061015840

119899(119909) + 119906

119899(119909)

(20)

For this specific example (7) reduces to

(1199061015840

119888)119899(119909) + (119906

119888)119899(119909) = minus119906

119899(08119909) minus 119906

1015840

119899(119909) minus 119906

119899(119909) (21)

When applying the iteration formula (4) we start withthe following trivial solution as the initial function whichsatisfies the given initial conditions

1199060= 1 (22)

Using (4) and (21) the successive iteration results are

1199061= minus1 + 2119890

minus119909

1199062= 1 + 10119890

minus119909minus 10119890minus(45)119909

1199063= minus1 +

218

9119890minus119909

minus 50119890minus(45)119909

+250

9119890minus(1625)119909

1199064= 1 +

21490

549119890minus119909

minus1090

9119890minus(45)119909

+1250

9119890minus(1625)119909

minus31250

549119890minus(64125)119909

(23)

Table 2 shows the PIA(1 1) solutions of (17) for iterationnumbers 119899 = 5 6 7 8 9 As stated above we do not know theexact solution therefore we estimate a solution and observea convergence

A comparison of the results obtained by several differentmethods is shown in Table 3 The numerical results areconsistent

33 Illustrative Example 3 Consider the first-order nonlin-ear pantograph differential equation [29]

1199061015840(119909) = minus2119906

2(119909

2) + 1 0 le 119909 le 1 (24)

Journal of Applied Mathematics 5

Table 2 Convergence of the PIA(1 1) solution for example 2

119909119894

1199065(119909) 119906

6(119909) 119906

7(119909) 119906

8(119909) 119906

9(119909)

0 1 1 1 1 1

02 06646909 06646910 06646910 06646910 0664691004 04335604 04335607 04335607 04335607 0433560706 02764790 02764824 02764823 02764823 0276482308 01714677 01714846 01714841 01714841 017148411 01026142 01026723 01026700 01026701 01026701

with the initial condition

119906 (0) = 0 (25)

which has the exact solution 119906(119909) = sin(119909)Equation (24) can be written in the following form

1199061015840(119909) + 120576119906

2(119909

2) minus 1 = 0 (26)

where 120576 = 2 is assumed to be the perturbation parameterThenonzero terms of (7) in PIA(1 1) algorithm are

1198651199061015840 = 1

119865120576= 1199062

119899(119909

2)

119865 = 1199061015840

119899(119909) minus 1

(27)

The 119899th correction term of the perturbation expansion in (4)is

(1199061015840

119888)119899(119909) = minus119906

2

119899(119909

2) minus

1199061015840

119899(119909) minus 1

2 (28)

In this example we start with the initial guess

1199060= 119909 (29)

Thus the successive approximations are

1199061= 119909 minus

1

61199093

1199062= 119909 minus

1

61199093+

1

1201199095minus

1

80641199097

1199063= 119909 minus

1

61199093+

1

1201199095minus

1

50401199097+

61

232243201199099

minus67

340623360011990911+

1

1288175616011990913

minus1

799065243648011990915

(30)

Subsequently we determine the absolute maximum error for119906119899as

119864119899=1003817100381710038171003817119906119899 (119909) minus 119906 (119909)

1003817100381710038171003817infin

= max 1003816100381610038161003816119906119899 (119909) minus 119906 (119909)1003816100381610038161003816 0 le 119909 le 1

(31)

04 08 12 16 2 24 28 32 36 4 440x

minus1

minus08

minus06

minus04

minus02

002040608

1

u(x)

Exact solutionFirst iteration solution

Second iteration solution

Third iteration solutionof the PIA(1 1)

of the PIA(1 1)

of the PIA(1 1)

Figure 1 Comparison of the first three iteration solutions of thePIA(1 1)method with the exact solution for example 3

The absolute maximum errors 119864119899for different values of 119899 are

given in Table 4 Note that the error decreases continually as119899 increases

The same problem is solved in various studies and itis seen that the absolute maximum error for solving thisexample via Homotopy Asymptotic Method is 12 times 10

minus6

[30] and via Optimal Homotopy Asymptotic Method is 4 times10minus8 [29] while the absolute maximum error obtained by

the PIA(1 1) method is 535 times 10minus20 This affirmed that

the presented method is more accurate than the other twomethods

In Figure 1 the first three iteration solutions (119899 = 1 2 3)

of the PIA(1 1) method contrasted with the exact solutionIt is revealed that even when the solution interval of theproblem is expanded PIA(1 1) method goes on to yieldconvergent numerical solutions

34 Illustrative Example 4 Consider the initial value prob-lem of second-order differential equation with pantographdelay [31]

11990610158401015840(119909) = minus119906 (

119909

2) minus 1199062(119909) + sin4 (119909) + sin2 (119909) + 8

119909 ge 0

(32)

with initial conditions

119906 (0) = 2

1199061015840(0) = 0

(33)

which has the exact solution 119906(119909) = (5 minus cos 2119909)2

341 PIA(1 1) Solution Equation (32) can be written in thefollowing form

11990610158401015840(119909) + 120576 (119906 (

119909

2) + 1199062(119909))

= sin4 (119909) + sin2 (119909) + 8

(34)

6 Journal of Applied Mathematics

Table 3 Comparison of the PIA(1 1) solution with other numerical methods for example 2

119909119894 Walsh series method [26]

DUSF series method[27]

Hermite seriesmethod [3]

Taylor seriesmethod [4]

Laguerre matrixmethod [28] PIA(1 1)

119898 = 100ℎ = 001

119873 = 8 119873 = 8 119873 = 8 119899 = 8

0 1 1 1 1 1 102 0665621 0664677 0664691 0664691 06646910 0664691004 0432426 0433540 0433561 0433561 04335607 0433560706 0275140 0276460 0276482 0276482 02764831 0276482308 0170320 0171464 0171484 0171484 01714942 017148411 0100856 0102652 0102670 0102744 01027437 01026701

Table 4 The absolute maximum errors 119864119899for example 3

119899 1 2 3 4 5 6119864119899

813119890 minus 3 716119890 minus 5 123119890 minus 7 438119890 minus 11 291119890 minus 15 535119890 minus 20

where 120576 = 1 is the artificially introduced perturbationparameter Using (3) (34) returns to

119865 (11990610158401015840 1199061015840 119906 119906120572 120576) = 119906

10158401015840(119909) + 120576 (119906 (

119909

2) + 1199062(119909))

minus (sin4 (119909) + sin2 (119909) + 8) = 0

(35)

The nonzero terms of (7) in PIA(1 1) algorithm are

11986511990610158401015840 = 1

119865120576= 119906119899(119909

2) + 1199062

119899(119909)

119865 = 11990610158401015840

119899(119909) minus sin4 (119909) minus sin2 (119909) minus 8

(36)

Substituting (36) (32) reduces to

(11990610158401015840

119888)119899(119909) = minus119906

119899(119909

2) minus 11990610158401015840

119899(119909) minus 119906

2

119899(119909) + sin4 (119909)

+ sin2 (119909) + 8

(37)

The initial trial function is selected as

1199060= 2 (38)

and using (37) the approximate first iteration solution isobtained as

1199061=13

16minus

1

16sin4 (119909) + 3

16cos2 (119909) + 23119909

2

16

+ cos2 (1199092)

(39)

Even the first iteration of PIA(1 1) solution 1199061 gave the same

result as the first iteration solution obtained by the VIM [31]Still the second and third iteration solutions of the problemare better than those of the VIM solutions as observed inFigures 2 and 3

Exact solutionSecond iteration solution Second iteration solution of the VIM

2

22

24

26

28

3

u(x)

02 04 06 08 10x

of the PIA(1 1)

Figure 2 Comparison of the second iteration solutions of the VIMand PIA(1 1) with exact solution for example 4

342 PIA(1 2) Solution In order to apply the PIA(1 2)algorithm obtained by (9) we have to write (32) in thefollowing form

119865 (11990610158401015840 1199061015840 119906 119906120572 120576) = 119906

10158401015840(119909) + 120576119906

2(119909) + 120576

2119906 (

119909

2)

minus (sin4 (119909) + sin2 (119909) + 8) = 0

(40)

where 120576 is the artificially introduced perturbation parameterThe nonzero terms of (9) in PIA(1 2) algorithm are

11986511990610158401015840 = 1

119865119906120576= 2119906119899(119909)

119865120576= 1199062

119899(119909)

119865120576120576= 2119906119899(119909

2)

119865 = 11990610158401015840

119899(119909) minus sin4 (119909) minus sin2 (119909) minus 8

(41)

Journal of Applied Mathematics 7

Exact solution

Third iteration solution of the VIM

2

22

24

26

28

u(x)

02 04 06 08 10x

Third iteration solution of the PIA(1 1)

Figure 3The comparison of the third iteration solutions of theVIMand PIA(1 1) with exact solution for example 4

Note that introducing the small parameter 120576 = 1 as acoefficient of the pantograph term simplifies (40) and makesit easily solvable For this specific example (40) reads

(11990610158401015840

119888)119899(119909) + (2119906

119899(119909)) (119906

119888)119899(119909)

= minus119906119899(119909

2) minus 11990610158401015840

119899(119909) minus 119906

2

119899(119909) + sin4 (119909) + sin2 (119909)

+ 8

(42)

When applying the iteration formula (4) we select thefollowing initially assumed function

1199060= 2 (43)

and using (9) the approximate solution is

1199061=263

96minus49

96cos (2119909) minus 1

12cos4 (2119909) + 1

48cos2 (119909)

minus1

4119909 sin (119909) cos (119909) minus 1

6cos (119909)

(44)

First iteration of PIA(1 2) and second and third iterationsof VIM solutions are compared with the exact solutionAs can be seen from Figure 4 even the first iteration ofPIA(1 2) yields much better results than the first threeiteration solutions of the VIM

35 Illustrative Example 5 Consider the initial value prob-lem of nonlinear variable coefficient differential equationwith pantograph delay [6]

11990610158401015840(119909) = 119906 (119909) minus

8

11990921199062(119909

2) 119909 ge 0 (45)

with initial conditions

119906 (0) = 0

1199061015840(0) = 1

(46)

which has the exact solution 119906(119909) = 119909119890minus119909

2

22

24

26

28

3

u(x)

02 04 06 08 10x

of the VIM

of the VIM

Exact solutionFirst iteration solution

Second iteration solution

Third iteration solutionof the PIA(1 2)

Figure 4 The comparison of second and third iteration solutionsof the VIM and first iteration solution of the PIA(1 2) with exactsolution of example 4

Equation (45) can be written in the following form

11990610158401015840(119909) + 120576 (

8

11990921199062(119909

2) minus 119906 (119909)) = 0 (47)

where 120576 = 1 is the artificially introduced perturbationparameter Using (3) (45) returns to

119865 (11990610158401015840 1199061015840 119906 119906120572 120576) = 119906

10158401015840(119909) + 120576 (

8

11990921199062(119909

2) minus 119906 (119909))

= 0

(48)

The nonzero terms in PIA(1 1) algorithm are

11986511990610158401015840 = 1

119865120576=

8

11990921199062

119899(119909

2) minus 119906119899(119909)

119865 = 11990610158401015840

119899(119909)

(49)

Substituting (49) (45) reduces to

(11990610158401015840

119888)119899(119909) = minus

8

11990921199062

119899(119909

2) + 119906119899(119909) minus 119906

10158401015840

119899(119909) (50)

The initial trial function is selected as

1199060(119909) = 119909 (51)

and using (50) the approximate first iteration solution isobtained as

1199061= 119909 minus 119909

2+1

61199093

1199062= 119909 minus 119909

2+1

21199093minus

5

361199094+

1

801199095minus

1

86401199096

(52)

The absolute maximum errors 119864119899for different values of 119899 are

given in Table 5 and it is revealed that the error decreasescontinually as 119899 increases

8 Journal of Applied Mathematics

Table 5 The absolute maximum errors 119864119899for example 5

119899 1 2 3 4 5 6 7119864119899

201119890 minus 1 561119890 minus 3 159119890 minus 5 275119890 minus 7 368119890 minus 9 676119890 minus 11 494119890 minus 13

Exact solution8th iteration solution 7th iteration solution 6th iteration solution 5th iteration solution

0

01

02

03

04

06 12 18 24 3 36 42 480

of the PIA(1 1)

of the PIA(1 1)

of the PIA(1 1)

of the PIA(1 1)

Figure 5 Comparison of the iteration solutions of the PIA(1 1)method with the exact solution for example 5

Figure 5 illustrates the PIA(1 1) solution of this examplein the interval [0 5] In this example we enlarge the intervalso as to compare our solutionmore thoroughly with the exactsolution The present method yields pretty good results

36 Illustrative Example 6 Consider the second-order non-linear pantograph functional differential equation [7 8]

11990610158401015840(119909) = ([119906 (119909)]

2+ |119906 (119909)|

3) 119906 (

119909

2) 119909 ge 0 (53)

with initial conditions

119906 (0) = 1 119906 (1) =1

2(54)

which has the exact solution 119906(119909) = 1(1 + 119909)Equation (45) can be written in the following form

11990610158401015840(119909) minus 120576 (([119906 (119909)]

2+ |119906 (119909)|

3) 119906 (

119909

2)) = 0 (55)

where 120576 is the artificially introduced small parameter Using(3) (45) returns to

119865 (11990610158401015840 1199061015840 119906 119906120572 120576)

= 11990610158401015840(119909) minus 120576 (([119906 (119909)]

2+ |119906 (119909)|

3) 119906 (

119909

2)) = 0

(56)

The nonzero terms in PIA(1 1) algorithm are

11986511990610158401015840 = 1

119865120576= minus(([119906 (119909)]

2+1003816100381610038161003816119906119899 (119909)

10038161003816100381610038163

) 119906119899(119909

2))

119865 = 11990610158401015840

119899(119909)

(57)

First iteration solution Second iteration solution Third iteration solution

01 02 03 04 05 06 07 08 09 10x

04

05

06

07

08

09

1

u(x)

of the PIA(1 1)

of the PIA(1 1)

of the PIA(1 1)

Figure 6 Comparison of the first three iteration solutions of thePIA(1 1)method with the exact solution for example 6

Substituting (49) (45) reduces to

(11990610158401015840

119888)119899(119909) = minus119906

10158401015840

119899(119909)

+ (([119906 (119909)]2+1003816100381610038161003816119906119899 (119909)

10038161003816100381610038163

) 119906119899(119909

2))

(58)

The initial trial function is selected as

1199060(119909) = 1 minus

119909

2 (59)

and using (50) the approximate first iteration solution isobtained as

1199061(119909) = 1 minus

1073

960119909 + 1199092minus1

21199093+13

961199094minus

3

1601199095

+1

9601199096

(60)

In Figure 6 first three iteration solutions obtained by thePIA(1 1) algorithm are compared with the exact solutionThe convergence of iteration solutions to the exact solutionis obvious

Figure 7 illustrates the absolute error functions of thefirst three iteration solutions of the PIA(1 1) method Themaximum absolute error decreases from 003013 in the firstiteration to 000262 in the third iteration

4 Conclusion

The new perturbation-iteration method is employed for thefirst time to numerically solve the pantograph equationsThe comparative results showed that the method is highly

Journal of Applied Mathematics 9

Absolute error function for first iteration solutionAbsolute error function for second iteration solutionAbsolute error function for third iteration solution

00005

0010015

0020025

0030035

004

01 02 03 04 05 06 07 08 09 10

Figure 7 Comparison of the absolute error functions of the firstthree iteration solutions of the PIA(1 1) method with the exactsolution for example 6

efficient and reliable in both linear and nonlinear problemsWhile for the linear problems studied PIA(1 1) algorithmgave almost the same results as Variational Iteration Methodit gave better results for the nonlinear problems On theother hand the results obtained in the first iteration of thePIA(1 2) algorithm are more convergent than those obtainedin the third iteration of the Variational IterationMethod thisvalidates the performance of the present method Althoughthe equations of PIA(1 2) algorithm are more entangledthan those of PIA(1 1) faster convergence is achieved withPIA(1 2) when compared to PIA(1 1)

It is conceived that the present method can be useful indeveloping new algorithms since they can be generated invarious forms according to the number of correction terms inperturbation expansion and the order of derivation in Taylorexpansion

The disadvantage of the method is that the number ofterms in the solution function increases seriously

In this study the solution intervals of problems arechosen generally the same as those in the references forcomparison purposes In three of the examples the solutionintervals are extended moderately yet for a wider solutioninterval the method should be modified such that the Taylorseries expansions are made around points other than zeroThe method can also be extended to other types of delaydifferential equations but some modifications are required

Conflict of Interests

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

References

[1] MZLiuandDLildquoProperties of analytic solution andnumericalsolution of multi-pantograph equationrdquo Applied Mathematicsand Computation vol 155 no 3 pp 853ndash871 2004

[2] Y Muroya E Ishiwata and H Brunner ldquoOn the attain-able order of collocation methods for pantograph integro-differential equationsrdquo Journal of Computational and AppliedMathematics vol 152 no 1-2 pp 347ndash366 2003

[3] S Yalcınbas M Aynigul and M Sezer ldquoA collocation methodusing Hermite polynomials for approximate solution of panto-graph equationsrdquo Journal of the Franklin Institute vol 348 no 6pp 1128ndash1139 2011

[4] M Sezer andAAkyuz-Dascıoglu ldquoATaylormethod for numer-ical solution of generalized pantograph equations with linearfunctional argumentrdquo Journal of Computational and AppliedMathematics vol 200 no 1 pp 217ndash225 2007

[5] S Sedaghat Y Ordokhani and M Dehghan ldquoNumerical solu-tion of the delay differential equations of pantograph type viaChebyshev polynomialsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 12 pp 4815ndash4830 2012

[6] B Benhammouda H Vazquez-Leal and L Hernandez-Martinez ldquoProcedure for exact solutions of nonlinear panto-graph delay differential equationsrdquo British Journal ofMathemat-ics amp Computer Science vol 4 no 19 pp 2738ndash2751 2014

[7] S Abbasbandy and C Bervillier ldquoAnalytic continuation ofTaylor series and the boundary value problems of some nonlin-ear ordinary differential equationsrdquo Applied Mathematics andComputation vol 218 no 5 pp 2178ndash2199 2011

[8] M A Z Raja ldquoNumerical treatment for boundary value prob-lems of Pantograph functional differential equation using com-putational intelligence algorithmsrdquo Applied Soft Computingvol 24 pp 806ndash821 2014

[9] D Trif ldquoDirect operatorial tau method for pantograph-typeequationsrdquo Applied Mathematics and Computation vol 219 no4 pp 2194ndash2203 2012

[10] A H Nayfeh Introduction to Perturbation Techniques JohnWiley amp Sons New York NY USA 1981

[11] J-H He ldquoIteration perturbation method for strongly nonlinearoscillationsrdquo Journal of Vibration and Control vol 7 no 5 pp631ndash642 2001

[12] R E Mickens ldquoIterationmethod solutions for conservative andlimit-cycle x13 force oscillatorsrdquo Journal of Sound and Vibra-tion vol 292 no 3ndash5 pp 964ndash968 2006

[13] Y Aksoy M Pakdemirli S Abbasbandy and H Boyacı ldquoNewperturbation-iteration solutions for nonlinear heat transferequationsrdquo International Journal of Numerical Methods for Heatamp Fluid Flow vol 22 no 7 pp 814ndash828 2012

[14] T Ozis and A Yıldırım ldquoGenerating the periodic solutions forforcing van der Pol oscillators by the iteration perturbationmethodrdquo Nonlinear Analysis Real World Applications vol 10no 4 pp 1984ndash1989 2009

[15] DDGanji S Karimpour and S S Ganji ldquoHersquos iteration pertur-bation method to nonlinear oscillations of mechanical systemswith single-degree-of freedomrdquo International Journal ofModernPhysics B vol 23 no 11 pp 2469ndash2477 2009

[16] VMarinca andN Herisanu ldquoAmodified iteration perturbationmethod for somenonlinear oscillation problemsrdquoActaMechan-ica vol 184 no 1ndash4 pp 231ndash242 2006

[17] M Pakdemirli and H Boyacı ldquoGeneration of root findingalgorithms via perturbation theory and some formulasrdquoAppliedMathematics and Computation vol 184 no 2 pp 783ndash7882007

[18] M Pakdemirli H Boyacı and H A Yurtsever ldquoPerturbativederivation and comparisons of root-finding algorithms withfourth order derivativesrdquoMathematicalampComputational Appli-cations vol 12 no 2 pp 117ndash124 2007

[19] M Pakdemirli H Boyacı and H A Yurtsever ldquoA root findingalgorithmwith fifth order derivativesrdquoMathematical ampCompu-tational Applications vol 13 no 2 pp 123ndash128 2008

10 Journal of Applied Mathematics

[20] Y Aksoy and M Pakdemirli ldquoNew perturbation-iterationsolutions for Bratu-type equationsrdquo Computers amp Mathematicswith Applications vol 59 no 8 pp 2802ndash2808 2010

[21] I T Dolapcı M Senol and M Pakdemirli ldquoNew perturbationiteration solutions for Fredholm and Volterra integral equa-tionsrdquo Journal of Applied Mathematics vol 2013 Article ID682537 5 pages 2013

[22] A Saadatmandi and M Dehghan ldquoVariational iterationmethod for solving a generalized pantograph equationrdquo Com-puters amp Mathematics with Applications vol 58 no 11-12 pp2190ndash2196 2009

[23] M Sezer S Yalcınbas and M Gulsu ldquoA Taylor polyno-mial approach for solving generalized pantograph equationswith nonhomogenous termrdquo International Journal of ComputerMathematics vol 85 no 7 pp 1055ndash1063 2008

[24] X Y Li and B Y Wu ldquoA continuous method for nonlocalfunctional differential equations with delayed or advancedargumentsrdquo Journal of Mathematical Analysis and Applicationsvol 409 no 1 pp 485ndash493 2014

[25] E H Doha A H Bhrawy D Baleanu and R M Hafez ldquoA newJacobi rationalmdashGauss collocation method for numerical solu-tion of generalized pantograph equationsrdquo Applied NumericalMathematics vol 77 pp 43ndash54 2014

[26] G P Rao and K R Palanisamy ldquoWalsh stretch matrices andfunctional-differential equationsrdquo IEEE Transactions on Auto-matic Control vol 27 no 1 pp 272ndash276 1982

[27] C Hwang ldquoSolution of a functional differential equation viadelayed unit step functionsrdquo International Journal of SystemsScience vol 14 no 9 pp 1065ndash1073 1983

[28] S Yuzbası E Gok andM Sezer ldquoLaguerrematrixmethodwiththe residual error estimation for solutions of a class of delaydifferential equationsrdquo Mathematical Methods in the AppliedSciences vol 37 no 4 pp 453ndash463 2014

[29] N R Anakira A K Alomari and I Hashim ldquoOptimal homo-topy asymptotic method for solving delay differential equa-tionsrdquoMathematical Problems in Engineering vol 2013 ArticleID 498902 11 pages 2013

[30] 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

[31] H Liu A Xiao and L Su ldquoConvergence of variational iterationmethod for second-order delay differential equationsrdquo Journalof Applied Mathematics vol 2013 Article ID 634670 9 pages2013

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

2 Journal of Applied Mathematics

by Pakdemirli and Boyacı [17] where they proposed new rootfinding algorithms using the new systematic approach Laterfourth-order [18] and fifth-order [19] derivative algorithmswere presented Aksoy and Pakdemirli [20] systematicallygenerated the present method for both linear and nonlinearsecond-order differential equations and applied to it Bratu-type equations Dolapci et al [21] used this method to solveFredholm and Volterra integral equations and argued that itcan be applied to both types of integral equations

The main purpose of the work presented in this paperis to apply the new perturbation-iteration solution for thepantograph-type delay differential equations To the authorsrsquoknowledge the application of perturbation-iteration algo-rithms to pantograph-type delay differential equations isnovel The two types of perturbation-iteration algorithmsPIA(1 1) and PIA(1 2) are introduced in the next sectionThen the method is applied to pantograph equations via sixexamples two of which are first-order linear one is first-ordernonlinear and the others are second-order nonlinear theyshowed excellent agreement with the published results andverified the accuracy and efficiency of the present methodThe results are discussed and commented on

The general form of pantograph-type delay differentialequations of both first- and second-order including (1) canbe stated as

119865 (11990610158401015840 1199061015840 119906 119906120572 120576 119909) = 0 (3)

where 119906 = 119906(119909) 119906120572

= 119906(120572119909) 0 lt 120572 lt 1 is a givenconstant and 120576 is the perturbation parameter In this studyweinvestigated a solution for (3) which is closed form Equa-tion (3) covers many different problems studied by variousresearchers

2 Perturbation-Iteration Algorithms

Perturbation-iteration algorithms are briefly called PIA(119899119898)Here 119899 denotes the number of correction terms in the per-turbation expansion and 119898 denotes the order of derivativesin the Taylor series expansion Generally 119899 is smaller thanor equal to 119898 otherwise the correction terms cannot becalculated In this study PIA(1 1) and PIA(1 2) algorithmsconstructed by taking correction term in the perturbationexpansion together with first- and second-order derivativesrespectively are employed

21 Perturbation-Iteration Algorithm PIA(1 1) In PIA(1 1)algorithm we propose a perturbation-iteration algorithm bytaking one correction term in the perturbation expansion andcorrection terms of only the first derivatives in the Taylorexpansion that is 119899 = 1 119898 = 1 Let us consider a second-order pantograph differential equation written in the formof (3) Taking only one correction term in the perturbationexpansion the straightforward expansion for the solution ofeach iteration can be written as follows

119906119899+1

= 119906119899+ 120576 (119906

119888)119899 (4)

where (119906119888)119899is the 119899th correction term of the perturbation-

iteration algorithm Then we substitute (4) into (3) andexpand it in a Taylor series with first derivatives to obtain

119865 (11990610158401015840

119899 1199061015840

119899 119906119899 (119906120572)119899 0)

+ 11986511990610158401015840 (11990610158401015840

119899 1199061015840

119899 119906119899 (119906120572)119899 0) 120576 (119906

10158401015840

119888)119899

+ 1198651199061015840 (11990610158401015840

119899 1199061015840

119899 119906119899 (119906120572)119899 0) 120576 (119906

1015840

119888)119899

+ 119865119906(11990610158401015840

119899 1199061015840

119899 119906119899 (119906120572)119899 0) 120576 (119906

119888)119899

+ 119865119906120572

(11990610158401015840

119899 1199061015840

119899 119906119899 (119906120572)119899 0) 120576 ((119906

120572)119888)119899

+ 119865120576(11990610158401015840

119899 1199061015840

119899 119906119899 (119906120572)119899 0) 120576 = 0

(5)

where ( )1015840 denotes differentiationwith respect to the indepen-dent variable and

11986511990610158401015840 =

120597119865

12059711990610158401015840

1198651199061015840 =

120597119865

1205971199061015840

119865119906=120597119865

120597119906

119865119906120572

=120597119865

120597119906120572

119865120576=120597119865

120597120576

(6)

Bearing in mind that all derivatives are evaluated at 120576 = 0 andrewriting the equation in the following more suitable form

11986511990610158401015840 (11990610158401015840

119888)119899+ 1198651199061015840 (1199061015840

119888)119899+ 119865119906(119906119888)119899+ 119865119906120572

((119906120572)119888)119899

= minus119865120576minus119865

120576

(7)

one may easily notice that (7) is a variable coefficientpantograph equation Starting with an initial guess 119906

0 first

we determine (119906119888)0from (7) and then substitute it into (4)

for calculating 1199061 This iteration procedure is repeated using

(4) and (7) until the approximation is sufficient within a user-defined threshold

22 Perturbation-Iteration Algorithm PIA(1 2) In PIA(1 2)algorithm we put forward a perturbation-iteration algorithmby taking one correction term in the perturbation expansionand correction terms of up to second derivatives in the Taylorexpansion that is 119899 = 1 119898 = 2 Only one correctionterm perturbation expansion was given before in (4) Now

Journal of Applied Mathematics 3

we substitute (4) into (3) and expand in a Taylor series of upto second-order derivatives to obtain

119865 + 11986511990610158401015840120576 (11990610158401015840

119888)119899+ 1198651199061015840120576 (1199061015840

119888)119899+ 119865119906120576 (119906119888)119899

+ 119865119906120572

120576 ((119906120572)119888)119899+ 119865120576120576 +

1

211986511990610158401015840119906101584010158401205762(11990610158401015840

119888)2

119899

+1

2119865119906101584011990610158401205762(1199061015840

119888)2

119899+1

21198651199061199061205762(119906119888)2

119899

+1

2119865119906120572119906120572

1205762((119906120572)119888)2

119899+1

21205762119865120576120576

+ 1198651199061015840101584011990610158401205762(11990610158401015840

119888)119899(1199061015840

119888)119899+ 119865119906101584010158401199061205762(11990610158401015840

119888)119899(119906119888)119899

+ 11986511990610158401015840119906120572

1205762(11990610158401015840

119888)119899((119906120572)119888)119899+ 119865119906101584010158401205761205762(11990610158401015840

119888)119899

+ 11986511990610158401199061205762(1199061015840

119888)119899(119906119888)119899+ 1198651199061015840119906120572

1205762(1199061015840

119888)119899((119906120572)119888)119899

+ 11986511990610158401205761205762(1199061015840

119888)119899+ 119865119906119906120572

1205762(119906119888)119899((119906120572)119888)119899

+ 1198651199061205761205762(119906119888)119899+ 1198651199061205721205761205762((119906120572)119888)119899= 0

(8)

Then bearing in mind that all derivatives are evaluated at120576 = 0 we rewrite this equation in the following more suitableform

(12057611986511990610158401015840 + 120576211986511990610158401015840120576) (11990610158401015840

119888)119899+ (1205761198651199061015840 + 12057621198651199061015840120576) (1199061015840

119888)119899

+ (120576119865119906+ 1205762119865119906120576) (119906119888)119899+ (120576119865119906120572

+1205762119865119906120572120576) ((119906120572)119888)119899

+1

212057621198651199061015840101584011990610158401015840 (11990610158401015840

119888)2

119899+1

2120576211986511990610158401199061015840 (1199061015840

119888)2

119899

+1

21205762119865119906119906(119906119888)2

119899+1

21205762119865119906120572119906120572

((119906120572)119888)2

119899

+ 1205762119865119906101584010158401199061015840 (11990610158401015840

119888)119899(1199061015840

119888)119899+ 120576211986511990610158401015840119906(11990610158401015840

119888)119899(119906119888)119899

+ 120576211986511990610158401015840119906120572

(11990610158401015840

119888)119899((119906120572)119888)119899+ 12057621198651199061015840119906(1199061015840

119888)119899(119906119888)119899

+ 12057621198651199061015840119906120572

(1199061015840

119888)119899((119906120572)119888)119899+ 1205762119865119906119906120572

(119906119888)119899((119906120572)119888)119899

= minus120576119865120576minus 119865 minus

1

21205762119865120576120576

(9)

Equation (9) is a variable coefficient pantograph equationof order two We repeat the previously mentioned iterationprocedure until the approximation is sufficient within a user-defined threshold

3 Application of theMethod to the Pantograph-TypeDelay Differential Equations

In order to illustrate the accuracy and applicability of thepresented method the new perturbation-iteration algorithmis applied to six pantograph-type delay differential equationsof different types For comparison purposes the solutionintervals of problems are chosen generally the same as thosein the references Yet the solution intervals are enlarged inillustrative examples 3 and 5 For a further wider intervalthe method should be modified such that the Taylor seriesexpansions are made around points other than zero

31 Illustrative Example 1 Consider the first-order linearpantograph differential equation [25]

1199061015840(119909) + 119906 (119909) minus

1

10119906 (

119909

5) = minus

1

10119890minus(15)119909

0 le 119909 le 1 (10)

with the initial condition

119906 (0) = 1 (11)

for which the exact solution is 119906(119909) = 119890minus119909 [25]

Equation (10) can be written in the following form

1199061015840(119909) + 119906 (119909) minus 120576119906 (

119909

5) +

1

10119890minus(15)119909

= 0 (12)

where 120576 = 01 is assumed to be the perturbation parameterThe nonzero terms of (7) in PIA(1 1) algorithm are

1198651199061015840 = 1

119865119906= 1

119865120576= minus119906119899(119909

5)

119865 = 1199061015840

119899(119909) + 119906

119899(119909) +

1

10119890minus(15)119909

(13)

Upon substituting the terms of (13) (7) reduces to

(1199061015840

119888)119899(119909) + (119906

119888)119899(119909)

= 119906119899(119909

5) minus

1199061015840

119899(119909) + 119906

119899(119909) + (110) 119890

minus(15)119909

01

(14)

When applying the iteration formula (4) we select an initiallyassumed function Here we start with the following trivialsolution which satisfies the given initial condition

1199060= 1 (15)

4 Journal of Applied Mathematics

Table 1 Comparison of the absolute errors of the PIA(1 1)method with other numerical methods for example 1

119909VIM [22] Taylor method [23] Continuous method [24]

JRC method120572 = 120573 = 05 [25] PIA(1 1)

119899 = 4 119873 = 64 119899 = 4

2minus1

240119890 minus 15 124119890 minus 10 333119890 minus 11 393119890 minus 14 143119890 minus 15

2minus2

791119890 minus 17 974119890 minus 11 413119890 minus 11 216119890 minus 14 791119890 minus 17

2minus3

253119890 minus 18 700119890 minus 11 462119890 minus 11 166119890 minus 15 253119890 minus 18

2minus4

803119890 minus 20 914119890 minus 11 489119890 minus 11 621119890 minus 15 803119890 minus 20

2minus5

252119890 minus 21 528119890 minus 11 505119890 minus 11 368119890 minus 14 252119890 minus 21

2minus6

700119890 minus 23 195119890 minus 11 474119890 minus 11 454119890 minus 14 792119890 minus 23

Using (4) and (14) the approximate solutions at each stepbecome

1199061=

1

10minus1

8119890minus(15)119909

+41

40119890minus119909

1199062=

1

102+

1

320119890minus(15)119909

minus5

384119890minus(125)119909

+9599

9600119890minus119909

1199063=

1

103minus

1

76800119890minus(15)119909

+1

3072119890minus(125)119909

minus125

95232119890minus(1125)119909

+11904001

11904000119890minus119909

1199064=

1

104+

1

95232000119890minus(15)119909

minus1

737280119890minus(125)119909

+25

761825119890minus(1125)119909

minus15625

118849536119890minus(1625)119909

+74280959999

74280960000119890minus119909

(16)

Table 1 gives a comparison of the absolute errors of thePIA(1 1) method with those of other numerical methods Itis apparent from the table that the absolute errors of PIA(1 1)method are smaller than those of the other methods for119899 = 4 and also considering the first four iteration solutionsgiven in (16) one may notice that the numerical solutionconverges to the exact solution Absolute errors obtained byPIA(1 1) method are almost the same as those obtained bythe Variational Iteration Method (VIM) [22] for 119899 = 4

32 Illustrative Example 2 Consider the first-order linearpantograph differential equation [3]

1199061015840(119909) + 119906 (119909) + 119906 (08119909) = 0 0 le 119909 le 1 (17)

with the initial condition

119906 (0) = 1 (18)

which does not have an exact solution Equation (17) can bewritten in the following form

1199061015840(119909) + 119906 (119909) + 120576119906 (08119909) = 0 (19)

where 120576 is the artificially introduced perturbation parameterThe nonzero terms of (7) in PIA(1 1) algorithm are

1198651199061015840 = 1

119865119906= 1

119865120576= 119906119899(08119909)

119865 = 1199061015840

119899(119909) + 119906

119899(119909)

(20)

For this specific example (7) reduces to

(1199061015840

119888)119899(119909) + (119906

119888)119899(119909) = minus119906

119899(08119909) minus 119906

1015840

119899(119909) minus 119906

119899(119909) (21)

When applying the iteration formula (4) we start withthe following trivial solution as the initial function whichsatisfies the given initial conditions

1199060= 1 (22)

Using (4) and (21) the successive iteration results are

1199061= minus1 + 2119890

minus119909

1199062= 1 + 10119890

minus119909minus 10119890minus(45)119909

1199063= minus1 +

218

9119890minus119909

minus 50119890minus(45)119909

+250

9119890minus(1625)119909

1199064= 1 +

21490

549119890minus119909

minus1090

9119890minus(45)119909

+1250

9119890minus(1625)119909

minus31250

549119890minus(64125)119909

(23)

Table 2 shows the PIA(1 1) solutions of (17) for iterationnumbers 119899 = 5 6 7 8 9 As stated above we do not know theexact solution therefore we estimate a solution and observea convergence

A comparison of the results obtained by several differentmethods is shown in Table 3 The numerical results areconsistent

33 Illustrative Example 3 Consider the first-order nonlin-ear pantograph differential equation [29]

1199061015840(119909) = minus2119906

2(119909

2) + 1 0 le 119909 le 1 (24)

Journal of Applied Mathematics 5

Table 2 Convergence of the PIA(1 1) solution for example 2

119909119894

1199065(119909) 119906

6(119909) 119906

7(119909) 119906

8(119909) 119906

9(119909)

0 1 1 1 1 1

02 06646909 06646910 06646910 06646910 0664691004 04335604 04335607 04335607 04335607 0433560706 02764790 02764824 02764823 02764823 0276482308 01714677 01714846 01714841 01714841 017148411 01026142 01026723 01026700 01026701 01026701

with the initial condition

119906 (0) = 0 (25)

which has the exact solution 119906(119909) = sin(119909)Equation (24) can be written in the following form

1199061015840(119909) + 120576119906

2(119909

2) minus 1 = 0 (26)

where 120576 = 2 is assumed to be the perturbation parameterThenonzero terms of (7) in PIA(1 1) algorithm are

1198651199061015840 = 1

119865120576= 1199062

119899(119909

2)

119865 = 1199061015840

119899(119909) minus 1

(27)

The 119899th correction term of the perturbation expansion in (4)is

(1199061015840

119888)119899(119909) = minus119906

2

119899(119909

2) minus

1199061015840

119899(119909) minus 1

2 (28)

In this example we start with the initial guess

1199060= 119909 (29)

Thus the successive approximations are

1199061= 119909 minus

1

61199093

1199062= 119909 minus

1

61199093+

1

1201199095minus

1

80641199097

1199063= 119909 minus

1

61199093+

1

1201199095minus

1

50401199097+

61

232243201199099

minus67

340623360011990911+

1

1288175616011990913

minus1

799065243648011990915

(30)

Subsequently we determine the absolute maximum error for119906119899as

119864119899=1003817100381710038171003817119906119899 (119909) minus 119906 (119909)

1003817100381710038171003817infin

= max 1003816100381610038161003816119906119899 (119909) minus 119906 (119909)1003816100381610038161003816 0 le 119909 le 1

(31)

04 08 12 16 2 24 28 32 36 4 440x

minus1

minus08

minus06

minus04

minus02

002040608

1

u(x)

Exact solutionFirst iteration solution

Second iteration solution

Third iteration solutionof the PIA(1 1)

of the PIA(1 1)

of the PIA(1 1)

Figure 1 Comparison of the first three iteration solutions of thePIA(1 1)method with the exact solution for example 3

The absolute maximum errors 119864119899for different values of 119899 are

given in Table 4 Note that the error decreases continually as119899 increases

The same problem is solved in various studies and itis seen that the absolute maximum error for solving thisexample via Homotopy Asymptotic Method is 12 times 10

minus6

[30] and via Optimal Homotopy Asymptotic Method is 4 times10minus8 [29] while the absolute maximum error obtained by

the PIA(1 1) method is 535 times 10minus20 This affirmed that

the presented method is more accurate than the other twomethods

In Figure 1 the first three iteration solutions (119899 = 1 2 3)

of the PIA(1 1) method contrasted with the exact solutionIt is revealed that even when the solution interval of theproblem is expanded PIA(1 1) method goes on to yieldconvergent numerical solutions

34 Illustrative Example 4 Consider the initial value prob-lem of second-order differential equation with pantographdelay [31]

11990610158401015840(119909) = minus119906 (

119909

2) minus 1199062(119909) + sin4 (119909) + sin2 (119909) + 8

119909 ge 0

(32)

with initial conditions

119906 (0) = 2

1199061015840(0) = 0

(33)

which has the exact solution 119906(119909) = (5 minus cos 2119909)2

341 PIA(1 1) Solution Equation (32) can be written in thefollowing form

11990610158401015840(119909) + 120576 (119906 (

119909

2) + 1199062(119909))

= sin4 (119909) + sin2 (119909) + 8

(34)

6 Journal of Applied Mathematics

Table 3 Comparison of the PIA(1 1) solution with other numerical methods for example 2

119909119894 Walsh series method [26]

DUSF series method[27]

Hermite seriesmethod [3]

Taylor seriesmethod [4]

Laguerre matrixmethod [28] PIA(1 1)

119898 = 100ℎ = 001

119873 = 8 119873 = 8 119873 = 8 119899 = 8

0 1 1 1 1 1 102 0665621 0664677 0664691 0664691 06646910 0664691004 0432426 0433540 0433561 0433561 04335607 0433560706 0275140 0276460 0276482 0276482 02764831 0276482308 0170320 0171464 0171484 0171484 01714942 017148411 0100856 0102652 0102670 0102744 01027437 01026701

Table 4 The absolute maximum errors 119864119899for example 3

119899 1 2 3 4 5 6119864119899

813119890 minus 3 716119890 minus 5 123119890 minus 7 438119890 minus 11 291119890 minus 15 535119890 minus 20

where 120576 = 1 is the artificially introduced perturbationparameter Using (3) (34) returns to

119865 (11990610158401015840 1199061015840 119906 119906120572 120576) = 119906

10158401015840(119909) + 120576 (119906 (

119909

2) + 1199062(119909))

minus (sin4 (119909) + sin2 (119909) + 8) = 0

(35)

The nonzero terms of (7) in PIA(1 1) algorithm are

11986511990610158401015840 = 1

119865120576= 119906119899(119909

2) + 1199062

119899(119909)

119865 = 11990610158401015840

119899(119909) minus sin4 (119909) minus sin2 (119909) minus 8

(36)

Substituting (36) (32) reduces to

(11990610158401015840

119888)119899(119909) = minus119906

119899(119909

2) minus 11990610158401015840

119899(119909) minus 119906

2

119899(119909) + sin4 (119909)

+ sin2 (119909) + 8

(37)

The initial trial function is selected as

1199060= 2 (38)

and using (37) the approximate first iteration solution isobtained as

1199061=13

16minus

1

16sin4 (119909) + 3

16cos2 (119909) + 23119909

2

16

+ cos2 (1199092)

(39)

Even the first iteration of PIA(1 1) solution 1199061 gave the same

result as the first iteration solution obtained by the VIM [31]Still the second and third iteration solutions of the problemare better than those of the VIM solutions as observed inFigures 2 and 3

Exact solutionSecond iteration solution Second iteration solution of the VIM

2

22

24

26

28

3

u(x)

02 04 06 08 10x

of the PIA(1 1)

Figure 2 Comparison of the second iteration solutions of the VIMand PIA(1 1) with exact solution for example 4

342 PIA(1 2) Solution In order to apply the PIA(1 2)algorithm obtained by (9) we have to write (32) in thefollowing form

119865 (11990610158401015840 1199061015840 119906 119906120572 120576) = 119906

10158401015840(119909) + 120576119906

2(119909) + 120576

2119906 (

119909

2)

minus (sin4 (119909) + sin2 (119909) + 8) = 0

(40)

where 120576 is the artificially introduced perturbation parameterThe nonzero terms of (9) in PIA(1 2) algorithm are

11986511990610158401015840 = 1

119865119906120576= 2119906119899(119909)

119865120576= 1199062

119899(119909)

119865120576120576= 2119906119899(119909

2)

119865 = 11990610158401015840

119899(119909) minus sin4 (119909) minus sin2 (119909) minus 8

(41)

Journal of Applied Mathematics 7

Exact solution

Third iteration solution of the VIM

2

22

24

26

28

u(x)

02 04 06 08 10x

Third iteration solution of the PIA(1 1)

Figure 3The comparison of the third iteration solutions of theVIMand PIA(1 1) with exact solution for example 4

Note that introducing the small parameter 120576 = 1 as acoefficient of the pantograph term simplifies (40) and makesit easily solvable For this specific example (40) reads

(11990610158401015840

119888)119899(119909) + (2119906

119899(119909)) (119906

119888)119899(119909)

= minus119906119899(119909

2) minus 11990610158401015840

119899(119909) minus 119906

2

119899(119909) + sin4 (119909) + sin2 (119909)

+ 8

(42)

When applying the iteration formula (4) we select thefollowing initially assumed function

1199060= 2 (43)

and using (9) the approximate solution is

1199061=263

96minus49

96cos (2119909) minus 1

12cos4 (2119909) + 1

48cos2 (119909)

minus1

4119909 sin (119909) cos (119909) minus 1

6cos (119909)

(44)

First iteration of PIA(1 2) and second and third iterationsof VIM solutions are compared with the exact solutionAs can be seen from Figure 4 even the first iteration ofPIA(1 2) yields much better results than the first threeiteration solutions of the VIM

35 Illustrative Example 5 Consider the initial value prob-lem of nonlinear variable coefficient differential equationwith pantograph delay [6]

11990610158401015840(119909) = 119906 (119909) minus

8

11990921199062(119909

2) 119909 ge 0 (45)

with initial conditions

119906 (0) = 0

1199061015840(0) = 1

(46)

which has the exact solution 119906(119909) = 119909119890minus119909

2

22

24

26

28

3

u(x)

02 04 06 08 10x

of the VIM

of the VIM

Exact solutionFirst iteration solution

Second iteration solution

Third iteration solutionof the PIA(1 2)

Figure 4 The comparison of second and third iteration solutionsof the VIM and first iteration solution of the PIA(1 2) with exactsolution of example 4

Equation (45) can be written in the following form

11990610158401015840(119909) + 120576 (

8

11990921199062(119909

2) minus 119906 (119909)) = 0 (47)

where 120576 = 1 is the artificially introduced perturbationparameter Using (3) (45) returns to

119865 (11990610158401015840 1199061015840 119906 119906120572 120576) = 119906

10158401015840(119909) + 120576 (

8

11990921199062(119909

2) minus 119906 (119909))

= 0

(48)

The nonzero terms in PIA(1 1) algorithm are

11986511990610158401015840 = 1

119865120576=

8

11990921199062

119899(119909

2) minus 119906119899(119909)

119865 = 11990610158401015840

119899(119909)

(49)

Substituting (49) (45) reduces to

(11990610158401015840

119888)119899(119909) = minus

8

11990921199062

119899(119909

2) + 119906119899(119909) minus 119906

10158401015840

119899(119909) (50)

The initial trial function is selected as

1199060(119909) = 119909 (51)

and using (50) the approximate first iteration solution isobtained as

1199061= 119909 minus 119909

2+1

61199093

1199062= 119909 minus 119909

2+1

21199093minus

5

361199094+

1

801199095minus

1

86401199096

(52)

The absolute maximum errors 119864119899for different values of 119899 are

given in Table 5 and it is revealed that the error decreasescontinually as 119899 increases

8 Journal of Applied Mathematics

Table 5 The absolute maximum errors 119864119899for example 5

119899 1 2 3 4 5 6 7119864119899

201119890 minus 1 561119890 minus 3 159119890 minus 5 275119890 minus 7 368119890 minus 9 676119890 minus 11 494119890 minus 13

Exact solution8th iteration solution 7th iteration solution 6th iteration solution 5th iteration solution

0

01

02

03

04

06 12 18 24 3 36 42 480

of the PIA(1 1)

of the PIA(1 1)

of the PIA(1 1)

of the PIA(1 1)

Figure 5 Comparison of the iteration solutions of the PIA(1 1)method with the exact solution for example 5

Figure 5 illustrates the PIA(1 1) solution of this examplein the interval [0 5] In this example we enlarge the intervalso as to compare our solutionmore thoroughly with the exactsolution The present method yields pretty good results

36 Illustrative Example 6 Consider the second-order non-linear pantograph functional differential equation [7 8]

11990610158401015840(119909) = ([119906 (119909)]

2+ |119906 (119909)|

3) 119906 (

119909

2) 119909 ge 0 (53)

with initial conditions

119906 (0) = 1 119906 (1) =1

2(54)

which has the exact solution 119906(119909) = 1(1 + 119909)Equation (45) can be written in the following form

11990610158401015840(119909) minus 120576 (([119906 (119909)]

2+ |119906 (119909)|

3) 119906 (

119909

2)) = 0 (55)

where 120576 is the artificially introduced small parameter Using(3) (45) returns to

119865 (11990610158401015840 1199061015840 119906 119906120572 120576)

= 11990610158401015840(119909) minus 120576 (([119906 (119909)]

2+ |119906 (119909)|

3) 119906 (

119909

2)) = 0

(56)

The nonzero terms in PIA(1 1) algorithm are

11986511990610158401015840 = 1

119865120576= minus(([119906 (119909)]

2+1003816100381610038161003816119906119899 (119909)

10038161003816100381610038163

) 119906119899(119909

2))

119865 = 11990610158401015840

119899(119909)

(57)

First iteration solution Second iteration solution Third iteration solution

01 02 03 04 05 06 07 08 09 10x

04

05

06

07

08

09

1

u(x)

of the PIA(1 1)

of the PIA(1 1)

of the PIA(1 1)

Figure 6 Comparison of the first three iteration solutions of thePIA(1 1)method with the exact solution for example 6

Substituting (49) (45) reduces to

(11990610158401015840

119888)119899(119909) = minus119906

10158401015840

119899(119909)

+ (([119906 (119909)]2+1003816100381610038161003816119906119899 (119909)

10038161003816100381610038163

) 119906119899(119909

2))

(58)

The initial trial function is selected as

1199060(119909) = 1 minus

119909

2 (59)

and using (50) the approximate first iteration solution isobtained as

1199061(119909) = 1 minus

1073

960119909 + 1199092minus1

21199093+13

961199094minus

3

1601199095

+1

9601199096

(60)

In Figure 6 first three iteration solutions obtained by thePIA(1 1) algorithm are compared with the exact solutionThe convergence of iteration solutions to the exact solutionis obvious

Figure 7 illustrates the absolute error functions of thefirst three iteration solutions of the PIA(1 1) method Themaximum absolute error decreases from 003013 in the firstiteration to 000262 in the third iteration

4 Conclusion

The new perturbation-iteration method is employed for thefirst time to numerically solve the pantograph equationsThe comparative results showed that the method is highly

Journal of Applied Mathematics 9

Absolute error function for first iteration solutionAbsolute error function for second iteration solutionAbsolute error function for third iteration solution

00005

0010015

0020025

0030035

004

01 02 03 04 05 06 07 08 09 10

Figure 7 Comparison of the absolute error functions of the firstthree iteration solutions of the PIA(1 1) method with the exactsolution for example 6

efficient and reliable in both linear and nonlinear problemsWhile for the linear problems studied PIA(1 1) algorithmgave almost the same results as Variational Iteration Methodit gave better results for the nonlinear problems On theother hand the results obtained in the first iteration of thePIA(1 2) algorithm are more convergent than those obtainedin the third iteration of the Variational IterationMethod thisvalidates the performance of the present method Althoughthe equations of PIA(1 2) algorithm are more entangledthan those of PIA(1 1) faster convergence is achieved withPIA(1 2) when compared to PIA(1 1)

It is conceived that the present method can be useful indeveloping new algorithms since they can be generated invarious forms according to the number of correction terms inperturbation expansion and the order of derivation in Taylorexpansion

The disadvantage of the method is that the number ofterms in the solution function increases seriously

In this study the solution intervals of problems arechosen generally the same as those in the references forcomparison purposes In three of the examples the solutionintervals are extended moderately yet for a wider solutioninterval the method should be modified such that the Taylorseries expansions are made around points other than zeroThe method can also be extended to other types of delaydifferential equations but some modifications are required

Conflict of Interests

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

References

[1] MZLiuandDLildquoProperties of analytic solution andnumericalsolution of multi-pantograph equationrdquo Applied Mathematicsand Computation vol 155 no 3 pp 853ndash871 2004

[2] Y Muroya E Ishiwata and H Brunner ldquoOn the attain-able order of collocation methods for pantograph integro-differential equationsrdquo Journal of Computational and AppliedMathematics vol 152 no 1-2 pp 347ndash366 2003

[3] S Yalcınbas M Aynigul and M Sezer ldquoA collocation methodusing Hermite polynomials for approximate solution of panto-graph equationsrdquo Journal of the Franklin Institute vol 348 no 6pp 1128ndash1139 2011

[4] M Sezer andAAkyuz-Dascıoglu ldquoATaylormethod for numer-ical solution of generalized pantograph equations with linearfunctional argumentrdquo Journal of Computational and AppliedMathematics vol 200 no 1 pp 217ndash225 2007

[5] S Sedaghat Y Ordokhani and M Dehghan ldquoNumerical solu-tion of the delay differential equations of pantograph type viaChebyshev polynomialsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 12 pp 4815ndash4830 2012

[6] B Benhammouda H Vazquez-Leal and L Hernandez-Martinez ldquoProcedure for exact solutions of nonlinear panto-graph delay differential equationsrdquo British Journal ofMathemat-ics amp Computer Science vol 4 no 19 pp 2738ndash2751 2014

[7] S Abbasbandy and C Bervillier ldquoAnalytic continuation ofTaylor series and the boundary value problems of some nonlin-ear ordinary differential equationsrdquo Applied Mathematics andComputation vol 218 no 5 pp 2178ndash2199 2011

[8] M A Z Raja ldquoNumerical treatment for boundary value prob-lems of Pantograph functional differential equation using com-putational intelligence algorithmsrdquo Applied Soft Computingvol 24 pp 806ndash821 2014

[9] D Trif ldquoDirect operatorial tau method for pantograph-typeequationsrdquo Applied Mathematics and Computation vol 219 no4 pp 2194ndash2203 2012

[10] A H Nayfeh Introduction to Perturbation Techniques JohnWiley amp Sons New York NY USA 1981

[11] J-H He ldquoIteration perturbation method for strongly nonlinearoscillationsrdquo Journal of Vibration and Control vol 7 no 5 pp631ndash642 2001

[12] R E Mickens ldquoIterationmethod solutions for conservative andlimit-cycle x13 force oscillatorsrdquo Journal of Sound and Vibra-tion vol 292 no 3ndash5 pp 964ndash968 2006

[13] Y Aksoy M Pakdemirli S Abbasbandy and H Boyacı ldquoNewperturbation-iteration solutions for nonlinear heat transferequationsrdquo International Journal of Numerical Methods for Heatamp Fluid Flow vol 22 no 7 pp 814ndash828 2012

[14] T Ozis and A Yıldırım ldquoGenerating the periodic solutions forforcing van der Pol oscillators by the iteration perturbationmethodrdquo Nonlinear Analysis Real World Applications vol 10no 4 pp 1984ndash1989 2009

[15] DDGanji S Karimpour and S S Ganji ldquoHersquos iteration pertur-bation method to nonlinear oscillations of mechanical systemswith single-degree-of freedomrdquo International Journal ofModernPhysics B vol 23 no 11 pp 2469ndash2477 2009

[16] VMarinca andN Herisanu ldquoAmodified iteration perturbationmethod for somenonlinear oscillation problemsrdquoActaMechan-ica vol 184 no 1ndash4 pp 231ndash242 2006

[17] M Pakdemirli and H Boyacı ldquoGeneration of root findingalgorithms via perturbation theory and some formulasrdquoAppliedMathematics and Computation vol 184 no 2 pp 783ndash7882007

[18] M Pakdemirli H Boyacı and H A Yurtsever ldquoPerturbativederivation and comparisons of root-finding algorithms withfourth order derivativesrdquoMathematicalampComputational Appli-cations vol 12 no 2 pp 117ndash124 2007

[19] M Pakdemirli H Boyacı and H A Yurtsever ldquoA root findingalgorithmwith fifth order derivativesrdquoMathematical ampCompu-tational Applications vol 13 no 2 pp 123ndash128 2008

10 Journal of Applied Mathematics

[20] Y Aksoy and M Pakdemirli ldquoNew perturbation-iterationsolutions for Bratu-type equationsrdquo Computers amp Mathematicswith Applications vol 59 no 8 pp 2802ndash2808 2010

[21] I T Dolapcı M Senol and M Pakdemirli ldquoNew perturbationiteration solutions for Fredholm and Volterra integral equa-tionsrdquo Journal of Applied Mathematics vol 2013 Article ID682537 5 pages 2013

[22] A Saadatmandi and M Dehghan ldquoVariational iterationmethod for solving a generalized pantograph equationrdquo Com-puters amp Mathematics with Applications vol 58 no 11-12 pp2190ndash2196 2009

[23] M Sezer S Yalcınbas and M Gulsu ldquoA Taylor polyno-mial approach for solving generalized pantograph equationswith nonhomogenous termrdquo International Journal of ComputerMathematics vol 85 no 7 pp 1055ndash1063 2008

[24] X Y Li and B Y Wu ldquoA continuous method for nonlocalfunctional differential equations with delayed or advancedargumentsrdquo Journal of Mathematical Analysis and Applicationsvol 409 no 1 pp 485ndash493 2014

[25] E H Doha A H Bhrawy D Baleanu and R M Hafez ldquoA newJacobi rationalmdashGauss collocation method for numerical solu-tion of generalized pantograph equationsrdquo Applied NumericalMathematics vol 77 pp 43ndash54 2014

[26] G P Rao and K R Palanisamy ldquoWalsh stretch matrices andfunctional-differential equationsrdquo IEEE Transactions on Auto-matic Control vol 27 no 1 pp 272ndash276 1982

[27] C Hwang ldquoSolution of a functional differential equation viadelayed unit step functionsrdquo International Journal of SystemsScience vol 14 no 9 pp 1065ndash1073 1983

[28] S Yuzbası E Gok andM Sezer ldquoLaguerrematrixmethodwiththe residual error estimation for solutions of a class of delaydifferential equationsrdquo Mathematical Methods in the AppliedSciences vol 37 no 4 pp 453ndash463 2014

[29] N R Anakira A K Alomari and I Hashim ldquoOptimal homo-topy asymptotic method for solving delay differential equa-tionsrdquoMathematical Problems in Engineering vol 2013 ArticleID 498902 11 pages 2013

[30] 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

[31] H Liu A Xiao and L Su ldquoConvergence of variational iterationmethod for second-order delay differential equationsrdquo Journalof Applied Mathematics vol 2013 Article ID 634670 9 pages2013

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Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

we substitute (4) into (3) and expand in a Taylor series of upto second-order derivatives to obtain

119865 + 11986511990610158401015840120576 (11990610158401015840

119888)119899+ 1198651199061015840120576 (1199061015840

119888)119899+ 119865119906120576 (119906119888)119899

+ 119865119906120572

120576 ((119906120572)119888)119899+ 119865120576120576 +

1

211986511990610158401015840119906101584010158401205762(11990610158401015840

119888)2

119899

+1

2119865119906101584011990610158401205762(1199061015840

119888)2

119899+1

21198651199061199061205762(119906119888)2

119899

+1

2119865119906120572119906120572

1205762((119906120572)119888)2

119899+1

21205762119865120576120576

+ 1198651199061015840101584011990610158401205762(11990610158401015840

119888)119899(1199061015840

119888)119899+ 119865119906101584010158401199061205762(11990610158401015840

119888)119899(119906119888)119899

+ 11986511990610158401015840119906120572

1205762(11990610158401015840

119888)119899((119906120572)119888)119899+ 119865119906101584010158401205761205762(11990610158401015840

119888)119899

+ 11986511990610158401199061205762(1199061015840

119888)119899(119906119888)119899+ 1198651199061015840119906120572

1205762(1199061015840

119888)119899((119906120572)119888)119899

+ 11986511990610158401205761205762(1199061015840

119888)119899+ 119865119906119906120572

1205762(119906119888)119899((119906120572)119888)119899

+ 1198651199061205761205762(119906119888)119899+ 1198651199061205721205761205762((119906120572)119888)119899= 0

(8)

Then bearing in mind that all derivatives are evaluated at120576 = 0 we rewrite this equation in the following more suitableform

(12057611986511990610158401015840 + 120576211986511990610158401015840120576) (11990610158401015840

119888)119899+ (1205761198651199061015840 + 12057621198651199061015840120576) (1199061015840

119888)119899

+ (120576119865119906+ 1205762119865119906120576) (119906119888)119899+ (120576119865119906120572

+1205762119865119906120572120576) ((119906120572)119888)119899

+1

212057621198651199061015840101584011990610158401015840 (11990610158401015840

119888)2

119899+1

2120576211986511990610158401199061015840 (1199061015840

119888)2

119899

+1

21205762119865119906119906(119906119888)2

119899+1

21205762119865119906120572119906120572

((119906120572)119888)2

119899

+ 1205762119865119906101584010158401199061015840 (11990610158401015840

119888)119899(1199061015840

119888)119899+ 120576211986511990610158401015840119906(11990610158401015840

119888)119899(119906119888)119899

+ 120576211986511990610158401015840119906120572

(11990610158401015840

119888)119899((119906120572)119888)119899+ 12057621198651199061015840119906(1199061015840

119888)119899(119906119888)119899

+ 12057621198651199061015840119906120572

(1199061015840

119888)119899((119906120572)119888)119899+ 1205762119865119906119906120572

(119906119888)119899((119906120572)119888)119899

= minus120576119865120576minus 119865 minus

1

21205762119865120576120576

(9)

Equation (9) is a variable coefficient pantograph equationof order two We repeat the previously mentioned iterationprocedure until the approximation is sufficient within a user-defined threshold

3 Application of theMethod to the Pantograph-TypeDelay Differential Equations

In order to illustrate the accuracy and applicability of thepresented method the new perturbation-iteration algorithmis applied to six pantograph-type delay differential equationsof different types For comparison purposes the solutionintervals of problems are chosen generally the same as thosein the references Yet the solution intervals are enlarged inillustrative examples 3 and 5 For a further wider intervalthe method should be modified such that the Taylor seriesexpansions are made around points other than zero

31 Illustrative Example 1 Consider the first-order linearpantograph differential equation [25]

1199061015840(119909) + 119906 (119909) minus

1

10119906 (

119909

5) = minus

1

10119890minus(15)119909

0 le 119909 le 1 (10)

with the initial condition

119906 (0) = 1 (11)

for which the exact solution is 119906(119909) = 119890minus119909 [25]

Equation (10) can be written in the following form

1199061015840(119909) + 119906 (119909) minus 120576119906 (

119909

5) +

1

10119890minus(15)119909

= 0 (12)

where 120576 = 01 is assumed to be the perturbation parameterThe nonzero terms of (7) in PIA(1 1) algorithm are

1198651199061015840 = 1

119865119906= 1

119865120576= minus119906119899(119909

5)

119865 = 1199061015840

119899(119909) + 119906

119899(119909) +

1

10119890minus(15)119909

(13)

Upon substituting the terms of (13) (7) reduces to

(1199061015840

119888)119899(119909) + (119906

119888)119899(119909)

= 119906119899(119909

5) minus

1199061015840

119899(119909) + 119906

119899(119909) + (110) 119890

minus(15)119909

01

(14)

When applying the iteration formula (4) we select an initiallyassumed function Here we start with the following trivialsolution which satisfies the given initial condition

1199060= 1 (15)

4 Journal of Applied Mathematics

Table 1 Comparison of the absolute errors of the PIA(1 1)method with other numerical methods for example 1

119909VIM [22] Taylor method [23] Continuous method [24]

JRC method120572 = 120573 = 05 [25] PIA(1 1)

119899 = 4 119873 = 64 119899 = 4

2minus1

240119890 minus 15 124119890 minus 10 333119890 minus 11 393119890 minus 14 143119890 minus 15

2minus2

791119890 minus 17 974119890 minus 11 413119890 minus 11 216119890 minus 14 791119890 minus 17

2minus3

253119890 minus 18 700119890 minus 11 462119890 minus 11 166119890 minus 15 253119890 minus 18

2minus4

803119890 minus 20 914119890 minus 11 489119890 minus 11 621119890 minus 15 803119890 minus 20

2minus5

252119890 minus 21 528119890 minus 11 505119890 minus 11 368119890 minus 14 252119890 minus 21

2minus6

700119890 minus 23 195119890 minus 11 474119890 minus 11 454119890 minus 14 792119890 minus 23

Using (4) and (14) the approximate solutions at each stepbecome

1199061=

1

10minus1

8119890minus(15)119909

+41

40119890minus119909

1199062=

1

102+

1

320119890minus(15)119909

minus5

384119890minus(125)119909

+9599

9600119890minus119909

1199063=

1

103minus

1

76800119890minus(15)119909

+1

3072119890minus(125)119909

minus125

95232119890minus(1125)119909

+11904001

11904000119890minus119909

1199064=

1

104+

1

95232000119890minus(15)119909

minus1

737280119890minus(125)119909

+25

761825119890minus(1125)119909

minus15625

118849536119890minus(1625)119909

+74280959999

74280960000119890minus119909

(16)

Table 1 gives a comparison of the absolute errors of thePIA(1 1) method with those of other numerical methods Itis apparent from the table that the absolute errors of PIA(1 1)method are smaller than those of the other methods for119899 = 4 and also considering the first four iteration solutionsgiven in (16) one may notice that the numerical solutionconverges to the exact solution Absolute errors obtained byPIA(1 1) method are almost the same as those obtained bythe Variational Iteration Method (VIM) [22] for 119899 = 4

32 Illustrative Example 2 Consider the first-order linearpantograph differential equation [3]

1199061015840(119909) + 119906 (119909) + 119906 (08119909) = 0 0 le 119909 le 1 (17)

with the initial condition

119906 (0) = 1 (18)

which does not have an exact solution Equation (17) can bewritten in the following form

1199061015840(119909) + 119906 (119909) + 120576119906 (08119909) = 0 (19)

where 120576 is the artificially introduced perturbation parameterThe nonzero terms of (7) in PIA(1 1) algorithm are

1198651199061015840 = 1

119865119906= 1

119865120576= 119906119899(08119909)

119865 = 1199061015840

119899(119909) + 119906

119899(119909)

(20)

For this specific example (7) reduces to

(1199061015840

119888)119899(119909) + (119906

119888)119899(119909) = minus119906

119899(08119909) minus 119906

1015840

119899(119909) minus 119906

119899(119909) (21)

When applying the iteration formula (4) we start withthe following trivial solution as the initial function whichsatisfies the given initial conditions

1199060= 1 (22)

Using (4) and (21) the successive iteration results are

1199061= minus1 + 2119890

minus119909

1199062= 1 + 10119890

minus119909minus 10119890minus(45)119909

1199063= minus1 +

218

9119890minus119909

minus 50119890minus(45)119909

+250

9119890minus(1625)119909

1199064= 1 +

21490

549119890minus119909

minus1090

9119890minus(45)119909

+1250

9119890minus(1625)119909

minus31250

549119890minus(64125)119909

(23)

Table 2 shows the PIA(1 1) solutions of (17) for iterationnumbers 119899 = 5 6 7 8 9 As stated above we do not know theexact solution therefore we estimate a solution and observea convergence

A comparison of the results obtained by several differentmethods is shown in Table 3 The numerical results areconsistent

33 Illustrative Example 3 Consider the first-order nonlin-ear pantograph differential equation [29]

1199061015840(119909) = minus2119906

2(119909

2) + 1 0 le 119909 le 1 (24)

Journal of Applied Mathematics 5

Table 2 Convergence of the PIA(1 1) solution for example 2

119909119894

1199065(119909) 119906

6(119909) 119906

7(119909) 119906

8(119909) 119906

9(119909)

0 1 1 1 1 1

02 06646909 06646910 06646910 06646910 0664691004 04335604 04335607 04335607 04335607 0433560706 02764790 02764824 02764823 02764823 0276482308 01714677 01714846 01714841 01714841 017148411 01026142 01026723 01026700 01026701 01026701

with the initial condition

119906 (0) = 0 (25)

which has the exact solution 119906(119909) = sin(119909)Equation (24) can be written in the following form

1199061015840(119909) + 120576119906

2(119909

2) minus 1 = 0 (26)

where 120576 = 2 is assumed to be the perturbation parameterThenonzero terms of (7) in PIA(1 1) algorithm are

1198651199061015840 = 1

119865120576= 1199062

119899(119909

2)

119865 = 1199061015840

119899(119909) minus 1

(27)

The 119899th correction term of the perturbation expansion in (4)is

(1199061015840

119888)119899(119909) = minus119906

2

119899(119909

2) minus

1199061015840

119899(119909) minus 1

2 (28)

In this example we start with the initial guess

1199060= 119909 (29)

Thus the successive approximations are

1199061= 119909 minus

1

61199093

1199062= 119909 minus

1

61199093+

1

1201199095minus

1

80641199097

1199063= 119909 minus

1

61199093+

1

1201199095minus

1

50401199097+

61

232243201199099

minus67

340623360011990911+

1

1288175616011990913

minus1

799065243648011990915

(30)

Subsequently we determine the absolute maximum error for119906119899as

119864119899=1003817100381710038171003817119906119899 (119909) minus 119906 (119909)

1003817100381710038171003817infin

= max 1003816100381610038161003816119906119899 (119909) minus 119906 (119909)1003816100381610038161003816 0 le 119909 le 1

(31)

04 08 12 16 2 24 28 32 36 4 440x

minus1

minus08

minus06

minus04

minus02

002040608

1

u(x)

Exact solutionFirst iteration solution

Second iteration solution

Third iteration solutionof the PIA(1 1)

of the PIA(1 1)

of the PIA(1 1)

Figure 1 Comparison of the first three iteration solutions of thePIA(1 1)method with the exact solution for example 3

The absolute maximum errors 119864119899for different values of 119899 are

given in Table 4 Note that the error decreases continually as119899 increases

The same problem is solved in various studies and itis seen that the absolute maximum error for solving thisexample via Homotopy Asymptotic Method is 12 times 10

minus6

[30] and via Optimal Homotopy Asymptotic Method is 4 times10minus8 [29] while the absolute maximum error obtained by

the PIA(1 1) method is 535 times 10minus20 This affirmed that

the presented method is more accurate than the other twomethods

In Figure 1 the first three iteration solutions (119899 = 1 2 3)

of the PIA(1 1) method contrasted with the exact solutionIt is revealed that even when the solution interval of theproblem is expanded PIA(1 1) method goes on to yieldconvergent numerical solutions

34 Illustrative Example 4 Consider the initial value prob-lem of second-order differential equation with pantographdelay [31]

11990610158401015840(119909) = minus119906 (

119909

2) minus 1199062(119909) + sin4 (119909) + sin2 (119909) + 8

119909 ge 0

(32)

with initial conditions

119906 (0) = 2

1199061015840(0) = 0

(33)

which has the exact solution 119906(119909) = (5 minus cos 2119909)2

341 PIA(1 1) Solution Equation (32) can be written in thefollowing form

11990610158401015840(119909) + 120576 (119906 (

119909

2) + 1199062(119909))

= sin4 (119909) + sin2 (119909) + 8

(34)

6 Journal of Applied Mathematics

Table 3 Comparison of the PIA(1 1) solution with other numerical methods for example 2

119909119894 Walsh series method [26]

DUSF series method[27]

Hermite seriesmethod [3]

Taylor seriesmethod [4]

Laguerre matrixmethod [28] PIA(1 1)

119898 = 100ℎ = 001

119873 = 8 119873 = 8 119873 = 8 119899 = 8

0 1 1 1 1 1 102 0665621 0664677 0664691 0664691 06646910 0664691004 0432426 0433540 0433561 0433561 04335607 0433560706 0275140 0276460 0276482 0276482 02764831 0276482308 0170320 0171464 0171484 0171484 01714942 017148411 0100856 0102652 0102670 0102744 01027437 01026701

Table 4 The absolute maximum errors 119864119899for example 3

119899 1 2 3 4 5 6119864119899

813119890 minus 3 716119890 minus 5 123119890 minus 7 438119890 minus 11 291119890 minus 15 535119890 minus 20

where 120576 = 1 is the artificially introduced perturbationparameter Using (3) (34) returns to

119865 (11990610158401015840 1199061015840 119906 119906120572 120576) = 119906

10158401015840(119909) + 120576 (119906 (

119909

2) + 1199062(119909))

minus (sin4 (119909) + sin2 (119909) + 8) = 0

(35)

The nonzero terms of (7) in PIA(1 1) algorithm are

11986511990610158401015840 = 1

119865120576= 119906119899(119909

2) + 1199062

119899(119909)

119865 = 11990610158401015840

119899(119909) minus sin4 (119909) minus sin2 (119909) minus 8

(36)

Substituting (36) (32) reduces to

(11990610158401015840

119888)119899(119909) = minus119906

119899(119909

2) minus 11990610158401015840

119899(119909) minus 119906

2

119899(119909) + sin4 (119909)

+ sin2 (119909) + 8

(37)

The initial trial function is selected as

1199060= 2 (38)

and using (37) the approximate first iteration solution isobtained as

1199061=13

16minus

1

16sin4 (119909) + 3

16cos2 (119909) + 23119909

2

16

+ cos2 (1199092)

(39)

Even the first iteration of PIA(1 1) solution 1199061 gave the same

result as the first iteration solution obtained by the VIM [31]Still the second and third iteration solutions of the problemare better than those of the VIM solutions as observed inFigures 2 and 3

Exact solutionSecond iteration solution Second iteration solution of the VIM

2

22

24

26

28

3

u(x)

02 04 06 08 10x

of the PIA(1 1)

Figure 2 Comparison of the second iteration solutions of the VIMand PIA(1 1) with exact solution for example 4

342 PIA(1 2) Solution In order to apply the PIA(1 2)algorithm obtained by (9) we have to write (32) in thefollowing form

119865 (11990610158401015840 1199061015840 119906 119906120572 120576) = 119906

10158401015840(119909) + 120576119906

2(119909) + 120576

2119906 (

119909

2)

minus (sin4 (119909) + sin2 (119909) + 8) = 0

(40)

where 120576 is the artificially introduced perturbation parameterThe nonzero terms of (9) in PIA(1 2) algorithm are

11986511990610158401015840 = 1

119865119906120576= 2119906119899(119909)

119865120576= 1199062

119899(119909)

119865120576120576= 2119906119899(119909

2)

119865 = 11990610158401015840

119899(119909) minus sin4 (119909) minus sin2 (119909) minus 8

(41)

Journal of Applied Mathematics 7

Exact solution

Third iteration solution of the VIM

2

22

24

26

28

u(x)

02 04 06 08 10x

Third iteration solution of the PIA(1 1)

Figure 3The comparison of the third iteration solutions of theVIMand PIA(1 1) with exact solution for example 4

Note that introducing the small parameter 120576 = 1 as acoefficient of the pantograph term simplifies (40) and makesit easily solvable For this specific example (40) reads

(11990610158401015840

119888)119899(119909) + (2119906

119899(119909)) (119906

119888)119899(119909)

= minus119906119899(119909

2) minus 11990610158401015840

119899(119909) minus 119906

2

119899(119909) + sin4 (119909) + sin2 (119909)

+ 8

(42)

When applying the iteration formula (4) we select thefollowing initially assumed function

1199060= 2 (43)

and using (9) the approximate solution is

1199061=263

96minus49

96cos (2119909) minus 1

12cos4 (2119909) + 1

48cos2 (119909)

minus1

4119909 sin (119909) cos (119909) minus 1

6cos (119909)

(44)

First iteration of PIA(1 2) and second and third iterationsof VIM solutions are compared with the exact solutionAs can be seen from Figure 4 even the first iteration ofPIA(1 2) yields much better results than the first threeiteration solutions of the VIM

35 Illustrative Example 5 Consider the initial value prob-lem of nonlinear variable coefficient differential equationwith pantograph delay [6]

11990610158401015840(119909) = 119906 (119909) minus

8

11990921199062(119909

2) 119909 ge 0 (45)

with initial conditions

119906 (0) = 0

1199061015840(0) = 1

(46)

which has the exact solution 119906(119909) = 119909119890minus119909

2

22

24

26

28

3

u(x)

02 04 06 08 10x

of the VIM

of the VIM

Exact solutionFirst iteration solution

Second iteration solution

Third iteration solutionof the PIA(1 2)

Figure 4 The comparison of second and third iteration solutionsof the VIM and first iteration solution of the PIA(1 2) with exactsolution of example 4

Equation (45) can be written in the following form

11990610158401015840(119909) + 120576 (

8

11990921199062(119909

2) minus 119906 (119909)) = 0 (47)

where 120576 = 1 is the artificially introduced perturbationparameter Using (3) (45) returns to

119865 (11990610158401015840 1199061015840 119906 119906120572 120576) = 119906

10158401015840(119909) + 120576 (

8

11990921199062(119909

2) minus 119906 (119909))

= 0

(48)

The nonzero terms in PIA(1 1) algorithm are

11986511990610158401015840 = 1

119865120576=

8

11990921199062

119899(119909

2) minus 119906119899(119909)

119865 = 11990610158401015840

119899(119909)

(49)

Substituting (49) (45) reduces to

(11990610158401015840

119888)119899(119909) = minus

8

11990921199062

119899(119909

2) + 119906119899(119909) minus 119906

10158401015840

119899(119909) (50)

The initial trial function is selected as

1199060(119909) = 119909 (51)

and using (50) the approximate first iteration solution isobtained as

1199061= 119909 minus 119909

2+1

61199093

1199062= 119909 minus 119909

2+1

21199093minus

5

361199094+

1

801199095minus

1

86401199096

(52)

The absolute maximum errors 119864119899for different values of 119899 are

given in Table 5 and it is revealed that the error decreasescontinually as 119899 increases

8 Journal of Applied Mathematics

Table 5 The absolute maximum errors 119864119899for example 5

119899 1 2 3 4 5 6 7119864119899

201119890 minus 1 561119890 minus 3 159119890 minus 5 275119890 minus 7 368119890 minus 9 676119890 minus 11 494119890 minus 13

Exact solution8th iteration solution 7th iteration solution 6th iteration solution 5th iteration solution

0

01

02

03

04

06 12 18 24 3 36 42 480

of the PIA(1 1)

of the PIA(1 1)

of the PIA(1 1)

of the PIA(1 1)

Figure 5 Comparison of the iteration solutions of the PIA(1 1)method with the exact solution for example 5

Figure 5 illustrates the PIA(1 1) solution of this examplein the interval [0 5] In this example we enlarge the intervalso as to compare our solutionmore thoroughly with the exactsolution The present method yields pretty good results

36 Illustrative Example 6 Consider the second-order non-linear pantograph functional differential equation [7 8]

11990610158401015840(119909) = ([119906 (119909)]

2+ |119906 (119909)|

3) 119906 (

119909

2) 119909 ge 0 (53)

with initial conditions

119906 (0) = 1 119906 (1) =1

2(54)

which has the exact solution 119906(119909) = 1(1 + 119909)Equation (45) can be written in the following form

11990610158401015840(119909) minus 120576 (([119906 (119909)]

2+ |119906 (119909)|

3) 119906 (

119909

2)) = 0 (55)

where 120576 is the artificially introduced small parameter Using(3) (45) returns to

119865 (11990610158401015840 1199061015840 119906 119906120572 120576)

= 11990610158401015840(119909) minus 120576 (([119906 (119909)]

2+ |119906 (119909)|

3) 119906 (

119909

2)) = 0

(56)

The nonzero terms in PIA(1 1) algorithm are

11986511990610158401015840 = 1

119865120576= minus(([119906 (119909)]

2+1003816100381610038161003816119906119899 (119909)

10038161003816100381610038163

) 119906119899(119909

2))

119865 = 11990610158401015840

119899(119909)

(57)

First iteration solution Second iteration solution Third iteration solution

01 02 03 04 05 06 07 08 09 10x

04

05

06

07

08

09

1

u(x)

of the PIA(1 1)

of the PIA(1 1)

of the PIA(1 1)

Figure 6 Comparison of the first three iteration solutions of thePIA(1 1)method with the exact solution for example 6

Substituting (49) (45) reduces to

(11990610158401015840

119888)119899(119909) = minus119906

10158401015840

119899(119909)

+ (([119906 (119909)]2+1003816100381610038161003816119906119899 (119909)

10038161003816100381610038163

) 119906119899(119909

2))

(58)

The initial trial function is selected as

1199060(119909) = 1 minus

119909

2 (59)

and using (50) the approximate first iteration solution isobtained as

1199061(119909) = 1 minus

1073

960119909 + 1199092minus1

21199093+13

961199094minus

3

1601199095

+1

9601199096

(60)

In Figure 6 first three iteration solutions obtained by thePIA(1 1) algorithm are compared with the exact solutionThe convergence of iteration solutions to the exact solutionis obvious

Figure 7 illustrates the absolute error functions of thefirst three iteration solutions of the PIA(1 1) method Themaximum absolute error decreases from 003013 in the firstiteration to 000262 in the third iteration

4 Conclusion

The new perturbation-iteration method is employed for thefirst time to numerically solve the pantograph equationsThe comparative results showed that the method is highly

Journal of Applied Mathematics 9

Absolute error function for first iteration solutionAbsolute error function for second iteration solutionAbsolute error function for third iteration solution

00005

0010015

0020025

0030035

004

01 02 03 04 05 06 07 08 09 10

Figure 7 Comparison of the absolute error functions of the firstthree iteration solutions of the PIA(1 1) method with the exactsolution for example 6

efficient and reliable in both linear and nonlinear problemsWhile for the linear problems studied PIA(1 1) algorithmgave almost the same results as Variational Iteration Methodit gave better results for the nonlinear problems On theother hand the results obtained in the first iteration of thePIA(1 2) algorithm are more convergent than those obtainedin the third iteration of the Variational IterationMethod thisvalidates the performance of the present method Althoughthe equations of PIA(1 2) algorithm are more entangledthan those of PIA(1 1) faster convergence is achieved withPIA(1 2) when compared to PIA(1 1)

It is conceived that the present method can be useful indeveloping new algorithms since they can be generated invarious forms according to the number of correction terms inperturbation expansion and the order of derivation in Taylorexpansion

The disadvantage of the method is that the number ofterms in the solution function increases seriously

In this study the solution intervals of problems arechosen generally the same as those in the references forcomparison purposes In three of the examples the solutionintervals are extended moderately yet for a wider solutioninterval the method should be modified such that the Taylorseries expansions are made around points other than zeroThe method can also be extended to other types of delaydifferential equations but some modifications are required

Conflict of Interests

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

References

[1] MZLiuandDLildquoProperties of analytic solution andnumericalsolution of multi-pantograph equationrdquo Applied Mathematicsand Computation vol 155 no 3 pp 853ndash871 2004

[2] Y Muroya E Ishiwata and H Brunner ldquoOn the attain-able order of collocation methods for pantograph integro-differential equationsrdquo Journal of Computational and AppliedMathematics vol 152 no 1-2 pp 347ndash366 2003

[3] S Yalcınbas M Aynigul and M Sezer ldquoA collocation methodusing Hermite polynomials for approximate solution of panto-graph equationsrdquo Journal of the Franklin Institute vol 348 no 6pp 1128ndash1139 2011

[4] M Sezer andAAkyuz-Dascıoglu ldquoATaylormethod for numer-ical solution of generalized pantograph equations with linearfunctional argumentrdquo Journal of Computational and AppliedMathematics vol 200 no 1 pp 217ndash225 2007

[5] S Sedaghat Y Ordokhani and M Dehghan ldquoNumerical solu-tion of the delay differential equations of pantograph type viaChebyshev polynomialsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 12 pp 4815ndash4830 2012

[6] B Benhammouda H Vazquez-Leal and L Hernandez-Martinez ldquoProcedure for exact solutions of nonlinear panto-graph delay differential equationsrdquo British Journal ofMathemat-ics amp Computer Science vol 4 no 19 pp 2738ndash2751 2014

[7] S Abbasbandy and C Bervillier ldquoAnalytic continuation ofTaylor series and the boundary value problems of some nonlin-ear ordinary differential equationsrdquo Applied Mathematics andComputation vol 218 no 5 pp 2178ndash2199 2011

[8] M A Z Raja ldquoNumerical treatment for boundary value prob-lems of Pantograph functional differential equation using com-putational intelligence algorithmsrdquo Applied Soft Computingvol 24 pp 806ndash821 2014

[9] D Trif ldquoDirect operatorial tau method for pantograph-typeequationsrdquo Applied Mathematics and Computation vol 219 no4 pp 2194ndash2203 2012

[10] A H Nayfeh Introduction to Perturbation Techniques JohnWiley amp Sons New York NY USA 1981

[11] J-H He ldquoIteration perturbation method for strongly nonlinearoscillationsrdquo Journal of Vibration and Control vol 7 no 5 pp631ndash642 2001

[12] R E Mickens ldquoIterationmethod solutions for conservative andlimit-cycle x13 force oscillatorsrdquo Journal of Sound and Vibra-tion vol 292 no 3ndash5 pp 964ndash968 2006

[13] Y Aksoy M Pakdemirli S Abbasbandy and H Boyacı ldquoNewperturbation-iteration solutions for nonlinear heat transferequationsrdquo International Journal of Numerical Methods for Heatamp Fluid Flow vol 22 no 7 pp 814ndash828 2012

[14] T Ozis and A Yıldırım ldquoGenerating the periodic solutions forforcing van der Pol oscillators by the iteration perturbationmethodrdquo Nonlinear Analysis Real World Applications vol 10no 4 pp 1984ndash1989 2009

[15] DDGanji S Karimpour and S S Ganji ldquoHersquos iteration pertur-bation method to nonlinear oscillations of mechanical systemswith single-degree-of freedomrdquo International Journal ofModernPhysics B vol 23 no 11 pp 2469ndash2477 2009

[16] VMarinca andN Herisanu ldquoAmodified iteration perturbationmethod for somenonlinear oscillation problemsrdquoActaMechan-ica vol 184 no 1ndash4 pp 231ndash242 2006

[17] M Pakdemirli and H Boyacı ldquoGeneration of root findingalgorithms via perturbation theory and some formulasrdquoAppliedMathematics and Computation vol 184 no 2 pp 783ndash7882007

[18] M Pakdemirli H Boyacı and H A Yurtsever ldquoPerturbativederivation and comparisons of root-finding algorithms withfourth order derivativesrdquoMathematicalampComputational Appli-cations vol 12 no 2 pp 117ndash124 2007

[19] M Pakdemirli H Boyacı and H A Yurtsever ldquoA root findingalgorithmwith fifth order derivativesrdquoMathematical ampCompu-tational Applications vol 13 no 2 pp 123ndash128 2008

10 Journal of Applied Mathematics

[20] Y Aksoy and M Pakdemirli ldquoNew perturbation-iterationsolutions for Bratu-type equationsrdquo Computers amp Mathematicswith Applications vol 59 no 8 pp 2802ndash2808 2010

[21] I T Dolapcı M Senol and M Pakdemirli ldquoNew perturbationiteration solutions for Fredholm and Volterra integral equa-tionsrdquo Journal of Applied Mathematics vol 2013 Article ID682537 5 pages 2013

[22] A Saadatmandi and M Dehghan ldquoVariational iterationmethod for solving a generalized pantograph equationrdquo Com-puters amp Mathematics with Applications vol 58 no 11-12 pp2190ndash2196 2009

[23] M Sezer S Yalcınbas and M Gulsu ldquoA Taylor polyno-mial approach for solving generalized pantograph equationswith nonhomogenous termrdquo International Journal of ComputerMathematics vol 85 no 7 pp 1055ndash1063 2008

[24] X Y Li and B Y Wu ldquoA continuous method for nonlocalfunctional differential equations with delayed or advancedargumentsrdquo Journal of Mathematical Analysis and Applicationsvol 409 no 1 pp 485ndash493 2014

[25] E H Doha A H Bhrawy D Baleanu and R M Hafez ldquoA newJacobi rationalmdashGauss collocation method for numerical solu-tion of generalized pantograph equationsrdquo Applied NumericalMathematics vol 77 pp 43ndash54 2014

[26] G P Rao and K R Palanisamy ldquoWalsh stretch matrices andfunctional-differential equationsrdquo IEEE Transactions on Auto-matic Control vol 27 no 1 pp 272ndash276 1982

[27] C Hwang ldquoSolution of a functional differential equation viadelayed unit step functionsrdquo International Journal of SystemsScience vol 14 no 9 pp 1065ndash1073 1983

[28] S Yuzbası E Gok andM Sezer ldquoLaguerrematrixmethodwiththe residual error estimation for solutions of a class of delaydifferential equationsrdquo Mathematical Methods in the AppliedSciences vol 37 no 4 pp 453ndash463 2014

[29] N R Anakira A K Alomari and I Hashim ldquoOptimal homo-topy asymptotic method for solving delay differential equa-tionsrdquoMathematical Problems in Engineering vol 2013 ArticleID 498902 11 pages 2013

[30] 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

[31] H Liu A Xiao and L Su ldquoConvergence of variational iterationmethod for second-order delay differential equationsrdquo Journalof Applied Mathematics vol 2013 Article ID 634670 9 pages2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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 Journal of Applied Mathematics

Table 1 Comparison of the absolute errors of the PIA(1 1)method with other numerical methods for example 1

119909VIM [22] Taylor method [23] Continuous method [24]

JRC method120572 = 120573 = 05 [25] PIA(1 1)

119899 = 4 119873 = 64 119899 = 4

2minus1

240119890 minus 15 124119890 minus 10 333119890 minus 11 393119890 minus 14 143119890 minus 15

2minus2

791119890 minus 17 974119890 minus 11 413119890 minus 11 216119890 minus 14 791119890 minus 17

2minus3

253119890 minus 18 700119890 minus 11 462119890 minus 11 166119890 minus 15 253119890 minus 18

2minus4

803119890 minus 20 914119890 minus 11 489119890 minus 11 621119890 minus 15 803119890 minus 20

2minus5

252119890 minus 21 528119890 minus 11 505119890 minus 11 368119890 minus 14 252119890 minus 21

2minus6

700119890 minus 23 195119890 minus 11 474119890 minus 11 454119890 minus 14 792119890 minus 23

Using (4) and (14) the approximate solutions at each stepbecome

1199061=

1

10minus1

8119890minus(15)119909

+41

40119890minus119909

1199062=

1

102+

1

320119890minus(15)119909

minus5

384119890minus(125)119909

+9599

9600119890minus119909

1199063=

1

103minus

1

76800119890minus(15)119909

+1

3072119890minus(125)119909

minus125

95232119890minus(1125)119909

+11904001

11904000119890minus119909

1199064=

1

104+

1

95232000119890minus(15)119909

minus1

737280119890minus(125)119909

+25

761825119890minus(1125)119909

minus15625

118849536119890minus(1625)119909

+74280959999

74280960000119890minus119909

(16)

Table 1 gives a comparison of the absolute errors of thePIA(1 1) method with those of other numerical methods Itis apparent from the table that the absolute errors of PIA(1 1)method are smaller than those of the other methods for119899 = 4 and also considering the first four iteration solutionsgiven in (16) one may notice that the numerical solutionconverges to the exact solution Absolute errors obtained byPIA(1 1) method are almost the same as those obtained bythe Variational Iteration Method (VIM) [22] for 119899 = 4

32 Illustrative Example 2 Consider the first-order linearpantograph differential equation [3]

1199061015840(119909) + 119906 (119909) + 119906 (08119909) = 0 0 le 119909 le 1 (17)

with the initial condition

119906 (0) = 1 (18)

which does not have an exact solution Equation (17) can bewritten in the following form

1199061015840(119909) + 119906 (119909) + 120576119906 (08119909) = 0 (19)

where 120576 is the artificially introduced perturbation parameterThe nonzero terms of (7) in PIA(1 1) algorithm are

1198651199061015840 = 1

119865119906= 1

119865120576= 119906119899(08119909)

119865 = 1199061015840

119899(119909) + 119906

119899(119909)

(20)

For this specific example (7) reduces to

(1199061015840

119888)119899(119909) + (119906

119888)119899(119909) = minus119906

119899(08119909) minus 119906

1015840

119899(119909) minus 119906

119899(119909) (21)

When applying the iteration formula (4) we start withthe following trivial solution as the initial function whichsatisfies the given initial conditions

1199060= 1 (22)

Using (4) and (21) the successive iteration results are

1199061= minus1 + 2119890

minus119909

1199062= 1 + 10119890

minus119909minus 10119890minus(45)119909

1199063= minus1 +

218

9119890minus119909

minus 50119890minus(45)119909

+250

9119890minus(1625)119909

1199064= 1 +

21490

549119890minus119909

minus1090

9119890minus(45)119909

+1250

9119890minus(1625)119909

minus31250

549119890minus(64125)119909

(23)

Table 2 shows the PIA(1 1) solutions of (17) for iterationnumbers 119899 = 5 6 7 8 9 As stated above we do not know theexact solution therefore we estimate a solution and observea convergence

A comparison of the results obtained by several differentmethods is shown in Table 3 The numerical results areconsistent

33 Illustrative Example 3 Consider the first-order nonlin-ear pantograph differential equation [29]

1199061015840(119909) = minus2119906

2(119909

2) + 1 0 le 119909 le 1 (24)

Journal of Applied Mathematics 5

Table 2 Convergence of the PIA(1 1) solution for example 2

119909119894

1199065(119909) 119906

6(119909) 119906

7(119909) 119906

8(119909) 119906

9(119909)

0 1 1 1 1 1

02 06646909 06646910 06646910 06646910 0664691004 04335604 04335607 04335607 04335607 0433560706 02764790 02764824 02764823 02764823 0276482308 01714677 01714846 01714841 01714841 017148411 01026142 01026723 01026700 01026701 01026701

with the initial condition

119906 (0) = 0 (25)

which has the exact solution 119906(119909) = sin(119909)Equation (24) can be written in the following form

1199061015840(119909) + 120576119906

2(119909

2) minus 1 = 0 (26)

where 120576 = 2 is assumed to be the perturbation parameterThenonzero terms of (7) in PIA(1 1) algorithm are

1198651199061015840 = 1

119865120576= 1199062

119899(119909

2)

119865 = 1199061015840

119899(119909) minus 1

(27)

The 119899th correction term of the perturbation expansion in (4)is

(1199061015840

119888)119899(119909) = minus119906

2

119899(119909

2) minus

1199061015840

119899(119909) minus 1

2 (28)

In this example we start with the initial guess

1199060= 119909 (29)

Thus the successive approximations are

1199061= 119909 minus

1

61199093

1199062= 119909 minus

1

61199093+

1

1201199095minus

1

80641199097

1199063= 119909 minus

1

61199093+

1

1201199095minus

1

50401199097+

61

232243201199099

minus67

340623360011990911+

1

1288175616011990913

minus1

799065243648011990915

(30)

Subsequently we determine the absolute maximum error for119906119899as

119864119899=1003817100381710038171003817119906119899 (119909) minus 119906 (119909)

1003817100381710038171003817infin

= max 1003816100381610038161003816119906119899 (119909) minus 119906 (119909)1003816100381610038161003816 0 le 119909 le 1

(31)

04 08 12 16 2 24 28 32 36 4 440x

minus1

minus08

minus06

minus04

minus02

002040608

1

u(x)

Exact solutionFirst iteration solution

Second iteration solution

Third iteration solutionof the PIA(1 1)

of the PIA(1 1)

of the PIA(1 1)

Figure 1 Comparison of the first three iteration solutions of thePIA(1 1)method with the exact solution for example 3

The absolute maximum errors 119864119899for different values of 119899 are

given in Table 4 Note that the error decreases continually as119899 increases

The same problem is solved in various studies and itis seen that the absolute maximum error for solving thisexample via Homotopy Asymptotic Method is 12 times 10

minus6

[30] and via Optimal Homotopy Asymptotic Method is 4 times10minus8 [29] while the absolute maximum error obtained by

the PIA(1 1) method is 535 times 10minus20 This affirmed that

the presented method is more accurate than the other twomethods

In Figure 1 the first three iteration solutions (119899 = 1 2 3)

of the PIA(1 1) method contrasted with the exact solutionIt is revealed that even when the solution interval of theproblem is expanded PIA(1 1) method goes on to yieldconvergent numerical solutions

34 Illustrative Example 4 Consider the initial value prob-lem of second-order differential equation with pantographdelay [31]

11990610158401015840(119909) = minus119906 (

119909

2) minus 1199062(119909) + sin4 (119909) + sin2 (119909) + 8

119909 ge 0

(32)

with initial conditions

119906 (0) = 2

1199061015840(0) = 0

(33)

which has the exact solution 119906(119909) = (5 minus cos 2119909)2

341 PIA(1 1) Solution Equation (32) can be written in thefollowing form

11990610158401015840(119909) + 120576 (119906 (

119909

2) + 1199062(119909))

= sin4 (119909) + sin2 (119909) + 8

(34)

6 Journal of Applied Mathematics

Table 3 Comparison of the PIA(1 1) solution with other numerical methods for example 2

119909119894 Walsh series method [26]

DUSF series method[27]

Hermite seriesmethod [3]

Taylor seriesmethod [4]

Laguerre matrixmethod [28] PIA(1 1)

119898 = 100ℎ = 001

119873 = 8 119873 = 8 119873 = 8 119899 = 8

0 1 1 1 1 1 102 0665621 0664677 0664691 0664691 06646910 0664691004 0432426 0433540 0433561 0433561 04335607 0433560706 0275140 0276460 0276482 0276482 02764831 0276482308 0170320 0171464 0171484 0171484 01714942 017148411 0100856 0102652 0102670 0102744 01027437 01026701

Table 4 The absolute maximum errors 119864119899for example 3

119899 1 2 3 4 5 6119864119899

813119890 minus 3 716119890 minus 5 123119890 minus 7 438119890 minus 11 291119890 minus 15 535119890 minus 20

where 120576 = 1 is the artificially introduced perturbationparameter Using (3) (34) returns to

119865 (11990610158401015840 1199061015840 119906 119906120572 120576) = 119906

10158401015840(119909) + 120576 (119906 (

119909

2) + 1199062(119909))

minus (sin4 (119909) + sin2 (119909) + 8) = 0

(35)

The nonzero terms of (7) in PIA(1 1) algorithm are

11986511990610158401015840 = 1

119865120576= 119906119899(119909

2) + 1199062

119899(119909)

119865 = 11990610158401015840

119899(119909) minus sin4 (119909) minus sin2 (119909) minus 8

(36)

Substituting (36) (32) reduces to

(11990610158401015840

119888)119899(119909) = minus119906

119899(119909

2) minus 11990610158401015840

119899(119909) minus 119906

2

119899(119909) + sin4 (119909)

+ sin2 (119909) + 8

(37)

The initial trial function is selected as

1199060= 2 (38)

and using (37) the approximate first iteration solution isobtained as

1199061=13

16minus

1

16sin4 (119909) + 3

16cos2 (119909) + 23119909

2

16

+ cos2 (1199092)

(39)

Even the first iteration of PIA(1 1) solution 1199061 gave the same

result as the first iteration solution obtained by the VIM [31]Still the second and third iteration solutions of the problemare better than those of the VIM solutions as observed inFigures 2 and 3

Exact solutionSecond iteration solution Second iteration solution of the VIM

2

22

24

26

28

3

u(x)

02 04 06 08 10x

of the PIA(1 1)

Figure 2 Comparison of the second iteration solutions of the VIMand PIA(1 1) with exact solution for example 4

342 PIA(1 2) Solution In order to apply the PIA(1 2)algorithm obtained by (9) we have to write (32) in thefollowing form

119865 (11990610158401015840 1199061015840 119906 119906120572 120576) = 119906

10158401015840(119909) + 120576119906

2(119909) + 120576

2119906 (

119909

2)

minus (sin4 (119909) + sin2 (119909) + 8) = 0

(40)

where 120576 is the artificially introduced perturbation parameterThe nonzero terms of (9) in PIA(1 2) algorithm are

11986511990610158401015840 = 1

119865119906120576= 2119906119899(119909)

119865120576= 1199062

119899(119909)

119865120576120576= 2119906119899(119909

2)

119865 = 11990610158401015840

119899(119909) minus sin4 (119909) minus sin2 (119909) minus 8

(41)

Journal of Applied Mathematics 7

Exact solution

Third iteration solution of the VIM

2

22

24

26

28

u(x)

02 04 06 08 10x

Third iteration solution of the PIA(1 1)

Figure 3The comparison of the third iteration solutions of theVIMand PIA(1 1) with exact solution for example 4

Note that introducing the small parameter 120576 = 1 as acoefficient of the pantograph term simplifies (40) and makesit easily solvable For this specific example (40) reads

(11990610158401015840

119888)119899(119909) + (2119906

119899(119909)) (119906

119888)119899(119909)

= minus119906119899(119909

2) minus 11990610158401015840

119899(119909) minus 119906

2

119899(119909) + sin4 (119909) + sin2 (119909)

+ 8

(42)

When applying the iteration formula (4) we select thefollowing initially assumed function

1199060= 2 (43)

and using (9) the approximate solution is

1199061=263

96minus49

96cos (2119909) minus 1

12cos4 (2119909) + 1

48cos2 (119909)

minus1

4119909 sin (119909) cos (119909) minus 1

6cos (119909)

(44)

First iteration of PIA(1 2) and second and third iterationsof VIM solutions are compared with the exact solutionAs can be seen from Figure 4 even the first iteration ofPIA(1 2) yields much better results than the first threeiteration solutions of the VIM

35 Illustrative Example 5 Consider the initial value prob-lem of nonlinear variable coefficient differential equationwith pantograph delay [6]

11990610158401015840(119909) = 119906 (119909) minus

8

11990921199062(119909

2) 119909 ge 0 (45)

with initial conditions

119906 (0) = 0

1199061015840(0) = 1

(46)

which has the exact solution 119906(119909) = 119909119890minus119909

2

22

24

26

28

3

u(x)

02 04 06 08 10x

of the VIM

of the VIM

Exact solutionFirst iteration solution

Second iteration solution

Third iteration solutionof the PIA(1 2)

Figure 4 The comparison of second and third iteration solutionsof the VIM and first iteration solution of the PIA(1 2) with exactsolution of example 4

Equation (45) can be written in the following form

11990610158401015840(119909) + 120576 (

8

11990921199062(119909

2) minus 119906 (119909)) = 0 (47)

where 120576 = 1 is the artificially introduced perturbationparameter Using (3) (45) returns to

119865 (11990610158401015840 1199061015840 119906 119906120572 120576) = 119906

10158401015840(119909) + 120576 (

8

11990921199062(119909

2) minus 119906 (119909))

= 0

(48)

The nonzero terms in PIA(1 1) algorithm are

11986511990610158401015840 = 1

119865120576=

8

11990921199062

119899(119909

2) minus 119906119899(119909)

119865 = 11990610158401015840

119899(119909)

(49)

Substituting (49) (45) reduces to

(11990610158401015840

119888)119899(119909) = minus

8

11990921199062

119899(119909

2) + 119906119899(119909) minus 119906

10158401015840

119899(119909) (50)

The initial trial function is selected as

1199060(119909) = 119909 (51)

and using (50) the approximate first iteration solution isobtained as

1199061= 119909 minus 119909

2+1

61199093

1199062= 119909 minus 119909

2+1

21199093minus

5

361199094+

1

801199095minus

1

86401199096

(52)

The absolute maximum errors 119864119899for different values of 119899 are

given in Table 5 and it is revealed that the error decreasescontinually as 119899 increases

8 Journal of Applied Mathematics

Table 5 The absolute maximum errors 119864119899for example 5

119899 1 2 3 4 5 6 7119864119899

201119890 minus 1 561119890 minus 3 159119890 minus 5 275119890 minus 7 368119890 minus 9 676119890 minus 11 494119890 minus 13

Exact solution8th iteration solution 7th iteration solution 6th iteration solution 5th iteration solution

0

01

02

03

04

06 12 18 24 3 36 42 480

of the PIA(1 1)

of the PIA(1 1)

of the PIA(1 1)

of the PIA(1 1)

Figure 5 Comparison of the iteration solutions of the PIA(1 1)method with the exact solution for example 5

Figure 5 illustrates the PIA(1 1) solution of this examplein the interval [0 5] In this example we enlarge the intervalso as to compare our solutionmore thoroughly with the exactsolution The present method yields pretty good results

36 Illustrative Example 6 Consider the second-order non-linear pantograph functional differential equation [7 8]

11990610158401015840(119909) = ([119906 (119909)]

2+ |119906 (119909)|

3) 119906 (

119909

2) 119909 ge 0 (53)

with initial conditions

119906 (0) = 1 119906 (1) =1

2(54)

which has the exact solution 119906(119909) = 1(1 + 119909)Equation (45) can be written in the following form

11990610158401015840(119909) minus 120576 (([119906 (119909)]

2+ |119906 (119909)|

3) 119906 (

119909

2)) = 0 (55)

where 120576 is the artificially introduced small parameter Using(3) (45) returns to

119865 (11990610158401015840 1199061015840 119906 119906120572 120576)

= 11990610158401015840(119909) minus 120576 (([119906 (119909)]

2+ |119906 (119909)|

3) 119906 (

119909

2)) = 0

(56)

The nonzero terms in PIA(1 1) algorithm are

11986511990610158401015840 = 1

119865120576= minus(([119906 (119909)]

2+1003816100381610038161003816119906119899 (119909)

10038161003816100381610038163

) 119906119899(119909

2))

119865 = 11990610158401015840

119899(119909)

(57)

First iteration solution Second iteration solution Third iteration solution

01 02 03 04 05 06 07 08 09 10x

04

05

06

07

08

09

1

u(x)

of the PIA(1 1)

of the PIA(1 1)

of the PIA(1 1)

Figure 6 Comparison of the first three iteration solutions of thePIA(1 1)method with the exact solution for example 6

Substituting (49) (45) reduces to

(11990610158401015840

119888)119899(119909) = minus119906

10158401015840

119899(119909)

+ (([119906 (119909)]2+1003816100381610038161003816119906119899 (119909)

10038161003816100381610038163

) 119906119899(119909

2))

(58)

The initial trial function is selected as

1199060(119909) = 1 minus

119909

2 (59)

and using (50) the approximate first iteration solution isobtained as

1199061(119909) = 1 minus

1073

960119909 + 1199092minus1

21199093+13

961199094minus

3

1601199095

+1

9601199096

(60)

In Figure 6 first three iteration solutions obtained by thePIA(1 1) algorithm are compared with the exact solutionThe convergence of iteration solutions to the exact solutionis obvious

Figure 7 illustrates the absolute error functions of thefirst three iteration solutions of the PIA(1 1) method Themaximum absolute error decreases from 003013 in the firstiteration to 000262 in the third iteration

4 Conclusion

The new perturbation-iteration method is employed for thefirst time to numerically solve the pantograph equationsThe comparative results showed that the method is highly

Journal of Applied Mathematics 9

Absolute error function for first iteration solutionAbsolute error function for second iteration solutionAbsolute error function for third iteration solution

00005

0010015

0020025

0030035

004

01 02 03 04 05 06 07 08 09 10

Figure 7 Comparison of the absolute error functions of the firstthree iteration solutions of the PIA(1 1) method with the exactsolution for example 6

efficient and reliable in both linear and nonlinear problemsWhile for the linear problems studied PIA(1 1) algorithmgave almost the same results as Variational Iteration Methodit gave better results for the nonlinear problems On theother hand the results obtained in the first iteration of thePIA(1 2) algorithm are more convergent than those obtainedin the third iteration of the Variational IterationMethod thisvalidates the performance of the present method Althoughthe equations of PIA(1 2) algorithm are more entangledthan those of PIA(1 1) faster convergence is achieved withPIA(1 2) when compared to PIA(1 1)

It is conceived that the present method can be useful indeveloping new algorithms since they can be generated invarious forms according to the number of correction terms inperturbation expansion and the order of derivation in Taylorexpansion

The disadvantage of the method is that the number ofterms in the solution function increases seriously

In this study the solution intervals of problems arechosen generally the same as those in the references forcomparison purposes In three of the examples the solutionintervals are extended moderately yet for a wider solutioninterval the method should be modified such that the Taylorseries expansions are made around points other than zeroThe method can also be extended to other types of delaydifferential equations but some modifications are required

Conflict of Interests

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

References

[1] MZLiuandDLildquoProperties of analytic solution andnumericalsolution of multi-pantograph equationrdquo Applied Mathematicsand Computation vol 155 no 3 pp 853ndash871 2004

[2] Y Muroya E Ishiwata and H Brunner ldquoOn the attain-able order of collocation methods for pantograph integro-differential equationsrdquo Journal of Computational and AppliedMathematics vol 152 no 1-2 pp 347ndash366 2003

[3] S Yalcınbas M Aynigul and M Sezer ldquoA collocation methodusing Hermite polynomials for approximate solution of panto-graph equationsrdquo Journal of the Franklin Institute vol 348 no 6pp 1128ndash1139 2011

[4] M Sezer andAAkyuz-Dascıoglu ldquoATaylormethod for numer-ical solution of generalized pantograph equations with linearfunctional argumentrdquo Journal of Computational and AppliedMathematics vol 200 no 1 pp 217ndash225 2007

[5] S Sedaghat Y Ordokhani and M Dehghan ldquoNumerical solu-tion of the delay differential equations of pantograph type viaChebyshev polynomialsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 12 pp 4815ndash4830 2012

[6] B Benhammouda H Vazquez-Leal and L Hernandez-Martinez ldquoProcedure for exact solutions of nonlinear panto-graph delay differential equationsrdquo British Journal ofMathemat-ics amp Computer Science vol 4 no 19 pp 2738ndash2751 2014

[7] S Abbasbandy and C Bervillier ldquoAnalytic continuation ofTaylor series and the boundary value problems of some nonlin-ear ordinary differential equationsrdquo Applied Mathematics andComputation vol 218 no 5 pp 2178ndash2199 2011

[8] M A Z Raja ldquoNumerical treatment for boundary value prob-lems of Pantograph functional differential equation using com-putational intelligence algorithmsrdquo Applied Soft Computingvol 24 pp 806ndash821 2014

[9] D Trif ldquoDirect operatorial tau method for pantograph-typeequationsrdquo Applied Mathematics and Computation vol 219 no4 pp 2194ndash2203 2012

[10] A H Nayfeh Introduction to Perturbation Techniques JohnWiley amp Sons New York NY USA 1981

[11] J-H He ldquoIteration perturbation method for strongly nonlinearoscillationsrdquo Journal of Vibration and Control vol 7 no 5 pp631ndash642 2001

[12] R E Mickens ldquoIterationmethod solutions for conservative andlimit-cycle x13 force oscillatorsrdquo Journal of Sound and Vibra-tion vol 292 no 3ndash5 pp 964ndash968 2006

[13] Y Aksoy M Pakdemirli S Abbasbandy and H Boyacı ldquoNewperturbation-iteration solutions for nonlinear heat transferequationsrdquo International Journal of Numerical Methods for Heatamp Fluid Flow vol 22 no 7 pp 814ndash828 2012

[14] T Ozis and A Yıldırım ldquoGenerating the periodic solutions forforcing van der Pol oscillators by the iteration perturbationmethodrdquo Nonlinear Analysis Real World Applications vol 10no 4 pp 1984ndash1989 2009

[15] DDGanji S Karimpour and S S Ganji ldquoHersquos iteration pertur-bation method to nonlinear oscillations of mechanical systemswith single-degree-of freedomrdquo International Journal ofModernPhysics B vol 23 no 11 pp 2469ndash2477 2009

[16] VMarinca andN Herisanu ldquoAmodified iteration perturbationmethod for somenonlinear oscillation problemsrdquoActaMechan-ica vol 184 no 1ndash4 pp 231ndash242 2006

[17] M Pakdemirli and H Boyacı ldquoGeneration of root findingalgorithms via perturbation theory and some formulasrdquoAppliedMathematics and Computation vol 184 no 2 pp 783ndash7882007

[18] M Pakdemirli H Boyacı and H A Yurtsever ldquoPerturbativederivation and comparisons of root-finding algorithms withfourth order derivativesrdquoMathematicalampComputational Appli-cations vol 12 no 2 pp 117ndash124 2007

[19] M Pakdemirli H Boyacı and H A Yurtsever ldquoA root findingalgorithmwith fifth order derivativesrdquoMathematical ampCompu-tational Applications vol 13 no 2 pp 123ndash128 2008

10 Journal of Applied Mathematics

[20] Y Aksoy and M Pakdemirli ldquoNew perturbation-iterationsolutions for Bratu-type equationsrdquo Computers amp Mathematicswith Applications vol 59 no 8 pp 2802ndash2808 2010

[21] I T Dolapcı M Senol and M Pakdemirli ldquoNew perturbationiteration solutions for Fredholm and Volterra integral equa-tionsrdquo Journal of Applied Mathematics vol 2013 Article ID682537 5 pages 2013

[22] A Saadatmandi and M Dehghan ldquoVariational iterationmethod for solving a generalized pantograph equationrdquo Com-puters amp Mathematics with Applications vol 58 no 11-12 pp2190ndash2196 2009

[23] M Sezer S Yalcınbas and M Gulsu ldquoA Taylor polyno-mial approach for solving generalized pantograph equationswith nonhomogenous termrdquo International Journal of ComputerMathematics vol 85 no 7 pp 1055ndash1063 2008

[24] X Y Li and B Y Wu ldquoA continuous method for nonlocalfunctional differential equations with delayed or advancedargumentsrdquo Journal of Mathematical Analysis and Applicationsvol 409 no 1 pp 485ndash493 2014

[25] E H Doha A H Bhrawy D Baleanu and R M Hafez ldquoA newJacobi rationalmdashGauss collocation method for numerical solu-tion of generalized pantograph equationsrdquo Applied NumericalMathematics vol 77 pp 43ndash54 2014

[26] G P Rao and K R Palanisamy ldquoWalsh stretch matrices andfunctional-differential equationsrdquo IEEE Transactions on Auto-matic Control vol 27 no 1 pp 272ndash276 1982

[27] C Hwang ldquoSolution of a functional differential equation viadelayed unit step functionsrdquo International Journal of SystemsScience vol 14 no 9 pp 1065ndash1073 1983

[28] S Yuzbası E Gok andM Sezer ldquoLaguerrematrixmethodwiththe residual error estimation for solutions of a class of delaydifferential equationsrdquo Mathematical Methods in the AppliedSciences vol 37 no 4 pp 453ndash463 2014

[29] N R Anakira A K Alomari and I Hashim ldquoOptimal homo-topy asymptotic method for solving delay differential equa-tionsrdquoMathematical Problems in Engineering vol 2013 ArticleID 498902 11 pages 2013

[30] 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

[31] H Liu A Xiao and L Su ldquoConvergence of variational iterationmethod for second-order delay differential equationsrdquo Journalof Applied Mathematics vol 2013 Article ID 634670 9 pages2013

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

Journal of Applied Mathematics 5

Table 2 Convergence of the PIA(1 1) solution for example 2

119909119894

1199065(119909) 119906

6(119909) 119906

7(119909) 119906

8(119909) 119906

9(119909)

0 1 1 1 1 1

02 06646909 06646910 06646910 06646910 0664691004 04335604 04335607 04335607 04335607 0433560706 02764790 02764824 02764823 02764823 0276482308 01714677 01714846 01714841 01714841 017148411 01026142 01026723 01026700 01026701 01026701

with the initial condition

119906 (0) = 0 (25)

which has the exact solution 119906(119909) = sin(119909)Equation (24) can be written in the following form

1199061015840(119909) + 120576119906

2(119909

2) minus 1 = 0 (26)

where 120576 = 2 is assumed to be the perturbation parameterThenonzero terms of (7) in PIA(1 1) algorithm are

1198651199061015840 = 1

119865120576= 1199062

119899(119909

2)

119865 = 1199061015840

119899(119909) minus 1

(27)

The 119899th correction term of the perturbation expansion in (4)is

(1199061015840

119888)119899(119909) = minus119906

2

119899(119909

2) minus

1199061015840

119899(119909) minus 1

2 (28)

In this example we start with the initial guess

1199060= 119909 (29)

Thus the successive approximations are

1199061= 119909 minus

1

61199093

1199062= 119909 minus

1

61199093+

1

1201199095minus

1

80641199097

1199063= 119909 minus

1

61199093+

1

1201199095minus

1

50401199097+

61

232243201199099

minus67

340623360011990911+

1

1288175616011990913

minus1

799065243648011990915

(30)

Subsequently we determine the absolute maximum error for119906119899as

119864119899=1003817100381710038171003817119906119899 (119909) minus 119906 (119909)

1003817100381710038171003817infin

= max 1003816100381610038161003816119906119899 (119909) minus 119906 (119909)1003816100381610038161003816 0 le 119909 le 1

(31)

04 08 12 16 2 24 28 32 36 4 440x

minus1

minus08

minus06

minus04

minus02

002040608

1

u(x)

Exact solutionFirst iteration solution

Second iteration solution

Third iteration solutionof the PIA(1 1)

of the PIA(1 1)

of the PIA(1 1)

Figure 1 Comparison of the first three iteration solutions of thePIA(1 1)method with the exact solution for example 3

The absolute maximum errors 119864119899for different values of 119899 are

given in Table 4 Note that the error decreases continually as119899 increases

The same problem is solved in various studies and itis seen that the absolute maximum error for solving thisexample via Homotopy Asymptotic Method is 12 times 10

minus6

[30] and via Optimal Homotopy Asymptotic Method is 4 times10minus8 [29] while the absolute maximum error obtained by

the PIA(1 1) method is 535 times 10minus20 This affirmed that

the presented method is more accurate than the other twomethods

In Figure 1 the first three iteration solutions (119899 = 1 2 3)

of the PIA(1 1) method contrasted with the exact solutionIt is revealed that even when the solution interval of theproblem is expanded PIA(1 1) method goes on to yieldconvergent numerical solutions

34 Illustrative Example 4 Consider the initial value prob-lem of second-order differential equation with pantographdelay [31]

11990610158401015840(119909) = minus119906 (

119909

2) minus 1199062(119909) + sin4 (119909) + sin2 (119909) + 8

119909 ge 0

(32)

with initial conditions

119906 (0) = 2

1199061015840(0) = 0

(33)

which has the exact solution 119906(119909) = (5 minus cos 2119909)2

341 PIA(1 1) Solution Equation (32) can be written in thefollowing form

11990610158401015840(119909) + 120576 (119906 (

119909

2) + 1199062(119909))

= sin4 (119909) + sin2 (119909) + 8

(34)

6 Journal of Applied Mathematics

Table 3 Comparison of the PIA(1 1) solution with other numerical methods for example 2

119909119894 Walsh series method [26]

DUSF series method[27]

Hermite seriesmethod [3]

Taylor seriesmethod [4]

Laguerre matrixmethod [28] PIA(1 1)

119898 = 100ℎ = 001

119873 = 8 119873 = 8 119873 = 8 119899 = 8

0 1 1 1 1 1 102 0665621 0664677 0664691 0664691 06646910 0664691004 0432426 0433540 0433561 0433561 04335607 0433560706 0275140 0276460 0276482 0276482 02764831 0276482308 0170320 0171464 0171484 0171484 01714942 017148411 0100856 0102652 0102670 0102744 01027437 01026701

Table 4 The absolute maximum errors 119864119899for example 3

119899 1 2 3 4 5 6119864119899

813119890 minus 3 716119890 minus 5 123119890 minus 7 438119890 minus 11 291119890 minus 15 535119890 minus 20

where 120576 = 1 is the artificially introduced perturbationparameter Using (3) (34) returns to

119865 (11990610158401015840 1199061015840 119906 119906120572 120576) = 119906

10158401015840(119909) + 120576 (119906 (

119909

2) + 1199062(119909))

minus (sin4 (119909) + sin2 (119909) + 8) = 0

(35)

The nonzero terms of (7) in PIA(1 1) algorithm are

11986511990610158401015840 = 1

119865120576= 119906119899(119909

2) + 1199062

119899(119909)

119865 = 11990610158401015840

119899(119909) minus sin4 (119909) minus sin2 (119909) minus 8

(36)

Substituting (36) (32) reduces to

(11990610158401015840

119888)119899(119909) = minus119906

119899(119909

2) minus 11990610158401015840

119899(119909) minus 119906

2

119899(119909) + sin4 (119909)

+ sin2 (119909) + 8

(37)

The initial trial function is selected as

1199060= 2 (38)

and using (37) the approximate first iteration solution isobtained as

1199061=13

16minus

1

16sin4 (119909) + 3

16cos2 (119909) + 23119909

2

16

+ cos2 (1199092)

(39)

Even the first iteration of PIA(1 1) solution 1199061 gave the same

result as the first iteration solution obtained by the VIM [31]Still the second and third iteration solutions of the problemare better than those of the VIM solutions as observed inFigures 2 and 3

Exact solutionSecond iteration solution Second iteration solution of the VIM

2

22

24

26

28

3

u(x)

02 04 06 08 10x

of the PIA(1 1)

Figure 2 Comparison of the second iteration solutions of the VIMand PIA(1 1) with exact solution for example 4

342 PIA(1 2) Solution In order to apply the PIA(1 2)algorithm obtained by (9) we have to write (32) in thefollowing form

119865 (11990610158401015840 1199061015840 119906 119906120572 120576) = 119906

10158401015840(119909) + 120576119906

2(119909) + 120576

2119906 (

119909

2)

minus (sin4 (119909) + sin2 (119909) + 8) = 0

(40)

where 120576 is the artificially introduced perturbation parameterThe nonzero terms of (9) in PIA(1 2) algorithm are

11986511990610158401015840 = 1

119865119906120576= 2119906119899(119909)

119865120576= 1199062

119899(119909)

119865120576120576= 2119906119899(119909

2)

119865 = 11990610158401015840

119899(119909) minus sin4 (119909) minus sin2 (119909) minus 8

(41)

Journal of Applied Mathematics 7

Exact solution

Third iteration solution of the VIM

2

22

24

26

28

u(x)

02 04 06 08 10x

Third iteration solution of the PIA(1 1)

Figure 3The comparison of the third iteration solutions of theVIMand PIA(1 1) with exact solution for example 4

Note that introducing the small parameter 120576 = 1 as acoefficient of the pantograph term simplifies (40) and makesit easily solvable For this specific example (40) reads

(11990610158401015840

119888)119899(119909) + (2119906

119899(119909)) (119906

119888)119899(119909)

= minus119906119899(119909

2) minus 11990610158401015840

119899(119909) minus 119906

2

119899(119909) + sin4 (119909) + sin2 (119909)

+ 8

(42)

When applying the iteration formula (4) we select thefollowing initially assumed function

1199060= 2 (43)

and using (9) the approximate solution is

1199061=263

96minus49

96cos (2119909) minus 1

12cos4 (2119909) + 1

48cos2 (119909)

minus1

4119909 sin (119909) cos (119909) minus 1

6cos (119909)

(44)

First iteration of PIA(1 2) and second and third iterationsof VIM solutions are compared with the exact solutionAs can be seen from Figure 4 even the first iteration ofPIA(1 2) yields much better results than the first threeiteration solutions of the VIM

35 Illustrative Example 5 Consider the initial value prob-lem of nonlinear variable coefficient differential equationwith pantograph delay [6]

11990610158401015840(119909) = 119906 (119909) minus

8

11990921199062(119909

2) 119909 ge 0 (45)

with initial conditions

119906 (0) = 0

1199061015840(0) = 1

(46)

which has the exact solution 119906(119909) = 119909119890minus119909

2

22

24

26

28

3

u(x)

02 04 06 08 10x

of the VIM

of the VIM

Exact solutionFirst iteration solution

Second iteration solution

Third iteration solutionof the PIA(1 2)

Figure 4 The comparison of second and third iteration solutionsof the VIM and first iteration solution of the PIA(1 2) with exactsolution of example 4

Equation (45) can be written in the following form

11990610158401015840(119909) + 120576 (

8

11990921199062(119909

2) minus 119906 (119909)) = 0 (47)

where 120576 = 1 is the artificially introduced perturbationparameter Using (3) (45) returns to

119865 (11990610158401015840 1199061015840 119906 119906120572 120576) = 119906

10158401015840(119909) + 120576 (

8

11990921199062(119909

2) minus 119906 (119909))

= 0

(48)

The nonzero terms in PIA(1 1) algorithm are

11986511990610158401015840 = 1

119865120576=

8

11990921199062

119899(119909

2) minus 119906119899(119909)

119865 = 11990610158401015840

119899(119909)

(49)

Substituting (49) (45) reduces to

(11990610158401015840

119888)119899(119909) = minus

8

11990921199062

119899(119909

2) + 119906119899(119909) minus 119906

10158401015840

119899(119909) (50)

The initial trial function is selected as

1199060(119909) = 119909 (51)

and using (50) the approximate first iteration solution isobtained as

1199061= 119909 minus 119909

2+1

61199093

1199062= 119909 minus 119909

2+1

21199093minus

5

361199094+

1

801199095minus

1

86401199096

(52)

The absolute maximum errors 119864119899for different values of 119899 are

given in Table 5 and it is revealed that the error decreasescontinually as 119899 increases

8 Journal of Applied Mathematics

Table 5 The absolute maximum errors 119864119899for example 5

119899 1 2 3 4 5 6 7119864119899

201119890 minus 1 561119890 minus 3 159119890 minus 5 275119890 minus 7 368119890 minus 9 676119890 minus 11 494119890 minus 13

Exact solution8th iteration solution 7th iteration solution 6th iteration solution 5th iteration solution

0

01

02

03

04

06 12 18 24 3 36 42 480

of the PIA(1 1)

of the PIA(1 1)

of the PIA(1 1)

of the PIA(1 1)

Figure 5 Comparison of the iteration solutions of the PIA(1 1)method with the exact solution for example 5

Figure 5 illustrates the PIA(1 1) solution of this examplein the interval [0 5] In this example we enlarge the intervalso as to compare our solutionmore thoroughly with the exactsolution The present method yields pretty good results

36 Illustrative Example 6 Consider the second-order non-linear pantograph functional differential equation [7 8]

11990610158401015840(119909) = ([119906 (119909)]

2+ |119906 (119909)|

3) 119906 (

119909

2) 119909 ge 0 (53)

with initial conditions

119906 (0) = 1 119906 (1) =1

2(54)

which has the exact solution 119906(119909) = 1(1 + 119909)Equation (45) can be written in the following form

11990610158401015840(119909) minus 120576 (([119906 (119909)]

2+ |119906 (119909)|

3) 119906 (

119909

2)) = 0 (55)

where 120576 is the artificially introduced small parameter Using(3) (45) returns to

119865 (11990610158401015840 1199061015840 119906 119906120572 120576)

= 11990610158401015840(119909) minus 120576 (([119906 (119909)]

2+ |119906 (119909)|

3) 119906 (

119909

2)) = 0

(56)

The nonzero terms in PIA(1 1) algorithm are

11986511990610158401015840 = 1

119865120576= minus(([119906 (119909)]

2+1003816100381610038161003816119906119899 (119909)

10038161003816100381610038163

) 119906119899(119909

2))

119865 = 11990610158401015840

119899(119909)

(57)

First iteration solution Second iteration solution Third iteration solution

01 02 03 04 05 06 07 08 09 10x

04

05

06

07

08

09

1

u(x)

of the PIA(1 1)

of the PIA(1 1)

of the PIA(1 1)

Figure 6 Comparison of the first three iteration solutions of thePIA(1 1)method with the exact solution for example 6

Substituting (49) (45) reduces to

(11990610158401015840

119888)119899(119909) = minus119906

10158401015840

119899(119909)

+ (([119906 (119909)]2+1003816100381610038161003816119906119899 (119909)

10038161003816100381610038163

) 119906119899(119909

2))

(58)

The initial trial function is selected as

1199060(119909) = 1 minus

119909

2 (59)

and using (50) the approximate first iteration solution isobtained as

1199061(119909) = 1 minus

1073

960119909 + 1199092minus1

21199093+13

961199094minus

3

1601199095

+1

9601199096

(60)

In Figure 6 first three iteration solutions obtained by thePIA(1 1) algorithm are compared with the exact solutionThe convergence of iteration solutions to the exact solutionis obvious

Figure 7 illustrates the absolute error functions of thefirst three iteration solutions of the PIA(1 1) method Themaximum absolute error decreases from 003013 in the firstiteration to 000262 in the third iteration

4 Conclusion

The new perturbation-iteration method is employed for thefirst time to numerically solve the pantograph equationsThe comparative results showed that the method is highly

Journal of Applied Mathematics 9

Absolute error function for first iteration solutionAbsolute error function for second iteration solutionAbsolute error function for third iteration solution

00005

0010015

0020025

0030035

004

01 02 03 04 05 06 07 08 09 10

Figure 7 Comparison of the absolute error functions of the firstthree iteration solutions of the PIA(1 1) method with the exactsolution for example 6

efficient and reliable in both linear and nonlinear problemsWhile for the linear problems studied PIA(1 1) algorithmgave almost the same results as Variational Iteration Methodit gave better results for the nonlinear problems On theother hand the results obtained in the first iteration of thePIA(1 2) algorithm are more convergent than those obtainedin the third iteration of the Variational IterationMethod thisvalidates the performance of the present method Althoughthe equations of PIA(1 2) algorithm are more entangledthan those of PIA(1 1) faster convergence is achieved withPIA(1 2) when compared to PIA(1 1)

It is conceived that the present method can be useful indeveloping new algorithms since they can be generated invarious forms according to the number of correction terms inperturbation expansion and the order of derivation in Taylorexpansion

The disadvantage of the method is that the number ofterms in the solution function increases seriously

In this study the solution intervals of problems arechosen generally the same as those in the references forcomparison purposes In three of the examples the solutionintervals are extended moderately yet for a wider solutioninterval the method should be modified such that the Taylorseries expansions are made around points other than zeroThe method can also be extended to other types of delaydifferential equations but some modifications are required

Conflict of Interests

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

References

[1] MZLiuandDLildquoProperties of analytic solution andnumericalsolution of multi-pantograph equationrdquo Applied Mathematicsand Computation vol 155 no 3 pp 853ndash871 2004

[2] Y Muroya E Ishiwata and H Brunner ldquoOn the attain-able order of collocation methods for pantograph integro-differential equationsrdquo Journal of Computational and AppliedMathematics vol 152 no 1-2 pp 347ndash366 2003

[3] S Yalcınbas M Aynigul and M Sezer ldquoA collocation methodusing Hermite polynomials for approximate solution of panto-graph equationsrdquo Journal of the Franklin Institute vol 348 no 6pp 1128ndash1139 2011

[4] M Sezer andAAkyuz-Dascıoglu ldquoATaylormethod for numer-ical solution of generalized pantograph equations with linearfunctional argumentrdquo Journal of Computational and AppliedMathematics vol 200 no 1 pp 217ndash225 2007

[5] S Sedaghat Y Ordokhani and M Dehghan ldquoNumerical solu-tion of the delay differential equations of pantograph type viaChebyshev polynomialsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 12 pp 4815ndash4830 2012

[6] B Benhammouda H Vazquez-Leal and L Hernandez-Martinez ldquoProcedure for exact solutions of nonlinear panto-graph delay differential equationsrdquo British Journal ofMathemat-ics amp Computer Science vol 4 no 19 pp 2738ndash2751 2014

[7] S Abbasbandy and C Bervillier ldquoAnalytic continuation ofTaylor series and the boundary value problems of some nonlin-ear ordinary differential equationsrdquo Applied Mathematics andComputation vol 218 no 5 pp 2178ndash2199 2011

[8] M A Z Raja ldquoNumerical treatment for boundary value prob-lems of Pantograph functional differential equation using com-putational intelligence algorithmsrdquo Applied Soft Computingvol 24 pp 806ndash821 2014

[9] D Trif ldquoDirect operatorial tau method for pantograph-typeequationsrdquo Applied Mathematics and Computation vol 219 no4 pp 2194ndash2203 2012

[10] A H Nayfeh Introduction to Perturbation Techniques JohnWiley amp Sons New York NY USA 1981

[11] J-H He ldquoIteration perturbation method for strongly nonlinearoscillationsrdquo Journal of Vibration and Control vol 7 no 5 pp631ndash642 2001

[12] R E Mickens ldquoIterationmethod solutions for conservative andlimit-cycle x13 force oscillatorsrdquo Journal of Sound and Vibra-tion vol 292 no 3ndash5 pp 964ndash968 2006

[13] Y Aksoy M Pakdemirli S Abbasbandy and H Boyacı ldquoNewperturbation-iteration solutions for nonlinear heat transferequationsrdquo International Journal of Numerical Methods for Heatamp Fluid Flow vol 22 no 7 pp 814ndash828 2012

[14] T Ozis and A Yıldırım ldquoGenerating the periodic solutions forforcing van der Pol oscillators by the iteration perturbationmethodrdquo Nonlinear Analysis Real World Applications vol 10no 4 pp 1984ndash1989 2009

[15] DDGanji S Karimpour and S S Ganji ldquoHersquos iteration pertur-bation method to nonlinear oscillations of mechanical systemswith single-degree-of freedomrdquo International Journal ofModernPhysics B vol 23 no 11 pp 2469ndash2477 2009

[16] VMarinca andN Herisanu ldquoAmodified iteration perturbationmethod for somenonlinear oscillation problemsrdquoActaMechan-ica vol 184 no 1ndash4 pp 231ndash242 2006

[17] M Pakdemirli and H Boyacı ldquoGeneration of root findingalgorithms via perturbation theory and some formulasrdquoAppliedMathematics and Computation vol 184 no 2 pp 783ndash7882007

[18] M Pakdemirli H Boyacı and H A Yurtsever ldquoPerturbativederivation and comparisons of root-finding algorithms withfourth order derivativesrdquoMathematicalampComputational Appli-cations vol 12 no 2 pp 117ndash124 2007

[19] M Pakdemirli H Boyacı and H A Yurtsever ldquoA root findingalgorithmwith fifth order derivativesrdquoMathematical ampCompu-tational Applications vol 13 no 2 pp 123ndash128 2008

10 Journal of Applied Mathematics

[20] Y Aksoy and M Pakdemirli ldquoNew perturbation-iterationsolutions for Bratu-type equationsrdquo Computers amp Mathematicswith Applications vol 59 no 8 pp 2802ndash2808 2010

[21] I T Dolapcı M Senol and M Pakdemirli ldquoNew perturbationiteration solutions for Fredholm and Volterra integral equa-tionsrdquo Journal of Applied Mathematics vol 2013 Article ID682537 5 pages 2013

[22] A Saadatmandi and M Dehghan ldquoVariational iterationmethod for solving a generalized pantograph equationrdquo Com-puters amp Mathematics with Applications vol 58 no 11-12 pp2190ndash2196 2009

[23] M Sezer S Yalcınbas and M Gulsu ldquoA Taylor polyno-mial approach for solving generalized pantograph equationswith nonhomogenous termrdquo International Journal of ComputerMathematics vol 85 no 7 pp 1055ndash1063 2008

[24] X Y Li and B Y Wu ldquoA continuous method for nonlocalfunctional differential equations with delayed or advancedargumentsrdquo Journal of Mathematical Analysis and Applicationsvol 409 no 1 pp 485ndash493 2014

[25] E H Doha A H Bhrawy D Baleanu and R M Hafez ldquoA newJacobi rationalmdashGauss collocation method for numerical solu-tion of generalized pantograph equationsrdquo Applied NumericalMathematics vol 77 pp 43ndash54 2014

[26] G P Rao and K R Palanisamy ldquoWalsh stretch matrices andfunctional-differential equationsrdquo IEEE Transactions on Auto-matic Control vol 27 no 1 pp 272ndash276 1982

[27] C Hwang ldquoSolution of a functional differential equation viadelayed unit step functionsrdquo International Journal of SystemsScience vol 14 no 9 pp 1065ndash1073 1983

[28] S Yuzbası E Gok andM Sezer ldquoLaguerrematrixmethodwiththe residual error estimation for solutions of a class of delaydifferential equationsrdquo Mathematical Methods in the AppliedSciences vol 37 no 4 pp 453ndash463 2014

[29] N R Anakira A K Alomari and I Hashim ldquoOptimal homo-topy asymptotic method for solving delay differential equa-tionsrdquoMathematical Problems in Engineering vol 2013 ArticleID 498902 11 pages 2013

[30] 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

[31] H Liu A Xiao and L Su ldquoConvergence of variational iterationmethod for second-order delay differential equationsrdquo Journalof Applied Mathematics vol 2013 Article ID 634670 9 pages2013

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 Journal of Applied Mathematics

Table 3 Comparison of the PIA(1 1) solution with other numerical methods for example 2

119909119894 Walsh series method [26]

DUSF series method[27]

Hermite seriesmethod [3]

Taylor seriesmethod [4]

Laguerre matrixmethod [28] PIA(1 1)

119898 = 100ℎ = 001

119873 = 8 119873 = 8 119873 = 8 119899 = 8

0 1 1 1 1 1 102 0665621 0664677 0664691 0664691 06646910 0664691004 0432426 0433540 0433561 0433561 04335607 0433560706 0275140 0276460 0276482 0276482 02764831 0276482308 0170320 0171464 0171484 0171484 01714942 017148411 0100856 0102652 0102670 0102744 01027437 01026701

Table 4 The absolute maximum errors 119864119899for example 3

119899 1 2 3 4 5 6119864119899

813119890 minus 3 716119890 minus 5 123119890 minus 7 438119890 minus 11 291119890 minus 15 535119890 minus 20

where 120576 = 1 is the artificially introduced perturbationparameter Using (3) (34) returns to

119865 (11990610158401015840 1199061015840 119906 119906120572 120576) = 119906

10158401015840(119909) + 120576 (119906 (

119909

2) + 1199062(119909))

minus (sin4 (119909) + sin2 (119909) + 8) = 0

(35)

The nonzero terms of (7) in PIA(1 1) algorithm are

11986511990610158401015840 = 1

119865120576= 119906119899(119909

2) + 1199062

119899(119909)

119865 = 11990610158401015840

119899(119909) minus sin4 (119909) minus sin2 (119909) minus 8

(36)

Substituting (36) (32) reduces to

(11990610158401015840

119888)119899(119909) = minus119906

119899(119909

2) minus 11990610158401015840

119899(119909) minus 119906

2

119899(119909) + sin4 (119909)

+ sin2 (119909) + 8

(37)

The initial trial function is selected as

1199060= 2 (38)

and using (37) the approximate first iteration solution isobtained as

1199061=13

16minus

1

16sin4 (119909) + 3

16cos2 (119909) + 23119909

2

16

+ cos2 (1199092)

(39)

Even the first iteration of PIA(1 1) solution 1199061 gave the same

result as the first iteration solution obtained by the VIM [31]Still the second and third iteration solutions of the problemare better than those of the VIM solutions as observed inFigures 2 and 3

Exact solutionSecond iteration solution Second iteration solution of the VIM

2

22

24

26

28

3

u(x)

02 04 06 08 10x

of the PIA(1 1)

Figure 2 Comparison of the second iteration solutions of the VIMand PIA(1 1) with exact solution for example 4

342 PIA(1 2) Solution In order to apply the PIA(1 2)algorithm obtained by (9) we have to write (32) in thefollowing form

119865 (11990610158401015840 1199061015840 119906 119906120572 120576) = 119906

10158401015840(119909) + 120576119906

2(119909) + 120576

2119906 (

119909

2)

minus (sin4 (119909) + sin2 (119909) + 8) = 0

(40)

where 120576 is the artificially introduced perturbation parameterThe nonzero terms of (9) in PIA(1 2) algorithm are

11986511990610158401015840 = 1

119865119906120576= 2119906119899(119909)

119865120576= 1199062

119899(119909)

119865120576120576= 2119906119899(119909

2)

119865 = 11990610158401015840

119899(119909) minus sin4 (119909) minus sin2 (119909) minus 8

(41)

Journal of Applied Mathematics 7

Exact solution

Third iteration solution of the VIM

2

22

24

26

28

u(x)

02 04 06 08 10x

Third iteration solution of the PIA(1 1)

Figure 3The comparison of the third iteration solutions of theVIMand PIA(1 1) with exact solution for example 4

Note that introducing the small parameter 120576 = 1 as acoefficient of the pantograph term simplifies (40) and makesit easily solvable For this specific example (40) reads

(11990610158401015840

119888)119899(119909) + (2119906

119899(119909)) (119906

119888)119899(119909)

= minus119906119899(119909

2) minus 11990610158401015840

119899(119909) minus 119906

2

119899(119909) + sin4 (119909) + sin2 (119909)

+ 8

(42)

When applying the iteration formula (4) we select thefollowing initially assumed function

1199060= 2 (43)

and using (9) the approximate solution is

1199061=263

96minus49

96cos (2119909) minus 1

12cos4 (2119909) + 1

48cos2 (119909)

minus1

4119909 sin (119909) cos (119909) minus 1

6cos (119909)

(44)

First iteration of PIA(1 2) and second and third iterationsof VIM solutions are compared with the exact solutionAs can be seen from Figure 4 even the first iteration ofPIA(1 2) yields much better results than the first threeiteration solutions of the VIM

35 Illustrative Example 5 Consider the initial value prob-lem of nonlinear variable coefficient differential equationwith pantograph delay [6]

11990610158401015840(119909) = 119906 (119909) minus

8

11990921199062(119909

2) 119909 ge 0 (45)

with initial conditions

119906 (0) = 0

1199061015840(0) = 1

(46)

which has the exact solution 119906(119909) = 119909119890minus119909

2

22

24

26

28

3

u(x)

02 04 06 08 10x

of the VIM

of the VIM

Exact solutionFirst iteration solution

Second iteration solution

Third iteration solutionof the PIA(1 2)

Figure 4 The comparison of second and third iteration solutionsof the VIM and first iteration solution of the PIA(1 2) with exactsolution of example 4

Equation (45) can be written in the following form

11990610158401015840(119909) + 120576 (

8

11990921199062(119909

2) minus 119906 (119909)) = 0 (47)

where 120576 = 1 is the artificially introduced perturbationparameter Using (3) (45) returns to

119865 (11990610158401015840 1199061015840 119906 119906120572 120576) = 119906

10158401015840(119909) + 120576 (

8

11990921199062(119909

2) minus 119906 (119909))

= 0

(48)

The nonzero terms in PIA(1 1) algorithm are

11986511990610158401015840 = 1

119865120576=

8

11990921199062

119899(119909

2) minus 119906119899(119909)

119865 = 11990610158401015840

119899(119909)

(49)

Substituting (49) (45) reduces to

(11990610158401015840

119888)119899(119909) = minus

8

11990921199062

119899(119909

2) + 119906119899(119909) minus 119906

10158401015840

119899(119909) (50)

The initial trial function is selected as

1199060(119909) = 119909 (51)

and using (50) the approximate first iteration solution isobtained as

1199061= 119909 minus 119909

2+1

61199093

1199062= 119909 minus 119909

2+1

21199093minus

5

361199094+

1

801199095minus

1

86401199096

(52)

The absolute maximum errors 119864119899for different values of 119899 are

given in Table 5 and it is revealed that the error decreasescontinually as 119899 increases

8 Journal of Applied Mathematics

Table 5 The absolute maximum errors 119864119899for example 5

119899 1 2 3 4 5 6 7119864119899

201119890 minus 1 561119890 minus 3 159119890 minus 5 275119890 minus 7 368119890 minus 9 676119890 minus 11 494119890 minus 13

Exact solution8th iteration solution 7th iteration solution 6th iteration solution 5th iteration solution

0

01

02

03

04

06 12 18 24 3 36 42 480

of the PIA(1 1)

of the PIA(1 1)

of the PIA(1 1)

of the PIA(1 1)

Figure 5 Comparison of the iteration solutions of the PIA(1 1)method with the exact solution for example 5

Figure 5 illustrates the PIA(1 1) solution of this examplein the interval [0 5] In this example we enlarge the intervalso as to compare our solutionmore thoroughly with the exactsolution The present method yields pretty good results

36 Illustrative Example 6 Consider the second-order non-linear pantograph functional differential equation [7 8]

11990610158401015840(119909) = ([119906 (119909)]

2+ |119906 (119909)|

3) 119906 (

119909

2) 119909 ge 0 (53)

with initial conditions

119906 (0) = 1 119906 (1) =1

2(54)

which has the exact solution 119906(119909) = 1(1 + 119909)Equation (45) can be written in the following form

11990610158401015840(119909) minus 120576 (([119906 (119909)]

2+ |119906 (119909)|

3) 119906 (

119909

2)) = 0 (55)

where 120576 is the artificially introduced small parameter Using(3) (45) returns to

119865 (11990610158401015840 1199061015840 119906 119906120572 120576)

= 11990610158401015840(119909) minus 120576 (([119906 (119909)]

2+ |119906 (119909)|

3) 119906 (

119909

2)) = 0

(56)

The nonzero terms in PIA(1 1) algorithm are

11986511990610158401015840 = 1

119865120576= minus(([119906 (119909)]

2+1003816100381610038161003816119906119899 (119909)

10038161003816100381610038163

) 119906119899(119909

2))

119865 = 11990610158401015840

119899(119909)

(57)

First iteration solution Second iteration solution Third iteration solution

01 02 03 04 05 06 07 08 09 10x

04

05

06

07

08

09

1

u(x)

of the PIA(1 1)

of the PIA(1 1)

of the PIA(1 1)

Figure 6 Comparison of the first three iteration solutions of thePIA(1 1)method with the exact solution for example 6

Substituting (49) (45) reduces to

(11990610158401015840

119888)119899(119909) = minus119906

10158401015840

119899(119909)

+ (([119906 (119909)]2+1003816100381610038161003816119906119899 (119909)

10038161003816100381610038163

) 119906119899(119909

2))

(58)

The initial trial function is selected as

1199060(119909) = 1 minus

119909

2 (59)

and using (50) the approximate first iteration solution isobtained as

1199061(119909) = 1 minus

1073

960119909 + 1199092minus1

21199093+13

961199094minus

3

1601199095

+1

9601199096

(60)

In Figure 6 first three iteration solutions obtained by thePIA(1 1) algorithm are compared with the exact solutionThe convergence of iteration solutions to the exact solutionis obvious

Figure 7 illustrates the absolute error functions of thefirst three iteration solutions of the PIA(1 1) method Themaximum absolute error decreases from 003013 in the firstiteration to 000262 in the third iteration

4 Conclusion

The new perturbation-iteration method is employed for thefirst time to numerically solve the pantograph equationsThe comparative results showed that the method is highly

Journal of Applied Mathematics 9

Absolute error function for first iteration solutionAbsolute error function for second iteration solutionAbsolute error function for third iteration solution

00005

0010015

0020025

0030035

004

01 02 03 04 05 06 07 08 09 10

Figure 7 Comparison of the absolute error functions of the firstthree iteration solutions of the PIA(1 1) method with the exactsolution for example 6

efficient and reliable in both linear and nonlinear problemsWhile for the linear problems studied PIA(1 1) algorithmgave almost the same results as Variational Iteration Methodit gave better results for the nonlinear problems On theother hand the results obtained in the first iteration of thePIA(1 2) algorithm are more convergent than those obtainedin the third iteration of the Variational IterationMethod thisvalidates the performance of the present method Althoughthe equations of PIA(1 2) algorithm are more entangledthan those of PIA(1 1) faster convergence is achieved withPIA(1 2) when compared to PIA(1 1)

It is conceived that the present method can be useful indeveloping new algorithms since they can be generated invarious forms according to the number of correction terms inperturbation expansion and the order of derivation in Taylorexpansion

The disadvantage of the method is that the number ofterms in the solution function increases seriously

In this study the solution intervals of problems arechosen generally the same as those in the references forcomparison purposes In three of the examples the solutionintervals are extended moderately yet for a wider solutioninterval the method should be modified such that the Taylorseries expansions are made around points other than zeroThe method can also be extended to other types of delaydifferential equations but some modifications are required

Conflict of Interests

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

References

[1] MZLiuandDLildquoProperties of analytic solution andnumericalsolution of multi-pantograph equationrdquo Applied Mathematicsand Computation vol 155 no 3 pp 853ndash871 2004

[2] Y Muroya E Ishiwata and H Brunner ldquoOn the attain-able order of collocation methods for pantograph integro-differential equationsrdquo Journal of Computational and AppliedMathematics vol 152 no 1-2 pp 347ndash366 2003

[3] S Yalcınbas M Aynigul and M Sezer ldquoA collocation methodusing Hermite polynomials for approximate solution of panto-graph equationsrdquo Journal of the Franklin Institute vol 348 no 6pp 1128ndash1139 2011

[4] M Sezer andAAkyuz-Dascıoglu ldquoATaylormethod for numer-ical solution of generalized pantograph equations with linearfunctional argumentrdquo Journal of Computational and AppliedMathematics vol 200 no 1 pp 217ndash225 2007

[5] S Sedaghat Y Ordokhani and M Dehghan ldquoNumerical solu-tion of the delay differential equations of pantograph type viaChebyshev polynomialsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 12 pp 4815ndash4830 2012

[6] B Benhammouda H Vazquez-Leal and L Hernandez-Martinez ldquoProcedure for exact solutions of nonlinear panto-graph delay differential equationsrdquo British Journal ofMathemat-ics amp Computer Science vol 4 no 19 pp 2738ndash2751 2014

[7] S Abbasbandy and C Bervillier ldquoAnalytic continuation ofTaylor series and the boundary value problems of some nonlin-ear ordinary differential equationsrdquo Applied Mathematics andComputation vol 218 no 5 pp 2178ndash2199 2011

[8] M A Z Raja ldquoNumerical treatment for boundary value prob-lems of Pantograph functional differential equation using com-putational intelligence algorithmsrdquo Applied Soft Computingvol 24 pp 806ndash821 2014

[9] D Trif ldquoDirect operatorial tau method for pantograph-typeequationsrdquo Applied Mathematics and Computation vol 219 no4 pp 2194ndash2203 2012

[10] A H Nayfeh Introduction to Perturbation Techniques JohnWiley amp Sons New York NY USA 1981

[11] J-H He ldquoIteration perturbation method for strongly nonlinearoscillationsrdquo Journal of Vibration and Control vol 7 no 5 pp631ndash642 2001

[12] R E Mickens ldquoIterationmethod solutions for conservative andlimit-cycle x13 force oscillatorsrdquo Journal of Sound and Vibra-tion vol 292 no 3ndash5 pp 964ndash968 2006

[13] Y Aksoy M Pakdemirli S Abbasbandy and H Boyacı ldquoNewperturbation-iteration solutions for nonlinear heat transferequationsrdquo International Journal of Numerical Methods for Heatamp Fluid Flow vol 22 no 7 pp 814ndash828 2012

[14] T Ozis and A Yıldırım ldquoGenerating the periodic solutions forforcing van der Pol oscillators by the iteration perturbationmethodrdquo Nonlinear Analysis Real World Applications vol 10no 4 pp 1984ndash1989 2009

[15] DDGanji S Karimpour and S S Ganji ldquoHersquos iteration pertur-bation method to nonlinear oscillations of mechanical systemswith single-degree-of freedomrdquo International Journal ofModernPhysics B vol 23 no 11 pp 2469ndash2477 2009

[16] VMarinca andN Herisanu ldquoAmodified iteration perturbationmethod for somenonlinear oscillation problemsrdquoActaMechan-ica vol 184 no 1ndash4 pp 231ndash242 2006

[17] M Pakdemirli and H Boyacı ldquoGeneration of root findingalgorithms via perturbation theory and some formulasrdquoAppliedMathematics and Computation vol 184 no 2 pp 783ndash7882007

[18] M Pakdemirli H Boyacı and H A Yurtsever ldquoPerturbativederivation and comparisons of root-finding algorithms withfourth order derivativesrdquoMathematicalampComputational Appli-cations vol 12 no 2 pp 117ndash124 2007

[19] M Pakdemirli H Boyacı and H A Yurtsever ldquoA root findingalgorithmwith fifth order derivativesrdquoMathematical ampCompu-tational Applications vol 13 no 2 pp 123ndash128 2008

10 Journal of Applied Mathematics

[20] Y Aksoy and M Pakdemirli ldquoNew perturbation-iterationsolutions for Bratu-type equationsrdquo Computers amp Mathematicswith Applications vol 59 no 8 pp 2802ndash2808 2010

[21] I T Dolapcı M Senol and M Pakdemirli ldquoNew perturbationiteration solutions for Fredholm and Volterra integral equa-tionsrdquo Journal of Applied Mathematics vol 2013 Article ID682537 5 pages 2013

[22] A Saadatmandi and M Dehghan ldquoVariational iterationmethod for solving a generalized pantograph equationrdquo Com-puters amp Mathematics with Applications vol 58 no 11-12 pp2190ndash2196 2009

[23] M Sezer S Yalcınbas and M Gulsu ldquoA Taylor polyno-mial approach for solving generalized pantograph equationswith nonhomogenous termrdquo International Journal of ComputerMathematics vol 85 no 7 pp 1055ndash1063 2008

[24] X Y Li and B Y Wu ldquoA continuous method for nonlocalfunctional differential equations with delayed or advancedargumentsrdquo Journal of Mathematical Analysis and Applicationsvol 409 no 1 pp 485ndash493 2014

[25] E H Doha A H Bhrawy D Baleanu and R M Hafez ldquoA newJacobi rationalmdashGauss collocation method for numerical solu-tion of generalized pantograph equationsrdquo Applied NumericalMathematics vol 77 pp 43ndash54 2014

[26] G P Rao and K R Palanisamy ldquoWalsh stretch matrices andfunctional-differential equationsrdquo IEEE Transactions on Auto-matic Control vol 27 no 1 pp 272ndash276 1982

[27] C Hwang ldquoSolution of a functional differential equation viadelayed unit step functionsrdquo International Journal of SystemsScience vol 14 no 9 pp 1065ndash1073 1983

[28] S Yuzbası E Gok andM Sezer ldquoLaguerrematrixmethodwiththe residual error estimation for solutions of a class of delaydifferential equationsrdquo Mathematical Methods in the AppliedSciences vol 37 no 4 pp 453ndash463 2014

[29] N R Anakira A K Alomari and I Hashim ldquoOptimal homo-topy asymptotic method for solving delay differential equa-tionsrdquoMathematical Problems in Engineering vol 2013 ArticleID 498902 11 pages 2013

[30] 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

[31] H Liu A Xiao and L Su ldquoConvergence of variational iterationmethod for second-order delay differential equationsrdquo Journalof Applied Mathematics vol 2013 Article ID 634670 9 pages2013

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

Journal of Applied Mathematics 7

Exact solution

Third iteration solution of the VIM

2

22

24

26

28

u(x)

02 04 06 08 10x

Third iteration solution of the PIA(1 1)

Figure 3The comparison of the third iteration solutions of theVIMand PIA(1 1) with exact solution for example 4

Note that introducing the small parameter 120576 = 1 as acoefficient of the pantograph term simplifies (40) and makesit easily solvable For this specific example (40) reads

(11990610158401015840

119888)119899(119909) + (2119906

119899(119909)) (119906

119888)119899(119909)

= minus119906119899(119909

2) minus 11990610158401015840

119899(119909) minus 119906

2

119899(119909) + sin4 (119909) + sin2 (119909)

+ 8

(42)

When applying the iteration formula (4) we select thefollowing initially assumed function

1199060= 2 (43)

and using (9) the approximate solution is

1199061=263

96minus49

96cos (2119909) minus 1

12cos4 (2119909) + 1

48cos2 (119909)

minus1

4119909 sin (119909) cos (119909) minus 1

6cos (119909)

(44)

First iteration of PIA(1 2) and second and third iterationsof VIM solutions are compared with the exact solutionAs can be seen from Figure 4 even the first iteration ofPIA(1 2) yields much better results than the first threeiteration solutions of the VIM

35 Illustrative Example 5 Consider the initial value prob-lem of nonlinear variable coefficient differential equationwith pantograph delay [6]

11990610158401015840(119909) = 119906 (119909) minus

8

11990921199062(119909

2) 119909 ge 0 (45)

with initial conditions

119906 (0) = 0

1199061015840(0) = 1

(46)

which has the exact solution 119906(119909) = 119909119890minus119909

2

22

24

26

28

3

u(x)

02 04 06 08 10x

of the VIM

of the VIM

Exact solutionFirst iteration solution

Second iteration solution

Third iteration solutionof the PIA(1 2)

Figure 4 The comparison of second and third iteration solutionsof the VIM and first iteration solution of the PIA(1 2) with exactsolution of example 4

Equation (45) can be written in the following form

11990610158401015840(119909) + 120576 (

8

11990921199062(119909

2) minus 119906 (119909)) = 0 (47)

where 120576 = 1 is the artificially introduced perturbationparameter Using (3) (45) returns to

119865 (11990610158401015840 1199061015840 119906 119906120572 120576) = 119906

10158401015840(119909) + 120576 (

8

11990921199062(119909

2) minus 119906 (119909))

= 0

(48)

The nonzero terms in PIA(1 1) algorithm are

11986511990610158401015840 = 1

119865120576=

8

11990921199062

119899(119909

2) minus 119906119899(119909)

119865 = 11990610158401015840

119899(119909)

(49)

Substituting (49) (45) reduces to

(11990610158401015840

119888)119899(119909) = minus

8

11990921199062

119899(119909

2) + 119906119899(119909) minus 119906

10158401015840

119899(119909) (50)

The initial trial function is selected as

1199060(119909) = 119909 (51)

and using (50) the approximate first iteration solution isobtained as

1199061= 119909 minus 119909

2+1

61199093

1199062= 119909 minus 119909

2+1

21199093minus

5

361199094+

1

801199095minus

1

86401199096

(52)

The absolute maximum errors 119864119899for different values of 119899 are

given in Table 5 and it is revealed that the error decreasescontinually as 119899 increases

8 Journal of Applied Mathematics

Table 5 The absolute maximum errors 119864119899for example 5

119899 1 2 3 4 5 6 7119864119899

201119890 minus 1 561119890 minus 3 159119890 minus 5 275119890 minus 7 368119890 minus 9 676119890 minus 11 494119890 minus 13

Exact solution8th iteration solution 7th iteration solution 6th iteration solution 5th iteration solution

0

01

02

03

04

06 12 18 24 3 36 42 480

of the PIA(1 1)

of the PIA(1 1)

of the PIA(1 1)

of the PIA(1 1)

Figure 5 Comparison of the iteration solutions of the PIA(1 1)method with the exact solution for example 5

Figure 5 illustrates the PIA(1 1) solution of this examplein the interval [0 5] In this example we enlarge the intervalso as to compare our solutionmore thoroughly with the exactsolution The present method yields pretty good results

36 Illustrative Example 6 Consider the second-order non-linear pantograph functional differential equation [7 8]

11990610158401015840(119909) = ([119906 (119909)]

2+ |119906 (119909)|

3) 119906 (

119909

2) 119909 ge 0 (53)

with initial conditions

119906 (0) = 1 119906 (1) =1

2(54)

which has the exact solution 119906(119909) = 1(1 + 119909)Equation (45) can be written in the following form

11990610158401015840(119909) minus 120576 (([119906 (119909)]

2+ |119906 (119909)|

3) 119906 (

119909

2)) = 0 (55)

where 120576 is the artificially introduced small parameter Using(3) (45) returns to

119865 (11990610158401015840 1199061015840 119906 119906120572 120576)

= 11990610158401015840(119909) minus 120576 (([119906 (119909)]

2+ |119906 (119909)|

3) 119906 (

119909

2)) = 0

(56)

The nonzero terms in PIA(1 1) algorithm are

11986511990610158401015840 = 1

119865120576= minus(([119906 (119909)]

2+1003816100381610038161003816119906119899 (119909)

10038161003816100381610038163

) 119906119899(119909

2))

119865 = 11990610158401015840

119899(119909)

(57)

First iteration solution Second iteration solution Third iteration solution

01 02 03 04 05 06 07 08 09 10x

04

05

06

07

08

09

1

u(x)

of the PIA(1 1)

of the PIA(1 1)

of the PIA(1 1)

Figure 6 Comparison of the first three iteration solutions of thePIA(1 1)method with the exact solution for example 6

Substituting (49) (45) reduces to

(11990610158401015840

119888)119899(119909) = minus119906

10158401015840

119899(119909)

+ (([119906 (119909)]2+1003816100381610038161003816119906119899 (119909)

10038161003816100381610038163

) 119906119899(119909

2))

(58)

The initial trial function is selected as

1199060(119909) = 1 minus

119909

2 (59)

and using (50) the approximate first iteration solution isobtained as

1199061(119909) = 1 minus

1073

960119909 + 1199092minus1

21199093+13

961199094minus

3

1601199095

+1

9601199096

(60)

In Figure 6 first three iteration solutions obtained by thePIA(1 1) algorithm are compared with the exact solutionThe convergence of iteration solutions to the exact solutionis obvious

Figure 7 illustrates the absolute error functions of thefirst three iteration solutions of the PIA(1 1) method Themaximum absolute error decreases from 003013 in the firstiteration to 000262 in the third iteration

4 Conclusion

The new perturbation-iteration method is employed for thefirst time to numerically solve the pantograph equationsThe comparative results showed that the method is highly

Journal of Applied Mathematics 9

Absolute error function for first iteration solutionAbsolute error function for second iteration solutionAbsolute error function for third iteration solution

00005

0010015

0020025

0030035

004

01 02 03 04 05 06 07 08 09 10

Figure 7 Comparison of the absolute error functions of the firstthree iteration solutions of the PIA(1 1) method with the exactsolution for example 6

efficient and reliable in both linear and nonlinear problemsWhile for the linear problems studied PIA(1 1) algorithmgave almost the same results as Variational Iteration Methodit gave better results for the nonlinear problems On theother hand the results obtained in the first iteration of thePIA(1 2) algorithm are more convergent than those obtainedin the third iteration of the Variational IterationMethod thisvalidates the performance of the present method Althoughthe equations of PIA(1 2) algorithm are more entangledthan those of PIA(1 1) faster convergence is achieved withPIA(1 2) when compared to PIA(1 1)

It is conceived that the present method can be useful indeveloping new algorithms since they can be generated invarious forms according to the number of correction terms inperturbation expansion and the order of derivation in Taylorexpansion

The disadvantage of the method is that the number ofterms in the solution function increases seriously

In this study the solution intervals of problems arechosen generally the same as those in the references forcomparison purposes In three of the examples the solutionintervals are extended moderately yet for a wider solutioninterval the method should be modified such that the Taylorseries expansions are made around points other than zeroThe method can also be extended to other types of delaydifferential equations but some modifications are required

Conflict of Interests

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

References

[1] MZLiuandDLildquoProperties of analytic solution andnumericalsolution of multi-pantograph equationrdquo Applied Mathematicsand Computation vol 155 no 3 pp 853ndash871 2004

[2] Y Muroya E Ishiwata and H Brunner ldquoOn the attain-able order of collocation methods for pantograph integro-differential equationsrdquo Journal of Computational and AppliedMathematics vol 152 no 1-2 pp 347ndash366 2003

[3] S Yalcınbas M Aynigul and M Sezer ldquoA collocation methodusing Hermite polynomials for approximate solution of panto-graph equationsrdquo Journal of the Franklin Institute vol 348 no 6pp 1128ndash1139 2011

[4] M Sezer andAAkyuz-Dascıoglu ldquoATaylormethod for numer-ical solution of generalized pantograph equations with linearfunctional argumentrdquo Journal of Computational and AppliedMathematics vol 200 no 1 pp 217ndash225 2007

[5] S Sedaghat Y Ordokhani and M Dehghan ldquoNumerical solu-tion of the delay differential equations of pantograph type viaChebyshev polynomialsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 12 pp 4815ndash4830 2012

[6] B Benhammouda H Vazquez-Leal and L Hernandez-Martinez ldquoProcedure for exact solutions of nonlinear panto-graph delay differential equationsrdquo British Journal ofMathemat-ics amp Computer Science vol 4 no 19 pp 2738ndash2751 2014

[7] S Abbasbandy and C Bervillier ldquoAnalytic continuation ofTaylor series and the boundary value problems of some nonlin-ear ordinary differential equationsrdquo Applied Mathematics andComputation vol 218 no 5 pp 2178ndash2199 2011

[8] M A Z Raja ldquoNumerical treatment for boundary value prob-lems of Pantograph functional differential equation using com-putational intelligence algorithmsrdquo Applied Soft Computingvol 24 pp 806ndash821 2014

[9] D Trif ldquoDirect operatorial tau method for pantograph-typeequationsrdquo Applied Mathematics and Computation vol 219 no4 pp 2194ndash2203 2012

[10] A H Nayfeh Introduction to Perturbation Techniques JohnWiley amp Sons New York NY USA 1981

[11] J-H He ldquoIteration perturbation method for strongly nonlinearoscillationsrdquo Journal of Vibration and Control vol 7 no 5 pp631ndash642 2001

[12] R E Mickens ldquoIterationmethod solutions for conservative andlimit-cycle x13 force oscillatorsrdquo Journal of Sound and Vibra-tion vol 292 no 3ndash5 pp 964ndash968 2006

[13] Y Aksoy M Pakdemirli S Abbasbandy and H Boyacı ldquoNewperturbation-iteration solutions for nonlinear heat transferequationsrdquo International Journal of Numerical Methods for Heatamp Fluid Flow vol 22 no 7 pp 814ndash828 2012

[14] T Ozis and A Yıldırım ldquoGenerating the periodic solutions forforcing van der Pol oscillators by the iteration perturbationmethodrdquo Nonlinear Analysis Real World Applications vol 10no 4 pp 1984ndash1989 2009

[15] DDGanji S Karimpour and S S Ganji ldquoHersquos iteration pertur-bation method to nonlinear oscillations of mechanical systemswith single-degree-of freedomrdquo International Journal ofModernPhysics B vol 23 no 11 pp 2469ndash2477 2009

[16] VMarinca andN Herisanu ldquoAmodified iteration perturbationmethod for somenonlinear oscillation problemsrdquoActaMechan-ica vol 184 no 1ndash4 pp 231ndash242 2006

[17] M Pakdemirli and H Boyacı ldquoGeneration of root findingalgorithms via perturbation theory and some formulasrdquoAppliedMathematics and Computation vol 184 no 2 pp 783ndash7882007

[18] M Pakdemirli H Boyacı and H A Yurtsever ldquoPerturbativederivation and comparisons of root-finding algorithms withfourth order derivativesrdquoMathematicalampComputational Appli-cations vol 12 no 2 pp 117ndash124 2007

[19] M Pakdemirli H Boyacı and H A Yurtsever ldquoA root findingalgorithmwith fifth order derivativesrdquoMathematical ampCompu-tational Applications vol 13 no 2 pp 123ndash128 2008

10 Journal of Applied Mathematics

[20] Y Aksoy and M Pakdemirli ldquoNew perturbation-iterationsolutions for Bratu-type equationsrdquo Computers amp Mathematicswith Applications vol 59 no 8 pp 2802ndash2808 2010

[21] I T Dolapcı M Senol and M Pakdemirli ldquoNew perturbationiteration solutions for Fredholm and Volterra integral equa-tionsrdquo Journal of Applied Mathematics vol 2013 Article ID682537 5 pages 2013

[22] A Saadatmandi and M Dehghan ldquoVariational iterationmethod for solving a generalized pantograph equationrdquo Com-puters amp Mathematics with Applications vol 58 no 11-12 pp2190ndash2196 2009

[23] M Sezer S Yalcınbas and M Gulsu ldquoA Taylor polyno-mial approach for solving generalized pantograph equationswith nonhomogenous termrdquo International Journal of ComputerMathematics vol 85 no 7 pp 1055ndash1063 2008

[24] X Y Li and B Y Wu ldquoA continuous method for nonlocalfunctional differential equations with delayed or advancedargumentsrdquo Journal of Mathematical Analysis and Applicationsvol 409 no 1 pp 485ndash493 2014

[25] E H Doha A H Bhrawy D Baleanu and R M Hafez ldquoA newJacobi rationalmdashGauss collocation method for numerical solu-tion of generalized pantograph equationsrdquo Applied NumericalMathematics vol 77 pp 43ndash54 2014

[26] G P Rao and K R Palanisamy ldquoWalsh stretch matrices andfunctional-differential equationsrdquo IEEE Transactions on Auto-matic Control vol 27 no 1 pp 272ndash276 1982

[27] C Hwang ldquoSolution of a functional differential equation viadelayed unit step functionsrdquo International Journal of SystemsScience vol 14 no 9 pp 1065ndash1073 1983

[28] S Yuzbası E Gok andM Sezer ldquoLaguerrematrixmethodwiththe residual error estimation for solutions of a class of delaydifferential equationsrdquo Mathematical Methods in the AppliedSciences vol 37 no 4 pp 453ndash463 2014

[29] N R Anakira A K Alomari and I Hashim ldquoOptimal homo-topy asymptotic method for solving delay differential equa-tionsrdquoMathematical Problems in Engineering vol 2013 ArticleID 498902 11 pages 2013

[30] 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

[31] H Liu A Xiao and L Su ldquoConvergence of variational iterationmethod for second-order delay differential equationsrdquo Journalof Applied Mathematics vol 2013 Article ID 634670 9 pages2013

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 Journal of Applied Mathematics

Table 5 The absolute maximum errors 119864119899for example 5

119899 1 2 3 4 5 6 7119864119899

201119890 minus 1 561119890 minus 3 159119890 minus 5 275119890 minus 7 368119890 minus 9 676119890 minus 11 494119890 minus 13

Exact solution8th iteration solution 7th iteration solution 6th iteration solution 5th iteration solution

0

01

02

03

04

06 12 18 24 3 36 42 480

of the PIA(1 1)

of the PIA(1 1)

of the PIA(1 1)

of the PIA(1 1)

Figure 5 Comparison of the iteration solutions of the PIA(1 1)method with the exact solution for example 5

Figure 5 illustrates the PIA(1 1) solution of this examplein the interval [0 5] In this example we enlarge the intervalso as to compare our solutionmore thoroughly with the exactsolution The present method yields pretty good results

36 Illustrative Example 6 Consider the second-order non-linear pantograph functional differential equation [7 8]

11990610158401015840(119909) = ([119906 (119909)]

2+ |119906 (119909)|

3) 119906 (

119909

2) 119909 ge 0 (53)

with initial conditions

119906 (0) = 1 119906 (1) =1

2(54)

which has the exact solution 119906(119909) = 1(1 + 119909)Equation (45) can be written in the following form

11990610158401015840(119909) minus 120576 (([119906 (119909)]

2+ |119906 (119909)|

3) 119906 (

119909

2)) = 0 (55)

where 120576 is the artificially introduced small parameter Using(3) (45) returns to

119865 (11990610158401015840 1199061015840 119906 119906120572 120576)

= 11990610158401015840(119909) minus 120576 (([119906 (119909)]

2+ |119906 (119909)|

3) 119906 (

119909

2)) = 0

(56)

The nonzero terms in PIA(1 1) algorithm are

11986511990610158401015840 = 1

119865120576= minus(([119906 (119909)]

2+1003816100381610038161003816119906119899 (119909)

10038161003816100381610038163

) 119906119899(119909

2))

119865 = 11990610158401015840

119899(119909)

(57)

First iteration solution Second iteration solution Third iteration solution

01 02 03 04 05 06 07 08 09 10x

04

05

06

07

08

09

1

u(x)

of the PIA(1 1)

of the PIA(1 1)

of the PIA(1 1)

Figure 6 Comparison of the first three iteration solutions of thePIA(1 1)method with the exact solution for example 6

Substituting (49) (45) reduces to

(11990610158401015840

119888)119899(119909) = minus119906

10158401015840

119899(119909)

+ (([119906 (119909)]2+1003816100381610038161003816119906119899 (119909)

10038161003816100381610038163

) 119906119899(119909

2))

(58)

The initial trial function is selected as

1199060(119909) = 1 minus

119909

2 (59)

and using (50) the approximate first iteration solution isobtained as

1199061(119909) = 1 minus

1073

960119909 + 1199092minus1

21199093+13

961199094minus

3

1601199095

+1

9601199096

(60)

In Figure 6 first three iteration solutions obtained by thePIA(1 1) algorithm are compared with the exact solutionThe convergence of iteration solutions to the exact solutionis obvious

Figure 7 illustrates the absolute error functions of thefirst three iteration solutions of the PIA(1 1) method Themaximum absolute error decreases from 003013 in the firstiteration to 000262 in the third iteration

4 Conclusion

The new perturbation-iteration method is employed for thefirst time to numerically solve the pantograph equationsThe comparative results showed that the method is highly

Journal of Applied Mathematics 9

Absolute error function for first iteration solutionAbsolute error function for second iteration solutionAbsolute error function for third iteration solution

00005

0010015

0020025

0030035

004

01 02 03 04 05 06 07 08 09 10

Figure 7 Comparison of the absolute error functions of the firstthree iteration solutions of the PIA(1 1) method with the exactsolution for example 6

efficient and reliable in both linear and nonlinear problemsWhile for the linear problems studied PIA(1 1) algorithmgave almost the same results as Variational Iteration Methodit gave better results for the nonlinear problems On theother hand the results obtained in the first iteration of thePIA(1 2) algorithm are more convergent than those obtainedin the third iteration of the Variational IterationMethod thisvalidates the performance of the present method Althoughthe equations of PIA(1 2) algorithm are more entangledthan those of PIA(1 1) faster convergence is achieved withPIA(1 2) when compared to PIA(1 1)

It is conceived that the present method can be useful indeveloping new algorithms since they can be generated invarious forms according to the number of correction terms inperturbation expansion and the order of derivation in Taylorexpansion

The disadvantage of the method is that the number ofterms in the solution function increases seriously

In this study the solution intervals of problems arechosen generally the same as those in the references forcomparison purposes In three of the examples the solutionintervals are extended moderately yet for a wider solutioninterval the method should be modified such that the Taylorseries expansions are made around points other than zeroThe method can also be extended to other types of delaydifferential equations but some modifications are required

Conflict of Interests

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

References

[1] MZLiuandDLildquoProperties of analytic solution andnumericalsolution of multi-pantograph equationrdquo Applied Mathematicsand Computation vol 155 no 3 pp 853ndash871 2004

[2] Y Muroya E Ishiwata and H Brunner ldquoOn the attain-able order of collocation methods for pantograph integro-differential equationsrdquo Journal of Computational and AppliedMathematics vol 152 no 1-2 pp 347ndash366 2003

[3] S Yalcınbas M Aynigul and M Sezer ldquoA collocation methodusing Hermite polynomials for approximate solution of panto-graph equationsrdquo Journal of the Franklin Institute vol 348 no 6pp 1128ndash1139 2011

[4] M Sezer andAAkyuz-Dascıoglu ldquoATaylormethod for numer-ical solution of generalized pantograph equations with linearfunctional argumentrdquo Journal of Computational and AppliedMathematics vol 200 no 1 pp 217ndash225 2007

[5] S Sedaghat Y Ordokhani and M Dehghan ldquoNumerical solu-tion of the delay differential equations of pantograph type viaChebyshev polynomialsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 12 pp 4815ndash4830 2012

[6] B Benhammouda H Vazquez-Leal and L Hernandez-Martinez ldquoProcedure for exact solutions of nonlinear panto-graph delay differential equationsrdquo British Journal ofMathemat-ics amp Computer Science vol 4 no 19 pp 2738ndash2751 2014

[7] S Abbasbandy and C Bervillier ldquoAnalytic continuation ofTaylor series and the boundary value problems of some nonlin-ear ordinary differential equationsrdquo Applied Mathematics andComputation vol 218 no 5 pp 2178ndash2199 2011

[8] M A Z Raja ldquoNumerical treatment for boundary value prob-lems of Pantograph functional differential equation using com-putational intelligence algorithmsrdquo Applied Soft Computingvol 24 pp 806ndash821 2014

[9] D Trif ldquoDirect operatorial tau method for pantograph-typeequationsrdquo Applied Mathematics and Computation vol 219 no4 pp 2194ndash2203 2012

[10] A H Nayfeh Introduction to Perturbation Techniques JohnWiley amp Sons New York NY USA 1981

[11] J-H He ldquoIteration perturbation method for strongly nonlinearoscillationsrdquo Journal of Vibration and Control vol 7 no 5 pp631ndash642 2001

[12] R E Mickens ldquoIterationmethod solutions for conservative andlimit-cycle x13 force oscillatorsrdquo Journal of Sound and Vibra-tion vol 292 no 3ndash5 pp 964ndash968 2006

[13] Y Aksoy M Pakdemirli S Abbasbandy and H Boyacı ldquoNewperturbation-iteration solutions for nonlinear heat transferequationsrdquo International Journal of Numerical Methods for Heatamp Fluid Flow vol 22 no 7 pp 814ndash828 2012

[14] T Ozis and A Yıldırım ldquoGenerating the periodic solutions forforcing van der Pol oscillators by the iteration perturbationmethodrdquo Nonlinear Analysis Real World Applications vol 10no 4 pp 1984ndash1989 2009

[15] DDGanji S Karimpour and S S Ganji ldquoHersquos iteration pertur-bation method to nonlinear oscillations of mechanical systemswith single-degree-of freedomrdquo International Journal ofModernPhysics B vol 23 no 11 pp 2469ndash2477 2009

[16] VMarinca andN Herisanu ldquoAmodified iteration perturbationmethod for somenonlinear oscillation problemsrdquoActaMechan-ica vol 184 no 1ndash4 pp 231ndash242 2006

[17] M Pakdemirli and H Boyacı ldquoGeneration of root findingalgorithms via perturbation theory and some formulasrdquoAppliedMathematics and Computation vol 184 no 2 pp 783ndash7882007

[18] M Pakdemirli H Boyacı and H A Yurtsever ldquoPerturbativederivation and comparisons of root-finding algorithms withfourth order derivativesrdquoMathematicalampComputational Appli-cations vol 12 no 2 pp 117ndash124 2007

[19] M Pakdemirli H Boyacı and H A Yurtsever ldquoA root findingalgorithmwith fifth order derivativesrdquoMathematical ampCompu-tational Applications vol 13 no 2 pp 123ndash128 2008

10 Journal of Applied Mathematics

[20] Y Aksoy and M Pakdemirli ldquoNew perturbation-iterationsolutions for Bratu-type equationsrdquo Computers amp Mathematicswith Applications vol 59 no 8 pp 2802ndash2808 2010

[21] I T Dolapcı M Senol and M Pakdemirli ldquoNew perturbationiteration solutions for Fredholm and Volterra integral equa-tionsrdquo Journal of Applied Mathematics vol 2013 Article ID682537 5 pages 2013

[22] A Saadatmandi and M Dehghan ldquoVariational iterationmethod for solving a generalized pantograph equationrdquo Com-puters amp Mathematics with Applications vol 58 no 11-12 pp2190ndash2196 2009

[23] M Sezer S Yalcınbas and M Gulsu ldquoA Taylor polyno-mial approach for solving generalized pantograph equationswith nonhomogenous termrdquo International Journal of ComputerMathematics vol 85 no 7 pp 1055ndash1063 2008

[24] X Y Li and B Y Wu ldquoA continuous method for nonlocalfunctional differential equations with delayed or advancedargumentsrdquo Journal of Mathematical Analysis and Applicationsvol 409 no 1 pp 485ndash493 2014

[25] E H Doha A H Bhrawy D Baleanu and R M Hafez ldquoA newJacobi rationalmdashGauss collocation method for numerical solu-tion of generalized pantograph equationsrdquo Applied NumericalMathematics vol 77 pp 43ndash54 2014

[26] G P Rao and K R Palanisamy ldquoWalsh stretch matrices andfunctional-differential equationsrdquo IEEE Transactions on Auto-matic Control vol 27 no 1 pp 272ndash276 1982

[27] C Hwang ldquoSolution of a functional differential equation viadelayed unit step functionsrdquo International Journal of SystemsScience vol 14 no 9 pp 1065ndash1073 1983

[28] S Yuzbası E Gok andM Sezer ldquoLaguerrematrixmethodwiththe residual error estimation for solutions of a class of delaydifferential equationsrdquo Mathematical Methods in the AppliedSciences vol 37 no 4 pp 453ndash463 2014

[29] N R Anakira A K Alomari and I Hashim ldquoOptimal homo-topy asymptotic method for solving delay differential equa-tionsrdquoMathematical Problems in Engineering vol 2013 ArticleID 498902 11 pages 2013

[30] 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

[31] H Liu A Xiao and L Su ldquoConvergence of variational iterationmethod for second-order delay differential equationsrdquo Journalof Applied Mathematics vol 2013 Article ID 634670 9 pages2013

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

Journal of Applied Mathematics 9

Absolute error function for first iteration solutionAbsolute error function for second iteration solutionAbsolute error function for third iteration solution

00005

0010015

0020025

0030035

004

01 02 03 04 05 06 07 08 09 10

Figure 7 Comparison of the absolute error functions of the firstthree iteration solutions of the PIA(1 1) method with the exactsolution for example 6

efficient and reliable in both linear and nonlinear problemsWhile for the linear problems studied PIA(1 1) algorithmgave almost the same results as Variational Iteration Methodit gave better results for the nonlinear problems On theother hand the results obtained in the first iteration of thePIA(1 2) algorithm are more convergent than those obtainedin the third iteration of the Variational IterationMethod thisvalidates the performance of the present method Althoughthe equations of PIA(1 2) algorithm are more entangledthan those of PIA(1 1) faster convergence is achieved withPIA(1 2) when compared to PIA(1 1)

It is conceived that the present method can be useful indeveloping new algorithms since they can be generated invarious forms according to the number of correction terms inperturbation expansion and the order of derivation in Taylorexpansion

The disadvantage of the method is that the number ofterms in the solution function increases seriously

In this study the solution intervals of problems arechosen generally the same as those in the references forcomparison purposes In three of the examples the solutionintervals are extended moderately yet for a wider solutioninterval the method should be modified such that the Taylorseries expansions are made around points other than zeroThe method can also be extended to other types of delaydifferential equations but some modifications are required

Conflict of Interests

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

References

[1] MZLiuandDLildquoProperties of analytic solution andnumericalsolution of multi-pantograph equationrdquo Applied Mathematicsand Computation vol 155 no 3 pp 853ndash871 2004

[2] Y Muroya E Ishiwata and H Brunner ldquoOn the attain-able order of collocation methods for pantograph integro-differential equationsrdquo Journal of Computational and AppliedMathematics vol 152 no 1-2 pp 347ndash366 2003

[3] S Yalcınbas M Aynigul and M Sezer ldquoA collocation methodusing Hermite polynomials for approximate solution of panto-graph equationsrdquo Journal of the Franklin Institute vol 348 no 6pp 1128ndash1139 2011

[4] M Sezer andAAkyuz-Dascıoglu ldquoATaylormethod for numer-ical solution of generalized pantograph equations with linearfunctional argumentrdquo Journal of Computational and AppliedMathematics vol 200 no 1 pp 217ndash225 2007

[5] S Sedaghat Y Ordokhani and M Dehghan ldquoNumerical solu-tion of the delay differential equations of pantograph type viaChebyshev polynomialsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 12 pp 4815ndash4830 2012

[6] B Benhammouda H Vazquez-Leal and L Hernandez-Martinez ldquoProcedure for exact solutions of nonlinear panto-graph delay differential equationsrdquo British Journal ofMathemat-ics amp Computer Science vol 4 no 19 pp 2738ndash2751 2014

[7] S Abbasbandy and C Bervillier ldquoAnalytic continuation ofTaylor series and the boundary value problems of some nonlin-ear ordinary differential equationsrdquo Applied Mathematics andComputation vol 218 no 5 pp 2178ndash2199 2011

[8] M A Z Raja ldquoNumerical treatment for boundary value prob-lems of Pantograph functional differential equation using com-putational intelligence algorithmsrdquo Applied Soft Computingvol 24 pp 806ndash821 2014

[9] D Trif ldquoDirect operatorial tau method for pantograph-typeequationsrdquo Applied Mathematics and Computation vol 219 no4 pp 2194ndash2203 2012

[10] A H Nayfeh Introduction to Perturbation Techniques JohnWiley amp Sons New York NY USA 1981

[11] J-H He ldquoIteration perturbation method for strongly nonlinearoscillationsrdquo Journal of Vibration and Control vol 7 no 5 pp631ndash642 2001

[12] R E Mickens ldquoIterationmethod solutions for conservative andlimit-cycle x13 force oscillatorsrdquo Journal of Sound and Vibra-tion vol 292 no 3ndash5 pp 964ndash968 2006

[13] Y Aksoy M Pakdemirli S Abbasbandy and H Boyacı ldquoNewperturbation-iteration solutions for nonlinear heat transferequationsrdquo International Journal of Numerical Methods for Heatamp Fluid Flow vol 22 no 7 pp 814ndash828 2012

[14] T Ozis and A Yıldırım ldquoGenerating the periodic solutions forforcing van der Pol oscillators by the iteration perturbationmethodrdquo Nonlinear Analysis Real World Applications vol 10no 4 pp 1984ndash1989 2009

[15] DDGanji S Karimpour and S S Ganji ldquoHersquos iteration pertur-bation method to nonlinear oscillations of mechanical systemswith single-degree-of freedomrdquo International Journal ofModernPhysics B vol 23 no 11 pp 2469ndash2477 2009

[16] VMarinca andN Herisanu ldquoAmodified iteration perturbationmethod for somenonlinear oscillation problemsrdquoActaMechan-ica vol 184 no 1ndash4 pp 231ndash242 2006

[17] M Pakdemirli and H Boyacı ldquoGeneration of root findingalgorithms via perturbation theory and some formulasrdquoAppliedMathematics and Computation vol 184 no 2 pp 783ndash7882007

[18] M Pakdemirli H Boyacı and H A Yurtsever ldquoPerturbativederivation and comparisons of root-finding algorithms withfourth order derivativesrdquoMathematicalampComputational Appli-cations vol 12 no 2 pp 117ndash124 2007

[19] M Pakdemirli H Boyacı and H A Yurtsever ldquoA root findingalgorithmwith fifth order derivativesrdquoMathematical ampCompu-tational Applications vol 13 no 2 pp 123ndash128 2008

10 Journal of Applied Mathematics

[20] Y Aksoy and M Pakdemirli ldquoNew perturbation-iterationsolutions for Bratu-type equationsrdquo Computers amp Mathematicswith Applications vol 59 no 8 pp 2802ndash2808 2010

[21] I T Dolapcı M Senol and M Pakdemirli ldquoNew perturbationiteration solutions for Fredholm and Volterra integral equa-tionsrdquo Journal of Applied Mathematics vol 2013 Article ID682537 5 pages 2013

[22] A Saadatmandi and M Dehghan ldquoVariational iterationmethod for solving a generalized pantograph equationrdquo Com-puters amp Mathematics with Applications vol 58 no 11-12 pp2190ndash2196 2009

[23] M Sezer S Yalcınbas and M Gulsu ldquoA Taylor polyno-mial approach for solving generalized pantograph equationswith nonhomogenous termrdquo International Journal of ComputerMathematics vol 85 no 7 pp 1055ndash1063 2008

[24] X Y Li and B Y Wu ldquoA continuous method for nonlocalfunctional differential equations with delayed or advancedargumentsrdquo Journal of Mathematical Analysis and Applicationsvol 409 no 1 pp 485ndash493 2014

[25] E H Doha A H Bhrawy D Baleanu and R M Hafez ldquoA newJacobi rationalmdashGauss collocation method for numerical solu-tion of generalized pantograph equationsrdquo Applied NumericalMathematics vol 77 pp 43ndash54 2014

[26] G P Rao and K R Palanisamy ldquoWalsh stretch matrices andfunctional-differential equationsrdquo IEEE Transactions on Auto-matic Control vol 27 no 1 pp 272ndash276 1982

[27] C Hwang ldquoSolution of a functional differential equation viadelayed unit step functionsrdquo International Journal of SystemsScience vol 14 no 9 pp 1065ndash1073 1983

[28] S Yuzbası E Gok andM Sezer ldquoLaguerrematrixmethodwiththe residual error estimation for solutions of a class of delaydifferential equationsrdquo Mathematical Methods in the AppliedSciences vol 37 no 4 pp 453ndash463 2014

[29] N R Anakira A K Alomari and I Hashim ldquoOptimal homo-topy asymptotic method for solving delay differential equa-tionsrdquoMathematical Problems in Engineering vol 2013 ArticleID 498902 11 pages 2013

[30] 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

[31] H Liu A Xiao and L Su ldquoConvergence of variational iterationmethod for second-order delay differential equationsrdquo Journalof Applied Mathematics vol 2013 Article ID 634670 9 pages2013

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 Journal of Applied Mathematics

[20] Y Aksoy and M Pakdemirli ldquoNew perturbation-iterationsolutions for Bratu-type equationsrdquo Computers amp Mathematicswith Applications vol 59 no 8 pp 2802ndash2808 2010

[21] I T Dolapcı M Senol and M Pakdemirli ldquoNew perturbationiteration solutions for Fredholm and Volterra integral equa-tionsrdquo Journal of Applied Mathematics vol 2013 Article ID682537 5 pages 2013

[22] A Saadatmandi and M Dehghan ldquoVariational iterationmethod for solving a generalized pantograph equationrdquo Com-puters amp Mathematics with Applications vol 58 no 11-12 pp2190ndash2196 2009

[23] M Sezer S Yalcınbas and M Gulsu ldquoA Taylor polyno-mial approach for solving generalized pantograph equationswith nonhomogenous termrdquo International Journal of ComputerMathematics vol 85 no 7 pp 1055ndash1063 2008

[24] X Y Li and B Y Wu ldquoA continuous method for nonlocalfunctional differential equations with delayed or advancedargumentsrdquo Journal of Mathematical Analysis and Applicationsvol 409 no 1 pp 485ndash493 2014

[25] E H Doha A H Bhrawy D Baleanu and R M Hafez ldquoA newJacobi rationalmdashGauss collocation method for numerical solu-tion of generalized pantograph equationsrdquo Applied NumericalMathematics vol 77 pp 43ndash54 2014

[26] G P Rao and K R Palanisamy ldquoWalsh stretch matrices andfunctional-differential equationsrdquo IEEE Transactions on Auto-matic Control vol 27 no 1 pp 272ndash276 1982

[27] C Hwang ldquoSolution of a functional differential equation viadelayed unit step functionsrdquo International Journal of SystemsScience vol 14 no 9 pp 1065ndash1073 1983

[28] S Yuzbası E Gok andM Sezer ldquoLaguerrematrixmethodwiththe residual error estimation for solutions of a class of delaydifferential equationsrdquo Mathematical Methods in the AppliedSciences vol 37 no 4 pp 453ndash463 2014

[29] N R Anakira A K Alomari and I Hashim ldquoOptimal homo-topy asymptotic method for solving delay differential equa-tionsrdquoMathematical Problems in Engineering vol 2013 ArticleID 498902 11 pages 2013

[30] 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

[31] H Liu A Xiao and L Su ldquoConvergence of variational iterationmethod for second-order delay differential equationsrdquo Journalof Applied Mathematics vol 2013 Article ID 634670 9 pages2013

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