research article perturbation-iteration method for...

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 704137 6 pageshttpdxdoiorg1011552013704137

Research ArticlePerturbation-Iteration Method for First-Order DifferentialEquations and Systems

Mehmet Fenol1 Ehsan Timuccedilin DolapccedilJ2 YiLit Aksoy2 and Mehmet Pakdemirli2

1 Department of Mathematics Nevsehir University 50300 Nevsehir Turkey2Department of Mechanical Engineering Celal Bayar University Muradiye 45140 Manisa Turkey

Correspondence should be addressed to Mehmet Pakdemirli mpakcbuedutr

Received 16 March 2013 Revised 18 April 2013 Accepted 19 April 2013

Academic Editor Yisheng Song

Copyright copy 2013 Mehmet Senol et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The previously developed new perturbation-iteration algorithm has been applied to differential equation systems for the first timeThe iteration algorithm for systems is developed first The algorithm is tested for a single equation coupled two equations andcoupled three equations Solutions are compared with those of variational iteration method and numerical solutions and a goodagreement is found The method can be applied to differential equation systems with success

1 Introduction

Perturbation methods are classical methods which havebeen used over a century to obtain approximate analyticalsolutionsThemethod has been successfully applied to differ-ential equations integro differential equations and algebraicequations Many different perturbation techniques such asthe method of multiple scales the method of averaging therenormalizationmethod the Lindstedt-Poincaremethod themethod ofmatched asymptotic expansions and their variantswere developed [1 2]

The major limitation of the perturbation methods isthe requirement of a small parameter Sometimes the smallparameter may also be artificially introduced into the equa-tionsThe solutions therefore have a limited range of validityAlthough the solutions are valid for weakly nonlinear prob-lems they are not admissible usually for strongly nonlinearproblems

Several techniques have been proposed in the literaturerecently to obtain admissible solutions which do not requiresmall parameter assumption While a complete review ofthe attempts is beyond the scope of this work linearizedperturbation method parameter expanding method newtime transformations as modifications of Lindstedt-Poincaremethod and iteration methods can be mentioned as exam-ples [3ndash14]

Recently a class of alternative perturbation-iterationalgorithms has been proposed The fundamentals of thealgorithms were outlined for the first-order differential equa-tions by Pakdemirli et al [15] Several iteration algorithmscan be derived by taking different number of terms in theperturbation expansions and different order of correctionterms in the Taylor series expansions The perturbationiteration algorithm is called PIA(119899 119898) where 119899 representsthe correction terms in the perturbation expansion and 119898

represents the highest order derivative term in the Taylorseries This new method has been successfully implementedto Bratu-Type equations [16] Solutions obtained by this newmethod and those obtained by variational iteration method(VIM) were contrasted and it is shown that while PIA(1 1)algorithmusually produces identical resultswithVIM higherorder algorithms PIA(119899 119898) produce better results The newperturbation-iteration technique was applied to nonlinearheat transfer equations by Aksoy et al [17] very recentlyOne of the main advantages is that the new method does notrequire initial assumptions or transformation of the equationsto another form Actually the techniques were developed firstfor algebraic equations [18ndash20] and then adopted to ordinarydifferential equations [15ndash17]

In this study the iteration algorithms for single equationsare generalized to arbitrary number of first-order coupledequations An application of the algorithm to a single

2 Abstract and Applied Analysis

equation which is a degenerate case is treated first Thencoupled systems with two and three equations are solvedSolutions are contrasted with available other approximatesolutions or numerical solutions It is found that the methodcan be effectively applied to differential equation systems aswell

2 Perturbation-Iteration Algorithm PIA(1 119898)

In this section a perturbation-iteration algorithm PIA(1 119898)is constructed by taking one correction term in the perturba-tion expansion and correction terms of119898th-order derivativesin the Taylor series expansion

Consider the following system of first-order differentialequations

119865119896 (119896 119906119895 120576 119905) = 0 119896 = 1 2 119870 119895 = 1 2 119870 (1)

where 119870 represents the number of differential equations inthe system and the number of dependent variables119870 = 1 fora single equation In the open form the system of equationsis

1198651 = 1198651 (1 1199061 1199062 119906119870 120576 119905) = 0

1198652 = 1198652 (2 1199061 1199062 119906119870 120576 119905) = 0

119865119870 = 119865119870 (119870 1199061 1199062 119906119870 120576 119905) = 0

(2)

Assume an approximate solution of the system

119906119896119899+1 = 119906119896119899 + 120576119906119888

119896119899 (3)

with one correction term in the perturbation expansion Thesubscript 119899 represents the 119899th iteration over this approximatesolutionThe system can be approximatedwith a Taylor seriesexpansion in the neighborhood of 120576 = 0 as

119865119896 =

119872

sum

119898=0

1

119898

[(

119889

119889120576

)

119898

119865119896]

120576=0

120576119898 119896 = 1 2 119870 (4)

where

119889

119889120576

=

120597119896119899+1

120597120576

120597

120597119896119899+1

+

119870

sum

119895=1

(

120597119906119895119899+1

120597120576

120597

120597119906119895119899+1

) +

120597

120597120576

(5)

is defined for the 119899 + 1th iterative equation

119865119896 (119896119899+1 119906119895119899+1 120576 119905) = 0 (6)

Substituting (5) into (4) one obtains an iteration equation

119865119896 =

119872

sum

119898=0

1

119898

[

[

(119888

119896119899

120597

120597119896119899+1

+

119870

sum

119895=1

119906119888

119895119899

120597

120597119906119895119899+1

+

120597

120597120576

)

119898

119865119896]

]120576=0

times 120576119898

= 0 119896 = 1 2 119870

(7)

which is a first-order differential equation and can be solvedfor the correction terms 119906

119888

119896119899 Then using (3) the 119899 + 1th

iteration solution can be found Iterations are terminated aftera successful approximation is obtained

Note that for amore general algorithm 119899 correction termsinstead of one can be taken in expansion (3) which wouldthen be a PIA(119899 119898) algorithm The algorithm can also begeneralized to a differential equation system having arbitraryorder of derivatives

3 Applications

Applications of the theory developed will be outlined in thissection A first-order single equation and coupled systemswith two and three equations will be treated

Example 1 The following first-order differential equationarising in the cooling problem of a lumped system [21]

(1 + 120576119906)

119889119906

119889119905

+ 119906 = 0 119906 (0) = 1 (8)

will be treated with PIA(1 1) and PIA(1 2) algorithms Thespecific heat is assumed to be a linear function of temperatureand the equation is cast into nondimensional formas outlinedin [21]

(i) PIA(1 1) Algorithm For the equation considered taking119872 = 1 and119870 = 1 (7) reduces to

119888

1119899+ 119906119888

1119899= minus

1119899 + 1199061119899

120576

minus 11990611198991119899(9)

which is the determining iteration equation for the perturba-tion correction term Assuming an initial solution using then(9) and (3) successive iteration functions can be determined

An initial trial function

11990610 = 119890minus119905 (10)

which satisfies the boundary condition is selected Substitut-ing this trial function into (9) solving for the correction termand using (3) one has

11990611 = (1 + 120576) 119890minus119905

minus 120576119890minus2119905

(11)

Abstract and Applied Analysis 3

The successive iterations are

11990612 = (1 + 120576 +

1205762

2

+

1205763

6

) 119890minus119905

minus 120576 (1 + 2120576 + 1205762) 119890minus2119905

+

3

2

1205762(1 + 120576) 119890

minus3119905minus

2

3

1205763119890minus4119905

11990613 =

119890minus119905

840

(840 + 840120576 + 4201205762+ 140120576

3

+351205764+ 211205765+ 71205766+ 1205767)

minus

119890minus2119905

36

120576(6 + 6120576 + 31205762+ 1205763)

2

+

119890minus3119905

4

1205762(1 + 120576)

2(6 + 6120576 + 3120576

2+ 1205763)

minus

119890minus4119905

3

1205763(8 + 20120576 + 21120576

2+ 121205763+ 31205764)

+

5119890minus5119905

72

1205764(39 + 93120576 + 87120576

2+ 291205763)

minus

43119890minus6119905

20

1205765(1 + 120576)

2+

7119890minus7119905

6

1205766(1 + 120576) minus

16119890minus8119905

63

1205767

(12)

These results are the same with the results of the variationaliteration method given in [21](ii) PIA(1 2) Algorithm A higher iteration algorithm can beconstructed by taking119872 = 2 For this choice (7) reduces to

(1 + 1205761199061119899) 119888

1119899+ (1 + 1205761119899) 119906

119888

1119899= minus

1119899 + 1199061119899

120576

minus 11990611198991119899

(13)

Taking the same initial trial function as given in (10) thesuccessive iterations are

11990611 = 119890minus119905

+

2120576 (119890119905minus 1) + 120576

2(119890119905minus 119890minus119905)

2(119890119905+ 120576)2

11990612 = 119890minus119905

+

2120576 (119890119905minus 1) + 120576

2(119890119905minus 119890minus119905)

2(119890119905+ 120576)2

+

1205763

24(119890119905+ 120576)5119890minus119905(minus1 + 119890

119905)

2

times (minus161198902119905

+ 41198903119905

minus 15119890119905120576 minus 10119890

2119905120576

+1198903119905120576 minus 3120576

2minus 61198901199051205762minus 311989021199051205762)

(14)

In (13) during the iterations the resulting equation comesout to be a variable coefficient system For obtaining the lastiteration due to complexity 11990610 is taken instead of 11990611 in thecoefficients of the left-hand side The third iteration result isnot given here for brevity

Table 1 Comparison of percentage errors of PIA(1 2) with PIA(1 1)and VIM for 120576 = 1

t error for PIA(12) error for PIA(11) and VIM1199061

1199062

1199063

1199061

1199062

1199063

0 000 000 000 000 000 0001 177 019 002 587 171 3152 029 026 004 938 588 1533 398 030 000 1911 266 0634 631 000 003 2355 008 0995 736 021 003 2534 112 0976 778 030 003 2602 161 0937 794 034 002 2628 179 092 meanerror 455 020 002 1786 258 140

The error in Table 1 is defined as

error =

1003816100381610038161003816numerical solution minus approximate solution100381610038161003816

1003816

numerical solution

times 100

(15)

As can be seen from Table 1 PIA(1 2) performs better thanVIM and PIA(1 1)

Example 2 Two coupled stiff systemwill now be consideredSolutionswill be obtained by PIA(1 1) algorithmThe coupledsystem is [22]

1 = minus10021199061 + 10001199062

2

2 = 1199061 minus 1199062 minus 1199062

2

(16)

with the initial conditions

1199061 (0) = 1 1199062 (0) = 1 (17)

for which exact solutions are available as

1199061 = 119890minus2119905

1199062 = 119890minus119905 (18)

An artificial perturbation parameter is inserted as follows

1198651 = 1119899+1 + 10021199061119899+1 minus 12057610001199062

2119899+1= 0

1198652 = 2119899+1 + 1199062119899+1 minus 1199061119899+1 + 1205761199062

2119899+1= 0

(19)

For (19) (7) reduces to

120576119888

1119899+ 1002120576119906

119888

1119899= minus1119899 minus 10021199061119899 + 1205761000119906

2

2119899

120576119888

2119899+ 120576119906119888

2119899= minus2119899 minus 1199062119899 minus 120576119906

2

2119899+ 1199061119899+1

(20)

If the initial trial functions are taken as

11990610 = 1

11990620 = 1

(21)


2 4 6 8 10 12

5

10

15

20

119899 = 4

119899 = 5

119899 = 6

Num Sol

119905

1199061(119905

)

Figure 1 Comparison of iterations (119899 = 4 5 6) of PIA(1 1) andnumerical solutions for 1199061

the successive iterations are

11990611 =

500

501

+

119890minus1002119905

501

11990621 = minus

1

501

minus

119890minus1002119905

501501

+

1003119890minus119905

1001

11990612 =

500

125751501

minus

500119890minus2004119905

126003129753501

+

2006000119890minus1003119905

502002501

+

1006009119890minus2119905

1002001

minus

2006000119890minus119905

502002501

+ 119890minus1002119905

(minus

1007012008

251503253001

+

2000119905

251252001

)

11990622 = minus

1

125751501

+

119890minus2004119905

252132136759503

minus

1003119890minus1003119905

251252001

+ 119890minus1002119905

(

335670670

83918252084667

minus

2000119905

251503253001

)

+ 119890minus119905

(

504264776770742984

504264776776764003

+

2006119905

502002501

)

(22)

These results are identical with the results of the variationaliteration method given in [22]

Example 3 The problem of spreading of a nonfatal disease ina population which is assumed to have constant size over theperiod of the epidemic is considered in [23] The followingsystem determines the progress of the disease

1 = minus12057311990611199062

2 = 12057311990611199062 minus 1205741199062

3 = 1205741199062

1199061 (0) = 20 1199062 (0) = 15 1199063 (0) = 10

(23)

2 4 6 8 10 12

16

18

20

22

24

26

28

30119899 = 5

119899 = 4

119899 = 6

Num Sol

1199062(119905

)

119905


2 4 6 8 10 12

11

12

13

14

15

16

Num Sol

119899 = 6

119899 = 4

119899 = 5

119905

1199063(119905

)


The system is solved using PIA(1 1) Perturbation parameteris artificially introduced as

1198651 = 1119899+1 + 1205731205761199061119899+11199062119899+1 = 0

1198652 = 2119899+1 minus 1205731205761199061119899+11199062119899+1 + 1205741205761199062119899+1 = 0

1198653 = 3119899+1 minus 1205741205761199062119899+1 = 0

(24)

For the previous equations (7) reduces to

1119899 + 120576119888

1119899+ 12057312057611990611198991199062119899 = 0

2119899 + 120576119888

2119899+ 120576 (120574 minus 1205731199061119899) 1199062119899 = 0

3119899 + 120576119888

3119899minus 1205741205761199062119899 = 0

(25)

The initial trial functions are

11990610 = 20

11990620 = 15

11990630 = 10

(26)


The following iteration results are obtained for 120573 = 1100 and120574 = 150

11990611 = 20 minus 3119905

11990621 = 15 + 27119905

11990631 = 10 + 03119905

11990612 = 20 minus 3119905 minus 00451199052+ 0027119905

3

11990622 = 15 + 27119905 + 00181199052minus 0027119905

3

11990632 = 10 + 03119905 + 00271199052

11990613 = 20 minus 3119905 minus

9

2

times 10minus21199052+

561

2

times 10minus41199053

minus

621

8

times 10minus51199054minus

15309

5


minus

567

2


729

7


11990623 = 15 + 27 times 10minus2119905 +

9

5


minus 2817 times 10minus51199053minus

513

8


15309

5


567

2


729

7


11990633 = 10 + 3 times 10minus1119905 + 27 times 10

minus31199052

+

3

25


27

2


11990614 = 20 minus 3119905 minus

9

2


561

2


minus

6363

8


126603

4


97641

2


9913563

28

times 10minus11

1199057

+

112714011

112

times 10minus13

1199058minus

1398332889

56

times 10minus15

1199059

minus

13180077

2

times 10minus16

11990510

+

943578963

7

times 10minus18

11990511

+

334611

125

times 10minus15

11990512

minus

25187679

52

times 10minus18

11990513

+

59049

14

times 10minus18

11990514

+

177147

245

times 10minus18

11990515

11990624 = 15 +

27

10

119905 +

9

500

1199052minus 2817 times 10

minus51199053

minus

26181

4


127629

4


+

447381

4

times 10minus10

1199056minus

9936243

28

times 10minus11

1199057

minus

102798011

112

times 10minus13

1199058+

1398332889

56

times 10minus15

1199059

+

13180077

2

times 10minus16

11990510

minus

943578963

7

times 10minus18

11990511

minus

334611

125

times 10minus15

11990512

+

25187679

52

times 10minus18

11990513

+

59049

14

times 10minus18

11990514

minus

177147

245

times 10minus18

11990515

11990634 = 10 +

3

10

119905 +

27

1000

1199052+

3

25000

1199053minus

2817

2


minus

513

5


5103

5


+ 81 times 10minus10

1199057minus

729

28

times 10minus10

1199058

(27)

The previous solutions are the same as those obtained fromvariational iteration method [23]

Functions 1199061minus3 are plotted in Figures 1 2 and 3 Thehigher iterations that is 119899 = 4 5 6 calculated by symbolicprograms are compared with the numerical solutions Asthe number of iterations increase the approximate analyticalsolutions converge to the numerical solutions

4 Concluding Remarks

The newly developed perturbation-iteration algorithm isapplied to systems of equations for the first timeThe theory isdeveloped first and then applied to three different problemsBased on this study and on the previous work [17] one canconclude that while PIA(1 1) algorithm produces compatibleresults with the VIM method PIA(1 2) produces betterresults than the PIA(1 1) and the VIM

References

[1] A H Nayfeh Perturbation Methods Wiley-Interscience NewYork NY USA 1973

[2] A V Skorokhod F C Hoppensteadt and H Salehi RandomPerturbationMethods with Applications in Science and Engineer-ing Springer New York NY USA 2002

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

[4] R E Mickens ldquoIteration procedure for determining approx-imate solutions to nonlinear oscillator equationsrdquo Journal ofSound and Vibration vol 116 no 1 pp 185ndash187 1987

[5] R EMickens ldquoA generalized iteration procedure for calculatingapproximations to periodic solutions of lsquotruly nonlinear oscilla-torsrsquordquo Journal of Sound and Vibration vol 287 no 4-5 pp 1045ndash1052 2005

[6] R E Mickens ldquoIterationmethod solutions for conservative andlimit-cycle force oscillatorsrdquo Journal of Sound and Vibrationvol 292 no 3ndash5 pp 964ndash968 2006

[7] K Cooper and R E Mickens ldquoGeneralized harmonic bal-ancenumerical method for determining analytical approxima-tions to the periodic solutions of the potentialrdquo Journal of Soundand Vibration vol 250 no 5 pp 951ndash954 2002


[8] H Hu and Z-G Xiong ldquoOscillations in an 119909(2119898+1)(2119899+1) poten-

tialrdquo Journal of Sound andVibration vol 259 no 4 pp 977ndash9802003

[9] M A Abdou ldquoOn the variational iteration methodrdquo PhysicsLetters A vol 366 no 1-2 pp 61ndash68 2007

[10] S Q Wang and J H He ldquoNonlinear oscillator with discon-tinuity by parameter-expansion methodrdquo Chaos Solitons andFractals vol 35 no 4 pp 688ndash691 2008

[11] J-H He ldquoModified Lindstedt-Poincare methods for somestrongly non-linear oscillations part I expansion of a constantrdquoInternational Journal of Non-Linear Mechanics vol 37 no 2 pp309ndash314 2002

[12] J I Ramos ldquoOn Linstedt-Poincare technique for the quinticDuffing equationrdquo Applied Mathematics and Computation vol193 no 2 pp 303ndash310 2007

[13] T Ozis and A Yıldırım ldquoDetermination of periodic solutionfor a 119906

13 force by Hersquos modified Lindstedt-Poincare methodrdquoJournal of Sound and Vibration vol 301 no 1-2 pp 415ndash4192007

[14] R E Mickens ldquoOscillations in an 11990941199093 potentialrdquo Journal of

Sound and Vibration vol 246 no 2 pp 375ndash378 2001[15] M Pakdemirli Y Aksoy and H Boyacı ldquoA new perturbation-

iteration approach for first order differential equationsrdquoMathe-matical and Computational Applications vol 16 no 4 pp 890ndash899 2011

[16] Y Aksoy andM Pakdemirli ldquoNew perturbation-iteration solu-tions for Bratu-type equationsrdquo Computers and Mathematicswith Applications vol 59 no 8 pp 2802ndash2808 2010

[17] Y Aksoy M Pakdemirli and S Abbasbandy ldquoNew pertur-bation-iteration solutions for nonlinear heat transfer equa-tionsrdquo International Journal of Numerical Methods for Heat andFluid Flow vol 22 no 7 pp 814ndash828 2012

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

[19] M Pakdemirli H Boyacı and H A Yurtsever ldquoPerturbativederivation and comparisons of root-finding algorithms withfourth order derivativesrdquo Mathematical and ComputationalApplications vol 12 no 2 pp 117ndash124 2007

[20] M Pakdemirli H Boyacı and H A Yurtsever ldquoA root-finding algorithm with fifth order derivativesrdquo MathematicalandComputational Applications vol 13 no 2 pp 123ndash128 2008

[21] H Tari D D Ganji and H Babazadeh ldquoThe application of Hersquosvariational iteration method to nonlinear equations arising inheat transferrdquo Physics Letters A vol 363 no 3 pp 213ndash217 2007

[22] M T Darvishi F Khani and A A Soliman ldquoThe numericalsimulation for stiff systems of ordinary differential equationsrdquoComputers and Mathematics with Applications vol 54 no 7-8pp 1055ndash1063 2007

[23] M Rafei H Daniali and D D Ganji ldquoVariational interationmethod for solving the epidemic model and the prey andpredator problemrdquo Applied Mathematics and Computation vol186 no 2 pp 1701ndash1709 2007

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Stochastic AnalysisInternational Journal of


equation which is a degenerate case is treated first Thencoupled systems with two and three equations are solvedSolutions are contrasted with available other approximatesolutions or numerical solutions It is found that the methodcan be effectively applied to differential equation systems aswell

2 Perturbation-Iteration Algorithm PIA(1 119898)

In this section a perturbation-iteration algorithm PIA(1 119898)is constructed by taking one correction term in the perturba-tion expansion and correction terms of119898th-order derivativesin the Taylor series expansion

Consider the following system of first-order differentialequations

119865119896 (119896 119906119895 120576 119905) = 0 119896 = 1 2 119870 119895 = 1 2 119870 (1)

where 119870 represents the number of differential equations inthe system and the number of dependent variables119870 = 1 fora single equation In the open form the system of equationsis

1198651 = 1198651 (1 1199061 1199062 119906119870 120576 119905) = 0

1198652 = 1198652 (2 1199061 1199062 119906119870 120576 119905) = 0

119865119870 = 119865119870 (119870 1199061 1199062 119906119870 120576 119905) = 0

(2)

Assume an approximate solution of the system

119906119896119899+1 = 119906119896119899 + 120576119906119888

119896119899 (3)

with one correction term in the perturbation expansion Thesubscript 119899 represents the 119899th iteration over this approximatesolutionThe system can be approximatedwith a Taylor seriesexpansion in the neighborhood of 120576 = 0 as

119865119896 =

119872

sum

119898=0

1

119898

[(

119889

119889120576

)

119898

119865119896]

120576=0

120576119898 119896 = 1 2 119870 (4)

where

119889

119889120576

=

120597119896119899+1

120597120576

120597

120597119896119899+1

+

119870

sum

119895=1

(

120597119906119895119899+1

120597120576

120597

120597119906119895119899+1

) +

120597

120597120576

(5)

is defined for the 119899 + 1th iterative equation

119865119896 (119896119899+1 119906119895119899+1 120576 119905) = 0 (6)

Substituting (5) into (4) one obtains an iteration equation

119865119896 =

119872

sum

119898=0

1

119898

[

[

(119888

119896119899

120597

120597119896119899+1

+

119870

sum

119895=1

119906119888

119895119899

120597

120597119906119895119899+1

+

120597

120597120576

)

119898

119865119896]

]120576=0

times 120576119898

= 0 119896 = 1 2 119870

(7)

which is a first-order differential equation and can be solvedfor the correction terms 119906

119888

119896119899 Then using (3) the 119899 + 1th

iteration solution can be found Iterations are terminated aftera successful approximation is obtained

Note that for amore general algorithm 119899 correction termsinstead of one can be taken in expansion (3) which wouldthen be a PIA(119899 119898) algorithm The algorithm can also begeneralized to a differential equation system having arbitraryorder of derivatives

3 Applications

Applications of the theory developed will be outlined in thissection A first-order single equation and coupled systemswith two and three equations will be treated

Example 1 The following first-order differential equationarising in the cooling problem of a lumped system [21]

(1 + 120576119906)

119889119906

119889119905

+ 119906 = 0 119906 (0) = 1 (8)

will be treated with PIA(1 1) and PIA(1 2) algorithms Thespecific heat is assumed to be a linear function of temperatureand the equation is cast into nondimensional formas outlinedin [21]

(i) PIA(1 1) Algorithm For the equation considered taking119872 = 1 and119870 = 1 (7) reduces to

119888

1119899+ 119906119888

1119899= minus

1119899 + 1199061119899

120576

minus 11990611198991119899(9)

which is the determining iteration equation for the perturba-tion correction term Assuming an initial solution using then(9) and (3) successive iteration functions can be determined

An initial trial function

11990610 = 119890minus119905 (10)

which satisfies the boundary condition is selected Substitut-ing this trial function into (9) solving for the correction termand using (3) one has

11990611 = (1 + 120576) 119890minus119905

minus 120576119890minus2119905

(11)



11990612 = (1 + 120576 +

1205762

2

+

1205763

6

) 119890minus119905

minus 120576 (1 + 2120576 + 1205762) 119890minus2119905

+

3

2

1205762(1 + 120576) 119890

minus3119905minus

2

3

1205763119890minus4119905

11990613 =

119890minus119905

840

(840 + 840120576 + 4201205762+ 140120576

3

+351205764+ 211205765+ 71205766+ 1205767)

minus

119890minus2119905

36

120576(6 + 6120576 + 31205762+ 1205763)

2

+

119890minus3119905

4

1205762(1 + 120576)

2(6 + 6120576 + 3120576

2+ 1205763)

minus

119890minus4119905

3

1205763(8 + 20120576 + 21120576

2+ 121205763+ 31205764)

+

5119890minus5119905

72

1205764(39 + 93120576 + 87120576

2+ 291205763)

minus

43119890minus6119905

20

1205765(1 + 120576)

2+

7119890minus7119905

6

1205766(1 + 120576) minus

16119890minus8119905

63

1205767

(12)


(1 + 1205761199061119899) 119888

1119899+ (1 + 1205761119899) 119906

119888

1119899= minus

1119899 + 1199061119899

120576

minus 11990611198991119899

(13)


11990611 = 119890minus119905

+

2120576 (119890119905minus 1) + 120576

2(119890119905minus 119890minus119905)

2(119890119905+ 120576)2

11990612 = 119890minus119905

+

2120576 (119890119905minus 1) + 120576

2(119890119905minus 119890minus119905)

2(119890119905+ 120576)2

+

1205763

24(119890119905+ 120576)5119890minus119905(minus1 + 119890

119905)

2

times (minus161198902119905

+ 41198903119905

minus 15119890119905120576 minus 10119890

2119905120576

+1198903119905120576 minus 3120576

2minus 61198901199051205762minus 311989021199051205762)

(14)




1199062

1199063

1199061

1199062

1199063

0 000 000 000 000 000 0001 177 019 002 587 171 3152 029 026 004 938 588 1533 398 030 000 1911 266 0634 631 000 003 2355 008 0995 736 021 003 2534 112 0976 778 030 003 2602 161 0937 794 034 002 2628 179 092 meanerror 455 020 002 1786 258 140


error =


1003816

numerical solution

times 100

(15)



1 = minus10021199061 + 10001199062

2

2 = 1199061 minus 1199062 minus 1199062

2

(16)


1199061 (0) = 1 1199062 (0) = 1 (17)


1199061 = 119890minus2119905

1199062 = 119890minus119905 (18)


1198651 = 1119899+1 + 10021199061119899+1 minus 12057610001199062

2119899+1= 0

1198652 = 2119899+1 + 1199062119899+1 minus 1199061119899+1 + 1205761199062

2119899+1= 0

(19)


120576119888

1119899+ 1002120576119906

119888

1119899= minus1119899 minus 10021199061119899 + 1205761000119906

2

2119899

120576119888

2119899+ 120576119906119888


2

2119899+ 1199061119899+1

(20)


11990610 = 1

11990620 = 1

(21)


2 4 6 8 10 12

5

10

15

20

119899 = 4

119899 = 5

119899 = 6

Num Sol

119905

1199061(119905

)



11990611 =

500

501

+

119890minus1002119905

501

11990621 = minus

1

501

minus

119890minus1002119905

501501

+

1003119890minus119905

1001

11990612 =

500

125751501

minus

500119890minus2004119905

126003129753501

+

2006000119890minus1003119905

502002501

+

1006009119890minus2119905

1002001

minus

2006000119890minus119905

502002501

+ 119890minus1002119905

(minus

1007012008

251503253001

+

2000119905

251252001

)

11990622 = minus

1

125751501

+

119890minus2004119905

252132136759503

minus

1003119890minus1003119905

251252001

+ 119890minus1002119905

(

335670670

83918252084667

minus

2000119905

251503253001

)

+ 119890minus119905

(

504264776770742984

504264776776764003

+

2006119905

502002501

)

(22)



1 = minus12057311990611199062

2 = 12057311990611199062 minus 1205741199062

3 = 1205741199062

1199061 (0) = 20 1199062 (0) = 15 1199063 (0) = 10

(23)

2 4 6 8 10 12

16

18

20

22

24

26

28

30119899 = 5

119899 = 4

119899 = 6

Num Sol

1199062(119905

)

119905


2 4 6 8 10 12

11

12

13

14

15

16

Num Sol

119899 = 6

119899 = 4

119899 = 5

119905

1199063(119905

)



1198651 = 1119899+1 + 1205731205761199061119899+11199062119899+1 = 0

1198652 = 2119899+1 minus 1205731205761199061119899+11199062119899+1 + 1205741205761199062119899+1 = 0

1198653 = 3119899+1 minus 1205741205761199062119899+1 = 0

(24)


1119899 + 120576119888

1119899+ 12057312057611990611198991199062119899 = 0

2119899 + 120576119888

2119899+ 120576 (120574 minus 1205731199061119899) 1199062119899 = 0

3119899 + 120576119888

3119899minus 1205741205761199062119899 = 0

(25)


11990610 = 20

11990620 = 15

11990630 = 10

(26)



11990611 = 20 minus 3119905

11990621 = 15 + 27119905

11990631 = 10 + 03119905

11990612 = 20 minus 3119905 minus 00451199052+ 0027119905

3

11990622 = 15 + 27119905 + 00181199052minus 0027119905

3

11990632 = 10 + 03119905 + 00271199052

11990613 = 20 minus 3119905 minus

9

2


561

2


minus

621

8


15309

5


minus

567

2


729

7


11990623 = 15 + 27 times 10minus2119905 +

9

5



513

8


15309

5


567

2


729

7



minus31199052

+

3

25


27

2


11990614 = 20 minus 3119905 minus

9

2


561

2


minus

6363

8


126603

4


97641

2


9913563

28

times 10minus11

1199057

+

112714011

112

times 10minus13

1199058minus

1398332889

56

times 10minus15

1199059

minus

13180077

2

times 10minus16

11990510

+

943578963

7

times 10minus18

11990511

+

334611

125

times 10minus15

11990512

minus

25187679

52

times 10minus18

11990513

+

59049

14

times 10minus18

11990514

+

177147

245

times 10minus18

11990515

11990624 = 15 +

27

10

119905 +

9

500

1199052minus 2817 times 10

minus51199053

minus

26181

4


127629

4


+

447381

4

times 10minus10

1199056minus

9936243

28

times 10minus11

1199057

minus

102798011

112

times 10minus13

1199058+

1398332889

56

times 10minus15

1199059

+

13180077

2

times 10minus16

11990510

minus

943578963

7

times 10minus18

11990511

minus

334611

125

times 10minus15

11990512

+

25187679

52

times 10minus18

11990513

+

59049

14

times 10minus18

11990514

minus

177147

245

times 10minus18

11990515

11990634 = 10 +

3

10

119905 +

27

1000

1199052+

3

25000

1199053minus

2817

2


minus

513

5


5103

5



1199057minus

729

28

times 10minus10

1199058

(27)





References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











11990612 = (1 + 120576 +

1205762

2

+

1205763

6

) 119890minus119905

minus 120576 (1 + 2120576 + 1205762) 119890minus2119905

+

3

2

1205762(1 + 120576) 119890

minus3119905minus

2

3

1205763119890minus4119905

11990613 =

119890minus119905

840

(840 + 840120576 + 4201205762+ 140120576

3

+351205764+ 211205765+ 71205766+ 1205767)

minus

119890minus2119905

36

120576(6 + 6120576 + 31205762+ 1205763)

2

+

119890minus3119905

4

1205762(1 + 120576)

2(6 + 6120576 + 3120576

2+ 1205763)

minus

119890minus4119905

3

1205763(8 + 20120576 + 21120576

2+ 121205763+ 31205764)

+

5119890minus5119905

72

1205764(39 + 93120576 + 87120576

2+ 291205763)

minus

43119890minus6119905

20

1205765(1 + 120576)

2+

7119890minus7119905

6

1205766(1 + 120576) minus

16119890minus8119905

63

1205767

(12)


(1 + 1205761199061119899) 119888

1119899+ (1 + 1205761119899) 119906

119888

1119899= minus

1119899 + 1199061119899

120576

minus 11990611198991119899

(13)


11990611 = 119890minus119905

+

2120576 (119890119905minus 1) + 120576

2(119890119905minus 119890minus119905)

2(119890119905+ 120576)2

11990612 = 119890minus119905

+

2120576 (119890119905minus 1) + 120576

2(119890119905minus 119890minus119905)

2(119890119905+ 120576)2

+

1205763

24(119890119905+ 120576)5119890minus119905(minus1 + 119890

119905)

2

times (minus161198902119905

+ 41198903119905

minus 15119890119905120576 minus 10119890

2119905120576

+1198903119905120576 minus 3120576

2minus 61198901199051205762minus 311989021199051205762)

(14)




1199062

1199063

1199061

1199062

1199063

0 000 000 000 000 000 0001 177 019 002 587 171 3152 029 026 004 938 588 1533 398 030 000 1911 266 0634 631 000 003 2355 008 0995 736 021 003 2534 112 0976 778 030 003 2602 161 0937 794 034 002 2628 179 092 meanerror 455 020 002 1786 258 140


error =


1003816

numerical solution

times 100

(15)



1 = minus10021199061 + 10001199062

2

2 = 1199061 minus 1199062 minus 1199062

2

(16)


1199061 (0) = 1 1199062 (0) = 1 (17)


1199061 = 119890minus2119905

1199062 = 119890minus119905 (18)


1198651 = 1119899+1 + 10021199061119899+1 minus 12057610001199062

2119899+1= 0

1198652 = 2119899+1 + 1199062119899+1 minus 1199061119899+1 + 1205761199062

2119899+1= 0

(19)


120576119888

1119899+ 1002120576119906

119888

1119899= minus1119899 minus 10021199061119899 + 1205761000119906

2

2119899

120576119888

2119899+ 120576119906119888


2

2119899+ 1199061119899+1

(20)


11990610 = 1

11990620 = 1

(21)


2 4 6 8 10 12

5

10

15

20

119899 = 4

119899 = 5

119899 = 6

Num Sol

119905

1199061(119905

)



11990611 =

500

501

+

119890minus1002119905

501

11990621 = minus

1

501

minus

119890minus1002119905

501501

+

1003119890minus119905

1001

11990612 =

500

125751501

minus

500119890minus2004119905

126003129753501

+

2006000119890minus1003119905

502002501

+

1006009119890minus2119905

1002001

minus

2006000119890minus119905

502002501

+ 119890minus1002119905

(minus

1007012008

251503253001

+

2000119905

251252001

)

11990622 = minus

1

125751501

+

119890minus2004119905

252132136759503

minus

1003119890minus1003119905

251252001

+ 119890minus1002119905

(

335670670

83918252084667

minus

2000119905

251503253001

)

+ 119890minus119905

(

504264776770742984

504264776776764003

+

2006119905

502002501

)

(22)



1 = minus12057311990611199062

2 = 12057311990611199062 minus 1205741199062

3 = 1205741199062

1199061 (0) = 20 1199062 (0) = 15 1199063 (0) = 10

(23)

2 4 6 8 10 12

16

18

20

22

24

26

28

30119899 = 5

119899 = 4

119899 = 6

Num Sol

1199062(119905

)

119905


2 4 6 8 10 12

11

12

13

14

15

16

Num Sol

119899 = 6

119899 = 4

119899 = 5

119905

1199063(119905

)



1198651 = 1119899+1 + 1205731205761199061119899+11199062119899+1 = 0

1198652 = 2119899+1 minus 1205731205761199061119899+11199062119899+1 + 1205741205761199062119899+1 = 0

1198653 = 3119899+1 minus 1205741205761199062119899+1 = 0

(24)


1119899 + 120576119888

1119899+ 12057312057611990611198991199062119899 = 0

2119899 + 120576119888

2119899+ 120576 (120574 minus 1205731199061119899) 1199062119899 = 0

3119899 + 120576119888

3119899minus 1205741205761199062119899 = 0

(25)


11990610 = 20

11990620 = 15

11990630 = 10

(26)



11990611 = 20 minus 3119905

11990621 = 15 + 27119905

11990631 = 10 + 03119905

11990612 = 20 minus 3119905 minus 00451199052+ 0027119905

3

11990622 = 15 + 27119905 + 00181199052minus 0027119905

3

11990632 = 10 + 03119905 + 00271199052

11990613 = 20 minus 3119905 minus

9

2


561

2


minus

621

8


15309

5


minus

567

2


729

7


11990623 = 15 + 27 times 10minus2119905 +

9

5



513

8


15309

5


567

2


729

7



minus31199052

+

3

25


27

2


11990614 = 20 minus 3119905 minus

9

2


561

2


minus

6363

8


126603

4


97641

2


9913563

28

times 10minus11

1199057

+

112714011

112

times 10minus13

1199058minus

1398332889

56

times 10minus15

1199059

minus

13180077

2

times 10minus16

11990510

+

943578963

7

times 10minus18

11990511

+

334611

125

times 10minus15

11990512

minus

25187679

52

times 10minus18

11990513

+

59049

14

times 10minus18

11990514

+

177147

245

times 10minus18

11990515

11990624 = 15 +

27

10

119905 +

9

500

1199052minus 2817 times 10

minus51199053

minus

26181

4


127629

4


+

447381

4

times 10minus10

1199056minus

9936243

28

times 10minus11

1199057

minus

102798011

112

times 10minus13

1199058+

1398332889

56

times 10minus15

1199059

+

13180077

2

times 10minus16

11990510

minus

943578963

7

times 10minus18

11990511

minus

334611

125

times 10minus15

11990512

+

25187679

52

times 10minus18

11990513

+

59049

14

times 10minus18

11990514

minus

177147

245

times 10minus18

11990515

11990634 = 10 +

3

10

119905 +

27

1000

1199052+

3

25000

1199053minus

2817

2


minus

513

5


5103

5



1199057minus

729

28

times 10minus10

1199058

(27)





References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










2 4 6 8 10 12

5

10

15

20

119899 = 4

119899 = 5

119899 = 6

Num Sol

119905

1199061(119905

)



11990611 =

500

501

+

119890minus1002119905

501

11990621 = minus

1

501

minus

119890minus1002119905

501501

+

1003119890minus119905

1001

11990612 =

500

125751501

minus

500119890minus2004119905

126003129753501

+

2006000119890minus1003119905

502002501

+

1006009119890minus2119905

1002001

minus

2006000119890minus119905

502002501

+ 119890minus1002119905

(minus

1007012008

251503253001

+

2000119905

251252001

)

11990622 = minus

1

125751501

+

119890minus2004119905

252132136759503

minus

1003119890minus1003119905

251252001

+ 119890minus1002119905

(

335670670

83918252084667

minus

2000119905

251503253001

)

+ 119890minus119905

(

504264776770742984

504264776776764003

+

2006119905

502002501

)

(22)



1 = minus12057311990611199062

2 = 12057311990611199062 minus 1205741199062

3 = 1205741199062

1199061 (0) = 20 1199062 (0) = 15 1199063 (0) = 10

(23)

2 4 6 8 10 12

16

18

20

22

24

26

28

30119899 = 5

119899 = 4

119899 = 6

Num Sol

1199062(119905

)

119905


2 4 6 8 10 12

11

12

13

14

15

16

Num Sol

119899 = 6

119899 = 4

119899 = 5

119905

1199063(119905

)



1198651 = 1119899+1 + 1205731205761199061119899+11199062119899+1 = 0

1198652 = 2119899+1 minus 1205731205761199061119899+11199062119899+1 + 1205741205761199062119899+1 = 0

1198653 = 3119899+1 minus 1205741205761199062119899+1 = 0

(24)


1119899 + 120576119888

1119899+ 12057312057611990611198991199062119899 = 0

2119899 + 120576119888

2119899+ 120576 (120574 minus 1205731199061119899) 1199062119899 = 0

3119899 + 120576119888

3119899minus 1205741205761199062119899 = 0

(25)


11990610 = 20

11990620 = 15

11990630 = 10

(26)



11990611 = 20 minus 3119905

11990621 = 15 + 27119905

11990631 = 10 + 03119905

11990612 = 20 minus 3119905 minus 00451199052+ 0027119905

3

11990622 = 15 + 27119905 + 00181199052minus 0027119905

3

11990632 = 10 + 03119905 + 00271199052

11990613 = 20 minus 3119905 minus

9

2


561

2


minus

621

8


15309

5


minus

567

2


729

7


11990623 = 15 + 27 times 10minus2119905 +

9

5



513

8


15309

5


567

2


729

7



minus31199052

+

3

25


27

2


11990614 = 20 minus 3119905 minus

9

2


561

2


minus

6363

8


126603

4


97641

2


9913563

28

times 10minus11

1199057

+

112714011

112

times 10minus13

1199058minus

1398332889

56

times 10minus15

1199059

minus

13180077

2

times 10minus16

11990510

+

943578963

7

times 10minus18

11990511

+

334611

125

times 10minus15

11990512

minus

25187679

52

times 10minus18

11990513

+

59049

14

times 10minus18

11990514

+

177147

245

times 10minus18

11990515

11990624 = 15 +

27

10

119905 +

9

500

1199052minus 2817 times 10

minus51199053

minus

26181

4


127629

4


+

447381

4

times 10minus10

1199056minus

9936243

28

times 10minus11

1199057

minus

102798011

112

times 10minus13

1199058+

1398332889

56

times 10minus15

1199059

+

13180077

2

times 10minus16

11990510

minus

943578963

7

times 10minus18

11990511

minus

334611

125

times 10minus15

11990512

+

25187679

52

times 10minus18

11990513

+

59049

14

times 10minus18

11990514

minus

177147

245

times 10minus18

11990515

11990634 = 10 +

3

10

119905 +

27

1000

1199052+

3

25000

1199053minus

2817

2


minus

513

5


5103

5



1199057minus

729

28

times 10minus10

1199058

(27)





References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











11990611 = 20 minus 3119905

11990621 = 15 + 27119905

11990631 = 10 + 03119905

11990612 = 20 minus 3119905 minus 00451199052+ 0027119905

3

11990622 = 15 + 27119905 + 00181199052minus 0027119905

3

11990632 = 10 + 03119905 + 00271199052

11990613 = 20 minus 3119905 minus

9

2


561

2


minus

621

8


15309

5


minus

567

2


729

7


11990623 = 15 + 27 times 10minus2119905 +

9

5



513

8


15309

5


567

2


729

7



minus31199052

+

3

25


27

2


11990614 = 20 minus 3119905 minus

9

2


561

2


minus

6363

8


126603

4


97641

2


9913563

28

times 10minus11

1199057

+

112714011

112

times 10minus13

1199058minus

1398332889

56

times 10minus15

1199059

minus

13180077

2

times 10minus16

11990510

+

943578963

7

times 10minus18

11990511

+

334611

125

times 10minus15

11990512

minus

25187679

52

times 10minus18

11990513

+

59049

14

times 10minus18

11990514

+

177147

245

times 10minus18

11990515

11990624 = 15 +

27

10

119905 +

9

500

1199052minus 2817 times 10

minus51199053

minus

26181

4


127629

4


+

447381

4

times 10minus10

1199056minus

9936243

28

times 10minus11

1199057

minus

102798011

112

times 10minus13

1199058+

1398332889

56

times 10minus15

1199059

+

13180077

2

times 10minus16

11990510

minus

943578963

7

times 10minus18

11990511

minus

334611

125

times 10minus15

11990512

+

25187679

52

times 10minus18

11990513

+

59049

14

times 10minus18

11990514

minus

177147

245

times 10minus18

11990515

11990634 = 10 +

3

10

119905 +

27

1000

1199052+

3

25000

1199053minus

2817

2


minus

513

5


5103

5



1199057minus

729

28

times 10minus10

1199058

(27)





References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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