research article the solution of fourth order boundary...

Research ArticleThe Solution of Fourth Order BoundaryValue Problem Arising out of the Beam-ColumnTheory Using Adomian Decomposition Method

Omer Kelesoglu

Department of Civil Engineering Technology Faculty Firat University 23119 Elazig Turkey

Correspondence should be addressed to Omer Kelesoglu okelesoglufiratedutr

Received 23 November 2013 Accepted 2 March 2014 Published 30 March 2014

Academic Editor Sergii V Kavun

Copyright copy 2014 Omer Kelesoglu 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

Adomian decomposition method (ADM) is applied to linear nonhomogeneous boundary value problem arising from the beam-column theory The obtained results are expressed in tables and graphs We obtain rapidly converging results to exact solution byusing the ADMThis situation indicates that the method is appropriate and reliable for such problems

1 Introduction

It is possible to model many of the physical events thattake place in nature using linear and nonlinear differentialequations This modeling enables us to understand andinterpret the particular event in a much better manner Thusfinding the analytical and approximate solutions of suchmodels with initial and boundary conditions gain impor-tance Differential equations have had an important placein engineering since many years Scientists and engineersgenerally examine systems that undergo changes

Many methods have been developed to determine theanalytical and approximate solutions of linear and nonlineardifferential equations with initial and boundary value condi-tions and among these methods the ADM [1ndash6] homotopyperturbation method [7ndash11] variational iteration method[12ndash19] and homotopy analysismethod [20ndash25] can be listedPackage software such as Matematica Maple or Matlab isused to overcome tedious algebraic operations when usingthese methods The determination of the analytical andapproximate solutions of linear and nonlinear differentialequations is an important topic for civil engineering becausethese equations are the mathematical models of complexevents that occur in engineering

In this study the approximate solution of the fourth orderlinear nonhomogeneous differential equation with initial

and boundary conditions that arises in the beam-columntheory will be determined via the ADMand comparisons willbe made with the existing results in the literature

The ADMwas first put forth in the 1980s by an Americanscientist named Adomian [26] This method is based on thedecomposition of the unknown function Using this methodit is possible to determine the approximate solutions forthe linear and nonlinear ordinary and partial differentialequationsThe ADM has been used effectively by researchersduring 1990 and 2007 especially for the solution of differentialand integral equations [27ndash32] Using this method a nonlin-ear problem can be applied directly without discretizationand linearization Hence this is a method that is preferredby researchers

In this study the analytical and approximate solutions ofa fourth order linear boundary value problemwere calculatedusing the ADM In addition the fourth order linear ordinarydifferential equation set used in the beam-column theorywas solved under specific boundary conditions The ADMused provides realistic solutions without changing the linearor nonlinear differential equation model Numerical resultsclose to the real solution can be found by calculating the termsof the limited number of decomposition series It is possibleto find the numerical solution of a differential equation usingthis method without the necessity of indexing

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 649471 6 pageshttpdxdoiorg1011552014649471

2 Mathematical Problems in Engineering

2 Adomian Decomposition Method

The method has many advantages in comparison with manyother traditional methods such as Finite differences Finiteelements and Galerkin method [33ndash35] The method givesconverging results adapted to the problems It is a disadvan-tage that convergence interval is small and the calculation ofthe ADM polynomials occurs in nonlinear problems in theADM

We now consider second order nonlinear ordinary differ-ential equations with initial conditions This equation can bewritten in an operator by

119871119906 + 119877119906 + 119873119906 = 119891 (1)

where 119871 is the lower-order derivative which is assumed to beinvertible 119877 is a linear differential operator of order greaterthan119871119873 is a nonlinear differential operator and119891 is a sourceterm We next apply the inverse operator 119871minus1 to both sides of(1) and use the given initial conditions to get

119906 (119909) = 119906 (0) + 1199091199061015840

(0) + 119871minus1

(119892) minus 119871minus1

(119877119906) minus 119871minus1

(119873119906)

(2)

where

119871 =1198892

1198891199092 119871

minus1

= int

119909

0

int

119909

0

119889119909 119889119909 (3)

The ADM consists in decomposing the unknown func-tion 119906(119909) of any equation into a sum of infinite number ofcomponents given by the decomposition series

119906 (119909) =

infin

sum

119899=0

119906119899

(119909) (4)

where the components 119906119899

(119909) 119899 ge 0 are to be determinedin a recursive manner The nonlinear term 119873119906 will bedecomposed by the infinite series of polynomials given by

119873119906 =

infin

sum

119899=0

119860119899

(5)

where 119860119899

are Adomian polynomials [1 36] Substituting (3)and (4) into (2) gives

infin

sum

119899=0

119906119899

= 1199060

(119909) minus 119871minus1

119877(

infin

sum

119899=0

119906119899

(119909) minus 119871minus1

infin

sum

119899=0

119860119899

) (6)

where

1199060

(119909) = 119906 (0) + 1199091199061015840

(0) + 119871minus1

(119892) (7)

The various components 119906119899

of the solution 119906 can be easilydetermined by using the recursive relation

1199060

(119909) = 119906 (0) + 1199091199061015840

(0) + 119871minus1

119892 (119909)

119906119896+1

(119909) = minus 119871minus1

(119877119906119896

) minus 119871minus1

(119860119896

) 119896 ge 0

(8)

In order to obtain the numerical solutions of 119906(119909) closedsolution function using the decomposition method

120593119899

=

119899

sum

119896=0

119906119896

(119909) 119896 ge 0 (9)

being the term

119906 (119909) = lim120593119899

119899rarrinfin

(10)

can be calculated by taking into account the reduction for-mula (8) In addition the series solution of the decompositionwritten as (10) generally yields results that rapidly convergefor physical problems The convergence of the decomposi-tion series has been examined by many researchers in theliterature The convergence of the decomposition series hasbeen examined theoretically by Cherruault [37] In additionto these studies Abbaoui and Cherruault have suggesteda new approach in determining the convergence of thedecomposition series [38] These authors have determinedthe convergence of the decomposition series method bygiving new conditions

3 The Application of the ADM to the LinearProblem with Boundary Condition ThatArises in Beam-Column Theory

31 Problem Definition Fourth order differential equationsconsist of various physical problems that are related tothe elastic stability theory Differential relations should beestablished between the effects of various cross-section effectsin order to understand beam-column problems better

When a cross section of distance 119889119909 shown in Figure 1(b)is taken from a beam-column subject to both the 119875 axial loadand the 119902 spread load perpendicular to the axis as shownin Figure 1(a) internal forces arise in the element When theequilibrium equation in the 119910 direction is written with

119902 = minus119889119881

119889119909 (11)

the following ordinary differential equation is found [39ndash42]

minus119881 + 119902119889119909 + (119881 + 119889119881) = 0 (12)

The algebraic sums of the forces acting on both surfacesof the cross-section element are the same due to equilibriumConsider

119872 + 119902119889119909119889119909

2+ (119881 + 119889119881) 119889119909 minus (119872 + 119889119872) + 119875

119889119910

119889119909= 0

(13)

Here 119881 is the shear force acting on the surface of theelement whereas119872 is the bendingmoment that tries to bendthe cross-section element

If rotations are assumed to be small and the second orderterms in terms of 119889119909 are neglected then (13) becomes

119881 =119889119872

119889119909minus 119875

119889119910

119889119909 (14)

Mathematical Problems in Engineering 3

PP O

y

q(x)

xdx

(a)

q

VM

n

P

P

dx

V + dV

dydx

M+ dM

(b)

Figure 1 Cross-sectional analysis of thecolumn-beam element

Since rotations are assumed to be small if 11988921198891199092 = minus119872119864119868then (14) becomes

minus119881 = 1198641198681198893

119910

1198891199093+ 119875

119889119910

119889119909 (15)

Here 119864119868 represents bending rigidity If the derivative ofboth sides of (15) is taken in terms of 119909 then the fourth orderlinear differential equation for the elastic curve is found assuch

1198641198681198894

119910

1198891199094+ 119875

1198892

119910

1198891199092= 119902 (119909) (16)

32 Application of the ADM to the Problem

Example 1 We consider the fourth order linear nonhomoge-neous differential equation [43]

1198894

119910

1198891199094minus 2

1198892

119910

1198891199092+ 119910 = minus8119890

119909

119909 isin [0 1] (17)

with the boundary conditions of

119910 (0) = 11991010158401015840

(0) = 1 1199101015840

(1) = 11991010158401015840

(1) = minus119890 (18)

If (18) is written out in operator form we obtain

119871119910 = 211991010158401015840

minus 119910 minus 8119890119909

(19)

Here

119871 =1198894

1198891199094 119871

minus1

(sdot) = int

119909

0

int

119909

0

int

119909

0

int

119909

0

(sdot) 119889119909 119889119909 119889119909 119889119909

(20)

are the derivative and integral operators If the 119871minus1 inverse

operator is applied to (19) and initial conditions are taken wefind

119910 (119909) = 119860119909 +1

31198611199093

minus 8119871minus1

(119890119909

) + 2119871minus1

(11991010158401015840


(119910)

(21)

where 119860 = 1199101015840

(0) and 119861 = 119910101584010158401015840

(0) If we use (3) in (21) thenwe find

infin

sum

119899=0

119910119899

(119909) = 119860119909 +1

31198611199093

minus 8119871minus1

119890119909

+ 2119871minus1

(

infin

sum

119899=0

11991010158401015840

119899

(119909)) minus 119871minus1

(

infin

sum

119899=0

119910119899

(119909))

(22)

whereas the reduction formula given below can be writtenusing (22)

1199100

(119909) = 119860119909 +1

31198611199093

minus 8119871minus1

(119890119909

)

119910119899+1

(119909) = 2119871minus1

(11991010158401015840

119899


(119910119899

)

119899 ge 0

(23)

The 119860 and 119861 constants in the reduction formula will bedetermined using boundary conditions (18) after finding thedecomposition series From the reduction relation we canobtain the solution terms of the decomposition series as

1199100

(119909) = 8 minus 8119890119909

+ (8 + 119860) 119909 + 41199092

+1

6(8 + 119861) 119909

3

1199101

(119909) = 8 minus 8119890119909

+ 8119909 + 41199092

+41199093

3+

1199094

3

minus1

120(minus8 + 119860 minus 2119861) 119909

5

minus1199096

90minus

(8 + 119861) 1199097

5040

1199102

(119909) = 8 minus 8119890119909

+ 8119909 + 41199092

+41199093

3+

1199094

3minus

1199095

15+

1199096

90

minus(119860 minus 2 (2 + 119861)) 119909

7

2520minus

1199098

1680

+(119860 minus 4 (6 + 119861)) 119909

9

362880+

11990910

453600+

(8 + 119861) 11990911

39916800

1199103

(119909) = 8 minus 8119890119909

+ 8119909 + 41199092

+41199093

3+

1199094

3+

1199095

15

+1199096

90+

1199097

630+

1199098

5040minus

(minus2 + 119860 minus 2119861) 1199099

90720

minus11990910

64800+

(minus14 + 119860 minus 3119861) 11990911

9979200


+11990912

11975040minus

(minus40 + 119860 minus 6119861) 11990913

6227020800

minus11990914

10897286400minus

(8 + 119861) 11990915

1307674368000

1199104

(119909) = 8 minus 8119890119909

+ 8119909 + 41199092

+41199093

3+

1199094

3+

1199095

15

+1199096

90+

1199097

630+

1199098

5040+

1199099

45360

+11990910

453600minus

(minus1 + 119860 minus 2119861) 11990911

4989600minus

11990912

3991680

+(minus30 + 3119860 minus 8119861) 119909

13

1556755200+

1711990914

10897286400

minus(minus6119861 + 3119860 minus 12119861) 119909

15

653837184000minus

11990916

373621248000

+(119860 minus 8 (7 + 119861)) 119909

17

355687428096000+

11990918

800296713216000

+(8 + 119861) 119909

19

121645100408832000

(24)

If these terms are placed in (4) we obtain the approximatesolution obtained via the ADM using the five terms of (17)and problem (18) can be written as

119910 (119909) =

4

sum

119894=0

119910119894

(119909) = 1199100

(119909) + 1199101

(119909) + 1199102

(119909) + 1199103

(119909) + 1199104

(119909)

(25)

119910 (119909) = 40 minus 40119890119909

+ (40 + 119860) 119909 + 201199092

+1

6(40 + 119861) 119909

3

+41199094

3minus

1

120(119860 minus 2 (16 + 119861)) 119909

5

+1199096

45


7

5040minus

1199098

5040

minus(8 + 3119860 minus 4119861) 119909

9

362880minus

11990910

90720minus

(40 + 4119860 minus 5119861) 11990911

39916800

minus11990912

5987520+

(minus80 + 11119860 minus 26119861) 11990913

6227020800+

11990914

681080400


15

1307674368000minus

11990916

373621248000

+(119860 minus 8 (7 + 119861)) 119909

17

355687428096000+

11990918

800296713216000

+(8 + 119861) 119909

19

121645100408832000

(26)

0 02 04 06 08 10

01

02

03

04

AnalyticalADM

Figure 2 Graph showing the ADMand analytical solution obtainedusing the five terms of the decomposition series

Table 1 Numerical results obtained using the five terms of thedecomposition series

119909Analytical

solution 119910(119909)

Approximatesolution 0

5

Error|119910(119909) minus 0

5

|

0 0 0 001 0099465 0099465 292821 times 10minus15

02 0195424 0195424 minus216493 times 10minus15

03 0283470 0283470 610623 times 10minus16







10 0 210884 times 10minus8 minus210884 times 10minus8

If we use boundary conditions (18) in solutions series(26) then we have 119860 = 1 and 119861 = minus3 The analytical solutionof problem (17) and (18) is [43]

119910 (119909) = 119909 (1 minus 119909) 119890119909

(27)

The numerical results and graphs obtained for five termsusing the solution series in (26) have been given below

4 Results and Discussion

In this study the approximate solution of the fourth orderboundary value problem arising in beam-column theory hasbeen determined using theADMThismethod can be appliedon differential equations without the need for discretizationindexing or linearization

Nonhomogeneous problem has been handled regardingthe topic and the obtained results have been given in Table 1and Figure 2 As can be seen from the table and figure themethod used has given results that converge rapidly to theanalytical solution with the removal of a few terms from the


Table 2 Comparison of the absolute error obtained using the splinemethod (SM) and the ADM

119909SM

ℎ = 15

SMℎ = 110

ADM05

CPU time

0 0 0 0 054202 1960119864 minus 5 1228119864 minus 6 minus2164119864 minus 15 084604 3211119864 minus 5 2012119864 minus 6 minus1304119864 minus 13 087006 3568119864 minus 5 2235119864 minus 6 minus2587119864 minus 11 087508 2683119864 minus 5 1681119864 minus 6 minus1123119864 minus 9 08911 0 0 minus2108119864 minus 8 0902

solution series This shows that the method is suitable andreliable for such problems In addition the handled examplehas been compared with the results obtained by Chen andAtsuta [41 42] using nonpolynomial spline method As seenin Table 2 values close to those obtained in this study werefound only in case when the ldquoℎrdquo step was small New termscan be added to the solution series of the ADM to obtainresults that are much better than those of the spline method

In Figure 2 five terms of the decompositionmethod havebeen taken from the analytical solution and decompositionseries of example have been drawn in two dimensionalgraphs As seen in these graphs the analytical and approxi-mate solutions cannot be distinguished

In conclusion it has been determined that the ADM canbe applied to the linear homogeneous and nonhomogeneousboundary value problems that arise in civil engineering in thebeam-column theory Solutions that converge rapidly to theanalytical solution can be found without changing the natureof the physical phenomenon In addition the calculationsfor this method can be carried out using software such asMathematica Maple and Matlab

Conflict of Interests

The author declares that there is no conflict of interests

References

[1] G Adomian Solving Frontier Problems of Physics The Decom-position Method Kluwer Academic Publishers Boston MassUSA 1994

[2] G Adomian ldquoA review of the decomposition method andsome recent results for nonlinear equationsrdquoMathematical andComputer Modelling vol 13 no 7 pp 17ndash43 1990

[3] G Adomian and R Rach ldquoEquality of partial solutions in thedecomposition method for linear or nonlinear partial differ-ential equationsrdquo Computers amp Mathematics with Applicationsvol 19 no 12 pp 9ndash12 1990

[4] M Inc ldquoOn numerical solutions of partial differential equationsby the decomposition methodrdquo Kragujevac Journal of Mathe-matics vol 26 pp 153ndash164 2004

[5] M Inc ldquoDecomposition method for solving parabolic equa-tions in finite domainsrdquo Journal of ZhejiangUniversity SCIENCEA vol 6 no 10 pp 1058ndash1064 2005

[6] M Inc Y Cherruault and K Abbaoui ldquoA computationalapproach to the wave equations an application of the decom-position methodrdquo Kybernetes vol 33 no 1 pp 80ndash97 2004

[7] Z M Odibat ldquoA new modification of the homotopy pertur-bation method for linear and nonlinear operatorsrdquo AppliedMathematics and Computation vol 189 no 1 pp 746ndash753 2007

[8] S R S Alizadeh G G Domairry and S Karimpour ldquoAnapproximation of the analytical solution of the linear and non-linear integro-differential equations by homotopy perturbationmethodrdquo Acta Applicandae Mathematicae vol 104 no 3 pp355ndash366 2008

[9] Y-G Wang H-F Song and D Li ldquoSolving two-point bound-ary value problems using combined homotopy perturbationmethod and Greenrsquos function methodrdquo Applied Mathematicsand Computation vol 212 no 2 pp 366ndash376 2009

[10] J Biazar and H Ghazvini ldquoConvergence of the homotopy per-turbation method for partial differential equationsrdquo NonlinearAnalysis Real World Applications vol 10 no 5 pp 2633ndash26402009

[11] C-S Liu ldquoThe essence of the homotopy analysis methodrdquoAppliedMathematics and Computation vol 216 no 4 pp 1299ndash1303 2010

[12] S Momani and S Abuasad ldquoApplication of Hersquos variationaliteration method to Helmholtz equationrdquo Chaos Solitons ampFractals vol 27 no 5 pp 1119ndash1123 2006

[13] EM Abulwafa M A Abdou and A AMahmoud ldquoNonlinearfluid flows in pipe-like domain problem using variational-iteration methodrdquo Chaos Solitons amp Fractals vol 32 no 4 pp1384ndash1397 2007

[14] NH Sweilam andMMKhader ldquoVariational iterationmethodfor one dimensional nonlinear thermoelasticityrdquoChaos Solitonsamp Fractals vol 32 no 1 pp 145ndash149 2007

[15] L Xu ldquoVariational iteration method for solving integral equa-tionsrdquo Computers amp Mathematics with Applications vol 54 no7-8 pp 1071ndash1078 2007

[16] J-H He A-M Wazwaz and L Xu ldquoThe variational iterationmethod reliable efficient and promisingrdquo Computers amp Math-ematics with Applications vol 54 no 7-8 pp 879ndash880 2007

[17] L Xu J-H He and A-M Wazwaz ldquoPreface variationaliteration methodmdashreality potential and challengesrdquo Journal ofComputational and Applied Mathematics vol 207 no 1 pp 1ndash22007

[18] S B Coskun and M T Atay ldquoAnalysis of convective straightand radial fins with temperature-dependent thermal conduc-tivity using variational iteration method with comparison withrespect to finite element analysisrdquo Mathematical Problems inEngineering vol 2007 Article ID 42072 15 pages 2007

[19] M T Atay and S B Coskun ldquoEffects of nonlinearity on thevariational iteration solutions of nonlinear two-point boundaryvalue problems with comparison with respect to finite elementanalysisrdquo Mathematical Problems in Engineering vol 2008Article ID 857296 10 pages 2008

[20] M Inc and Y Ugurlu ldquoNumerical simulation of the regularizedlong wave equation by Hersquos homotopy perturbation methodrdquoPhysics Letters A General Atomic and Solid State Physics vol369 no 3 pp 173ndash179 2007

[21] A S Bataineh M S M Noorani and I Hashim ldquoHomotopyanalysis method for singular IVPs of Emden-Fowler typerdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 14 no 4 pp 1121ndash1131 2009


[22] S Abbasbandy E Babolian and M Ashtiani ldquoNumericalsolution of the generalized Zakharov equation by homotopyanalysis methodrdquo Communications in Nonlinear Science andNumerical Simulation vol 14 no 12 pp 4114ndash4121 2009

[23] M M Rashidi and S Dinarvand ldquoPurely analytic approximatesolutions for steady three-dimensional problem of conden-sation film on inclined rotating disk by homotopy analysismethodrdquo Nonlinear Analysis Real World Applications vol 10no 4 pp 2346ndash2356 2009

[24] L Song and H Zhang ldquoSolving the fractional BBM-Burgersequation using the homotopy analysis methodrdquo Chaos Solitonsamp Fractals vol 40 no 4 pp 1616ndash1622 2009

[25] S Liao ldquoOn the relationship between the homotopy analysismethod and Euler transformrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 6 pp 1421ndash14312010

[26] G Adomian Nonlinear Stochastic Operator Equations Aca-demic Press San Diego Calif USA 1986

[27] M Inc andM Isık ldquoAdomian decompositionmethod for three-dimensional parabolic equation with non-classic boundaryconditionsrdquo Journal of Analysis vol 11 pp 43ndash51 2003

[28] F Abdelwahid ldquoA mathematical model of Adomian polynomi-alsrdquoAppliedMathematics and Computation vol 141 no 2-3 pp447ndash453 2003

[29] E Babolian and S Javadi ldquoNew method for calculating Ado-mian polynomialsrdquoAppliedMathematics and Computation vol153 no 1 pp 253ndash259 2004

[30] Y Cherruault M Inc and K Abbaoui ldquoOn the solution of thenon-linear Korteweg-de Vries equation by the decompositionmethodrdquo Kybernetes vol 31 no 5 pp 766ndash772 2002

[31] AM A El-Sayed andM Gaber ldquoTheAdomian decompositionmethod for solving partial differential equations of fractal orderin finite domainsrdquo Physics Letters A vol 359 no 3 pp 175ndash1822006

[32] E Momoniat T A Selway and K Jina ldquoAnalysis of Adomiandecomposition applied to a third-order ordinary differentialequation from thin film flowrdquo Nonlinear Analysis TheoryMethods amp Applications vol 66 no 10 pp 2315ndash2324 2007

[33] L L Thompson and P M Pinsky ldquoA Galerkin least-squaresfinite element method for the two-dimensional Helmholtzequationrdquo International Journal for Numerical Methods in Engi-neering vol 38 no 3 pp 371ndash397 1995

[34] J Dolbow and T Belytschko ldquoNumerical integration of theGalerkin weak form in meshfree methodsrdquo ComputationalMechanics vol 23 no 3 pp 219ndash230 1999

[35] S N Atluri and T Zhu ldquoA new meshless local Petrov-Galerkin(MLPG) approach in computational mechanicsrdquo Computa-tional Mechanics vol 22 no 2 pp 117ndash127 1998

[36] A M Wazwaz Partial Differential Equations Methods andApplications Balkema Publishers RotterdamTheNetherlands2002

[37] Y Cherruault ldquoConvergence of AdomianrsquosmethodrdquoKybernetesvol 18 no 2 pp 31ndash38 1989

[38] K Abbaoui and Y Cherruault ldquoNew ideas for proving conver-gence of decomposition methodsrdquo Computers amp Mathematicswith Applications vol 29 no 7 pp 103ndash108 1995

[39] S P Timoshenko and JM Gere Theory of Elastic StabilityMcGraw-Hill New York NY USA 2nd edition 1961

[40] S P Timoshenko and JM Gere Theory of Elastic StabilityMcGraw-Hill New York NY USA 1985

[41] W F Chen and T AtsutaTheory of Beam-Columns vol 1 of In-plane Behavior and Design McGraw-Hill New York NY USA1976

[42] W F Chen and T Atsuta Theory ofBeam-Columns vol 2 ofSpace Behavior and Design McGraw-Hill New York NY USA1977

[43] O A Taiwo and O M Ogunlaran ldquoA non-polynomial splinemethod for solving linear fourth-order boundary-value prob-lemsrdquo International Journal of Physical Sciences vol 6 no 13pp 3246ndash3254 2011

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Hindawi Publishing Corporationhttpwwwhindawicom

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Stochastic AnalysisInternational Journal of


2 Adomian Decomposition Method

The method has many advantages in comparison with manyother traditional methods such as Finite differences Finiteelements and Galerkin method [33ndash35] The method givesconverging results adapted to the problems It is a disadvan-tage that convergence interval is small and the calculation ofthe ADM polynomials occurs in nonlinear problems in theADM

We now consider second order nonlinear ordinary differ-ential equations with initial conditions This equation can bewritten in an operator by

119871119906 + 119877119906 + 119873119906 = 119891 (1)

where 119871 is the lower-order derivative which is assumed to beinvertible 119877 is a linear differential operator of order greaterthan119871119873 is a nonlinear differential operator and119891 is a sourceterm We next apply the inverse operator 119871minus1 to both sides of(1) and use the given initial conditions to get

119906 (119909) = 119906 (0) + 1199091199061015840

(0) + 119871minus1

(119892) minus 119871minus1

(119877119906) minus 119871minus1

(119873119906)

(2)

where

119871 =1198892

1198891199092 119871

minus1

= int

119909

0

int

119909

0

119889119909 119889119909 (3)

The ADM consists in decomposing the unknown func-tion 119906(119909) of any equation into a sum of infinite number ofcomponents given by the decomposition series

119906 (119909) =

infin

sum

119899=0

119906119899

(119909) (4)

where the components 119906119899

(119909) 119899 ge 0 are to be determinedin a recursive manner The nonlinear term 119873119906 will bedecomposed by the infinite series of polynomials given by

119873119906 =

infin

sum

119899=0

119860119899

(5)

where 119860119899

are Adomian polynomials [1 36] Substituting (3)and (4) into (2) gives

infin

sum

119899=0

119906119899

= 1199060

(119909) minus 119871minus1

119877(

infin

sum

119899=0

119906119899

(119909) minus 119871minus1

infin

sum

119899=0

119860119899

) (6)

where

1199060

(119909) = 119906 (0) + 1199091199061015840

(0) + 119871minus1

(119892) (7)

The various components 119906119899

of the solution 119906 can be easilydetermined by using the recursive relation

1199060

(119909) = 119906 (0) + 1199091199061015840

(0) + 119871minus1

119892 (119909)

119906119896+1

(119909) = minus 119871minus1

(119877119906119896


(119860119896

) 119896 ge 0

(8)

In order to obtain the numerical solutions of 119906(119909) closedsolution function using the decomposition method

120593119899

=

119899

sum

119896=0

119906119896

(119909) 119896 ge 0 (9)

being the term

119906 (119909) = lim120593119899

119899rarrinfin

(10)

can be calculated by taking into account the reduction for-mula (8) In addition the series solution of the decompositionwritten as (10) generally yields results that rapidly convergefor physical problems The convergence of the decomposi-tion series has been examined by many researchers in theliterature The convergence of the decomposition series hasbeen examined theoretically by Cherruault [37] In additionto these studies Abbaoui and Cherruault have suggesteda new approach in determining the convergence of thedecomposition series [38] These authors have determinedthe convergence of the decomposition series method bygiving new conditions

3 The Application of the ADM to the LinearProblem with Boundary Condition ThatArises in Beam-Column Theory

31 Problem Definition Fourth order differential equationsconsist of various physical problems that are related tothe elastic stability theory Differential relations should beestablished between the effects of various cross-section effectsin order to understand beam-column problems better

When a cross section of distance 119889119909 shown in Figure 1(b)is taken from a beam-column subject to both the 119875 axial loadand the 119902 spread load perpendicular to the axis as shownin Figure 1(a) internal forces arise in the element When theequilibrium equation in the 119910 direction is written with

119902 = minus119889119881

119889119909 (11)

the following ordinary differential equation is found [39ndash42]

minus119881 + 119902119889119909 + (119881 + 119889119881) = 0 (12)

The algebraic sums of the forces acting on both surfacesof the cross-section element are the same due to equilibriumConsider

119872 + 119902119889119909119889119909

2+ (119881 + 119889119881) 119889119909 minus (119872 + 119889119872) + 119875

119889119910

119889119909= 0

(13)

Here 119881 is the shear force acting on the surface of theelement whereas119872 is the bendingmoment that tries to bendthe cross-section element

If rotations are assumed to be small and the second orderterms in terms of 119889119909 are neglected then (13) becomes

119881 =119889119872

119889119909minus 119875

119889119910

119889119909 (14)


PP O

y

q(x)

xdx

(a)

q

VM

n

P

P

dx

V + dV

dydx

M+ dM

(b)



minus119881 = 1198641198681198893

119910

1198891199093+ 119875

119889119910

119889119909 (15)


1198641198681198894

119910

1198891199094+ 119875

1198892

119910

1198891199092= 119902 (119909) (16)



1198894

119910

1198891199094minus 2

1198892

119910

1198891199092+ 119910 = minus8119890

119909

119909 isin [0 1] (17)


119910 (0) = 11991010158401015840

(0) = 1 1199101015840

(1) = 11991010158401015840

(1) = minus119890 (18)


119871119910 = 211991010158401015840

minus 119910 minus 8119890119909

(19)

Here

119871 =1198894

1198891199094 119871

minus1

(sdot) = int

119909

0

int

119909

0

int

119909

0

int

119909

0

(sdot) 119889119909 119889119909 119889119909 119889119909

(20)



119910 (119909) = 119860119909 +1

31198611199093

minus 8119871minus1

(119890119909

) + 2119871minus1

(11991010158401015840


(119910)

(21)

where 119860 = 1199101015840

(0) and 119861 = 119910101584010158401015840


infin

sum

119899=0

119910119899

(119909) = 119860119909 +1

31198611199093

minus 8119871minus1

119890119909

+ 2119871minus1

(

infin

sum

119899=0

11991010158401015840

119899

(119909)) minus 119871minus1

(

infin

sum

119899=0

119910119899

(119909))

(22)


1199100

(119909) = 119860119909 +1

31198611199093

minus 8119871minus1

(119890119909

)

119910119899+1

(119909) = 2119871minus1

(11991010158401015840

119899


(119910119899

)

119899 ge 0

(23)


1199100

(119909) = 8 minus 8119890119909

+ (8 + 119860) 119909 + 41199092

+1

6(8 + 119861) 119909

3

1199101

(119909) = 8 minus 8119890119909

+ 8119909 + 41199092

+41199093

3+

1199094

3

minus1

120(minus8 + 119860 minus 2119861) 119909

5

minus1199096

90minus

(8 + 119861) 1199097

5040

1199102

(119909) = 8 minus 8119890119909

+ 8119909 + 41199092

+41199093

3+

1199094

3minus

1199095

15+

1199096

90

minus(119860 minus 2 (2 + 119861)) 119909

7

2520minus

1199098

1680

+(119860 minus 4 (6 + 119861)) 119909

9

362880+

11990910

453600+

(8 + 119861) 11990911

39916800

1199103

(119909) = 8 minus 8119890119909

+ 8119909 + 41199092

+41199093

3+

1199094

3+

1199095

15

+1199096

90+

1199097

630+

1199098

5040minus

(minus2 + 119860 minus 2119861) 1199099

90720

minus11990910

64800+

(minus14 + 119860 minus 3119861) 11990911

9979200


+11990912

11975040minus

(minus40 + 119860 minus 6119861) 11990913

6227020800

minus11990914

10897286400minus

(8 + 119861) 11990915

1307674368000

1199104

(119909) = 8 minus 8119890119909

+ 8119909 + 41199092

+41199093

3+

1199094

3+

1199095

15

+1199096

90+

1199097

630+

1199098

5040+

1199099

45360

+11990910

453600minus

(minus1 + 119860 minus 2119861) 11990911

4989600minus

11990912

3991680

+(minus30 + 3119860 minus 8119861) 119909

13

1556755200+

1711990914

10897286400


15

653837184000minus

11990916

373621248000

+(119860 minus 8 (7 + 119861)) 119909

17

355687428096000+

11990918

800296713216000

+(8 + 119861) 119909

19

121645100408832000

(24)


119910 (119909) =

4

sum

119894=0

119910119894

(119909) = 1199100

(119909) + 1199101

(119909) + 1199102

(119909) + 1199103

(119909) + 1199104

(119909)

(25)

119910 (119909) = 40 minus 40119890119909

+ (40 + 119860) 119909 + 201199092

+1

6(40 + 119861) 119909

3

+41199094

3minus

1

120(119860 minus 2 (16 + 119861)) 119909

5

+1199096

45


7

5040minus

1199098

5040

minus(8 + 3119860 minus 4119861) 119909

9

362880minus

11990910

90720minus

(40 + 4119860 minus 5119861) 11990911

39916800

minus11990912

5987520+

(minus80 + 11119860 minus 26119861) 11990913

6227020800+

11990914

681080400


15

1307674368000minus

11990916

373621248000

+(119860 minus 8 (7 + 119861)) 119909

17

355687428096000+

11990918

800296713216000

+(8 + 119861) 119909

19

121645100408832000

(26)

0 02 04 06 08 10

01

02

03

04

AnalyticalADM



119909Analytical

solution 119910(119909)


5

Error|119910(119909) minus 0

5

|

0 0 0 001 0099465 0099465 292821 times 10minus15


03 0283470 0283470 610623 times 10minus16









119910 (119909) = 119909 (1 minus 119909) 119890119909

(27)







119909SM

ℎ = 15

SMℎ = 110

ADM05

CPU time







References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










PP O

y

q(x)

xdx

(a)

q

VM

n

P

P

dx

V + dV

dydx

M+ dM

(b)



minus119881 = 1198641198681198893

119910

1198891199093+ 119875

119889119910

119889119909 (15)


1198641198681198894

119910

1198891199094+ 119875

1198892

119910

1198891199092= 119902 (119909) (16)



1198894

119910

1198891199094minus 2

1198892

119910

1198891199092+ 119910 = minus8119890

119909

119909 isin [0 1] (17)


119910 (0) = 11991010158401015840

(0) = 1 1199101015840

(1) = 11991010158401015840

(1) = minus119890 (18)


119871119910 = 211991010158401015840

minus 119910 minus 8119890119909

(19)

Here

119871 =1198894

1198891199094 119871

minus1

(sdot) = int

119909

0

int

119909

0

int

119909

0

int

119909

0

(sdot) 119889119909 119889119909 119889119909 119889119909

(20)



119910 (119909) = 119860119909 +1

31198611199093

minus 8119871minus1

(119890119909

) + 2119871minus1

(11991010158401015840


(119910)

(21)

where 119860 = 1199101015840

(0) and 119861 = 119910101584010158401015840


infin

sum

119899=0

119910119899

(119909) = 119860119909 +1

31198611199093

minus 8119871minus1

119890119909

+ 2119871minus1

(

infin

sum

119899=0

11991010158401015840

119899

(119909)) minus 119871minus1

(

infin

sum

119899=0

119910119899

(119909))

(22)


1199100

(119909) = 119860119909 +1

31198611199093

minus 8119871minus1

(119890119909

)

119910119899+1

(119909) = 2119871minus1

(11991010158401015840

119899


(119910119899

)

119899 ge 0

(23)


1199100

(119909) = 8 minus 8119890119909

+ (8 + 119860) 119909 + 41199092

+1

6(8 + 119861) 119909

3

1199101

(119909) = 8 minus 8119890119909

+ 8119909 + 41199092

+41199093

3+

1199094

3

minus1

120(minus8 + 119860 minus 2119861) 119909

5

minus1199096

90minus

(8 + 119861) 1199097

5040

1199102

(119909) = 8 minus 8119890119909

+ 8119909 + 41199092

+41199093

3+

1199094

3minus

1199095

15+

1199096

90

minus(119860 minus 2 (2 + 119861)) 119909

7

2520minus

1199098

1680

+(119860 minus 4 (6 + 119861)) 119909

9

362880+

11990910

453600+

(8 + 119861) 11990911

39916800

1199103

(119909) = 8 minus 8119890119909

+ 8119909 + 41199092

+41199093

3+

1199094

3+

1199095

15

+1199096

90+

1199097

630+

1199098

5040minus

(minus2 + 119860 minus 2119861) 1199099

90720

minus11990910

64800+

(minus14 + 119860 minus 3119861) 11990911

9979200


+11990912

11975040minus

(minus40 + 119860 minus 6119861) 11990913

6227020800

minus11990914

10897286400minus

(8 + 119861) 11990915

1307674368000

1199104

(119909) = 8 minus 8119890119909

+ 8119909 + 41199092

+41199093

3+

1199094

3+

1199095

15

+1199096

90+

1199097

630+

1199098

5040+

1199099

45360

+11990910

453600minus

(minus1 + 119860 minus 2119861) 11990911

4989600minus

11990912

3991680

+(minus30 + 3119860 minus 8119861) 119909

13

1556755200+

1711990914

10897286400


15

653837184000minus

11990916

373621248000

+(119860 minus 8 (7 + 119861)) 119909

17

355687428096000+

11990918

800296713216000

+(8 + 119861) 119909

19

121645100408832000

(24)


119910 (119909) =

4

sum

119894=0

119910119894

(119909) = 1199100

(119909) + 1199101

(119909) + 1199102

(119909) + 1199103

(119909) + 1199104

(119909)

(25)

119910 (119909) = 40 minus 40119890119909

+ (40 + 119860) 119909 + 201199092

+1

6(40 + 119861) 119909

3

+41199094

3minus

1

120(119860 minus 2 (16 + 119861)) 119909

5

+1199096

45


7

5040minus

1199098

5040

minus(8 + 3119860 minus 4119861) 119909

9

362880minus

11990910

90720minus

(40 + 4119860 minus 5119861) 11990911

39916800

minus11990912

5987520+

(minus80 + 11119860 minus 26119861) 11990913

6227020800+

11990914

681080400


15

1307674368000minus

11990916

373621248000

+(119860 minus 8 (7 + 119861)) 119909

17

355687428096000+

11990918

800296713216000

+(8 + 119861) 119909

19

121645100408832000

(26)

0 02 04 06 08 10

01

02

03

04

AnalyticalADM



119909Analytical

solution 119910(119909)


5

Error|119910(119909) minus 0

5

|

0 0 0 001 0099465 0099465 292821 times 10minus15


03 0283470 0283470 610623 times 10minus16









119910 (119909) = 119909 (1 minus 119909) 119890119909

(27)







119909SM

ℎ = 15

SMℎ = 110

ADM05

CPU time







References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+11990912

11975040minus

(minus40 + 119860 minus 6119861) 11990913

6227020800

minus11990914

10897286400minus

(8 + 119861) 11990915

1307674368000

1199104

(119909) = 8 minus 8119890119909

+ 8119909 + 41199092

+41199093

3+

1199094

3+

1199095

15

+1199096

90+

1199097

630+

1199098

5040+

1199099

45360

+11990910

453600minus

(minus1 + 119860 minus 2119861) 11990911

4989600minus

11990912

3991680

+(minus30 + 3119860 minus 8119861) 119909

13

1556755200+

1711990914

10897286400


15

653837184000minus

11990916

373621248000

+(119860 minus 8 (7 + 119861)) 119909

17

355687428096000+

11990918

800296713216000

+(8 + 119861) 119909

19

121645100408832000

(24)


119910 (119909) =

4

sum

119894=0

119910119894

(119909) = 1199100

(119909) + 1199101

(119909) + 1199102

(119909) + 1199103

(119909) + 1199104

(119909)

(25)

119910 (119909) = 40 minus 40119890119909

+ (40 + 119860) 119909 + 201199092

+1

6(40 + 119861) 119909

3

+41199094

3minus

1

120(119860 minus 2 (16 + 119861)) 119909

5

+1199096

45


7

5040minus

1199098

5040

minus(8 + 3119860 minus 4119861) 119909

9

362880minus

11990910

90720minus

(40 + 4119860 minus 5119861) 11990911

39916800

minus11990912

5987520+

(minus80 + 11119860 minus 26119861) 11990913

6227020800+

11990914

681080400


15

1307674368000minus

11990916

373621248000

+(119860 minus 8 (7 + 119861)) 119909

17

355687428096000+

11990918

800296713216000

+(8 + 119861) 119909

19

121645100408832000

(26)

0 02 04 06 08 10

01

02

03

04

AnalyticalADM



119909Analytical

solution 119910(119909)


5

Error|119910(119909) minus 0

5

|

0 0 0 001 0099465 0099465 292821 times 10minus15


03 0283470 0283470 610623 times 10minus16









119910 (119909) = 119909 (1 minus 119909) 119890119909

(27)







119909SM

ℎ = 15

SMℎ = 110

ADM05

CPU time







References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119909SM

ℎ = 15

SMℎ = 110

ADM05

CPU time







References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article the solution of fourth order boundary...

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