Research Article Analytical Solution of Nonlinear …downloads.hindawi.com/journals/ijce/2014/825797.pdfResearch Article Analytical Solution of Nonlinear Dynamics of a Self-Igniting

Research ArticleAnalytical Solution of Nonlinear Dynamics ofa Self-Igniting Reaction-Diffusion System UsingModified Adomian Decomposition Method

Felicia Shirly Peace1 Narmatha Sathiyaseelan1 and Lakshmanan Rajendran2

1 Department of Mathematics Lady Doak College Madurai Tamil Nadu 625002 India2Department of Mathematics Madura College Madurai Tamil Nadu 625011 India

Correspondence should be addressed to Lakshmanan Rajendran raj smsrediffmailcom

Received 24 September 2013 Revised 18 December 2013 Accepted 23 January 2014 Published 24 April 2014

Academic Editor Jean-Pierre Corriou

Copyright copy 2014 Felicia Shirly Peace et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A mathematical model of the dynamics of the self-ignition of a reaction-diffusion system is studied in this paper An approximateanalytical method (modified Adomian decomposition method) is used to solve nonlinear differential equations under steady-state condition Analytical expressions for concentrations of the gas reactant and the temperature have been derived for Lewisnumber (Le) and parameters 120573 120574 and 1206012 Furthermore in this work the numerical simulation of the problem is also reportedusing MATLAB program An agreement between analytical and numerical results is noted

1 Introduction

Nonlinear dynamical phenomena in combustion processare an active area of experimental and theoretical researchMathematical models that describe this phenomenon canbe considered as nonlinear dynamical systems The dynamiccharacterization of such models was developed by Continilloet al [1] The detailed numerical simulation of autoigni-tion of coal stockpiles leads to the observation of steadyregimes To better investigate this phenomenon two sim-plified distributed-parameter models were discussed whichincorporate heat conduction mass diffusion and one-stepArrhenius exothermic chemical reaction Both model equa-tions were solved with straight forward finite-differenceschemes [2] The problem of spontaneous ignition of coalstockpiles is challenging for safety implications and for itstheoretical complexity a spontaneous combustion reactiontakes place in a bed of solid fuel while flow driven bynatural convection generated by the onset of temperaturegradients within the pile occurs Coal stockpiles self-ignitewhen reaction of coal with oxygen present in the atmospheregenerates heat that is not efficiently removed toward theexternal ambient [3] Continillo et al [4 5] have analyzed the

self-combustion of coal piles in the absence of natural convec-tion The three main phenomena in the self-ignition of coalstockpile are convection reaction anddiffusionOn the otherhand Continillo et al [6] studied the dynamic behavior of atwo-dimensional coal pile also by accounting for natural con-vection As part of a comprehensive study of self-heating ofcoal stockpiles a simple mathematical model has been devel-oped To our knowledge no rigorous analytical expressionsof gas reactant (Y) and temperature (119879) have been derivedfor all possible values of parameters under steady-stateconditionsThepurpose of this paper is to derive approximateanalytical expressions for gas reactant concentration andtemperature using the modified Adomian decompositionmethod

2 Mathematical Formulation ofthe Boundary Value Problem

The nonlinear differential equations are those of a distribut-ed-parameter dynamic model of heterogeneous reaction in aone-dimensional layerThe gaseous reactant diffuses throughthe reacting medium and a first order one-step exothermic

Hindawi Publishing CorporationInternational Journal of Chemical EngineeringVolume 2014 Article ID 825797 8 pageshttpdxdoiorg1011552014825797

2 International Journal of Chemical Engineering

chemical reaction takes place The reaction rate dependson the temperature through the Arrhenius exponentialThe Arrhenius rate equation is a mathematical expressionwhich relates the rate constant of a chemical reaction to theexponential value of the temperature The model nonlinearequations in dimensionless form are [1]

120597119884

120597119905= Le120597

2119884

1205971199092minus 1206012119884 exp(minus

120574

119879)

120597119879

120597119905=1205972119879

1205971199092+ 1205731206012119884 exp(minus

120574

119879)

(1)

where 119884 is the concentration of the gas reactant T is thetemperature Le is the Lewis number (the ratio betweenmass and heat diffusivities) 120573 is the dimensionless heatof reaction 120601 is the thermal Thiele modulus (the ratioof the time scale of the limiting transport mechanism tothe time scale of intrinsic reaction kinetics [7]) and 120574 isthe dimensionless activation energy (minimum amount ofenergy between reactant molecules for effective collisionsbetween them) The boundary conditions are

119879 (0 119905) = 119879 (1 119905) = 1 119884 (0 119905) = 1

120597119884

120597119909

10038161003816100381610038161003816100381610038161003816119909=1= 0 for 119905 gt 0

(2)

Under steady-state condition the equations become

1198892119884

1198891199092minus1206012

Le119884 exp(minus

120574

119879) = 0

1198892119879

1198891199092+ 1205731206012119884 exp(minus

120574

119879) = 0

(3)

with boundary conditions

119879 = 1 119884 = 1 at 119909 = 0

119879 = 1119889119884

119889119909= 0 at 119909 = 1

(4)

3 Analytical Solution of Nonlinear Dynamicsof a Self-Igniting Reaction-DiffusionSystem under Steady-State Condition UsingModified Adomian Decomposition Method

In the recent years much attention is devoted to the applica-tion of theAdomian decompositionmethod to the solution ofvarious scientific models [8] An efficient modification of thestandard Adomian decomposition method for solving initialvalue problem in the second order partial differential equa-tion yields the MADM The MADM without linearizationperturbation transformation or discretization gives an ana-lytical solution in terms of a rapidly convergent infinite powerseries with easily computable terms The results show thatthe rate of convergence of modified Adomian decompositionmethod is higher than standard Adomian decompositionmethod [9ndash13] Using this method (see Appendix A) we

1206012 = 1000

1206012 = 5000

1206012 = 10000

1206012 = 20000

1206012 = 25000

1206012 = 30000

1206012 = 35000

10

1

0807060504

088

086030201 09

09

092

094

096

098

Dimensionless spatial coordinate (x)

Dim

ensio

nles

s con

cent

ratio

n (Y

)

1206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206012 = 1000

120601222222222 = 5000

1206012 = 10000

1206012 = 20000

1206012 = 25000

1206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206012 = 30000

1206012 = 35000

Figure 1 Dimensionless concentration Y versus dimensionlessspatial coordinate x using (5) for Le = 0233 120573 = 4287 120574 = 136and various values of 1206012 Solid lines represent the analytical solutionwhereas the dotted lines are for the numerical solution

obtain an approximate analytical expression of concentrationof gas reactant (119884) and temperature (119879) (see Appendix B) asfollows

119884 (119909) = 1 +1206012

Leexp (minus120574)(119909

2

2minus 119909) +

1206014

Leexp (minus2120574)

times [1

Le(1199094

24minus1199093

6+119909

3) minus

120574120573

2(1199094

12minus1199093

6+119909

6)]

(5)

119879 (119909) = 1 minus 1205731206012 exp (minus120574)(1199092

2minus119909

2) minus 1205731206014 exp (minus2120574)

times [1

Le(1199094

24minus1199093

6+119909

8) minus

120574120573

2(1199094

12minus1199093

6+119909

12)]

(6)

4 Numerical Simulation

Nonlinear diffusion equation (3) for the boundary condition(4) is also solved numerically We have used the functionpdex1 in MATLAB software to solve the initial-boundaryvalue problems for the nonlinear differential equationsnumerically This numerical solution is compared with ouranalytical results in Figures 1ndash4 Upon comparison it givesa satisfactory agreement for all values of the dimensionlessparameters 120573 120574 and 1206012TheMATLAB program is also givenin Algorithm 1

5 Discussion

Equations (5) and (6) represent the simplest form of approx-imate analytical expressions for the concentration of gasreactant and temperature for all values of parameters 120572120573 120574 and 1206012 The Thiele number (thermal) is the ratio oflayer thickness (L) and thermal diffusivity (120572) Equation (5)

International Journal of Chemical Engineering 3

1206012 = 1000

1206012 = 5000

1206012 = 10000

1206012 = 20000

1206012 = 25000

1206012 = 30000

1206012 = 35000

10

1

1005

101

102

103

104

1045

1015

1025

1035

09950807060504030201 09


Dim

ensio

nles

s tem

pera

ture

(T)

Figure 2 Dimensionless temperature T versus dimensionless spa-tial coordinate using (6) for Le = 0233 120573 = 4287 120574 =136 andvarious values of 1206012 Solid lines represent the analytical solutionwhereas the dotted lines are for the numerical solution

Dim

ensio

nles

s con

cent

ratio

n (Y

)

0 1

1

098

096

094

092

08801 02 03 04 05 06 07 08 09

09


120574 = 11

120574 = 10120574 = 12

120574 = 13

120574 = 115

120574 = 105 120574 = 125

120574 = 135


represents the new approximate analytical expression ofconcentration of gas reactant The numerical solution iscompared with the analytical results in Figures 1ndash4 Thesefigures represent the analytical and numerical concentra-tion profiles of gas reactant and temperature for differentvalues of parameters 120573 120574 and 1206012 Figure 1 represents thedimensionless concentration Y versus dimensionless spatialcoordinate 119909 for 1206012 le 35000 From the figure it is inferredthat the value of Y decreases when the value of 1206012 or layerthickness increases Figure 2 illustrates the dimensionlessconcentrationY versus dimensionless spatial coordinate119909 for1206012 le 35000 and we infer that the dimensionless temperatureincreases with the increase in values of layer thicknessFigure 3 represents the dimensionless concentrationY versusdimensionless spatial coordinate 119909 for values of 120574 le 136From the figure it is inferred that the value of Y decreases

120574 = 135

120574 = 125

120574 = 115

120574 = 105

120574 = 13

120574 = 11 120574 = 10

120574 = 12

0 1

1

07

101

102

103

08 090605040302010995

1005

1015

1025

1035

Dim

ensio

nles

s tem

pera

ture

(T)


Figure 4 Dimensionless temperature T versus dimensionless spa-tial coordinate x using (6) for Le = 0233 120573 = 4287 1206012 = 1000 andvarious values of 120574 Solid lines represent the analytical solutionwhereas the dotted lines are for the numerical solution

Table 1 Comparison between analytical and numerical values ofdimensionless concentration 119884 (Le = 0233 120573 = 4287 120574 = 136and 1206012 = 30000)

119909 Numerical value of 119884 Analytical value of 119884 deviation0 1 1 002 09661 09713 05304 09398 09489 09706 09216 09329 12308 09113 09233 1321 09082 09201 131

when the value of dimensionless activation energy (120574)increases Figure 4 illustrates the dimensionless temperatureT versus dimensionless spatial coordinate 119909 for the valuesof 120574 le 136 the dimensionless temperature increases withincrease in values of dimensionless activation energy (120574)From Figures 2 and 4 it is evident that the maximum valuefor dimensionless concentration is 1 and that temperatureattains its maximum value when the spatial coordinate 119909 =05 Figure 5 confirms the results given by Figures 1 to4

Our analytical results are compared with the numericalresults for the dimensionless concentration Y in Table 1The maximum relative error between our analytical resultsand simulation results for the concentration Y is 13Also in Table 2 our analytical results are compared withthe numerical results for the dimensionless temperature TSatisfactory agreement is noted The maximum relative errorin this case is 04

6 Conclusion

The system of steady-state nonlinear differential equa-tions in different dynamic modes for the concentration of


function pdex4m = 0x = linspace(01)t = linspace(0100000)sol = pdepe(mpdex4pdepdex4icpdex4bcxt)u1 = sol(1)u2 = sol(2)figureplot(xu1(end))title(lsquou1(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou1(x2)rsquo)mdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashndashfigureplot(xu2(end))title(lsquou2(xt)rsquo)xlabel(lsquoDistance xrsquo)ylabel(lsquou2(x2)rsquo)mdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashndashfunction [cfs] = pdex4pde(xtuDuDx)c = [1 1]f = [1 1]lowast DuDxl = 0233p = 35000b = 4287g = 136F = minusplowastu(1)lowastexp(minusgu(2))lF1 = blowastplowastu(1)lowastexp(minusgu(2))s = [F F1]mdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashndashfunction u0 = pdex4ic(x)u0 = [1 1]mdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashndashfunction [plqlprqr] = pdex4bc(xlulxrurt)pl = [ul(1)minus1 ul(2)minus1]ql = [0 0]pr = [0 ur(2)minus1]qr = [1 0]

Algorithm 1 MATLAB program to find the numerical solution of nonlinear differential equation (1) when the time (119905) is large (or) (3)

Table 2 Comparison between analytical and numerical values ofdimensionless temperature 119879 (Le = 0233 120573 = 4287 120574 = 136 and1206012 = 30000)

119909 Analytical value of 119879 Numerical value of 119879 deviation0 1 1 002 10155 11013 02704 10235 10191 04206 10233 10191 04108 10152 10128 0241 1 1 0

gas reactant and temperature has been solved analyticallyThe model investigated the influence of parameters overthe temperature and concentration of gas reactant in thedynamicmodeThe approximate analytical expression for thesteady-state concentration of gas reactant and temperature

is obtained using the modified Adomian decompositionmethod A satisfactory agreement with the numerical resultis noted The analytical results will be useful to characterizethe model predictions for the various values of parametersThiele number (1206012) Lewis number (Le) layer thickness (L)and thermal diffusivity (120572)

Appendices

A Basic Concept of Modified AdomianDecomposition Method

Consider the nonlinear differential equation in the form

119871 (119910) + 119873 (119910) = 119892 (119909) (A1)

with initial condition

119910 (0) = 119860 1199101015840 (0) = 119861 (A2)


4

1

09

21

1

0

05

03

085

095

Thiele number 1206012 Dimen

sionles

s

spatial

coord

inate (x

)times104

Dim

ensio

nles

s con

cent

ratio

n (Y

)

(a)

10

12

14

1

05

0

1

09

085

095

Dimensionless

spatial coordinate (x)

Dimensionless activation

energy 120574

Dim

ensio

nles

s con

cent

ratio

n (Y

)(b)

4

1

102

104

106

2

0 0

05

098

1Thiele number 120601 2 Dimensionless


times104

Dim

ensio

nles

s tem

pera

ture

(T)

(c)

10

12

14

1

05

0

1

102

104

106

Dimensionless



energy 120574

Dim

ensio

nles

s tem

pera

ture

(T)

(d)

Figure 5 (a)The normalized three-dimensional concentration of gas reactantY versus dimensionless spatial coordinate 119909 andThiele number1206012 (b) the normalized three-dimensional concentration of gas reactant Y versus dimensionless spatial coordinate 119909 and dimensionlessactivation energy 120574 (c) the normalized three-dimensional temperature T versus dimensionless spatial coordinate 119909 and Thiele number 1206012and (d) the normalized three-dimensional temperature T versus dimensionless spatial coordinate 119909 and dimensionless activation energy 120574

where 119873(119910) is a nonlinear real function 119892(119909) is the givenfunction and 119860 and 119861 are constants We propose the newdifferential operator as follows

119871 = 119909minus1198991198892

1198891199092119909119899119910 (A3)

So problem (A1) can be written as

119871 (119910) = 119892 (119909) minus 119873 (119910) (A4)

The inverse operator 119871minus1 is therefore considered a twofoldintegral operator as follows

119871minus1 (sdot) = 119909minus119899∬119909

0

119909119899 (sdot) 119889119909 119889119909 (A5)

By operating 119871minus1 on (A4) we have

119910 (119909) = 119860 + 119861119909 + 119871minus1119892 (119909) minus 119871minus1119873(119910)

= 119860 + 119871minus1119892 (119909) minus 119871minus1119873(119910)

(A6)


The modified Adomian decomposition method introducessolution 119910(119909) and the nonlinear function 119873(119910) by infiniteseries

119910 (119909) =infin

sum119899=0

119910119899(119909) (A7)

119873(119910) =infin

sum119899=0

119860119899 (A8)

where the components 119910119899(119909) of the solution 119910(119909) will be

determined recurrently and the Adomian polynomials 119860119899

of 119865(119909 119910) are evaluated using the formula

119860119899(119909) =

1

119899

119889119899

119889120582119899119873(infin

sum119899=0

(120582119899119910119899))

1003816100381610038161003816100381610038161003816100381610038161003816120582=0 (A9)

By substituting (A7) and (A8) into (A6)

infin

sum119899=0

119910119899(119909) = 119860 + 119871minus1119892 (119909) minus 119871

minus1

infin

sum119899=0

119860119899 (A10)

By using the modified Adomian decomposition method thecomponents 119910

119899(119909) can be determined as

1199100(119909) = 119860 + 119871minus1119892 (119909) 119910

119899+1(119909)

= minus119871minus1 (119860119899) 119899 ge 0

(A11)

which gives

1199100(119909) = 119860 + 119871minus1119892 (119909)

1199101(119909) = minus119871

minus1 (1198600)

1199102(119909) = minus119871

minus1 (1198601)

1199103(119909) = minus119871

minus1 (1198602)

(A12)

From (A9) and (A12) we can determine the components119910119899(119909) and hence the series solution of 119910(119909) in (A7) can be

immediately obtained

B Analytical Solution of Nonlinear Reaction-Diffusion Equation (3) Using ModifiedAdomian Decomposition Method

In this appendix we derive the general solution of nonlinearequation (3) by using modified Adomian decompositionmethod We write (3) in the operator form as

119871 (119884 (119909)) = 119896119873 (119884 (119909))

where 119871 = 119909minus11198892

1198891199092119909

=1206012

Le119884 exp(minus

120574

119879)

(B1)

119871 (119879 (119909)) = 119896119873 (119879 (119909))

= minus1205731206012119884 exp (minus120574

119879)

(B2)

Applying the inverse operator 119871minus1(sdot) = 119909minus1∬119909

0119909(sdot)119889119909 119889119909 on

both sides of (B1) and (B2) yields

119884 (119909) = 119860119909 + 119861 + 119871minus1 (

1206012

Le119884 exp(

minus120574

119879)) (B3)

119879 (119909) = 119862119909 + 119863 + 119871minus1 (minus1205731206012119884 exp (minus120574

119879)) (B4)

where 119860 119861 119862 and119863 are constants of integrationLet

119884 (119909) =infin

sum119899=0

119884119899(119909) (B5)

119879 (119909) =infin

sum119899=0

119879119899(119909) (B6)

119873[119884 (119909)] =infin

sum119899=0

119860119899 (B7)

where

119873[119884 (119909)] =1206012

Le119884 exp(minus

120574

119879) (B8)

119873[119879 (119909)] =infin

sum119899=0

119861119899 (B9)

where

119873[119879 (119909)] = minus1205731206012119884 exp(minus

120574

119879) (B10)

In view of (B5) (B7) and (B8) (B3) givesinfin

sum119899=0

119884119899(119909) = 119860119909 + 119861 + 119871

minus1 (infin

sum119899=0

119860119899) (B11)

We identify the zeroth component as

1198840(119909) = 119860119909 + 119861 (B12)


Using initial condition (4) we get 119860 = 0 and 119861 = 1

there4 1198840= 1 (B13)

and the remaining components are as the recurrence relation

119884119899+1

(119909) = 119871minus1 (119860119899) 119899 ge 0 (B14)

where 119860119899are the Adomian polynomials of 119884

1 1198842 119884119899 We

can find 119860119899as follows

1198600= 119873 (119884

0)

1198601=

119889

119889120582119873 (1198840+ 1205821198841)

(B15)


sum119899=0

119879119899(119909) = 119862119909 + 119863 + 119871minus1 (

infin

sum119899=0

119861119899) (B16)


1198790(119909) = 119862119909 + 119863 (B17)

Using initial condition (4) we get 119862 = 0 and119863 = 1

there4 1198790= 1 (B18)


119879119899+1

(119909) = 119871minus1 (119861119899) 119899 ge 0 (B19)


1 1198792 119879

119899 We


1198610= 119873 (119879

0)

1198611=

119889

119889120582119873 (1198790+ 1205821198791)

(B20)

Hence solving (B14) and (B19)we get the following solutions

1198841=1206012

Leexp (minus120574)(119909

2

2minus 119909) (B21)

1198791= minus1205731206012 exp (minus120574)(119909

2

2minus119909

2) (B22)

1198842=1206014

Leexp (minus2120574)( 1

Le(1199094

24minus1199093

6+119909

3)

minus120574120573

2(1199094

12minus1199093

6+119909

6))

(B23)

1198792= minus1205731206014 exp (minus2120574)( 1

Le(1199094

24minus1199093

6+119909

8)

minus120574120573

2(1199094

12minus1199093

6+119909

12))

(B24)

Adding (B13) (B21) and (B23) we get (5) in the text Adding(B18) (B22) and (B24) we get (6) in the text

C

See Algorithm 1

Notations

Symbols

1198880 Concentration of the free stream

119888119901 Specific heat

119863 Mass diffusivity119864 Activation energy1198700 Preexponential factor

119871 layer thicknessLe Lewis number119863120572119877 Gas reactant119879 Dimensionless temperature

0

Temperature0 Free stream temperature

119905 Dimensionless time 1205721198712 Time119884 Dimensionless concentration 119888119888

0

Greek letters

120572 Thermal diffusivity120573 Dimensionless heat reaction (minusΔ119867119888

0)(1205881198881199010)

120574 Dimensionless activation energy 119864(1198770)

Δ119867 Enthalpy of reaction120588 DensityΦ Thiele numberradic119896

01198712120572

Conflict of Interests

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

Acknowledgments

This work is supported by the University Grant Commission(UGC) Minor Project no F MRP-412212 (MRPUGC-SERO) Hyderabad Government of India The authors arethankful to Shri S Natanagopal secretary at Madura CollegeBoard and Dr R Murali principal at Madura College(autonomous) Madurai Tamil Nadu India for their con-stant encouragement

References

[1] G Continillo V Faraoni P L Maffettone and S CrescitellildquoNon-linear dynamics of a self-igniting reaction-diffusion sys-temrdquo Chemical Engineering Science vol 55 no 2 pp 303ndash3092000

[2] G Continillo P L Maffettone and S Crescitelli ldquoOn thenumerical simulation of the chaotic behavior of some distrib-uted-parameter systemsrdquo in Proceedings of the 1st National


Conference Chaos and Fractals in Chemical Engineering pp 218ndash226 World Scientific Singapore May 1994

[3] C Gaetano and G Giovanni Characterization of ChaoticDynamics in the Spontaneous Combustion of Coal Stock-piles Istituto di Ricerchesulla Combustione (IRC) ConsiglioNazionaledelle Ricerche Naples Italy 1996

[4] G Continillo P L Maffettone and S Crescitelli First Con-ference on Chemical and Process Engineering Paper no 22Firenze Italy 1993

[5] G Continillo P L Maffettone and S Crescitelli ldquoOn thenumerical simulation of the chaotic behaviour of somedistributed-parameter systemsrdquo inChaos and Fractals in Chem-ical Engineering G Biardi M Giona and A R Giona Eds pp218ndash226 World Scientific Singapore 1995

[6] G Continillo P L Maffettone and S Crescitelli in ICheaP-2C T Eris Ed vol 1 of AIDIC Conference Series pp 415ndash424Firenze Italy 1995

[7] W B J Zimmerman Microfluidics History Theory amp Applica-tions Springer New York NY USA 2006

[8] G Adomian ldquoConvergent series solution of nonlinear equa-tionsrdquo Journal of Computational and Applied Mathematics vol11 no 2 pp 225ndash230 1984

[9] Y Q Hasan and L M Zhu ldquoModified Adomian decompositionmethod for singular initial value problems in the second-orderordinary differential equationsrdquo Surveys in Mathematics and ItsApplications vol 3 pp 183ndash193 2008

[10] Y Q Hasan and L M Zhu ldquoSolving singular boundary valueproblems of higher-order ordinary differential equations bymodified Adomian decomposition methodrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 14 no 6pp 2592ndash2596 2009

[11] A-M Wazwaz ldquoA reliable modification of Adomian decompo-sitionmethodrdquoAppliedMathematics and Computation vol 102no 1 pp 77ndash86 1999

[12] A-M Wazwaz ldquoAnalytical approximations and pade approx-imants for volterrarsquos population modelrdquo Applied Mathematicsand Computation vol 100 no 1 pp 13ndash25 1999

[13] A-MWazwaz ldquoA newmethod for solving singular initial valueproblems in the second-order ordinary differential equationsrdquoAppliedMathematics and Computation vol 128 no 1 pp 45ndash572002

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



chemical reaction takes place The reaction rate dependson the temperature through the Arrhenius exponentialThe Arrhenius rate equation is a mathematical expressionwhich relates the rate constant of a chemical reaction to theexponential value of the temperature The model nonlinearequations in dimensionless form are [1]

120597119884

120597119905= Le120597

2119884

1205971199092minus 1206012119884 exp(minus

120574

119879)

120597119879

120597119905=1205972119879

1205971199092+ 1205731206012119884 exp(minus

120574

119879)

(1)

where 119884 is the concentration of the gas reactant T is thetemperature Le is the Lewis number (the ratio betweenmass and heat diffusivities) 120573 is the dimensionless heatof reaction 120601 is the thermal Thiele modulus (the ratioof the time scale of the limiting transport mechanism tothe time scale of intrinsic reaction kinetics [7]) and 120574 isthe dimensionless activation energy (minimum amount ofenergy between reactant molecules for effective collisionsbetween them) The boundary conditions are

119879 (0 119905) = 119879 (1 119905) = 1 119884 (0 119905) = 1

120597119884

120597119909

10038161003816100381610038161003816100381610038161003816119909=1= 0 for 119905 gt 0

(2)

Under steady-state condition the equations become

1198892119884

1198891199092minus1206012

Le119884 exp(minus

120574

119879) = 0

1198892119879

1198891199092+ 1205731206012119884 exp(minus

120574

119879) = 0

(3)

with boundary conditions

119879 = 1 119884 = 1 at 119909 = 0

119879 = 1119889119884

119889119909= 0 at 119909 = 1

(4)

3 Analytical Solution of Nonlinear Dynamicsof a Self-Igniting Reaction-DiffusionSystem under Steady-State Condition UsingModified Adomian Decomposition Method

In the recent years much attention is devoted to the applica-tion of theAdomian decompositionmethod to the solution ofvarious scientific models [8] An efficient modification of thestandard Adomian decomposition method for solving initialvalue problem in the second order partial differential equa-tion yields the MADM The MADM without linearizationperturbation transformation or discretization gives an ana-lytical solution in terms of a rapidly convergent infinite powerseries with easily computable terms The results show thatthe rate of convergence of modified Adomian decompositionmethod is higher than standard Adomian decompositionmethod [9ndash13] Using this method (see Appendix A) we

1206012 = 1000

1206012 = 5000

1206012 = 10000

1206012 = 20000

1206012 = 25000

1206012 = 30000

1206012 = 35000

10

1

0807060504

088

086030201 09

09

092

094

096

098


Dim

ensio

nles

s con

cent

ratio

n (Y

)

1206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206012 = 1000

120601222222222 = 5000

1206012 = 10000

1206012 = 20000

1206012 = 25000

1206011206011206011206011206011206011206011206011206011206011206011206011206011206011206011206012 = 30000

1206012 = 35000


obtain an approximate analytical expression of concentrationof gas reactant (119884) and temperature (119879) (see Appendix B) asfollows

119884 (119909) = 1 +1206012

Leexp (minus120574)(119909

2

2minus 119909) +

1206014

Leexp (minus2120574)

times [1

Le(1199094

24minus1199093

6+119909

3) minus

120574120573

2(1199094

12minus1199093

6+119909

6)]

(5)

119879 (119909) = 1 minus 1205731206012 exp (minus120574)(1199092

2minus119909

2) minus 1205731206014 exp (minus2120574)

times [1

Le(1199094

24minus1199093

6+119909

8) minus

120574120573

2(1199094

12minus1199093

6+119909

12)]

(6)

4 Numerical Simulation

Nonlinear diffusion equation (3) for the boundary condition(4) is also solved numerically We have used the functionpdex1 in MATLAB software to solve the initial-boundaryvalue problems for the nonlinear differential equationsnumerically This numerical solution is compared with ouranalytical results in Figures 1ndash4 Upon comparison it givesa satisfactory agreement for all values of the dimensionlessparameters 120573 120574 and 1206012TheMATLAB program is also givenin Algorithm 1

5 Discussion

Equations (5) and (6) represent the simplest form of approx-imate analytical expressions for the concentration of gasreactant and temperature for all values of parameters 120572120573 120574 and 1206012 The Thiele number (thermal) is the ratio oflayer thickness (L) and thermal diffusivity (120572) Equation (5)


1206012 = 1000

1206012 = 5000

1206012 = 10000

1206012 = 20000

1206012 = 25000

1206012 = 30000

1206012 = 35000

10

1

1005

101

102

103

104

1045

1015

1025

1035

09950807060504030201 09


Dim

ensio

nles

s tem

pera

ture

(T)


Dim

ensio

nles

s con

cent

ratio

n (Y

)

0 1

1

098

096

094

092

08801 02 03 04 05 06 07 08 09

09


120574 = 11

120574 = 10120574 = 12

120574 = 13

120574 = 115

120574 = 105 120574 = 125

120574 = 135



120574 = 135

120574 = 125

120574 = 115

120574 = 105

120574 = 13

120574 = 11 120574 = 10

120574 = 12

0 1

1

07

101

102

103

08 090605040302010995

1005

1015

1025

1035

Dim

ensio

nles

s tem

pera

ture

(T)







6 Conclusion









Appendices



119871 (119910) + 119873 (119910) = 119892 (119909) (A1)


119910 (0) = 119860 1199101015840 (0) = 119861 (A2)


4

1

09

21

1

0

05

03

085

095


sionles

s

spatial

coord

inate (x

)times104

Dim

ensio

nles

s con

cent

ratio

n (Y

)

(a)

10

12

14

1

05

0

1

09

085

095

Dimensionless



energy 120574

Dim

ensio

nles

s con

cent

ratio

n (Y

)(b)

4

1

102

104

106

2

0 0

05

098



times104

Dim

ensio

nles

s tem

pera

ture

(T)

(c)

10

12

14

1

05

0

1

102

104

106

Dimensionless



energy 120574

Dim

ensio

nles

s tem

pera

ture

(T)

(d)



119871 = 119909minus1198991198892

1198891199092119909119899119910 (A3)


119871 (119910) = 119892 (119909) minus 119873 (119910) (A4)


119871minus1 (sdot) = 119909minus119899∬119909

0

119909119899 (sdot) 119889119909 119889119909 (A5)




(A6)



119910 (119909) =infin

sum119899=0

119910119899(119909) (A7)

119873(119910) =infin

sum119899=0

119860119899 (A8)




119860119899(119909) =

1

119899

119889119899

119889120582119899119873(infin

sum119899=0

(120582119899119910119899))

1003816100381610038161003816100381610038161003816100381610038161003816120582=0 (A9)


infin

sum119899=0

119910119899(119909) = 119860 + 119871minus1119892 (119909) minus 119871

minus1

infin

sum119899=0

119860119899 (A10)



1199100(119909) = 119860 + 119871minus1119892 (119909) 119910

119899+1(119909)

= minus119871minus1 (119860119899) 119899 ge 0

(A11)

which gives

1199100(119909) = 119860 + 119871minus1119892 (119909)

1199101(119909) = minus119871

minus1 (1198600)

1199102(119909) = minus119871

minus1 (1198601)

1199103(119909) = minus119871

minus1 (1198602)

(A12)





119871 (119884 (119909)) = 119896119873 (119884 (119909))

where 119871 = 119909minus11198892

1198891199092119909

=1206012

Le119884 exp(minus

120574

119879)

(B1)

119871 (119879 (119909)) = 119896119873 (119879 (119909))

= minus1205731206012119884 exp (minus120574

119879)

(B2)


0119909(sdot)119889119909 119889119909 on


119884 (119909) = 119860119909 + 119861 + 119871minus1 (

1206012

Le119884 exp(

minus120574

119879)) (B3)


119879)) (B4)


119884 (119909) =infin

sum119899=0

119884119899(119909) (B5)

119879 (119909) =infin

sum119899=0

119879119899(119909) (B6)

119873[119884 (119909)] =infin

sum119899=0

119860119899 (B7)

where

119873[119884 (119909)] =1206012

Le119884 exp(minus

120574

119879) (B8)

119873[119879 (119909)] =infin

sum119899=0

119861119899 (B9)

where

119873[119879 (119909)] = minus1205731206012119884 exp(minus

120574

119879) (B10)


sum119899=0

119884119899(119909) = 119860119909 + 119861 + 119871

minus1 (infin

sum119899=0

119860119899) (B11)


1198840(119909) = 119860119909 + 119861 (B12)



there4 1198840= 1 (B13)


119884119899+1

(119909) = 119871minus1 (119860119899) 119899 ge 0 (B14)


1 1198842 119884119899 We


1198600= 119873 (119884

0)

1198601=

119889

119889120582119873 (1198840+ 1205821198841)

(B15)


sum119899=0

119879119899(119909) = 119862119909 + 119863 + 119871minus1 (

infin

sum119899=0

119861119899) (B16)


1198790(119909) = 119862119909 + 119863 (B17)


there4 1198790= 1 (B18)


119879119899+1

(119909) = 119871minus1 (119861119899) 119899 ge 0 (B19)


1 1198792 119879

119899 We


1198610= 119873 (119879

0)

1198611=

119889

119889120582119873 (1198790+ 1205821198791)

(B20)


1198841=1206012

Leexp (minus120574)(119909

2

2minus 119909) (B21)

1198791= minus1205731206012 exp (minus120574)(119909

2

2minus119909

2) (B22)

1198842=1206014


Le(1199094

24minus1199093

6+119909

3)

minus120574120573

2(1199094

12minus1199093

6+119909

6))

(B23)

1198792= minus1205731206014 exp (minus2120574)( 1

Le(1199094

24minus1199093

6+119909

8)

minus120574120573

2(1199094

12minus1199093

6+119909

12))

(B24)


C

See Algorithm 1

Notations

Symbols





0



0

Greek letters


0)(1205881198881199010)



01198712120572



Acknowledgments


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1206012 = 1000

1206012 = 5000

1206012 = 10000

1206012 = 20000

1206012 = 25000

1206012 = 30000

1206012 = 35000

10

1

1005

101

102

103

104

1045

1015

1025

1035

09950807060504030201 09


Dim

ensio

nles

s tem

pera

ture

(T)


Dim

ensio

nles

s con

cent

ratio

n (Y

)

0 1

1

098

096

094

092

08801 02 03 04 05 06 07 08 09

09


120574 = 11

120574 = 10120574 = 12

120574 = 13

120574 = 115

120574 = 105 120574 = 125

120574 = 135



120574 = 135

120574 = 125

120574 = 115

120574 = 105

120574 = 13

120574 = 11 120574 = 10

120574 = 12

0 1

1

07

101

102

103

08 090605040302010995

1005

1015

1025

1035

Dim

ensio

nles

s tem

pera

ture

(T)







6 Conclusion









Appendices



119871 (119910) + 119873 (119910) = 119892 (119909) (A1)


119910 (0) = 119860 1199101015840 (0) = 119861 (A2)


4

1

09

21

1

0

05

03

085

095


sionles

s

spatial

coord

inate (x

)times104

Dim

ensio

nles

s con

cent

ratio

n (Y

)

(a)

10

12

14

1

05

0

1

09

085

095

Dimensionless



energy 120574

Dim

ensio

nles

s con

cent

ratio

n (Y

)(b)

4

1

102

104

106

2

0 0

05

098



times104

Dim

ensio

nles

s tem

pera

ture

(T)

(c)

10

12

14

1

05

0

1

102

104

106

Dimensionless



energy 120574

Dim

ensio

nles

s tem

pera

ture

(T)

(d)



119871 = 119909minus1198991198892

1198891199092119909119899119910 (A3)


119871 (119910) = 119892 (119909) minus 119873 (119910) (A4)


119871minus1 (sdot) = 119909minus119899∬119909

0

119909119899 (sdot) 119889119909 119889119909 (A5)




(A6)



119910 (119909) =infin

sum119899=0

119910119899(119909) (A7)

119873(119910) =infin

sum119899=0

119860119899 (A8)




119860119899(119909) =

1

119899

119889119899

119889120582119899119873(infin

sum119899=0

(120582119899119910119899))

1003816100381610038161003816100381610038161003816100381610038161003816120582=0 (A9)


infin

sum119899=0

119910119899(119909) = 119860 + 119871minus1119892 (119909) minus 119871

minus1

infin

sum119899=0

119860119899 (A10)



1199100(119909) = 119860 + 119871minus1119892 (119909) 119910

119899+1(119909)

= minus119871minus1 (119860119899) 119899 ge 0

(A11)

which gives

1199100(119909) = 119860 + 119871minus1119892 (119909)

1199101(119909) = minus119871

minus1 (1198600)

1199102(119909) = minus119871

minus1 (1198601)

1199103(119909) = minus119871

minus1 (1198602)

(A12)





119871 (119884 (119909)) = 119896119873 (119884 (119909))

where 119871 = 119909minus11198892

1198891199092119909

=1206012

Le119884 exp(minus

120574

119879)

(B1)

119871 (119879 (119909)) = 119896119873 (119879 (119909))

= minus1205731206012119884 exp (minus120574

119879)

(B2)


0119909(sdot)119889119909 119889119909 on


119884 (119909) = 119860119909 + 119861 + 119871minus1 (

1206012

Le119884 exp(

minus120574

119879)) (B3)


119879)) (B4)


119884 (119909) =infin

sum119899=0

119884119899(119909) (B5)

119879 (119909) =infin

sum119899=0

119879119899(119909) (B6)

119873[119884 (119909)] =infin

sum119899=0

119860119899 (B7)

where

119873[119884 (119909)] =1206012

Le119884 exp(minus

120574

119879) (B8)

119873[119879 (119909)] =infin

sum119899=0

119861119899 (B9)

where

119873[119879 (119909)] = minus1205731206012119884 exp(minus

120574

119879) (B10)


sum119899=0

119884119899(119909) = 119860119909 + 119861 + 119871

minus1 (infin

sum119899=0

119860119899) (B11)


1198840(119909) = 119860119909 + 119861 (B12)



there4 1198840= 1 (B13)


119884119899+1

(119909) = 119871minus1 (119860119899) 119899 ge 0 (B14)


1 1198842 119884119899 We


1198600= 119873 (119884

0)

1198601=

119889

119889120582119873 (1198840+ 1205821198841)

(B15)


sum119899=0

119879119899(119909) = 119862119909 + 119863 + 119871minus1 (

infin

sum119899=0

119861119899) (B16)


1198790(119909) = 119862119909 + 119863 (B17)


there4 1198790= 1 (B18)


119879119899+1

(119909) = 119871minus1 (119861119899) 119899 ge 0 (B19)


1 1198792 119879

119899 We


1198610= 119873 (119879

0)

1198611=

119889

119889120582119873 (1198790+ 1205821198791)

(B20)


1198841=1206012

Leexp (minus120574)(119909

2

2minus 119909) (B21)

1198791= minus1205731206012 exp (minus120574)(119909

2

2minus119909

2) (B22)

1198842=1206014


Le(1199094

24minus1199093

6+119909

3)

minus120574120573

2(1199094

12minus1199093

6+119909

6))

(B23)

1198792= minus1205731206014 exp (minus2120574)( 1

Le(1199094

24minus1199093

6+119909

8)

minus120574120573

2(1199094

12minus1199093

6+119909

12))

(B24)


C

See Algorithm 1

Notations

Symbols





0



0

Greek letters


0)(1205881198881199010)



01198712120572



Acknowledgments


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















Appendices



119871 (119910) + 119873 (119910) = 119892 (119909) (A1)


119910 (0) = 119860 1199101015840 (0) = 119861 (A2)


4

1

09

21

1

0

05

03

085

095


sionles

s

spatial

coord

inate (x

)times104

Dim

ensio

nles

s con

cent

ratio

n (Y

)

(a)

10

12

14

1

05

0

1

09

085

095

Dimensionless



energy 120574

Dim

ensio

nles

s con

cent

ratio

n (Y

)(b)

4

1

102

104

106

2

0 0

05

098



times104

Dim

ensio

nles

s tem

pera

ture

(T)

(c)

10

12

14

1

05

0

1

102

104

106

Dimensionless



energy 120574

Dim

ensio

nles

s tem

pera

ture

(T)

(d)



119871 = 119909minus1198991198892

1198891199092119909119899119910 (A3)


119871 (119910) = 119892 (119909) minus 119873 (119910) (A4)


119871minus1 (sdot) = 119909minus119899∬119909

0

119909119899 (sdot) 119889119909 119889119909 (A5)




(A6)



119910 (119909) =infin

sum119899=0

119910119899(119909) (A7)

119873(119910) =infin

sum119899=0

119860119899 (A8)




119860119899(119909) =

1

119899

119889119899

119889120582119899119873(infin

sum119899=0

(120582119899119910119899))

1003816100381610038161003816100381610038161003816100381610038161003816120582=0 (A9)


infin

sum119899=0

119910119899(119909) = 119860 + 119871minus1119892 (119909) minus 119871

minus1

infin

sum119899=0

119860119899 (A10)



1199100(119909) = 119860 + 119871minus1119892 (119909) 119910

119899+1(119909)

= minus119871minus1 (119860119899) 119899 ge 0

(A11)

which gives

1199100(119909) = 119860 + 119871minus1119892 (119909)

1199101(119909) = minus119871

minus1 (1198600)

1199102(119909) = minus119871

minus1 (1198601)

1199103(119909) = minus119871

minus1 (1198602)

(A12)





119871 (119884 (119909)) = 119896119873 (119884 (119909))

where 119871 = 119909minus11198892

1198891199092119909

=1206012

Le119884 exp(minus

120574

119879)

(B1)

119871 (119879 (119909)) = 119896119873 (119879 (119909))

= minus1205731206012119884 exp (minus120574

119879)

(B2)


0119909(sdot)119889119909 119889119909 on


119884 (119909) = 119860119909 + 119861 + 119871minus1 (

1206012

Le119884 exp(

minus120574

119879)) (B3)


119879)) (B4)


119884 (119909) =infin

sum119899=0

119884119899(119909) (B5)

119879 (119909) =infin

sum119899=0

119879119899(119909) (B6)

119873[119884 (119909)] =infin

sum119899=0

119860119899 (B7)

where

119873[119884 (119909)] =1206012

Le119884 exp(minus

120574

119879) (B8)

119873[119879 (119909)] =infin

sum119899=0

119861119899 (B9)

where

119873[119879 (119909)] = minus1205731206012119884 exp(minus

120574

119879) (B10)


sum119899=0

119884119899(119909) = 119860119909 + 119861 + 119871

minus1 (infin

sum119899=0

119860119899) (B11)


1198840(119909) = 119860119909 + 119861 (B12)



there4 1198840= 1 (B13)


119884119899+1

(119909) = 119871minus1 (119860119899) 119899 ge 0 (B14)


1 1198842 119884119899 We


1198600= 119873 (119884

0)

1198601=

119889

119889120582119873 (1198840+ 1205821198841)

(B15)


sum119899=0

119879119899(119909) = 119862119909 + 119863 + 119871minus1 (

infin

sum119899=0

119861119899) (B16)


1198790(119909) = 119862119909 + 119863 (B17)


there4 1198790= 1 (B18)


119879119899+1

(119909) = 119871minus1 (119861119899) 119899 ge 0 (B19)


1 1198792 119879

119899 We


1198610= 119873 (119879

0)

1198611=

119889

119889120582119873 (1198790+ 1205821198791)

(B20)


1198841=1206012

Leexp (minus120574)(119909

2

2minus 119909) (B21)

1198791= minus1205731206012 exp (minus120574)(119909

2

2minus119909

2) (B22)

1198842=1206014


Le(1199094

24minus1199093

6+119909

3)

minus120574120573

2(1199094

12minus1199093

6+119909

6))

(B23)

1198792= minus1205731206014 exp (minus2120574)( 1

Le(1199094

24minus1199093

6+119909

8)

minus120574120573

2(1199094

12minus1199093

6+119909

12))

(B24)


C

See Algorithm 1

Notations

Symbols





0



0

Greek letters


0)(1205881198881199010)



01198712120572



Acknowledgments


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










4

1

09

21

1

0

05

03

085

095


sionles

s

spatial

coord

inate (x

)times104

Dim

ensio

nles

s con

cent

ratio

n (Y

)

(a)

10

12

14

1

05

0

1

09

085

095

Dimensionless



energy 120574

Dim

ensio

nles

s con

cent

ratio

n (Y

)(b)

4

1

102

104

106

2

0 0

05

098



times104

Dim

ensio

nles

s tem

pera

ture

(T)

(c)

10

12

14

1

05

0

1

102

104

106

Dimensionless



energy 120574

Dim

ensio

nles

s tem

pera

ture

(T)

(d)



119871 = 119909minus1198991198892

1198891199092119909119899119910 (A3)


119871 (119910) = 119892 (119909) minus 119873 (119910) (A4)


119871minus1 (sdot) = 119909minus119899∬119909

0

119909119899 (sdot) 119889119909 119889119909 (A5)




(A6)



119910 (119909) =infin

sum119899=0

119910119899(119909) (A7)

119873(119910) =infin

sum119899=0

119860119899 (A8)




119860119899(119909) =

1

119899

119889119899

119889120582119899119873(infin

sum119899=0

(120582119899119910119899))

1003816100381610038161003816100381610038161003816100381610038161003816120582=0 (A9)


infin

sum119899=0

119910119899(119909) = 119860 + 119871minus1119892 (119909) minus 119871

minus1

infin

sum119899=0

119860119899 (A10)



1199100(119909) = 119860 + 119871minus1119892 (119909) 119910

119899+1(119909)

= minus119871minus1 (119860119899) 119899 ge 0

(A11)

which gives

1199100(119909) = 119860 + 119871minus1119892 (119909)

1199101(119909) = minus119871

minus1 (1198600)

1199102(119909) = minus119871

minus1 (1198601)

1199103(119909) = minus119871

minus1 (1198602)

(A12)





119871 (119884 (119909)) = 119896119873 (119884 (119909))

where 119871 = 119909minus11198892

1198891199092119909

=1206012

Le119884 exp(minus

120574

119879)

(B1)

119871 (119879 (119909)) = 119896119873 (119879 (119909))

= minus1205731206012119884 exp (minus120574

119879)

(B2)


0119909(sdot)119889119909 119889119909 on


119884 (119909) = 119860119909 + 119861 + 119871minus1 (

1206012

Le119884 exp(

minus120574

119879)) (B3)


119879)) (B4)


119884 (119909) =infin

sum119899=0

119884119899(119909) (B5)

119879 (119909) =infin

sum119899=0

119879119899(119909) (B6)

119873[119884 (119909)] =infin

sum119899=0

119860119899 (B7)

where

119873[119884 (119909)] =1206012

Le119884 exp(minus

120574

119879) (B8)

119873[119879 (119909)] =infin

sum119899=0

119861119899 (B9)

where

119873[119879 (119909)] = minus1205731206012119884 exp(minus

120574

119879) (B10)


sum119899=0

119884119899(119909) = 119860119909 + 119861 + 119871

minus1 (infin

sum119899=0

119860119899) (B11)


1198840(119909) = 119860119909 + 119861 (B12)



there4 1198840= 1 (B13)


119884119899+1

(119909) = 119871minus1 (119860119899) 119899 ge 0 (B14)


1 1198842 119884119899 We


1198600= 119873 (119884

0)

1198601=

119889

119889120582119873 (1198840+ 1205821198841)

(B15)


sum119899=0

119879119899(119909) = 119862119909 + 119863 + 119871minus1 (

infin

sum119899=0

119861119899) (B16)


1198790(119909) = 119862119909 + 119863 (B17)


there4 1198790= 1 (B18)


119879119899+1

(119909) = 119871minus1 (119861119899) 119899 ge 0 (B19)


1 1198792 119879

119899 We


1198610= 119873 (119879

0)

1198611=

119889

119889120582119873 (1198790+ 1205821198791)

(B20)


1198841=1206012

Leexp (minus120574)(119909

2

2minus 119909) (B21)

1198791= minus1205731206012 exp (minus120574)(119909

2

2minus119909

2) (B22)

1198842=1206014


Le(1199094

24minus1199093

6+119909

3)

minus120574120573

2(1199094

12minus1199093

6+119909

6))

(B23)

1198792= minus1205731206014 exp (minus2120574)( 1

Le(1199094

24minus1199093

6+119909

8)

minus120574120573

2(1199094

12minus1199093

6+119909

12))

(B24)


C

See Algorithm 1

Notations

Symbols





0



0

Greek letters


0)(1205881198881199010)



01198712120572



Acknowledgments


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119910 (119909) =infin

sum119899=0

119910119899(119909) (A7)

119873(119910) =infin

sum119899=0

119860119899 (A8)




119860119899(119909) =

1

119899

119889119899

119889120582119899119873(infin

sum119899=0

(120582119899119910119899))

1003816100381610038161003816100381610038161003816100381610038161003816120582=0 (A9)


infin

sum119899=0

119910119899(119909) = 119860 + 119871minus1119892 (119909) minus 119871

minus1

infin

sum119899=0

119860119899 (A10)



1199100(119909) = 119860 + 119871minus1119892 (119909) 119910

119899+1(119909)

= minus119871minus1 (119860119899) 119899 ge 0

(A11)

which gives

1199100(119909) = 119860 + 119871minus1119892 (119909)

1199101(119909) = minus119871

minus1 (1198600)

1199102(119909) = minus119871

minus1 (1198601)

1199103(119909) = minus119871

minus1 (1198602)

(A12)





119871 (119884 (119909)) = 119896119873 (119884 (119909))

where 119871 = 119909minus11198892

1198891199092119909

=1206012

Le119884 exp(minus

120574

119879)

(B1)

119871 (119879 (119909)) = 119896119873 (119879 (119909))

= minus1205731206012119884 exp (minus120574

119879)

(B2)


0119909(sdot)119889119909 119889119909 on


119884 (119909) = 119860119909 + 119861 + 119871minus1 (

1206012

Le119884 exp(

minus120574

119879)) (B3)


119879)) (B4)


119884 (119909) =infin

sum119899=0

119884119899(119909) (B5)

119879 (119909) =infin

sum119899=0

119879119899(119909) (B6)

119873[119884 (119909)] =infin

sum119899=0

119860119899 (B7)

where

119873[119884 (119909)] =1206012

Le119884 exp(minus

120574

119879) (B8)

119873[119879 (119909)] =infin

sum119899=0

119861119899 (B9)

where

119873[119879 (119909)] = minus1205731206012119884 exp(minus

120574

119879) (B10)


sum119899=0

119884119899(119909) = 119860119909 + 119861 + 119871

minus1 (infin

sum119899=0

119860119899) (B11)


1198840(119909) = 119860119909 + 119861 (B12)



there4 1198840= 1 (B13)


119884119899+1

(119909) = 119871minus1 (119860119899) 119899 ge 0 (B14)


1 1198842 119884119899 We


1198600= 119873 (119884

0)

1198601=

119889

119889120582119873 (1198840+ 1205821198841)

(B15)


sum119899=0

119879119899(119909) = 119862119909 + 119863 + 119871minus1 (

infin

sum119899=0

119861119899) (B16)


1198790(119909) = 119862119909 + 119863 (B17)


there4 1198790= 1 (B18)


119879119899+1

(119909) = 119871minus1 (119861119899) 119899 ge 0 (B19)


1 1198792 119879

119899 We


1198610= 119873 (119879

0)

1198611=

119889

119889120582119873 (1198790+ 1205821198791)

(B20)


1198841=1206012

Leexp (minus120574)(119909

2

2minus 119909) (B21)

1198791= minus1205731206012 exp (minus120574)(119909

2

2minus119909

2) (B22)

1198842=1206014


Le(1199094

24minus1199093

6+119909

3)

minus120574120573

2(1199094

12minus1199093

6+119909

6))

(B23)

1198792= minus1205731206014 exp (minus2120574)( 1

Le(1199094

24minus1199093

6+119909

8)

minus120574120573

2(1199094

12minus1199093

6+119909

12))

(B24)


C

See Algorithm 1

Notations

Symbols





0



0

Greek letters


0)(1205881198881199010)



01198712120572



Acknowledgments


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











there4 1198840= 1 (B13)


119884119899+1

(119909) = 119871minus1 (119860119899) 119899 ge 0 (B14)


1 1198842 119884119899 We


1198600= 119873 (119884

0)

1198601=

119889

119889120582119873 (1198840+ 1205821198841)

(B15)


sum119899=0

119879119899(119909) = 119862119909 + 119863 + 119871minus1 (

infin

sum119899=0

119861119899) (B16)


1198790(119909) = 119862119909 + 119863 (B17)


there4 1198790= 1 (B18)


119879119899+1

(119909) = 119871minus1 (119861119899) 119899 ge 0 (B19)


1 1198792 119879

119899 We


1198610= 119873 (119879

0)

1198611=

119889

119889120582119873 (1198790+ 1205821198791)

(B20)


1198841=1206012

Leexp (minus120574)(119909

2

2minus 119909) (B21)

1198791= minus1205731206012 exp (minus120574)(119909

2

2minus119909

2) (B22)

1198842=1206014


Le(1199094

24minus1199093

6+119909

3)

minus120574120573

2(1199094

12minus1199093

6+119909

6))

(B23)

1198792= minus1205731206014 exp (minus2120574)( 1

Le(1199094

24minus1199093

6+119909

8)

minus120574120573

2(1199094

12minus1199093

6+119909

12))

(B24)


C

See Algorithm 1

Notations

Symbols





0



0

Greek letters


0)(1205881198881199010)



01198712120572



Acknowledgments


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article Analytical Solution of Nonlinear …downloads.hindawi.com/journals/ijce/2014/825797.pdfResearch Article Analytical Solution of Nonlinear Dynamics of a Self-Igniting