Research Article Approximate Analytical Solutions of the...

Research ArticleApproximate Analytical Solutions ofthe Fractional-Order Brusselator System Usingthe Polynomial Least Squares Method

Constantin Bota and Bogdan Csruntu

Department of Mathematics ldquoPolitehnicardquo University of Timisoara Piata Victoriei 2 300006 Timisoara Romania

Correspondence should be addressed to Bogdan Caruntu bogdancaruntuuptro

Received 31 July 2014 Revised 11 November 2014 Accepted 11 November 2014

Academic Editor Dumitru Baleanu

Copyright copy 2015 C Bota and B Caruntu 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

The paper presents a new method called the Polynomial Least Squares Method (PLSM) PLSM allows us to compute approximateanalytical solutions for the Brusselator system which is a fractional-order system of nonlinear differential equations

1 Introduction

In recent years in many practical applications in variousfields such as physicsmechanics chemistry and biology (seeeg [1ndash6]) the problems being studied are modeled usingfractional nonlinear equations

Formost of such fractional nonlinear equations the exactsolutions cannot be found and as a consequence a numericalsolution or if possible an analytical approximate solution ofthese equations is sought

Due to the complexity of this type of problems a generalapproximation algorithm does not exist and thus variousapproximation methods each with its strong and weakpoints were proposed including among others

(i) the Adomian Decomposition Method [7ndash9](ii) the Homotopy Analysis Method [10ndash12](iii) the Homotopy Perturbation Method [13 14](iv) the Laplace TransformMethod [15 16](v) the Fourier TransformMethod [17](vi) the Variational Iteration Method [18ndash20]

The objective of our paper is to present the PolynomialLeast Squares Method (PLSM) which allows us to computeapproximate analytical solutions for the Brusselator system

The fractional order Brusselator system was recently studiedby several authors [21ndash23] and can be expressed as follows

We consider the following Brusselator system

1198631205721

119905119909 (119905) = 119886 minus (120583 + 1) sdot 119909 (119905) + 119909 (119905)

2

sdot 119910 (119905)

1198631205722

119905119910 (119905) = 120583 sdot 119909 (119905) minus 119909 (119905)

2

sdot 119910 (119905)

(1)

together with the initial conditions

119909 (0) = 1198881 119910 (0) = 119888

2 (2)

where 119886 gt 0 120583 gt 0 0 lt 1205721le 1 0 lt 120572

2le 1 119888

1 1198882are real

constants and119863120572

119905denotes Caputorsquos fractional derivative [15]

119863120572

119905=

1

Γ (1 minus 120572)sdot int

119905

0

(119905 minus 120577)minus120572

sdot 1199091015840

(120577) 119889120577 0 lt 120572 le 1 (3)

In the next section we will introduce PLSM for theBrusselator system and in the third section we will comparethe approximate solutions obtained by using PLSM with theapproximate solutions from [20] The computations showthat the approximations computed by using our methodpresent an error smaller than the error of the correspondingsolutions from [20]

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2015 Article ID 450235 5 pageshttpdxdoiorg1011552015450235

2 Advances in Mathematical Physics

2 The Polynomial Least Squares Method

For problem ((1) (2)) we consider the remainder operators

D1(119909 (119905) 119910 (119905)) = 119863

1205721

119905119909 (119905)

minus [119886 minus (120583 + 1) sdot 119909 (119905) + 119909 (119905)2

sdot 119910 (119905)]

D2(119909 (119905) 119910 (119905)) = 119863

1205722

119905119909 (119905) minus [120583 sdot 119909 (119905) minus 119909 (119905)

2

sdot 119910 (119905)]

(4)

We will find approximate polynomial solutions 119909(119905) 119910(119905)of problem ((1) (2)) on the [0 119887] interval solutions whichsatisfy the following conditions

10038161003816100381610038161003816D119895(119909 (119905) 119910 (119905))

10038161003816100381610038161003816lt 120598 119895 = 1 2 120598 gt 0 (5)

119909 (0) = 1198881 119910 (0) = 119888

2 (6)

Definition 1 One calls an 120598-approximate polynomial solutionof problem ((1) (2)) an approximate polynomial solution(119909(119905) 119910(119905)) satisfying relations ((5) (6))

Definition 2 One calls a weak 120575-approximate polynomialsolution of problem ((1) (2)) an approximate polynomialsolution (119909(119905) 119910(119905)) satisfying the relations

int

119887

0

D2

119895(119909 (119905) 119910 (119905)) 119889119905 le 120575 119895 = 1 2 (7)

together with initial conditions (6)

Definition 3 One considers the sequence of polynomials119875119895

119898(119905) = 119886

119895

0+ 119886119895

1119905 + sdot sdot sdot + 119886

119895

119898119905119898 119886119895119894isin R 119894 = 0 1 119898 119895 = 1 2

satisfying the conditions

1198751

119898(0) = 119888

1 1198752

119898(0) = 119888

2 119898 gt 1 119898 isin N (8)

One calls the sequence of polynomials 119875119895119898(119905) convergent

to the solution of problem ((1) (2)) if lim119898rarrinfin

D119895(1198751

119898(119905)

1198752

119898(119905)) = 0

We will find weak 120598-polynomial solutions of the type

119909 (119905) =

119898

sum

119896=0

1198891

119896sdot 119905119896

119910 (119905) =

119898

sum

119896=0

1198892

119896sdot 119905119896

119898 gt 1 (9)

where the constants 119889119895

0 119889119895

1 119889

119895

119898 119895 = 1 2 are calculated

using the steps outlined as follows

(i) We attach to problem ((1) (2)) the following realfunctional

119869 (1198891

2 119889

1

119898 1198892

2 119889

2

119898) =

2

sum

119895=1

int

119887

0

D2

119895(119909 (119905) 119910 (119905)) 119889119905 (10)

where 11988910 1198892

0are computed as functions of 1198891

1 1198891

2

1198891

119898 1198892

1 1198892

2 119889

2

119898by using initial conditions (6)

(ii) We computes the values of 1198891

1 1198891

2 119889

1

119898 1198892

1 1198892

2

1198892

119898as the values which give the minimum of func-

tional (10) and the values of 1198891

0 1198892

0again as functions

of 1198891

1 1198891

2 119889

1

119898 1198892

1 1198892

2 119889

2

119898by using the initial

conditions(iii) Using the constants 119889

1

0 1198891

1 119889

1

119898 1198892

0 1198892

1 119889

2

119898thus

determined we consider the polynomials

1198791

119898(119905) =

119898

sum

119896=0

1198891

119896sdot 119905119896

1198792

119898(119905) =

119898

sum

119896=0

1198892

119896sdot 119905119896

119898 gt 1 (11)

The following convergence theorem holds

Theorem 4 The necessary condition for problem ((1) (2))to admit sequences of polynomials 119875

119895

119898(119905) convergent to the

solution of this problem is

lim119898rarrinfin

int

119887

0

D2

119895(1198791

119898(119905) 1198792

119898(119905)) 119889119905 = 0 (12)

Moreover forall120598 gt 0 exist1198980

isin N such that forall119898 isin N 119898 gt

1198980 it follows that 119879119895

119898(119905) 119895 = 1 2 are weak 120598-approximate

polynomial solutions of problem ((1) (2))

Proof Based on the way the coefficients of the polynomials119879119895

119898(119905) are computed and taking into account relations ((9)ndash

(11)) the following inequality holds

0 le int

119887

0

D2

119895(1198791

119898(119905) 1198792

119898(119905)) 119889119905

le int

119887

0

119899

sum

119895=1

D2

119895(1198751

119898(119905) 1198752

119898(119905)) 119889119905 forall119898 isin N

(13)

It follows that

0 le lim119898rarrinfin

int

119887

0

D2

119895(1198791

119898(119905) 1198792

119898(119905)) 119889119905

le lim119898rarrinfin

int

119887

0

119899

sum

119895=1

D2

119895(1198751

119898(119905) 1198752

119898(119905)) 119889119905 = 0 forall119898 isin N

(14)

We obtain

lim119898rarrinfin

int

119887

0

D2

119895(1198791

119898(119905) 1198792

119898(119905)) 119889119905 = 0 (15)

From this limit we obtain that forall120598 gt 0 exist1198980

isin N suchthat forall119898 isin N 119898 gt 119898

0 it follows that 119879119895

119898(119905) are weak 120598-

approximate polynomial solutions of problem ((1) (2)) 119895 =

1 2

Remark 5 Any 120598-approximate polynomial solutions of prob-lem ((1) (2)) are also weak 120598

2

sdot 119887-approximate polynomialsolutions but the opposite is not always true It follows thatthe set of weak approximate solutions of problem ((1) (2))also contains the approximate solutions of the system

Advances in Mathematical Physics 3

02 04 06 08 10

04

06

08

10

x(t) VIMx(t) PLSM

Figure 1 Approximations for 119909(119905) in the case 1205721= 1205722= 098

Taking into account the above remark in order to find120598-approximate polynomial solutions of problem ((1) (2)) byPLSM we will first determine weak approximate polynomialsolutions 119909(119905) 119910(119905) If |D

119895(119909(119905) 119910(119905))| lt 120598 119895 = 1 2 then

119909(119905) 119910(119905) are also 120598-approximate polynomial solutions of thesystem

3 Application the Fractional-OrderBrusselator System

We consider the following fractional-order Brusselator sys-tem [20]

1198631205721

119905119909 (119905) = minus2 sdot 119909 (119905) + 119909 (119905)

2

sdot 119910 (119905)

1198631205722

119905119910 (119905) = 119909 (119905) minus 119909 (119905)

2

sdot 119910 (119905)

(16)


119909 (0) = 1 119910 (0) = 1 (17)

In [20] approximate solutions of (17) are computed usingthe Variational Iteration Method (VIM) for the case 120572

1=

1205722

= 098 Also a comparison with numerical solutions ispresented for the particular case 120572

1= 1205722= 1 illustrating the

applicability of the method

31 The Case 1205721= 1205722= 098 For the case 120572

1= 1205722= 098

using PLSMwith119898 = 3 we obtain the following approximatepolynomial solutions

119909PLSM (119905) = 00243682 sdot 1199053

+ 0311138 sdot 1199052

minus 108655 sdot 119905 + 1

119910PLSM (119905) = minus 0184414 sdot 1199053

+ 0333424 sdot 1199052

+ 00349127 sdot 119905 + 1

(18)

In Figures 1 and 2 we compare these approximationswith the corresponding approximations of the same ordercomputed by VIM (relations (15) in [20]) obtaining a good

105

110

115

02 04 06 08 10

y(t) VIMy(t) PLSM


01

02

03

04

05

02 04 06 08 10

Remainder of (1) VIMRemainder of (1) PLSM

Figure 3The remainders corresponding to the first equation for thecase 120572

1= 1205722= 098

agreement In Figures 3 and 4 we compare the expressionsof remainders (4) obtained by replacing the approximatesolutions back in the equations It is easy to observe that theerrors obtained by using PLSM are smaller than the onesobtained by using VIM

32 The Case 1205721

= 1205722

= 1 For the case 1205721

= 1205722

= 1using PLSMwith119898 = 3we obtain the following approximatepolynomial solutions

119909PLSM (119905) = 00750974 sdot 1199053

+ 0201028 sdot 1199052

minus 102827 sdot 119905 + 1

119910PLSM (119905) = minus 0180088 sdot 1199053

+ 0334087 sdot 1199052

+ 00271107 sdot 119905 + 1

(19)

In this case both approximations (VIM and PLSM)consist of third-order polynomials

We omitted the figures which compare our approxima-tions with the ones given by VIM since they look almost thesame as the corresponding ones from the case 120572

1= 1205722= 098

However in the case 1205721= 1205722= 1 it is possible to compute

the absolute error corresponding to an approximate solution


02 04 06 08 10

minus005

minus010

minus015

minus020


Figure 4The remainders corresponding to the second equation forthe case 120572

1= 1205722= 098

02 04 06 08

02

04

06

08

10

Absolute error of x(t) VIMAbsolute error of x(t) PLSM

Figure 5The absolute errors corresponding to 119909(119905) for the case1205721=

1205722= 1

as the difference in absolute value between the approximatesolution and the numerical solution (in this case computedby using the Wolfram Mathematica software)

Figures 5 and 6 present the comparison between the abso-lute errors corresponding to the approximate solutions from[20] obtained by VIM and the absolute errors correspondingto our approximate solutions

Again it is easy to observe that the errors obtained byusing PLSMare smaller than the ones obtained by usingVIM

4 Conclusions

In this paper we present the Polynomial Least SquaresMethod which is a relatively straightforward and efficientmethod to compute approximate solutions for the fractional-order Brusselator system

The comparison with previous results illustrates theaccuracy of the method since we were able to compute moreprecise approximations than the previously computed ones

In closing we mention the fact that due to the nature ofthemethod it is relatively easy to extendPLSM for the general

0002

0004

0006

0008

0010

0012

0014

02 04 06 08 10

Absolute error of y(t) VIMAbsolute error of y(t) PLSM

Figure 6The absolute errors corresponding to119910(119905) for the case1205721=

1205722= 1

case of fractional systems of 119899 ge 3 nonlinear differentialequations

Conflict of Interests

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

References

[1] R Hilfer Applications of Fractional Calculus in Physics Aca-demic Press Orlando Fla USA 1999

[2] A M Spasic and M P Lazarevic ldquoElectroviscoelasticity ofliquidliquid interfaces fractional-order modelrdquo Journal ofColloid and Interface Science vol 282 no 1 pp 223ndash230 2005

[3] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics A Carpinteri and FMainardi Eds pp 291ndash348 Springer NewYork NYUSA 1997

[4] A Carpinteri and F Mainardi Fractals and Fractional Calculusin Continuum Mechanics Springer New York NY USA 1997

[5] R Goreno and F Mainardi ldquoFractional calculus integral anddifferential equations of fractional orderrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics A Carpinteri and FMainardi Eds pp 223ndash276 SpringerNewYorkNYUSA 1997

[6] AM Nagy and NH Sweilam ldquoAn efficient method for solvingfractional HodgkinndashHuxley modelrdquo Physics Letters A vol 378no 30-31 pp 1980ndash1984 2014

[7] V Daftardar-Gejji and H Jafari ldquoAdomian decomposition atool for solving a system of fractional differential equationsrdquoJournal of Mathematical Analysis and Applications vol 301 no2 pp 508ndash518 2005

[8] H Jafari and V Daftardar-Gejji ldquoSolving a system of nonlinearfractional differential equations using Adomian decomposi-tionrdquo Journal of Computational and Applied Mathematics vol196 no 2 pp 644ndash651 2006

[9] M Zurigat ldquoSolving nonlinear fractional differential equationusing a multi-step Laplace adomian decomposition methodrdquoAnnals of the University of Craiova Mathematics and ComputerScience Series vol 39 no 2 pp 200ndash210 2012


[10] H Jafari and S Seifi ldquoHomotopy analysis method for solvinglinear and nonlinear fractional diffusion-wave equationrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 5 pp 2006ndash2012 2009

[11] L Song and H Zhang ldquoApplication of homotopy analysismethod to fractional KdV-Burgers-Kuramoto equationrdquo Phy-sics Letters A vol 367 no 1-2 pp 88ndash94 2007

[12] M Zurigat S Momani Z Odibat and A Alawneh ldquoThehomotopy analysis method for handling systems of fractionaldifferential equationsrdquoAppliedMathematical Modelling vol 34no 1 pp 24ndash35 2010

[13] Z Odibat and S Momani ldquoModified homotopy perturbationmethod application to quadratic Riccati differential equationof fractional orderrdquo Chaos Solitons and Fractals vol 36 no 1pp 167ndash174 2008

[14] S Momani and Z Odibat ldquoNumerical approach to differentialequations of fractional orderrdquo Journal of Computational andApplied Mathematics vol 207 no 1 pp 96ndash110 2007

[15] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[16] K S Miller and B Ross An Introduction to the Fractional Cal-culus and Fractional Differential Equations Wiley New YorkNY USA 1993

[17] S Kemle and H Beyer ldquoGlobal and causal solutions of frac-tional differential equationsrdquo in Proceedings of the 2nd Interna-tional Workshop on Transform Methods and Special Functions(SCTP 97) pp 210ndash216 Singapore 1997

[18] S Dixit O P Singh and S Kumar ldquoAn analytic algorithm forsolving system of fractional differential equationsrdquo Journal ofModernMethods in Numerical Mathematics vol 1 no 1 pp 12ndash26 2010

[19] X-J Yang and D Baleanu ldquoFractal heat conduction problemsolved by local fractional variation iteration methodrdquo ThermalScience vol 17 no 2 pp 625ndash628 2013

[20] H Jafari A Kadem and D Baleanu ldquoVariational iterationmethod for a fractional-order Brusselator systemrdquoAbstract andApplied Analysis vol 2014 Article ID 496323 6 pages 2014

[21] Y Wang and C Li ldquoDoes the fractional Brusselator withefficient dimension less than 1 have a limit cyclerdquoPhysics LettersA vol 363 no 5-6 pp 414ndash419 2007

[22] V Gafiychuk and B Datsko ldquoStability analysis and limit cyclein fractional system with Brusselator nonlinearitiesrdquo PhysicsLetters Section A General Atomic and Solid State Physics vol372 no 29 pp 4902ndash4904 2008

[23] S Kumar Y Khan and A Yildirim ldquoA mathematical modelingarising in the chemical systems and its approximate numericalsolutionrdquo Asia-Pacific Journal of Chemical Engineering vol 7no 6 pp 835ndash840 2012

Submit your manuscripts athttpwwwhindawicom

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society



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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


2 The Polynomial Least Squares Method

For problem ((1) (2)) we consider the remainder operators

D1(119909 (119905) 119910 (119905)) = 119863

1205721

119905119909 (119905)

minus [119886 minus (120583 + 1) sdot 119909 (119905) + 119909 (119905)2

sdot 119910 (119905)]

D2(119909 (119905) 119910 (119905)) = 119863

1205722

119905119909 (119905) minus [120583 sdot 119909 (119905) minus 119909 (119905)

2

sdot 119910 (119905)]

(4)

We will find approximate polynomial solutions 119909(119905) 119910(119905)of problem ((1) (2)) on the [0 119887] interval solutions whichsatisfy the following conditions

10038161003816100381610038161003816D119895(119909 (119905) 119910 (119905))

10038161003816100381610038161003816lt 120598 119895 = 1 2 120598 gt 0 (5)

119909 (0) = 1198881 119910 (0) = 119888

2 (6)

Definition 1 One calls an 120598-approximate polynomial solutionof problem ((1) (2)) an approximate polynomial solution(119909(119905) 119910(119905)) satisfying relations ((5) (6))

Definition 2 One calls a weak 120575-approximate polynomialsolution of problem ((1) (2)) an approximate polynomialsolution (119909(119905) 119910(119905)) satisfying the relations

int

119887

0

D2

119895(119909 (119905) 119910 (119905)) 119889119905 le 120575 119895 = 1 2 (7)

together with initial conditions (6)

Definition 3 One considers the sequence of polynomials119875119895

119898(119905) = 119886

119895

0+ 119886119895

1119905 + sdot sdot sdot + 119886

119895

119898119905119898 119886119895119894isin R 119894 = 0 1 119898 119895 = 1 2

satisfying the conditions

1198751

119898(0) = 119888

1 1198752

119898(0) = 119888

2 119898 gt 1 119898 isin N (8)

One calls the sequence of polynomials 119875119895119898(119905) convergent

to the solution of problem ((1) (2)) if lim119898rarrinfin

D119895(1198751

119898(119905)

1198752

119898(119905)) = 0

We will find weak 120598-polynomial solutions of the type

119909 (119905) =

119898

sum

119896=0

1198891

119896sdot 119905119896

119910 (119905) =

119898

sum

119896=0

1198892

119896sdot 119905119896

119898 gt 1 (9)

where the constants 119889119895

0 119889119895

1 119889

119895

119898 119895 = 1 2 are calculated

using the steps outlined as follows

(i) We attach to problem ((1) (2)) the following realfunctional

119869 (1198891

2 119889

1

119898 1198892

2 119889

2

119898) =

2

sum

119895=1

int

119887

0

D2

119895(119909 (119905) 119910 (119905)) 119889119905 (10)

where 11988910 1198892

0are computed as functions of 1198891

1 1198891

2

1198891

119898 1198892

1 1198892

2 119889

2

119898by using initial conditions (6)

(ii) We computes the values of 1198891

1 1198891

2 119889

1

119898 1198892

1 1198892

2

1198892

119898as the values which give the minimum of func-

tional (10) and the values of 1198891

0 1198892

0again as functions

of 1198891

1 1198891

2 119889

1

119898 1198892

1 1198892

2 119889

2

119898by using the initial

conditions(iii) Using the constants 119889

1

0 1198891

1 119889

1

119898 1198892

0 1198892

1 119889

2

119898thus

determined we consider the polynomials

1198791

119898(119905) =

119898

sum

119896=0

1198891

119896sdot 119905119896

1198792

119898(119905) =

119898

sum

119896=0

1198892

119896sdot 119905119896

119898 gt 1 (11)

The following convergence theorem holds

Theorem 4 The necessary condition for problem ((1) (2))to admit sequences of polynomials 119875

119895

119898(119905) convergent to the

solution of this problem is

lim119898rarrinfin

int

119887

0

D2

119895(1198791

119898(119905) 1198792

119898(119905)) 119889119905 = 0 (12)

Moreover forall120598 gt 0 exist1198980

isin N such that forall119898 isin N 119898 gt

1198980 it follows that 119879119895

119898(119905) 119895 = 1 2 are weak 120598-approximate

polynomial solutions of problem ((1) (2))

Proof Based on the way the coefficients of the polynomials119879119895

119898(119905) are computed and taking into account relations ((9)ndash

(11)) the following inequality holds

0 le int

119887

0

D2

119895(1198791

119898(119905) 1198792

119898(119905)) 119889119905

le int

119887

0

119899

sum

119895=1

D2

119895(1198751

119898(119905) 1198752

119898(119905)) 119889119905 forall119898 isin N

(13)

It follows that

0 le lim119898rarrinfin

int

119887

0

D2

119895(1198791

119898(119905) 1198792

119898(119905)) 119889119905

le lim119898rarrinfin

int

119887

0

119899

sum

119895=1

D2

119895(1198751

119898(119905) 1198752

119898(119905)) 119889119905 = 0 forall119898 isin N

(14)

We obtain

lim119898rarrinfin

int

119887

0

D2

119895(1198791

119898(119905) 1198792

119898(119905)) 119889119905 = 0 (15)

From this limit we obtain that forall120598 gt 0 exist1198980

isin N suchthat forall119898 isin N 119898 gt 119898

0 it follows that 119879119895

119898(119905) are weak 120598-

approximate polynomial solutions of problem ((1) (2)) 119895 =

1 2

Remark 5 Any 120598-approximate polynomial solutions of prob-lem ((1) (2)) are also weak 120598

2

sdot 119887-approximate polynomialsolutions but the opposite is not always true It follows thatthe set of weak approximate solutions of problem ((1) (2))also contains the approximate solutions of the system


02 04 06 08 10

04

06

08

10

x(t) VIMx(t) PLSM



119895(119909(119905) 119910(119905))| lt 120598 119895 = 1 2 then




1198631205721

119905119909 (119905) = minus2 sdot 119909 (119905) + 119909 (119905)

2

sdot 119910 (119905)

1198631205722

119905119910 (119905) = 119909 (119905) minus 119909 (119905)

2

sdot 119910 (119905)

(16)


119909 (0) = 1 119910 (0) = 1 (17)


1=

1205722





1= 1205722= 098


119909PLSM (119905) = 00243682 sdot 1199053

+ 0311138 sdot 1199052

minus 108655 sdot 119905 + 1

119910PLSM (119905) = minus 0184414 sdot 1199053

+ 0333424 sdot 1199052

+ 00349127 sdot 119905 + 1

(18)


105

110

115

02 04 06 08 10

y(t) VIMy(t) PLSM


01

02

03

04

05

02 04 06 08 10



1= 1205722= 098


32 The Case 1205721

= 1205722


= 1205722


119909PLSM (119905) = 00750974 sdot 1199053

+ 0201028 sdot 1199052

minus 102827 sdot 119905 + 1

119910PLSM (119905) = minus 0180088 sdot 1199053

+ 0334087 sdot 1199052

+ 00271107 sdot 119905 + 1

(19)



1= 1205722= 098




02 04 06 08 10

minus005

minus010

minus015

minus020



1= 1205722= 098

02 04 06 08

02

04

06

08

10



1205722= 1




4 Conclusions




0002

0004

0006

0008

0010

0012

0014

02 04 06 08 10



1205722= 1




References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










02 04 06 08 10

04

06

08

10

x(t) VIMx(t) PLSM



119895(119909(119905) 119910(119905))| lt 120598 119895 = 1 2 then




1198631205721

119905119909 (119905) = minus2 sdot 119909 (119905) + 119909 (119905)

2

sdot 119910 (119905)

1198631205722

119905119910 (119905) = 119909 (119905) minus 119909 (119905)

2

sdot 119910 (119905)

(16)


119909 (0) = 1 119910 (0) = 1 (17)


1=

1205722





1= 1205722= 098


119909PLSM (119905) = 00243682 sdot 1199053

+ 0311138 sdot 1199052

minus 108655 sdot 119905 + 1

119910PLSM (119905) = minus 0184414 sdot 1199053

+ 0333424 sdot 1199052

+ 00349127 sdot 119905 + 1

(18)


105

110

115

02 04 06 08 10

y(t) VIMy(t) PLSM


01

02

03

04

05

02 04 06 08 10



1= 1205722= 098


32 The Case 1205721

= 1205722


= 1205722


119909PLSM (119905) = 00750974 sdot 1199053

+ 0201028 sdot 1199052

minus 102827 sdot 119905 + 1

119910PLSM (119905) = minus 0180088 sdot 1199053

+ 0334087 sdot 1199052

+ 00271107 sdot 119905 + 1

(19)



1= 1205722= 098




02 04 06 08 10

minus005

minus010

minus015

minus020



1= 1205722= 098

02 04 06 08

02

04

06

08

10



1205722= 1




4 Conclusions




0002

0004

0006

0008

0010

0012

0014

02 04 06 08 10



1205722= 1




References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










02 04 06 08 10

minus005

minus010

minus015

minus020



1= 1205722= 098

02 04 06 08

02

04

06

08

10



1205722= 1




4 Conclusions




0002

0004

0006

0008

0010

0012

0014

02 04 06 08 10



1205722= 1




References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Approximate Analytical Solutions of the...

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