nonlinear science letters a mathematics, physics … science letters a: mathematics, physics and...

Nonlinear Science Letters A Mathematics, Physics and Mechanics

Vol. 4, No.4 2013

CONTENTS

New exact solution to nonlinear Burger’s equation of fractional order using the fractional complex transformation and G’/G-expansion method

Muhammad Younis and Asim Zafar--------------------------------------------------------------------91-97 An analytic algorithm for fractional order (2+1) dimensional coupled Burgers’ system model

P. K. Singh, K. Vishal, V. S. Pandey and T. Som----------------------------------------------------98-111

On the random precision rough sets model

Chun-Zhen Zhong---------------------------------------------------------------------------------------112-116

A differential equation approach for stochastic multi-objective integer problem Y. M. Abdel-Azeem------------------------------------------------------------------------------------117-125

Nonlinear Science Letters A: Mathematics, Physics and Mechanics

Editors-in-chief

Syed Tauseef Mohyud-Din, HITEC University, Taxila Cantt Pakistan [email protected] Ji-Huan He, National Engineering Laboratory of Modern Silk, Soochow University, Suzhou, China [email protected]

Associate Editors

Prof. Xu, Chen College of Mathematics and Computational Science, Shenzhen University, Shenzhen Email: [email protected]

Kai-Teng Wu College of Mathematics and Information Sciences, Neijiang Normal University, Neijiang 641112, Sichuan, China

Xin-Wei Zhou Department of Mathematics, Kunming University, China [email protected]

Editorial Board

P. Donald Ariel Trinity Western Univ, Dept Math Sci, Langley, BC V2Y 1Y1 Canada Email: [email protected]

Eerdun Buhe Department of Mathematics, Hohhot University for Nationalities, 56 Tongdao Road, Hohhot 010051, China. Email: [email protected] Mehdi Dehghan Department of Applied Mathematics, Amirkabir University of Technology, Tehran, IRAN E-mail : [email protected]

Engui Fan School of Mathematics, Fudan University, Shanghai 204433, China E-mail: [email protected]

Jian-Wen Feng College of Mathematics & Computational Science, Shenzhen University, Shenzhen 518060, China Email: [email protected]

Naeem Faraz Modern Textile Institute Donghua University, Shanghai, China Haibin Li School of Mechanical,College of Science, Inner Mongolia University of Technology China [email protected]

Jun Liu College of Mathematics and Information Science, Qujing Normal University, Yunnan 655011, China

Yong Liu School of Textiles, Tianjin Polytechnic University, Tianjin, 300160, China Email: [email protected]

Hong-Cai Ma College of Science, Donghua University, Shanghai, China Email: [email protected]

Jian-Guo Ning State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing 100081, China

Wei-Dong Song State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing 100081, China, Email: [email protected]

Ahmet Yildirim Ege Univ, Dept Math, TR-35100 Bornova, Turkey [email protected]

Elcin Yusufoglu Dumlupinar University, Department of Mathematics , Kutahya, Turkey E-mail address: [email protected]

Sheng Zhang Department of Mathematics, Bohai University, Jinzhou 121000, China [email protected]

Ting Zhong Jishou Univ, Dept Math, Zhangjiajie 427000, Hunan, Peoples R China [email protected]

Xiao-ping Wang Department of Control Science and Engineering Huazhong University of Science and Technology Wuhan,Hubei,430074, China E-mail:[email protected]

Cheng-Bo Zheng Yanshan University Email: [email protected]

Instructions for Authors Purpose and Scope

Nonlinear Science Letters A(ISSN 2076-2275) offers a fast publication for novel and frontier nonlinear sciences. It encourages the submission of new research on:

1. New nonlinear phenomena( no mathematical analysis is needed). 2. New mathematical models (differential model, fractal model, fractional differential model,

differential-difference model , fuzzy model, stochastic model, and others) for various nonlinear problems.

3. New methods (analytical method, numerical method, optimization method, statistical method, allometric method, and others) for nonlinear equations. Generally one example should be given, sometimes only the solution procedure is enough.

4. New interpretation of a nonlinear phenomenon or a new solution of a nonlinear equation. 5. New theories for explanation of any nonlinear problems

Although the journal concentrates mainly on Letters within 6 pages, mini-review articles within 20 pages are invited by the Editor and will be published from time to time.

Manuscript Submission

Manuscripts must not have been copyrighted, classified, or submitted for publication elsewhere. Manuscript should be sent to the editorial office via email [email protected] in both MS-Word file ( *.doc instead of *.docx) and pdf file. Only original contributions, which are not published in any language and are not submitted for publication elsewhere, are accepted for peer review. Manuscripts should be carefully prepared for according to the journal’s template, please visit www.nonlinearscience.com for detailed information

Illustrations

All illustrations and photographs should be clear, sharp, black and white prints, with good contrast, on glossy paper. Color illustrations are encouraged, but authors will be asked by the publisher to cover the full costs incurred in color printing.

Copyright

Papers are considered for publication on the understanding that they have not been submitted to any other publisher. The submitted works must be original and not previously published elsewhere. The copyright of papers accepted for publication becomes the property of the publisher and reproduction of any part of the paper elsewhere is not allowed without the permission of the publisher. Authors who wish to reproduce illustrations which have already been published elsewhere must obtain the written permission of the copyright holder.

Page Charges

There is no page charge. But the author has to pay color illustrations, $ 200 per page, the author is also encouraged to pay a voluntary page charge of $300 per paper. The corresponding author will receive a soft copy of his paper in PDF file. No single off-print of the paper will be provided, the author can book an issue of the journal prior to printing. Subscription Information

Nonlinear Science Letters A (ISSN 2076-2275) ( www.NonLineScience.com) is a peer-reviewed journal published by Asian Academic Publisher Limited, Room 3208, Central Plaza, 18 Harbour Road, Wanchai, Hongkong, China. Subscription Rate for 2013 volume(4 issues): $400. Subscription orders should be sent via email [email protected].

M. Younis, A. Zafar, Nonlinear Sci. Lett. A, Vol.4, No.4, 91-97, 2013

Copyright © 2013 Asian Academic Publisher Ltd. Journal Homepage: www.NonlinearScience.com

New exact solution to nonlinear Burger’s equation of fractional order using the fractional complex transformation and G’/G-expansion method Muhammad Younis1 and Asim Zafar2 Centre for Undergraduate Studies, University of the Punjab Lahore 54590, Pakistan [email protected] Abstract

In this letter, a new application to find the exact solutions of nonlinear partial time-space fractional differential equation have been discussed. Firstly, the fractional complex transformation has been implemented to convert nonlinear partial fractional differential equations into nonlinear ordinary differential equations. Afterwards, the G’/G-expansion method has been implemented, to celebrate the exact solutions of these equations, in the sense of modified Riemann-Liouville derivative. As, an application the exact solutions of time-space fractional Burgers’ equation have been discussed. Keywords: exact solutions, complex transformation, G’/G-expansion method, nonlinear Burgers’ equation, fractional calculus theory

1 Introduction

Nonlinear partial differential equations have shown a variety of applications in almost every field.

Such as in electromagnetic, acoustics, electrochemistry, cosmology, biological and material science [1, 2, 3, 5]. Fractional differential equations can be considered as the generalization form of the differential equations, as they are involved with the derivatives of any real or complex order (for details see [3]).

In literature a lot of work has been done on nonlinear partial differential equations [4, 11, 12, 13] rather than nonlinear partial differential equations (NPDEs) of fractional order. Recently, some new techniques have been developed by different authors for NPDEs of fractional order [14, 15].

In this letter, the )/( ' GG -expansion method [4] has been applied in the sense of modified Riemann-Liouville derivative to find the exact solutions of space-time fractional nonlinear Burgers’ equation, which has the following form [9]:

1, ,<0 0,> 0,=2

2

≤∂∂

+∂∂

+∂∂ βαηω β

β

β

β

α

α

tx

ux

uutu

(1)

The rest of the letter is organized as follows, in section 2 the basic definitions and properties of the

1 [email protected] 2 [email protected]

ISSN 2076-2275: Nonlinear Science Letters A- Mathematics, Physics and Mechanics

fractional calculus are considered regrading to modified Riemann-Liouville derivative. In section 3, the )/( ' GG -expansion method has been proposed to find the exact solutions for NPDEs of fractional order with the help of fractional complex transformation. As an application, the new exact solutions of nonlinear Burgers’ equation have been discussed in section 4. In last section 5, the conclusion has been drawn.

2 Preliminaries and basic definitions

In this section, the method has been applied in the sense of the Jumarie’s modified Riemann-Liouville derivative [10, 16] of order α . For this, some basic definitions and properties of the fractional calculus theory are being considered (for details see [3]). Thus, the fractional integral and derivatives can be defined following [10, 16] as:

Definition 2.1

A real function 0>),( ssf , is said to be in the space RC ∈κκ , , if there exists a real number κ>p such that )(=)( 1 sfssf p , where )(0,)(1 ∞∈Csf , and it is said to be in the space mCκ if

NmCf m ∈∈ ,κ .

Definition 2.2 The Jumarie’s modified Riemann-Liouville derivative, of order α , can be defined by the following

expression:

⎪⎩

⎪⎨⎧

≥+≤

−−−Γ

−

−∫1.1,< ,))((

1,<<0 ,(0)))(()()(1

1=)(

)(

0

nnnsf

dffsdsd

sfDnn

s

s

α

αξξξα

α

αα

Moreover, some properties for the modified Riemann-Liouville derivative have also been given as follows:

,)(1

)(1= αα

α−

−+Γ+Γ rr

s sr

rsD

),()()()(=))()(( stDsgsgDsfsgsfD sssααα +

.))()](([=)()]([=)]([ '' αααα tgsgfDsgDsgfsgfD ssgs

3. Fractional complex transformation and )/( ' GG -expansion method for nonlinear fractional partial differential equations In this section, the )/( ' GG -expansion method [6, 7] has been discussed to obtain the solutions of nonlinear fractional partial differential equations.

For this, we consider the following NPDE of fractional order:

( ), , , ,..., , , , ,... = 0, 0 < , , < 1,t s x t t t s s s s xP u D u D u D u D D u D D u D D u D D u forα β γ α α α β β β β γ α β γ (2)


where u is an unknown function and P is a polynomial of u and its partial fractional derivatives along with the involvement of higher order derivatives and nonlinear terms.

To find the exact solutions, the )/( ' GG -expansion method can be performed using the following steps. Step 1: First, we convert the NFPDE into nonlinear ordinary differential equations using fractional complex transformation introduced by Li et al. [8].

The travelling wave variable

1)(1)(1)(

= ),(=),,(+Γ

++Γ

++Γ γβα

ξξγβα MyLxKtuyxtu

(3)

where LK , and M are non-zero arbitrary constants, permits us to reduce equation (3.2) to an ODE of )(= ξuu in the following form

0.=,...),,,( ''' ′′′ uuuuP (4)

If the possibility occurs, then equation (3.4) can be integrated term by term one or more times. Step 2: Suppose that the solution of equation (3.4) can be expressed as a polynomial of )/( ' GG in the form:

0, ,=)('

=≠⎟⎟

⎠

⎞⎜⎜⎝

⎛∑−

m

i

i

m

mi GGu ααξ (5)

where i′α s are constants and )(ξG satisfies the following second order linear ordinary differential equation.

0,=)()()( '' ξμξλξ GGG ++′ (6)

with λ and μ as constants. Step 3: The homogeneous balance can be used, to determine the positive integer m , between the highest order derivatives and the nonlinear terms appearing in (3.4).

Moreover, the degree of )(ξu can be defined as muD =)]([ ξ , which gives rise to the degree of the other expressions as follows:

,= qmqdudD

q

+⎥⎦

⎤⎢⎣

⎡ξ

).(= mqsmqqduduD

sqp ++

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ξ

Therefore, the value of m can be obtained for the equation (3.5). Step 4: After the substitution of (3.5) into (3.4) and using equation (3.6), we collect all the terms with


the same order of )/( ' GG together. Equate each coefficient of the obtained polynomial to zero, yields the set of algebraic equations for μλ ,,,, MLK and )2,...,1,0,=( mii ±±±α . Step 5: After solving the system of algebraic equations, and using the equation (3.6), the variety of exact solutions can be constructed. 4 Application of Burgers’ equation

In this section, the improved )/( ' GG -expansion method have been used to construct the exact solutions for nonlinear space-time fractional Burgers’ equation (1.1).

1. ,<0 0,> 0,=2

2

≤∂∂

+∂∂

+∂∂ βαβ

β

β

β

α

α

tx

ubx

uautu

(7)

It can be observed that the fractional complex transform

1)(1)(

= ),(=),(+Γ

++Γ αβ

ξξαβ LtKxutxu

(8)

where K and L are constants, permits to reduce the equation (4.7) into an ODE. After integrating once, we have the following form:

0,=2

'22 UbKKUaLUC +++

(9)

where C is a constant of integration. Now by considering the homogeneous balance between the highest order derivatives and nonlinear term presented in the above equation, we have the following form

0, ,=)( 1

1'

10

'

1 ≠⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛−

− ααααξGG

GGu

(10)

where K,,, 101 ααα− and L are arbitrary constants. To determine these constants substitute the

equation(4.10) into (4.9), and collecting all the terms with the same power of )/( ' GG together, equating each coefficient equal to zero, yields a set of algebraic equations.

0,=21

122

1 αα bKKa −

0,=12

011 λαααα bKKaL −+

0,=)(1)2(21

12

11200 μααααα −++++ −− bKaKLC

0,=12

011 −−− ++ λαααα bKKaL


0.=21

122

1 −− + αμα KbKa

After solving these algebraic equations with the help of software Maple, yields the following results. Case 1 For the values:

0.= = ,2= ,)(= 101

22

−−− ααα

λand

aKL

abK

abLbLCaK

Straightforward simplification yields the equation (4.10), the following equation

.2=)('

aKL

GG

abKu −⎟⎟

⎠

⎞⎜⎜⎝

⎛ξ

(11)

From the equations (3.6) and (4.11), we have the following travelling wave solutions. If 0>42 μλ − , then we have the following hyperbolic solution

.)4

2(cosh)4

2(sinh

)42

(sinh)42

(cosh4=)(

22

22

2

aKL

BA

BA

abKu −

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−+−

−+−−

μλξμλξ

μλξμλξ

μλξ

If 0<42 μλ − , then we have the following trigonometric solution

aKL

BA

BA

abKu −

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−+−

−+−−−

)42

(sin)42

(cos

)42

(cos)42

(sin4=)(

22

22

2

λμξλμξ

λμξλμξ

λμξ

and if 0=42 μλ − , then we have the following solution

,2=)(aKL

BAB

abKu −⎟⎟

⎠

⎞⎜⎜⎝

⎛+ ξ

ξ

where

.1)(1)(

)(=2

+Γ+

+Γ−

αβλξ

αβ LtabL

xbLCa

Case 2 For the values:

,2= ,)(

= 11

21

2101

20

2

aKbbK

C μαα

μαααλαμα−

−

−−− +−+−


0.= )(1= 12

0 αλα andbKLa

+−

Straightforward simplification yields the equation (4.10), the following equation

,2)(1=)(1'

2−

⎟⎟⎠

⎞⎜⎜⎝

⎛++−

GG

aKbbKL

au μλξ

(12)

From the equations (4.6) and (4.12), we have the following travelling wave solutions.

If 0>42 μλ − , then we have the following hyperbolic solution

.)4

2(cosh)4

2(sinh

)42

(sinh)42

(cosh)4()(1=)(

1

22

22

21

22

−

−

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−+−

−+−−++−

μλξμλξ

μλξμλξ

μλμλξBA

BA

aKbbKL

au

If 0<42 μλ − , then we have the following trigonometric solution

1

22

22

21

22

)42

(sin)42

(cos

)42

(cos)42

(sin)4()(1=)(

−

−

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−+−

−+−−−++−

μλξμλξ

μλξμλξ

μλμλξBA

BA

aKbbKL

au

and if 0=42 μλ − , then we have the following solution

,2)(1=)(1

2−

⎟⎟⎠

⎞⎜⎜⎝

⎛+

++−ξ

μλξBA

Ba

KbbKLa

u

where

.1)(1)(

=+Γ

++Γ αβ

ξαβ LtKx

where A and B are arbitrary constants. 5 Conclusion

The )/( ' GG -expansion method has been extended to solve the nonlinear partial differential equation of fractional order, in the sense of modified Riemann-Liouville derivative. First, the fractional complex transformation has been used to convert the fractional equations into ordinary differential equation. Then )/( ' GG -expansion method has been used to find exact solutions. As an application, the new exact solutions for the space-time fractional Burgers’ equations have been found. It can be concluded that this method is very simple, reliable and propose a variety of exact solutions to NPDEs of fractional order.


References

[1] R.S. Johnson, A non-linear equation incorporating damping and dispersion, J. Fluid Mech. 42 (1970) 49-60.

[2] W.G. Glöckle, T.F. Nonnenmacher, A fractional calculus approach to self-Similar protein dynamics, Biophys. J. 68 (1995) 46-53.

[3] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. [4] M. Wang, X. Li, J. Zhang, The )/( ' GG -expansion method and travelling wave soltions of

nonlinear evolution equations in mathematical physics, Phys. Lett. A 372 (2008) 417-423. [5] J.H. He, Some applications of nonlinear fractional differential equations and their applications,

Bull. Sci. Technol. 15(2) (1999) 86â€“90. [6] Z. Bin, )/( ' GG -expansion method for solving fractional partial differential equations in the

theory of mathematical physics, Commun. Theory. Phys. 58 (2012) 623-630. [7] K.A. Gepreel, S. Omran, Exact solutions for nonlinear partial fractioanl differential equations,

Chin. Phys. B 21(11) (2012) 110204. [8] Z.-B. Li, J.-H. He, Fractional complex transform for fractional differential equations, Math.

Comput. Applications 15(5) (2010) 970-973. [9] Q. Wang, Numerical solutions for fractional KDV-Burgers equation by Adomian decomposition

method, Appl. Math. Comput. 182 (2006) 1048-1055. [10] G. Jumarie, Modified Riemann-Liouville derivative and fractional Taylor series of

nondifferentiable functions further results, Comput. Math. Appl. 51 (2006) 1367-1624. [11] Z. Feng, On explicit excact solutions to the compound Burgers-KdV equation, Phys. Lett. A

293 (2002) 57-66. [12] S.K. Liu, Z.T. Fu, S.D. Liu, Q. Zhao, Jacobi elliptic function expansion method and periodic

wave solutions of nonlinear wave equations, Phys Lett A, 289 (2001) 69-74. [13] E.J. Parkes, B.R. Duffy, An automated tanh-function method for ï¬� nding solitary wave

solutions to non-linear evolution equations, Comput Phys Comm, 98 (1996) 288-300. [14] K.A. Gepreel, The homotopy perturbation method applied to the nonlinear fractional

Kolmogorov Petrovskii Piskunov equations, Appl. Math. Lett. 24(8) (2011) 1428-1434. [15] G.-C. Wu, A fractional characteristic method for solving fractional partial differential equations,

Appl. Math. Lett. 24(7) (2011) 1046-1050. [16] G. Jumarie, Laplaceâ transform of fractional order via the Mittag Leffler function and modified

Riemannâ€“Liouville derivative, Appl. Math. Lett. 22(11) (2009) 1659-1664.

P. K. Singh, K.Vishal, V.S.Pandey and T. Som, Nonlinear Sci. Lett. A, Vol.4, No.4, 98-111


An analytic algorithm for fractional order (2+1) dimensional coupled Burgers’ system model P. K. Singh, K. Vishal, V. S. Pandey and T. Som Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi – 221005, India Email: [email protected]

Abstract

In the present article an algorithm for the solution of fractional order (2+1) dimensional coupled Burgers’ equation is obtained using the Homotopy Perturbation Method and also the efficiency and accuracy of the algorithm are discussed. The present (2+1) dimensional fractional order coupled Burgers’ equation is solved with time fractional derivatives in Caputo sense after handling the non linear terms of the equation by He’s polynomials. The approximate analytical solutions of fractional order coupled equation are obtained for different fractional Brownian motions and also for standard motion .The approximate numerical solutions for different particular cases clearly exhibit that the present method is highly powerful and effective over the some other known methods even for fractional order systems. Keywords: Fractional order coupled Burgers’ system, Homotopy perturbation method, Fractional Brownian motion, Approximate solution. 1. Introduction

The investigation of exact solutions of nonlinear partial differential equations plays an important role in the study of nonlinear physical phenomena, for example in fluid mechanics, plasma physics, atmospheric science, optical fiber communications etc. In the past few decades, there has been significant progress in the development of different methods such as the inverse scattering method ([1]–[3]), Hirota’s bilinear method [4], B cklund transformation method ([5]-[7]), Darboux transformation method [8], similarity transformation method ([9], [10]), homogeneous balance method ([11],[12]), the sine-cosine method ([13],[14]) , tanh function method ([15],[16]), Jacobi elliptic function method ([17] - [19]), Painlev´e expansion method [20] etc.

The (2+1)-dimensional Burgers’ equation was first derived by Chen and Lou [21] when they used the inner parameter dependent symmetry constraint on the Lax pairs of the well-known modified Kadovtsev –Petviashvilli equation. Almost all nonlinear physical phenomena that appear in many areas of different scientific fields such as plasma physics, solid state physics, fluid dynamics, optical fibers, mathematical biology and chemical kinetics etc can be modelled by nonlinear partial differential equations (NLPDEs). The Homotopy Perturbation Method (HPM) was first proposed by the Chinese Mathematician Ji-Huan He ([22], [23]), for solving differential and integral equations with linear and nonlinear expression, which has been the subject of extensive analytical and numerical studies. The method, which is a coupling of the traditional perturbation method and homotopy in topology, deforms the original problem continuously to a simple problem, which can be easily solved. This method, which does not require a small parameter in its equation, has a significant advantage in the sense that provides an analytical approximate solution to a wide range of linear and nonlinear


99

problems in applied sciences. The HPM yields a very rapid convergence of the solution in series in most of the cases, usually only a few iterations lead to very accurate solutions. Thus HPM is a universal one which can solve various kinds of linear and nonlinear equations. Later Zurigat et al. [24] modified HPM. Their aim was to extend the application of the He’s HPM to solve linear and nonlinear systems of partial differential equations such as the systems of coupled Burgers’ equations in one and two dimensions. The obtained results confirmed the simplicity of the method to implement. We consider the following fractional model equations for (2+1) dimensional coupled Burgers’ system which are given by

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

2 2

2 2

2 2

2 2

, , , , , , , , , ,2 , , 2 , ,

, , , , , , , , , ,2 , , 2 , , ,

0 , 1.

u x y t u x y t u x y t u x y t u x y tu x y t v x y t

t x y x yv x y t v x y t v x y t v x y t v x y t

u x y t v x y tt x y x x

α

α

β

β

α β

∂ ∂ ∂ ∂ ∂= + + +

∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂= + + +

∂ ∂ ∂ ∂ ∂< <

(1)

Earlier an attempt had been taken by Hızel and Küçükarslan [25] to solve the above system in standard order system ( 1α β= = ) using HPM. But to the best of authors’ knowledge, the solution of the equations (1) for fractional order system is first of its kind. The salient feature of the article is the graphical presentations and numerical discussion of the field variables u (x, y, t) and v (x, y, t) for various fractional Brownian motions and also for standard motion in different particular cases.

2. Basic definitions In this section, we give the related definitions and properties of the fractional calculus from the book of I. Podlubny [26].

Definition 2.1. A real function ( ) , 0h t t > is said to be in the space , ,C Rμ μ∈ if there exists a real

number ,p μ> such that ( ) ( )1 ,ph t t h t= where ( ) ( )1 0, ,h t C∈ ∞ and it is said to be in the space

nCμ if and only if ( ),

nh Cμ∈ .n N∈

Definition 2.2. The Riemann–Liouville fractional integral operator ( )tJ α of order 0,α > of a function,

, 1,h Cμ μ∈ ≥ − is defined as

( )( )

( ) ( ) ( )1

0

1 0 ,t

tJ h t t h dα

α τ τ τ αα

−

= − >∫

( ) ( )0 ,tJ h t h t=

where ( )z is the well-known Gamma function.

Some of the properties of the operator tJ α , which we need here, are as follows: For , 1, , 0h Cμ μ α β∈ ≥ − > and 1:γ ≥ −

i. ( ) ( ) ,t t tJ J h t J h tα β α β+=

ii. ( ) ( ) ,t t t tJ J h t J J h tα β β α=


100

iii. ( )( )

11tJ t tα γ α γγ

α γ+⎛ ⎞+= ⎜ ⎟+ +⎝ ⎠

.

Definition 2.3. The fractional derivative tD α of ( )h t in the Caputo’s sense is defined as

( )( )

( ) ( ) ( )1

0

1 ,t n n

tD h t t h dn

αα τ τ τα

− −= −− ∫

for 11 , , 0, .nn n n N t h Cα −− < ≤ ∈ > ∈ The following are the two basic properties of the Caputo’s fractional derivative [27]: i. Let 1, .nh C n N−∈ ∈ Then , 0tD h nα α< ≤ is well defined and 1.tD h Cα

−∈

ii. Let 1 ,n n n Nα− < ≤ ∈ and , 1.nh Cμ μ∈ ≥ − This leads to

( ) ( ) ( ) ( ) ( )1

0

0!

knk

t tk

tJ D h t h t hk

α α−

+

=

= −∑ .

3. Analysis of Homotopy Perturbation Method (HPM) The Homotopy Perturbation Method (HPM), which provides approximate analytical solution, has applied to various nonlinear problems (see [28]-[31]) successfully. In this section, we introduce the reliable algorithm to handle the nonlinear system in a realistic and efficient way. The proposed algorithm will be used to investigate the system given by:

( ) ( ) ( )1 1 1 1, 21

, ...n

i i ni

D y t a t y g t y y yα

=

= +∑

( ) ( ) ( )2 2 2 1, 21

, ...n

i i ni


=

= +∑

( ) ( ) ( )3 3 3 1, 21

, ...n

i i ni


=

= +∑ ................................................... (2)

( ) ( ) ( )1, 21

, ...n

n ni i n ni


=

= +∑

subject to the initial conditions,

( )1 10y c= , ( )2 20y c= , ----, ( )0n ny c= , (3)

where ig is a nonlinear function for each 1, 2,...i n= . Using the homotopy perturbation method we can construct the following homotopy far as

( ) ( ) ( )1, 21

, ... ,n

t i ij j i nj

D y t a t y pg t y y yα

=

− =∑ (4)

where [ ]0,1p∈ is the homotopy parameter, which takes values from zero to unity. In case 0,p = equation (4) becomes a linear equation given by:


101

( ) ( )1

,n

i ij jj

D y t a t yα

=

= ∑ (5)

and when p = 1, equation (4) turns out to be the original equation given as system (2). Assuming that the solution of the system (2) is a power series in p given by

( ) ( ) ( ) ( ) ( )0 1 2 3 42 3 4 ...i i i i i iy y py p y p y p y= + + + + + (6)

Substituting equation (6) in equation (4), and equating the terms having identical powers of p, we obtain a series of linear equations in the form:

( ) ( ) ( ) ( ) ( ) ( )0 0 00,

1: , 0

n

i ij j i ij

p D y t a t y y cα

=

= =∑

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )1 1 1 0 11

1: , , 0 0

n

i ij j i ij

p D y t a t y g t y yα

=

= + =∑ ,

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )2 2 2 0 1 22

1: , , , 0 0

n

i ij j i ij

p D y t a t y g t y y yα

=

= + =∑ ,

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )3 3 3 0 1 2 33

1

: , , , , 0 0,n

i ij j i ij

p D y t a t y g t y y y yα

=

= + =∑

where the function ( ) ( ) ( )1 2 3, , ,...i i ig g g satisfy the equation ( ) ( ) ( ) ( ) ( ) ( ) ( )( )0 1 2 3 0 1 22 3 2

1 1 1 1, ..., ...i n n ng t y py p y p y y py p y+ + + + + + +

( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )0 0 0 2 0 0 0 1 1 111 2 1 2 1 2, , ,..., , , ,..., , , , ,...,i n i n ng t y y y pg t y y y t y y y= +

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )3 0 0 0 1 1 1 2 2 221 2 1 2 1 2, , ,..., , , ,..., , , ,..., ... ;i n n np g t y y y y y y y y y+ +

Now taking 1p = in the equation (4), it gives the solution of the system (2). It is obvious that the

above linear equations are easy to solve, and the components ( ) ,kiy 0,k ≥ of the homotopy

perturbation solution can be completely determined as a series solution.

Finally, we approximate the solution ( ) ( ) ( )0

ki i

k

y t y t∞

=

= ∑ by the truncated series

( ) ( ) ( )1

0

.N

ki i

k

t y tφ−

=

= ∑ (7)

It is also useful for the system ( )2 to construct the homotopy as for 1, 2,...,i n=

( ) ( ) ( )1, 21

, ... ,n

i ij j i nj

D y t p a t y pg t y y yα

=

− =∑ (8)


102

where [ ]0,1p∈ . In this case, the term ( ) ( )0

1

n

ij jj

a t y=∑

is combined with the component ( )1

iy and the

term ( ) ( )1

1

n

ij jj

a t y=∑ is combined with the component ( )2

iy and so on. This variation reduces the number

of terms in each component and also minimizes the size of calculations. By substituting equation (6) in equation (8), we obtain the following series of linear equations as ( ) ( ) ( ) ( )0 00

,: 0 , 0i i ip D y t y cα = =

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )1 0 1 0 11

1: , , 0 0

n

i ij j i ij

p D y t a t y g t y yα

=

= + =∑

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )2 1 2 0 1 22

1: , , , 0 0

n

i ij j i ij

p D y t a t y g t y y yα

=

= + =∑

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )3 2 3 0 1 2 33

1: , , , , 0 0

n

i ij j i ij

p D y t a t y g t y y y yα

=

= + =∑

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )4 3 4 0 1 2 3 44

1: , , , , , 0 0

n

i ij j i ij

p D y t a t y g t y y y y yα

=

= + =∑

and proceeding as before the solutions can be obtained by truncating the series given in equation (7) . 4. Solutions of the (2+1) dimensional fractional order coupled Burgers’ system using HPM In this section solutions of fractional ordered coupled Burgers’ system (1) are obtained by applying the Homotopy Perturbation Method taking a different pair of initial conditions. 4.1. Solitary solution of (2+1) dimensional fractional order coupled Burgers’ system In this section we take the initial conditions as

( ) ( )0 , , 0 1 tanh 2 1u x y x y= − − + + (9)

( ) ( )0 , , 0 1 2 tanh 2 1v x y x y= + − + + (10)

during the solutions of Burgers’ equation (1). Here we construct the Homotopy as

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

2 2

2 2

2 2

2 2

, , , , , , , , , ,2 , , 2 , ,

, , , , , , , , , ,2 , , 2 , ,

u x y t u x y t u x y t u x y t u x y tp u x y t v x y t

t x y x y

u x y t v x y t v x y t v x y t v x y tp u x y t v x y t

t x y x x

α

α

β

β

⎛ ⎞∂ ∂ ∂ ∂ ∂= + + +⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠

⎛ ⎞∂ ∂ ∂ ∂ ∂= + + +⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠

(11)

Assuming

( ) ( ) ( ) ( )20 1 2, , , , , , , , ...u x y t u x y t pu x y t p u x y t= + + + (12)

and


103

( ) ( ) ( ) ( )20 1 2, , , , , , , , ...v x y t v x y t pv x y t p v x y t= + + + (13)

and using equations (9) - (11) and also comparing the like powers of p , we get

( ) ( )

( ) ( )

0 00

0 0

, , , ,: 0 ,

, , , ,0 ,

u x y t u x y tp

t t

v x y t v x y tt t

α α

α α

β β

β β

∂ ∂− =

∂ ∂

∂ ∂− =

∂ ∂

(14)

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

2 21 0 0 01

02 2

0 00

2 21 0 0 0

02 2

0 00

, , , , , , , ,: 2 , ,

, , , ,2 , , ,

, , , , , , , ,2 , ,

, , , ,2 , , ,

u x y t u x y t u x y t u x y tp u x y t

t x y xu x y t u x y t

v x y ty t

v x y t v x y t v x y t v x y tv x y t

t x y yv x y t v x y t

u x y tx t

α

α

β

β

∂ ∂ ∂ ∂= + +

∂ ∂ ∂ ∂

∂ ∂+ +

∂ ∂

∂ ∂ ∂ ∂= + +

∂ ∂ ∂ ∂

∂ ∂+ +

∂ ∂

(15)

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( )

2 22 1 1 12

02 2

0 1 01 0 1

2 22 1 1 1

02 2

01 0

, , , , , , , ,: 2 , ,

, , , , , ,2 , , 2 , , 2 , , ,

, , , , , , , ,2 , ,

, ,2 , , 2 ,


t x y xu x y t u x y t u x y t

u x y t v x y t v x y tx y y


t x y yv x y t

v x y t u xy

α

α

β

β

∂ ∂ ∂ ∂= + +

∂ ∂ ∂ ∂

∂ ∂ ∂+ + +

∂ ∂ ∂

∂ ∂ ∂ ∂= + +

∂ ∂ ∂ ∂

∂+ +

∂( ) ( ) ( ) ( )1 0

1

, , , ,, 2 , , ,

v x y t v x y ty t u x y t

x x∂ ∂

+∂ ∂

(16)

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

2 23 2 2 23

02 2

1 01 2

2 1 00 1 2

, , , , , , , ,: 2 , ,

, , , ,2 , , 2 , ,

, , , , , ,2 , , 2 , , 2 , , ,


t x y xu x y t u x y t

u x y t u x y tx x

u x y t u x y t u x y tv x y t v x y t v x y t

y y y

α

α

∂ ∂ ∂ ∂= + +

∂ ∂ ∂ ∂

∂ ∂+ +

∂ ∂∂ ∂ ∂

+ + +∂ ∂ ∂


104

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

2 23 2 2 2

02 2

1 01 2

2 1 00 1 2

, , , , , , , ,2 , ,

, , , ,2 , , 2 , ,

, , , , , ,2 , , 2 , , 2 , ,


t x y yv x y t v x y t

v x y t v x y ty y

v x y t v x y t v x y tu x y t u x y t u x y t

x x x

β

β

∂ ∂ ∂ ∂= + +

∂ ∂ ∂ ∂

∂ ∂+ +

∂ ∂

∂ ∂ ∂+ + +

∂ ∂ ∂

(17)

Then the solutions of equations (14)-(17) are obtained as

( )( )

0

0

, , 1 tanh[1 2 ],

, , 1 2 tanh[1 2 ],

u x y t x y

v x y t x y

= − − +

= + − + (18)

( ) { ( ) ( )

( ) }( )

( ) { ( ) ( )

( )}

2 21

2

2 21

2

, , [2 sech [1 2 ] 1 tanh[1 2 ] 10 sech[1 2 ] tanh[1 2 ]

4 sech[1 2 ] 1 2 tanh[1 2 ] ] ,1

, , [ 4 sech[1 2 ] 1 tanh[1 2 ] 20 sech[1 2 ] tanh[1 2 ]

8 sech[1 2 ] 1 2 tanh[1 2 ]

u x y t x y x y x y x y

tx y x y

v x y t x y x y x y x y

tx y x y

α

β

α

= − + − − + + − + − +

− − + + − ++

= − − + − − + − − + − +

+ − + + − +( )

,1 β+

(19)

( ) ( )( )( )

( ) ( )( )( )

22

2

22

2

, , [8sech[1 2 ] tanh[1 2 ] ,1 2

, , [ 16sech[1 2 ] tanh[1 2 ] ,1 2

tu x y t x y x y

tv x y t x y x y

α

β

α

β

= − + − ++

= − − + − ++ (20)

( ) { ( )( )

( )}( )

( )( )

( )( )( )

2 23

34

34

2

, , [ 16 2sech[1 2 ] tanh[1 2 ]

sech[1 2 ] 1 10 tanh[1 2 ]1 3

1 280sech[1 2 ] tanh[1 2 ] ,

1 3 1

u x y t x y x y

tx y x y

tx y x y

α

α

α

α

α α

= − − + − +

+ − + − + − ++

+⎡ ⎤+ − + − +⎣ ⎦ + +

( ) { ( )( )

( )}( )

( )( )

( )( )( )

2 23

34

34

2

, , [32 2sech[1 2 ] tanh[1 2 ]

sech[1 2 ] 1 10 tanh[1 2 ]1 3

1 2160sech[1 2 ] tanh[1 2 ]

1 3 1

v x y t x y x y

tx y x y

tx y x y

β

β

β

β

β β

= − + − +

+ − + − + − ++

+⎡ ⎤+ − − + − +⎣ ⎦ + +

(21)

Finally, the approximate series solutions of the equation (1) are obtained as


105

( )1

0

, , ( , , )N

nn

u x y t u x y t−

=

= ∑ and 1

0

( , , ) ( , , )N

nn

v x y t v x y t−

=

= ∑ (22)

0.2 0.4 0.6 0.8 1.0

　2.0

　1.5

　1.0

　0.5

0.0

0.5

Figure 1. Plot of (x, y, t)u vs. time t for different values of α at β =1, x = 1, y =1.

0.2 0.4 0.6 0.8 1.0

4

5

6

7

Figure 2. Plot of (x, y, t)u vs. time t for different values of β at α = 1, x = 1, y = 1.

Figure 3. Plot of the (x, y,0.3)u in the interval 0 ≤ x ≤ 3 and 0 ≤ y ≤ 3 at α = β = 1.

(x, y, t)u

α=1

α=0.40

α =0.25

α=0.50

β=1

β=0.25

β=0.40

β=0.50

t

v(x, y, t)

t

y

x

exactu


106

Figure 4. Plot of v(x, y,0.3) in the interval 0 ≤ x ≤ 3 and 0 ≤ y ≤ 3 at α = β = 1.

4.2. A special solution of (2+1) dimensional fractional order coupled Burgers’ system In this section we find the solutions of the equation (1) under the initial conditions

( ) ( ) ( )( ) ( ) ( )

0

0

, ,0 2 tanh 1

, ,0 1 tanh 1

u x y x y x y

v x y x y x y

= − − + − − − +

= − − + − − − + (23)

Constructing the homotopy of the equation (1) as equation (11) and assuming u (x, y, t) and v (x, y, t) similar to equation (12) and (13) and proceeding as section 4.1, we get

( ) ( ) ( )( ) ( ) ( )

0

0

, , 2 tanh 1

, , 2 tanh 1

u x y t x y x y

v x y t x y x y

= − − + − − − +

= − − + − − − + (24)

( ) { ( )( )( )( )

( )}( )

( ) { ( )( )( )( )

21

2

2

21

2

2

, , 2 1 sech[1 ] 1 tanh[1 ]

2 1 sech[1 ] 2 tanh[1 ]

4 sech[1 ] tanh[1 ] ,1

, , 2 1 sech[1 ] 1 tanh[1 ]

2 1 sech[1 ] 2 tanh[1 ]

4 sech[1 ] tanh[

u x y t x y x y x y

x y x y x y

tx y x y

v x y t x y x y x y

x y x y x y

x y

α

α

= − + − − − − − − −

+ − + − − − − − − −

+ − − − −+

= − + − − − − − − −

+ − + − − − − − − −

+ − − ( )}( )

1 ] ,1

tx yβ

β− −

+

(25)

y

x

exactv


107

( ) { ( )( )

( ) ( )( )}( )

( ) { ( )( )

( )

2

4

22

2

4

2

, , 8 6 4 4 4 tanh[1 ]

2sech[1 ] 1 tanh[1 ]

3 2 2 sech[1 ] 4 3 2 2 tanh[1 ] ,1 2

, , 8 6 4 4 4 tanh[1 ]

2sech[1 ] 1 tanh[1 ]

3 2 2 sech[1 ] 4 3 2

u x y t x y x y

x y x y x y

tx y x y x y x y

v x y t x y x y

x y x y x y

x y x y x

α

α

= − − − − −

− − − − + + + − − +

− + + − − + − + + − −+

= − − − − −

− − − − + + + − − +

− + + − − + − +( )( )}( )

2

2 tanh[1 ] ,1 2

ty x yβ

β+ − −

+

(26)

( ) ( )( )

( )( )( )

{ ( ) ( )( ) ( )( )

2 23

3

6

2 2 2

3

, , 32sech[1 ] 2 tanh[1 ] 1 sech[1 ]

2 3 2 2 tanh[1 ]1 3

8 3 2 2 2 tanh[1 ] sech[1 ] 2 20 tanh[1 ]

2sech[1 ] 12 3 2 2 55 28 84 28 28 3 2

tanh[1 ] 15 8

u x y t x y x y x y

tx y x y

x y x y x y x y

x y x y x y y x y

x y x

α

α

= − − − − − − + − −

+ − + + − − ++

+ − + + + − − + − − + − − −

− − − + + + + − + + − +

− − + − + + ( ) ( )( )( )( )

( ) ( )

}( )

( )( )( )

2 3 2 2

2 4 3 2 3 2

2 2

33

2

46 36 8 12 3 2 46 72 24

tanh[1 ] sech[1 ] 13 8 46 36 8 12 3 2

46 72 24 4 5 2 2 tanh[1 ] 8 tanh[1 ]

1 21616 tanh[1 ] ,1 3 1

y y y x y x y y

x y x y x y y y x y

x y y x y x y x y

tx yα α

α α

− + + − + + − +

− − + − − − + + − + + − + +

− + + − + + − − − − −

+− − −

+ +

( ) ( )( )

( )( )( )

{ ( ) ( )( ) ( )( )

2 23

3

6

2 2 2

3

, , 32sech[1 ] 2tanh[1 ] 1 sech[1 ]

2 3 2 2 tanh[1 ]1 3

8 3 2 2 2tanh[1 ] sech[1 ] 2 20tanh[1 ]

2sech[1 ] 12 3 2 2 55 28 84 28 28 3 2

tanh[1 ] 15 8 4

v x y t x y x y x y

tx y x y

x y x y x y x y

x y x y x y y x y

x y x

β

β

= − − − − − − + − −

+ − + + − − ++

− + + + − − + − − + − − −

− − − + + + + − + + − +

− − + − + + ( ) ( )( )( )( )

( ) ( )

}( )

( )( )( )

2 3 2 2

2 4 3 2 3 2

2 2

33

2

6 36 8 12 3 2 46 72 24

tanh[1 ] sech[1 ] 13 8 46 36 8 12 3 2

46 72 24 4 5 2 2 tanh[1 ] 8tanh[1 ]

1 21616tanh[1 ]1 3 1

y y y x y x y y

x y x y x y y y x y

x y y x y x y x y

tx yβ β

β β

− + + − + + − +

− − + − − − + + − + + − + +

− + + − + + − − − − − −

+− −

+ +

(27)

Finally, the approximate solutions of (1) are obtained from equations (22).


108

0.2 0.4 0.6 0.8 1.0

　250

　200

　150

　100

　50

Figure 5. Plot of (x, y, t)u vs. t for different values of α at β =1, x =1,y=1 .

0.2 0.4 0.6 0.8 1.0

　250

　200

　150

　100

50

Figure 6. Plot of v(x, y, t) vs. t for different values of β at α =1, x =1,y=1.

Figure 7. The exact solution of the u(x, y,0.1) in the interval 1 ≤ x ≤ 3 and 1 ≤ y ≤ 3 at α = β = 1.

x

α=0.25α=0.40

α=0.50

α=1

β=0.50

β=1

β=0.40 β=0.25

(x, y, t)u

v(x, y, t)

t

t

y

exactu


109

Figure 8. The exact solution of the v(x, y,0.1) in the interval 1 ≤ x ≤ 3 and 1 ≤ y ≤ 3 at α = β = 1.

5. Numerical results and discussion In this section numerical results of the field variables (x, y, t)u and v(x, y, t) for different particular cases are calculated for the considered problem and are displayed through Figs. 1 - 4 and Fig. 5 – 8 for the initial conditions given in section 4.1 and section 4.2 respectively. Fig.1 shows the variations of

(x, y, t)u with time t for different values of α at 1x y= = . Here (x, y, t)u increases with increase in both α and t. Fig. 2 shows that v(x, y, t) decreases with increase of β and increases with increase of t when 1x y= = . The variations of (x, y, t)u and v(x, y, t) with x and y at t = 0.3 are depicted through Fig. 3 and Fig. 4 respectively when 1α β= = . The variations of (x, y, t)u for the section 4.2 depicted through Fig.5 are similar in nature to that of the section 4.1 with higher magnitude but variations of v(x, y, t) are opposite in nature (Fig.6) compared to Fig. 2 of the earlier section. Figure 7 and Figure 8 clearly illustrate the three dimensional variations of (x, y, t)u and v(x, y, t) at 0.1t = when 1α β= = . 6. Conclusion The article investigates the analytical approximate solutions of the fractional order Burgers’ model using the reliable and effective HPM. From the present algorithm it is seen that the solutions obtained are very accurate even taking few iterations. Thus it is found that HPM becomes more effective and producing desired results quickly than the solutions obtained by using other analytical as well as numerical methods viz., Tanh, VIM methods for an extensive range of a class of linear and non-linear fractional differential equations including (2+1) dimensional fractional order Burgers’ model. The method justifies more realistic series solutions that converge quickly in case of real physical problems.

exactv

xy


110

References [1] M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, Nonlinear evolution equations of

physical significance, Phys. Rev. Lett., 31 (1973) 125–127. [2] M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, The inverse scattering transform-

Fourier analysis for non-linear problems, Stud. Appl. Math., 35 (1974) 249–315. [3] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse

Scattering Cambridge (1991). [4] X. B. Hu and W. X. Ma, Application of Hirota;s bilinear formalism to the Toeplitz lattice-

some special solution-like solutions, Phys. Lett. A, 293(2002) 161–165. [5] Hu, K. Konno and M. Wadati, Simple derivation of Bäcklund transformation from Riccati

form of inverse method, Prog. Theor. Phys., 53 (1975) 1652–1656. [6] R. M. Miupa, Bäcklund Transformation, Springer Verlag, (1979). [7] C. Rogres and W. E. Shadwick, Backlund Transformations and Their Applications, Academic

Press, New York, (1982). [8] V. B. Matveev and M. A. Salle, Darboux transformations and Solitons, Springer Verlag,

Berlin (1991). [9] E. S. Suhubi, Isovector fields and similarity solutions for general balance equations, Int. J.

Engng. Sci., 29 (1991) 133–150. [10] S. K. Attallah, M. F. El-Sabbagh and A. T. Ali, Isovector fields and similarity solutions of

Einstein vacuum equations for rotating fields, Commun. Nonl. Sci. Numerical Simulation, 12 (2007) 1153–1561.

[11] M. L. Wang, Y. B. Zhou and Z. B. Li, Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Phys. Lett. A, 21 (1996) 67– 75.

[12] L. Yang, Z. Zhu and Y. Wang, Exact solutions of nonlinear equations, Phys. Lett., A, 260 (1999) 55– 59.

[13] Z. J. Yang, Travelling wave solutions to nonlinear evolution and wave equations, J. Phys. A: Math. Gen., 27 (1994) 2837–2855.

[14] C. T. Yan, A simple transformation for nonlinear waves, Phys. Lett. A, 224 (1996) 77-84. [15] E. G. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys.

Lett. A, 277 (2000) 212–218. [16] S. A. El-Wakil and M. A. Abdou, New exact travelling wave solutions using modified

extended tanh-function method, Chaos, Soliton and Fractals, 31 (2007) 840–852. [17] Z. Y. Yan, The new extended Jacobian elliptic function expansion algorithm and its

applications in nonlinear mathematical physics equations, Comput. Phys. Commun., 153 (2003) 145–154.

[18] M. F. El-Sabbagh and A. T. Ali, New exact solutions for (3+1)-dimensional Kadomtsev- Petviashvili equation and generalized (2+1)-dimensional Boussinesq equation, Int. J. Nonl. Sci. Numerical Simulation, 41(2005) 151–162.

[19] S. A. El-Wakil, A. Elgarayhi and A. Elhanbaly, Exact periodic wave solutions for some nonlinear partial differential equations, Chaos, Soliton and Fractals, 29 (2006) 1037–1044.

[20] J. Wies, M. Tabor and G. Garnevale, The Painlevé property for partial differential equations, J. Math. Phys., 24 (1983) 522-526.

[21] L. L. Chen and S. Y. Lou, Commun. Theor. Phys., 29 (1998) 313.


111

[22] J. H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Engng., 178 (1999) 257-262.

[23] J. H. He, A coupling method of homotopy technique and a perturbation technique for non-linear problems, Int. J. Non-Linear Mech., 35 (2000) 37-43.

[24] M. Zurigat, S. Momani and Z. Odibat, The Homotopy Analysis Method for Nonlinear Partial Differential Equations, Phys. Lett. A, 365 (2007) 345-350.

[25] E. Hızel and S. Küçükarslan, Homotopy perturbation method for (2+1) dimensional coupled Burgers’ system, Nonlinear Analysis: Real World Applications, 10 (2009) 1932-1938.

[26] I. Podlubny, Fractional Differential Equations, Academic Press, New York, (1999). [27] R. Gorenflo and F. Mainardi, Fractional Calculus: Integral and Differential Equations of

Fractional Order,Springer, Wien and New York, (1997). [28] S. Abbasbandy, Homotopy perturbation method for quadratic Riccati differential Equation and

comparison with Adomian’s decomposition methods, Appl. Math. Comp., 172 (2006) 485-490.

[29] Z. Odibat Z. and S. Momani, Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order, Chaos, Solitons & Fractals, 36 (2008) 167–174.

[30] J. H. He, Homtopy perturbation method: a new non-linear analytic technique, Appl. Math.Comp., 135 (2003) 73-79.

[31] J. H. He, Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitons & Fractals, 26 (2005) 695-700.

C.Z. Zhong, Nonlinear Sci. Lett.A, Vol.4, No.4, 112-116, 2013


On the random precision rough sets model Chun-Zhen Zhong Key Laboratory of Numerical Simulation of Sichuan Province College of Mathematics and Information Science, Neijiang Normal University Neijiang, Sichuan 641112, China; Email: [email protected]

Abstract Rough sets theory is a new mathematical tools to processing uncertain knowledge. The concept of random precision rough sets model based on a set-valued function is proposed with the two parameters and the properties of the model are discussed in this paper. Some useful results about the model are obtained. Keywords： Rough sets, Random rough sets, Random precision rough sets 1. Introduction

Rough sets theory (RST), proposed by Pawlak [1,2], is an extension of set theory for the study of intelligent systems characterized by insufficient and incomplete information. It provides a systematic approach for classification of objects through an indiscernability relation. Many examples of applications of the rough sets method to process control, economics, medical diagnosis, biochemistry, environmental science, biology, chemistry psychology, conflict analysis and other fields，such as Angiulli[3], Polkowski [4] and Zhong et al [5].

A lot of meaningful extensions of Pawlak rough sets are proposed by scholars. For example, variable precision rough sets model was proposed by Ziarko [6] in order to deal with a certain degree of “inclusion” and “belong to”. Zakowski et.al [7,8] extended Pawlak rough set to the covering rough by the covering in division. Zhang [9] further presented the random rough sets model based on set-valued function of x in order to expand rough sets application space and to generalize the classical rough sets. The overlapped part’s quantitative information of elementary set-valued function and object set is taken into account; the numbers of overlapping are used to characterize or approximate object set. At the same time, degree rough sets model based on random set is provided, and a series of important conclusions are also given by Yang et al [10].

In this paper, The concept of random precision rough sets model based on set-valued function of x is proposed with the two parameters α , β and the properties of the model are discussed. Some useful results about random precision rough sets model are got. And we also testify that this model has higher precision than general random rough sets model. 2. Preliminaries


113

Definition1. Zhang [9] Let U and W be two finite non-empty sets, ( ), 2 ,UU P be a probability

space. ( ), , ,A U W F P= is called random approximation space for any set-valued

function : 2WF U → . For any 2WX ∈ , { }( ) ( )F X x U F x X= ∈ ⊆

is called the random lower approximation of X . { }( ) ( )F X x U F x X= ∈ ≠∅∩

is called the random upper approximation of X . Definition 2. Let ( ), , ,A U W F P= be a random approximation space, For each X W⊆ ，

0 1β α≤ < ≤ ,

{ }| ( ) || ( )|( ) F x XF xF X x Uα α= ∈ ≥∩

is called the α random lower approximation of X . { }| ( ) |

| ( )|( ) F x XF xF X x Uβ β= ∈ >∩

is called the β random upper approximation of X . . ( ) ( )Fpos X F Xα β

α= is called the α ,β random positive region of X .

. ( ) ( )Fneg X F Xα ββ=

is called the α ,β random negative region of X . . ( ) ( ) ( )Fbn X F X F Xα β

β α= −

is called the α ,β random boundary region of X .When ( ) ( )F X F Xβ α= , it is called definable. Otherwise, it is called random precision rough sets. 3. The properties of random precision rough sets According to the Definition1and Definition2, we can easily get the theorem1. Theorem1. Let ( ), , ,A U W F P= be a random approximation space, For each X W⊆ ， 1α= and

0β = ,then (1) 1( ) ( )F X F X= ;

(2) 0 ( ) ( )F X F X= . Note1 : This theorem express that the model should be degraded to the general random rough sets

when 1α= and 0β = . Theorem 2. Let ( ), , ,A U W F P= be a random approximation space, For each X W⊆ ,

0 1β α≤ < ≤ , then (1) ( ) ( )F Fα β∅ = ∅ =∅ , ( ) ( )F U F U Uα β= = ;

(2) 1( ) ( )F X U F U Xα α−= − − , 1( ) ( )F X U F U Xβ β−= − − ;

(3) ( ) ( )F X F Xα β⊆ ;

(4) if X Y⊆ ,then ( ) ( )F X F Yα α⊆ , ( ) ( )F X F Yβ β⊆ ;


114

(5) ( ) ( ) ( )F X Y F X F Yα α α⊆∩ ∩ ,

( ) ( ) ( )F X Y F X F Yβ β β⊇∪ ∪ ;

(6) ( ) ( ) ( )F X Y F X F Yβ β β⊆∩ ∩ ,

( ) ( ) ( )F X Y F X F Yα α α⊇∪ ∪ ; (7) if 1 2α α≤ , 1 2β β≤ , then

2 1( ) ( )F X F Xα α⊆ ,

2 1( ) ( )F X F Xβ β⊆ .

Proof. (1) According to Definition 2, we have { }| ( ) |

| ( )|( ) F xF xF x Uα α∅∅ = ∈ ≥∩

{ }0x U α= ∈ ≥ =∅ { }| ( ) |

| ( )|( ) F xF xF x Uβ β∅∅ = ∈ >∩

{ }0x U β= ∈ > =∅ .

Similarly, we have ( ) ( )F U F U Uα β= = . (2) According to Definition2, we have

( )x F Xα∀ ∈ | ( ) || ( )|

F x XF x α⇔ ≥∩

⇔ | ( ) || ( )| 1F x XF x α¬ < −∩

1 ( )x F Xα−⇔ ∉ ¬

1 ( )x F Xα−⇔ ∈¬ ¬

1 ( )x U F U Xα−⇔ ∈ − − , So

1( ) ( )F X U F U Xα α−= − − . Similarly, we have

1( ) ( )F X U F U Xβ β−= − − .

(3) ( )x F Xα∀ ∈ , according to Definition2, we have | ( ) || ( )|

F x XF x α β≥ >∩ ,then ( )x F Xβ∈ ,that is

( ) ( )F X F Xα β⊆ .

(4) X Y⊆ ,we have | ( ) | | ( ) || ( )| | ( )|

F x X F x YF x F x≤∩ ∩ . ( )x F Xα∀ ∈ ,We can get inequality | ( ) |

| ( )|F x X

F x α≥∩ which

yields | ( ) | | ( ) || ( )| | ( )|

F x Y F x XF x F x α≥ ≥∩ ∩ these mean ( )x F Yα∈ . Therefore ( ) ( )F X F Yα α⊆ .

Similarly, we have ( ) ( )F X F Yβ β⊆ .

(5) Since X Y X⊆∩ and X Y Y⊆∩ ,according to properties(4),we have ( ) ( )F X Y F Xα α⊆∩ , ( ) ( )F X Y F Yα α⊆∩

which yields to ( ) ( ) ( )F X Y F X F Yα α α⊆∩ ∩ .

Similarly, we have ( ) ( ) ( )F X Y F X F Yβ β β⊇∪ ∪ .


115

(6) Since X Y X⊆∩ and X Y Y⊆∩ ,according to properties(4),we have ( ) ( )F X Y F Xα α⊆∩ , ( ) ( )F X Y F Yα α⊆∩ .

which yields to ( ) ( ) ( )F X Y F X F Yα α α⊆∩ ∩ .

Similarly, we have ( ) ( ) ( )F X Y F X F Yβ β β⊇∪ ∪ .

(7) For any 2( )x F Xα∈ , we have | ( ) |

2 1| ( )|F x X

F x α α≥ ≥∩ , which means 1( )x F Xα∈ .Therefore,

2 1( ) ( )F X F Xα α⊆ .

Similarly, we have

2 1( ) ( )F X F Xβ β⊆ .

Note: This theorem express that ( )F Xα and ( )F Xβ are both decreasing with the

α andβ increasing. So we can get the following two corollaries. Corollary 1. Let ( ), , ,A U W F P= be a random approximation space, For each X W⊆ ,

0 1β α≤ < ≤ , then

1( ) ( )F X F Xα⊆ , 0( ) ( )F X F Xβ ⊆ .

Corollary 2. Let ( ), , ,A U W F P= be a random approximation space, For each X W⊆ ,

0 1β α≤ < ≤ , ( ) ( ) ( ) ( )F X F X F X F Xα β⊆ ⊆ ⊆ .

Definition 3. Let ( ), , ,A U W F P= be a random approximation space, For each X W⊆ ,

0 1β α≤ < ≤ , Set

( ). ( ) F X

Ux αα βγ =

which is called the α ,β random approximation quality of X . Assume

( ).( )

( ) 1 F XF F X

X α

β

α βρ = −

which is called the α ,β random rough measure of X and ( ).( )

( ) F XF F X

X α

β

α βη =

is called theα ,β random approximation precision of X . We can easily get the following Theorem3 to Theorem5. Theorem3. Let ( ), , ,A U W F P= be a random approximation space, For each X W⊆ , 0 1β α≤ < ≤ , then

.0 ( ) 1C Xα βρ≤ ≤ ; .0 ( ) 1C Xα βη≤ ≤ .

Theorem4. Let ( ), , ,A U W F P= be a random approximation space, For each X W⊆ ,

0 1β α≤ < ≤ , if the random model is definable, then . ( ) 0C Xα βρ = ; . ( ) 1C Xα βη = .


116

Theorem5. Let ( ), , ,A U W F P= be a random approximation space, For each X W⊆ , 0 1β α≤ < ≤ , then

. ( ) ( )F FX Xα βρ ρ≤ ; . ( ) ( )F FX Xα βη η≥ . 4. Conclusions

In this paper, we have proposed the concept of the random precision rough sets model. we also

prove some useful properties and also testify that this model has higher precision than general random rough sets model.

References

[1] Z. Pawlak, Rough sets, International Journal of Computer and Information Science, 11 (1982)

341-356. [2] Z. Pawlak. Rough sets: Theoretical Aspects of Reasoning About Data, Kluwer Academic

Publishers, Boston, 1991. [3] F. Angiulli and C. Pizzuti, Outlier mining in large high-dimensional data sets, IEEE Trans. On

Knowledge and Data Engineering, 17 (2005) 203–215. [4] L. Polkowski, A. Skowron, Rough sets and current trends in computing. Springer, vol. 1424, 1998. [5] N. Zhong, Y. Yao, M. Ohshima, Peculiarity oriented multidatabase mining, IEEE Trans. On

Knowledge and Data Engineering, 15 (2003) 952–960. [6] W. Ziarko, Variable Precision Rough sets Model. Journal of Computer System Science, 46 (1993)

39-59. [7] W. Zakowski, Approximations in the space (u,π), Demonstratio Mathematica, 16 (1983) 761–769. [8] Z. Bonikowski, E. Bryniarski, U. Wybraniec, Extensions and intentions in the rough sets theory,

Information Sciences, 107 (1998) 149-167. [9] W. Zhang, W. Wu, J. Lian, The theory and method of rough sets, Beijing: Science Press, 2001. [10]T. Yang, J. Cai, X. Tang, Degree Rough sets Model Based on Random Sets and Applications,

Journal of Huaiyin Institute of Technology, 19 (2010) 14-16.

Y. M. Abdel-Azeem, Nonlinear Sci. Lett. A, Vol.4, No.4, 117-125, 2013


A differential equation approach for stochastic multi-objective integer problem Y. M. Abdel-Azeem Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City (11884), Cairo, Egypt. e-mail: [email protected] Abstract

This paper deals with stochastic multi-objective integer problems having stochastic parameters in the constraints (CHMOILP). In this paper we assume that a deterministic multi-objective problem is generated corresponding to (CHMOIP). The purpose of this paper is to present a technique for solving the (CHMOIP). The main features of this are based on applying the technique of differential equations approach, where the deterministic multi-objective integer problems corresponding to (CHMOIP) are transformed to nonlinear autonomous system of differential equations. We shall develop the relation between the critical points of differential system and local Pareto optimal solutions of the deterministic multi-objective integer problem to (CHMOIP). This method depends on using differential equations technique for solving multi-objective integer nonlinear programming problems which is very effective in finding many local Pareto optimal solutions. The behavior of the local solutions for slight perturbation of the parameters in the neighborhood of their chosen initial values in presented can be processed by using the technique of trajectory continuation. Keywords: Nonlinear Programming, Stochastic programming, Autonomous system of Differential Equation. 1. Introduction

The traditional way to evaluate any imprecision in the parameters of nonlinear programming models is through a post optimization analysis with the help of sensitivity analysis and parametric programming [6]. In this way we presented in earlier work [5] some results using differential equations approach. Another way to come to terms with imprecision is to apply some results achieved in the theory of fuzzy sets (see [3, 12, 13, 17]). Recently a technique of differential equations for fuzzy nonlinear programming is presented in [12]. The third way to handle with imprecision in the parameters of nonlinear programming models (NLP) is to model it with uncertainty and to apply stochastic programming. Some results were achieved for linear programming problems (see [15]). In this paper, we discuss on the third way to come to terms with impression by applying stochastic integer programming combined with differential equation approach. We deal with stochastic multi-objective integer problems having stochastic parameters in the constrains (CHMOIP) and assuming that deterministic multiobjective integer problems are generated corresponding to (CHMOIP). The deterministic multi-objective integer problem corresponding to (CHMOIP) is transformed to nonlinear integer autonomous system of differential equations. We shall develop the relation between the critical points of differential system and local Pareto optimal integer solutions of deterministic to (CHMOIP). A differential equations approach is presented as a new method for solving equality constrained


118

nonlinear programming problem in [8]. This technique is extended in [4] to be applicable for solving multi-objective nonlinear convex or non-convex programming problems with equality or inequality constraints. The multi-objective nonlinear programming problem is transformed also to a nonlinear autonomous system of differential equations. In the fact the asymptotically stable critical points of the differential system are constrained local Pareto optimal of the original optimization problem. 2. Problem Statement Solution Concept

Consider the multi-objective integer programming problem having random variables in the right-hand-side of the constraints of the following form: (CHMOIP) Maximize ( )F X Subject to

{ }: ( ) 0, l 1,2, ,qnlx H x h x∈ = ∈ ≤ =

( ){ }: ( ) , 0 1, 1,2, ,ni i i ix G x p g x i rν α α∈ = ∈ ≤ ≥ < < =

(2.1)

where ( )1 2( ) ( ), ( ), , ( )kF x f x f x f x= , 0jx ≥ and integer.

In addition, fi(x), i = 1, 2, ..., k, are real valued linear objective functions. M is the feasible set of constraints, p means probability and αi is a specified probability. This mean that the linear constraints may be violated some of the time, and at most for 100(1 − αi) of the time. For sake of simplicity, we assume that νi are random variables normally distributed with means E(νi) and variance σ2(νi). Thus the set of constraints G of problem (2.1) can be rewritten in the deterministic equivalent form as:

{ }2| ( ) ( ) ( ) , 1,2, , , 0, integerni i i i i jG x g x E k i r xν α σ ν β= ∈ ≤ + = = ≥

where ikα is the standard normal value, such that ( )ikαΦ = 1 and ( )αΦ represents the ”cumulative distribution function” of standard normal distribution evaluated at a. Therefore, problem (CHMOIP) can be understood as the following deterministic problem (MOIP) as follow: (MOIP) Maximize ( )1 2( ) ( ), ( ), , ( )mF x f x f x f x=

subject to

{ }: ( ) 0, 1,2, ,nlx H x h x l q∈ = ∈ ≤ =

{ }: ( ) , 1,2, ,ni ix G x g x i rβ∈ = ∈ ≤ = , (2.2)

where xj ≥ 0 and integer. The problem (2.2) will be treated by using ε -constraint method [2], we assume that the scalar objective and equality constrained corresponding to (MOIP) can be written in the following form:

( )kP ε Maximize ( )kf X ′ Subject to


119

{ }2| ( ) 0, 1,2, , 1, 1, ,j j jX F x f x s j k k mε′ ′∈ = + − = = − +

{ }2: ( ) 0, 1,2, ,l lX H x h x l qη′ ′ ′∈ = + = =

{ }2: ( ) 0, 1,2, ,i i iX G x g x l rε β′ ′′ ′∈ = + − = =

(2.3)

Here, ( ) ( )k kf x f x′ =

( )1 2 1 2 1 1 1 2 1 2 1 2, , , , , , , , , , , , , , , , , , , , ,n k k m q r iX x x x s s s s s η η η ε ε ε η η η− +′ = ,

( )1 2 1 1, , , , , , , 0Tk k m jxε ε ε ε ε ε− += ≥ and integer.

It is well known from [2] that the optimal solution *X ′ of ( )kP ε is an efficient solution of multi-objective problem if one of the following conditions is valid:

(i) *X ′ solves ( )kP ε for every k = 1, 2, · · · ,m.

(ii) *X ′ is the unique optimal solution of ( )kP ε .

Now, to solve the problem ( )( )kP ε we introduces the following autonomous system of differential equations.

( )1

1 2 3 2

3

( )T T T Tx kBX A A A f x

γγγ

′

⎡ ⎤⎢ ⎥′ + = −∇⎢ ⎥⎢ ⎥⎣ ⎦

(2.4)

( ) ( )1 2 3

TA A A X F H G′ ′ ′ ′= − . (2.5)

where B is a symmetric (n + m + q + r − 1) × (n + m + q + r + 1) matrix

A1 = x′F′, A2 = x′H′ and A3 = x′G′ (2.6)

The matrices A1,A2,A3 are of full ranks and γ1, γ2, γ3 are (m−1), (q), (r) dimensional vectors respectively. Assume that B is a nonsingular matrix, then the matrix

1 1 11 1 1 2 1 3

1 1 12 1 2 2 2 3

1 1 13 1 3 2 3 3

( )

T T T

T T T

T T T

A B A A B A A B AD x A B A A B A A B A

A B A A B A A B A

− − −

− − −

− − −

⎡ ⎤⎢ ⎥′ = ⎢ ⎥⎢ ⎥⎣ ⎦

(2.7)

in nonsingular and one can solve (2.4-2.6) for X′, γ1, γ2 and γ3 uniquely and obtain:


120

( ) ( )1 ( )Tx k

FX X PB f x P H

Gφ −

′

′⎡ ⎤⎢ ⎥′ ′ ′= = − ∇ − ⎢ ⎥

′⎢ ⎥⎣ ⎦

(2.8)

1 1 11 1 ( )T

x xD F D A B f xγ − − −′′= − ∇ , (2.9)

1 1 12 2 ( )T

x xD H D A B f xγ − − −′′= − ∇ , (2.10)

1 1 13 3 ( )T

x xD G D A B f xγ − − −′′= − ∇ , (2.11)

Here,

11P P= − (2.12)

( )1

1 11 1 2 3 2

3

T T T

AP B A A A D A

A

− −

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

(2.13)

( )1 11 2 3T T TP B A A A D− −= (2.14)

Also, we can prove the following useful identities: 2P P= , (2.15) 2

1P P= , (2.16)

[ ]1 2 3 0TA A A P = , (2.17)

[ ] [ ]1 2 3 1 1 2 3T TA A A P A A A= , (2.18)

[ ]1 2 3TA A A P I= , (2.19)

and

TBP P B= . (2.20)

Let ˆ( )M X be the tangent plane of the system of constraints at X̂ :

11

2

ˆ3

ˆ( ) : 0n m r

X X

AM X Y A Y

A

+ + −

′=

⎧ ⎫⎡ ⎤⎪ ⎪⎢ ⎥= ∈ =⎨ ⎬⎢ ⎥⎪ ⎪⎢ ⎥⎣ ⎦⎩ ⎭

(2.21)

From (2.15) the matrix ˆ( )P X is a projection operator which projects any vector in 1n m r+ + − onto ˆ( )M X .


121

3 Asymptotic Properties of the Fundamental Equations In this section, we derive asymptotic properties of differential equation (2.8) when t → ∞. Let us consider the solution of the system which passes through X′ = ξ at t = 0 by π(ξ, t) and the whole trajectory by C(ξ) i.e

C(ξ) = {π(ξ, t) : t T(ξ)}, T(ξ) = (ta(ξ), tb(ξ)) (3.1)

where, T(ξ) is the maximal interval of existence of the solution.

Theorem 3.1. If ( )*D X ′ is non-singular, then the necessary and sufficient condition that X′ is a

constrained stationary point is that *X ′ is a critical point of the system i.e. ( )* 0Xφ ′ = .

Proof. If *X ′ is a constrained stationary point, then there exist vectors * *1 2,γ γ and *

3γ which belong

to 1,m q− , and r respectively, such that

( ) *

*1*

1 2 3 2*3

( ) 0T T TX k X XX X

A A A f xγγγ

′ ′ ′=′=

⎡ ⎤⎢ ⎥⎡ ⎤ + ∇ =⎢ ⎥⎣ ⎦⎢ ⎥⎣ ⎦

(3.2)

* * *| 0, | 0, and | 0X X X X X X

F H G′ ′ ′ ′ ′ ′= = =

′ ′ ′= = = . (3.3)

Then (2.4-2.6) is satisfied by (X′) = = 0 at [γ1γ2γ3] = * * *1 2 3[ ]γ γ γ . Since ( )*D X ′ is non-singular,

(2.4-2.6) has the unique solution *X ′ and * * *1 2 3[ ]γ γ γ . It follows that ( )*Xφ ′ = 0. Conversely, if

( )*Xφ ′ = 0, then (3.2) and (3.3) are satisfied consequently, *X ′ is a constrained stationary point of *( )kf x′ .

Corollary 3.2. If the matrix

2

2 21 2 3

2

( ) [ ]T T Tk

FQ f x H

Gγ γ γ

′⎡ ⎤∇⎢ ⎥′= ∇ + ∇⎢ ⎥⎢ ⎥′∇⎣ ⎦

(3.4)


122

at the critical point *X ′ , is negative definite on the tangent plane ( )*M X ′ of the constraint

surface then *X ′ is strict local maximal point. The next Theorem concerns with the asymptotic stability on the set S, where S denote the set of point which satisfy equality constraints, i.e.

{ }1: 0, 0 0, n m rS X F H and G X + + −′ ′ ′= = = = ∈ (3.5)

Theorem 3.3. Let *X ′ be strict local maximal point. If there exist a neighbourhood U of *X ′ such

that for any X′ U ∩ S the matrix B( *X ′ ) is negative definite on the tangent plane M(X′), then ( *X ′ )

is a asymptotically stable on S. Proof. Since, S is invariant (i.e. the whole trajectory S(ξ) ⊂ S), it is sufficient to consider trajectories on S. Then for a solution X(t) ∈ U ∩ S we have:

( )

1

1

1 1

1 1

( )

0.

T

T

T T

T T

T

df X t df dXdt dX dt

Ff PB f P H

G

f PB f

fB P BPB ffB P BPB f

X BX

−

−

− −

− −

=

′⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥′= −∇ ∇ −⎢ ⎥⎢ ⎥⎢ ⎥′⎢ ⎥⎣ ⎦⎣ ⎦⎡ ⎤= −∇ ∇⎣ ⎦

= −∇ ∇

= −∇ ∇

′ ′= − >

(3.6)

We have from the assumption

( ) *( )0,

df X tX X

dt′

′ ′> ≠ (3.7)

( ) *( )0,

df X tX X

dt′

′ ′= = (3.8)

where, from the strict uniqueness of *X ′ there exists a neighbourhood V of X* such that f(X′) <

f( *X ′ ), X′ = *X ′ for any X′ V ∩ S where L(X′) = f(X′) − f( *X ′ ) satisfies L(X′) < 0, X′ = *X ′ ,


123

L(X′ ) = 0. *( ) 0,dL X X Xdt

′′< ≠ and

( ) 0dL Xdt

′= .

For X′∈ U ∩ V ∩ S. Thus the function L is a Liapunov’s function and *X ′ is asymptotic stable on S.

Clearly, a local maximal point *X ′ which satisfies the above theorem is stable, and is asymptotically

stable in the space 1n m r+ + − (See [8]). Thus, if the differential equation satisfies the assumption of Theorem 3, we get that any trajectory starting from a point within some sufficiently small

neighbourhood of the local maximal point *X ′ converges to *X ′ at t → ∞. 4 An Illustrative Example In the following we provide a numerical simple example to clarify the theory developed in the paper. (CHMOIP) Maximize ( )1 1 2 2 1 2( ) 2 , ( ) 2f x x x f x x x= + = − Subject to { }1 2 1 0.9,p x x b+ ≤ ≥

1 2, 0,x x ≥ and integer

1{ } 1E b = and 21{ } 25bσ =

(MOIP) Maximize ( )1 1 2 2 1 2( ) 2 , ( ) 2f x x x f x x x= + = − Subject to 1 2 7.25x x+ ≤ ,


By using the method of constraints [1], one can write

1( )P ε Maximize 1 1 2( ) 2f x x x= + Subject to 1 2 22x x ε− ≤

1 2 7.25x x+ ≤


start with 2 7.25ε ε= = and formulate 1( )P ε with equality constraints in the form

1( )P ε Maximize 1 1 2( ) 2f x x x= + subject to

21 2 1{ : 2 7.25}x F x x x s′ ′ ′∈ = − + =

21 2 1{ : 7.25}x G x x x η′ ′ ′∈ = + + =


124

1 2 1 1, , , 0,x x s η ≥ ( 1 2,x x are integer) where x′ = (x1, x2, s1, η1). Then we obtain

[ ]1 1( ) 1 2 2 0xA F s′ ′= ∇ = −

[ ]2 1( ) 1 1 0 2xA G η′ ′=∇ = choosing B = I, where I is the identity matrix, then

21

21

5 4 11 4s

Dη

⎡ ⎤+ −= ⎢ ⎥− +⎣ ⎦

2 2 2 21 1 1 1 2 1 1

2 2 2 2 2 21 1 1 1 1 1 1 1 1 1

1 2 2 21 1 1 1 1 1 1 1 1 1

2 2 2 21 1 1 1 1 1 1 1

9 4 8 4 6 8 12 88 4 9 16 4 6 16 6 81

6 8 6 16 8 (1 2 ) 412 8 6 8 4 4 (5 4 )

s s s ss s s s s

PD s s s s s s

s s s s

η η η ηη η η η η


⎡ ⎤+ − + + +⎢ ⎥− + + + − + +⎢ ⎥=⎢ ⎥+ − + +⎢ ⎥

+ + − +⎣ ⎦

1P I P= − and 2 21 1

21 1

1 1 12

1 1 1

3 4 2(3 2 )3 8 3 41

4 (1 2 ) 22 3 (5 4 )

sm s

PD s m s

s

ηηη

η η

⎡ ⎤+ +⎢ ⎥− − +⎢ ⎥=⎢ ⎥+⎢ ⎥

+⎣ ⎦

using the method of fourth order Rung-Kutta or Euler method with the initial point 0 0 0 01 2 1 1( , , , ) (0,0,0,0)x x s η = for solving the autonomous differential system which associated with

stochastic multi-objective integer optimization problem having chance constraints, we get the solution (7,0,0,0). References [1] Arenas, M. Bilbao, A. and Bodriguea, M. V., Solution of a multiobjective linear programming

problem, Europeam Journal of Operational Research, 119 (1999), 338-344. [2] V. Chan Kong and Y. Y. Haimes, Multiobjective Decision Making Theory and Methodology, Noth

Series in System Science and Engineering, New York, (1983). [3] D. Dubois and H. Parade, Fuzzy Sets and Systems, Theory and Applications, Academic Press, New

York, (1980). [4] Fatma M. Ali, Technique for solving multiobjective nonliner programming using differential

equation approach, J. information and optimization science, 18 (1999), 351-357. [5] Fatma M. Ali, Sensitivity analysis for parametric vector optimization problems using differential

equations approach, Int. J. math. & Math. Sci., 25:9 (2001), 621-628. [6] A. V. Fiacco, Introduction to sensitiviry and stability analysis in non-linear programming,

Academic Inc., USA, (1983). [7] Guddat, J. and Vosquez, F. G. and Tammer, K. and Wendler, K., Multiobjective and stochastic

optimization based on parametric optimization, Academic-Verleg, Berlin (1985). [8] Hiroshi Yamashira, A differential equation approach to nonlinear programming, Mathematical

Programming, 18 (1980), 155-168.


125

[9] Kelin, D. and Hanan, E., An algorithm for the multiobjective integer linear programming problem, European Journal of operational research 9 vol. 3, (1975), 378-385.

[10] J. Lasalle and S. Lefschetz, Stability theory by Liapunov’s Direct Method, Academic Press, London, (1961).

[11] W. Romisch and R. Schultz, Distribution sensitrity in stochastic programming, Mathematical Programming, 50(1991), 199-226.

[12] M. Sakawa and H. Yano, Interactive decision making for multiobjective nonlinear programming problems with fuzzy parameters, fuzzy setes and systems, 29 (1980), 315-326.

[13] M. Sakawa and H. Yano, Feasilbility and Pareto optimality for multiobjective nonlinear programming probles with fuzzy parameters, fuzzy sets and systems, 43 (1991), 1-16.

[14] Taha, H. A., Integer programming: Theory, Applications, and Computations, Academic Press, New York, (1975).

[15] R. Wets, Stochastic programming: solution techniques and approximation schemes in A. Bachem, M. Gotschel and B. Korte, Eds., Mathematical programming, (Springer Berlin 1983), 566-603.

[16] T. Yoshizawa, stability theory by Liapunov’s second method, Mathematical society of Japan, Tokyo, (1966).

[17] H. J. Zimmermann, Fuzzy sets theory and its applications, Kluwer Nijhoff, Dordrceht (1985). ical Society of Japan, Tokyo, (1966).



A differential equation approach for stochastic multi-objective integer problem Y. M. Abdel-Azeem Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City (11884), Cairo, Egypt. e-mail: [email protected] Abstract

This paper deals with stochastic multi-objective integer problems having stochastic parameters in the constraints (CHMOILP). In this paper we assume that a deterministic multi-objective problem is generated corresponding to (CHMOIP). The purpose of this paper is to present a technique for solving the (CHMOIP). The main features of this are based on applying the technique of differential equations approach, where the deterministic multi-objective integer problems corresponding to (CHMOIP) are transformed to nonlinear autonomous system of differential equations. We shall develop the relation between the critical points of differential system and local Pareto optimal solutions of the deterministic multi-objective integer problem to (CHMOIP). This method depends on using differential equations technique for solving multi-objective integer nonlinear programming problems which is very effective in finding many local Pareto optimal solutions. The behavior of the local solutions for slight perturbation of the parameters in the neighborhood of their chosen initial values in presented can be processed by using the technique of trajectory continuation. Keywords: Nonlinear Programming, Stochastic programming, Autonomous system of Differential Equation. 1. Introduction

The traditional way to evaluate any imprecision in the parameters of nonlinear programming models is through a post optimization analysis with the help of sensitivity analysis and parametric programming [6]. In this way we presented in earlier work [5] some results using differential equations approach. Another way to come to terms with imprecision is to apply some results achieved in the theory of fuzzy sets (see [3, 12, 13, 17]). Recently a technique of differential equations for fuzzy nonlinear programming is presented in [12]. The third way to handle with imprecision in the parameters of nonlinear programming models (NLP) is to model it with uncertainty and to apply stochastic programming. Some results were achieved for linear programming problems (see [15]). In this paper, we discuss on the third way to come to terms with impression by applying stochastic integer programming combined with differential equation approach. We deal with stochastic multi-objective integer problems having stochastic parameters in the constrains (CHMOIP) and assuming that deterministic multiobjective integer problems are generated corresponding to (CHMOIP). The deterministic multi-objective integer problem corresponding to (CHMOIP) is transformed to nonlinear integer autonomous system of differential equations. We shall develop the relation between the critical points of differential system and local Pareto optimal integer solutions of deterministic to (CHMOIP). A differential equations approach is presented as a new method for solving equality constrained


118

nonlinear programming problem in [8]. This technique is extended in [4] to be applicable for solving multi-objective nonlinear convex or non-convex programming problems with equality or inequality constraints. The multi-objective nonlinear programming problem is transformed also to a nonlinear autonomous system of differential equations. In the fact the asymptotically stable critical points of the differential system are constrained local Pareto optimal of the original optimization problem. 2. Problem Statement Solution Concept

Consider the multi-objective integer programming problem having random variables in the right-hand-side of the constraints of the following form: (CHMOIP) Maximize ( )F X Subject to

{ }: ( ) 0, l 1,2, ,qnlx H x h x∈ = ∈ ≤ =

( ){ }: ( ) , 0 1, 1,2, ,ni i i ix G x p g x i rν α α∈ = ∈ ≤ ≥ < < =

(2.1)

where ( )1 2( ) ( ), ( ), , ( )kF x f x f x f x= , 0jx ≥ and integer.

In addition, fi(x), i = 1, 2, ..., k, are real valued linear objective functions. M is the feasible set of constraints, p means probability and αi is a specified probability. This mean that the linear constraints may be violated some of the time, and at most for 100(1 − αi) of the time. For sake of simplicity, we assume that νi are random variables normally distributed with means E(νi) and variance σ2(νi). Thus the set of constraints G of problem (2.1) can be rewritten in the deterministic equivalent form as:

{ }2| ( ) ( ) ( ) , 1,2, , , 0, integerni i i i i jG x g x E k i r xν α σ ν β= ∈ ≤ + = = ≥

where ikα is the standard normal value, such that ( )ikαΦ = 1 and ( )αΦ represents the ”cumulative distribution function” of standard normal distribution evaluated at a. Therefore, problem (CHMOIP) can be understood as the following deterministic problem (MOIP) as follow: (MOIP) Maximize ( )1 2( ) ( ), ( ), , ( )mF x f x f x f x=

subject to

{ }: ( ) 0, 1,2, ,nlx H x h x l q∈ = ∈ ≤ =

{ }: ( ) , 1,2, ,ni ix G x g x i rβ∈ = ∈ ≤ = , (2.2)

where xj ≥ 0 and integer. The problem (2.2) will be treated by using ε -constraint method [2], we assume that the scalar objective and equality constrained corresponding to (MOIP) can be written in the following form:

( )kP ε Maximize ( )kf X ′ Subject to


119

{ }2| ( ) 0, 1,2, , 1, 1, ,j j jX F x f x s j k k mε′ ′∈ = + − = = − +

{ }2: ( ) 0, 1,2, ,l lX H x h x l qη′ ′ ′∈ = + = =

{ }2: ( ) 0, 1,2, ,i i iX G x g x l rε β′ ′′ ′∈ = + − = =

(2.3)

Here, ( ) ( )k kf x f x′ =

( )1 2 1 2 1 1 1 2 1 2 1 2, , , , , , , , , , , , , , , , , , , , ,n k k m q r iX x x x s s s s s η η η ε ε ε η η η− +′ = ,

( )1 2 1 1, , , , , , , 0Tk k m jxε ε ε ε ε ε− += ≥ and integer.

It is well known from [2] that the optimal solution *X ′ of ( )kP ε is an efficient solution of multi-objective problem if one of the following conditions is valid:

(i) *X ′ solves ( )kP ε for every k = 1, 2, · · · ,m.

(ii) *X ′ is the unique optimal solution of ( )kP ε .

Now, to solve the problem ( )( )kP ε we introduces the following autonomous system of differential equations.

( )1

1 2 3 2

3

( )T T T Tx kBX A A A f x

γγγ

′

⎡ ⎤⎢ ⎥′ + = −∇⎢ ⎥⎢ ⎥⎣ ⎦

(2.4)

( ) ( )1 2 3

TA A A X F H G′ ′ ′ ′= − . (2.5)

where B is a symmetric (n + m + q + r − 1) × (n + m + q + r + 1) matrix

A1 = x′F′, A2 = x′H′ and A3 = x′G′ (2.6)

The matrices A1,A2,A3 are of full ranks and γ1, γ2, γ3 are (m−1), (q), (r) dimensional vectors respectively. Assume that B is a nonsingular matrix, then the matrix

1 1 11 1 1 2 1 3

1 1 12 1 2 2 2 3

1 1 13 1 3 2 3 3

( )

T T T

T T T

T T T

A B A A B A A B AD x A B A A B A A B A

A B A A B A A B A

− − −

− − −

− − −

⎡ ⎤⎢ ⎥′ = ⎢ ⎥⎢ ⎥⎣ ⎦

(2.7)

in nonsingular and one can solve (2.4-2.6) for X′, γ1, γ2 and γ3 uniquely and obtain:


120

( ) ( )1 ( )Tx k

FX X PB f x P H

Gφ −

′

′⎡ ⎤⎢ ⎥′ ′ ′= = − ∇ − ⎢ ⎥

′⎢ ⎥⎣ ⎦

(2.8)

1 1 11 1 ( )T

x xD F D A B f xγ − − −′′= − ∇ , (2.9)

1 1 12 2 ( )T

x xD H D A B f xγ − − −′′= − ∇ , (2.10)

1 1 13 3 ( )T

x xD G D A B f xγ − − −′′= − ∇ , (2.11)

Here,

11P P= − (2.12)

( )1

1 11 1 2 3 2

3

T T T

AP B A A A D A

A

− −

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

(2.13)

( )1 11 2 3T T TP B A A A D− −= (2.14)

Also, we can prove the following useful identities: 2P P= , (2.15) 2

1P P= , (2.16)

[ ]1 2 3 0TA A A P = , (2.17)

[ ] [ ]1 2 3 1 1 2 3T TA A A P A A A= , (2.18)

[ ]1 2 3TA A A P I= , (2.19)

and

TBP P B= . (2.20)

Let ˆ( )M X be the tangent plane of the system of constraints at X̂ :

11

2

ˆ3

ˆ( ) : 0n m r

X X

AM X Y A Y

A

+ + −

′=

⎧ ⎫⎡ ⎤⎪ ⎪⎢ ⎥= ∈ =⎨ ⎬⎢ ⎥⎪ ⎪⎢ ⎥⎣ ⎦⎩ ⎭

(2.21)

From (2.15) the matrix ˆ( )P X is a projection operator which projects any vector in 1n m r+ + − onto ˆ( )M X .


121

3 Asymptotic Properties of the Fundamental Equations In this section, we derive asymptotic properties of differential equation (2.8) when t → ∞. Let us consider the solution of the system which passes through X′ = ξ at t = 0 by π(ξ, t) and the whole trajectory by C(ξ) i.e

C(ξ) = {π(ξ, t) : t T(ξ)}, T(ξ) = (ta(ξ), tb(ξ)) (3.1)

where, T(ξ) is the maximal interval of existence of the solution.

Theorem 3.1. If ( )*D X ′ is non-singular, then the necessary and sufficient condition that X′ is a

constrained stationary point is that *X ′ is a critical point of the system i.e. ( )* 0Xφ ′ = .

Proof. If *X ′ is a constrained stationary point, then there exist vectors * *1 2,γ γ and *

3γ which belong

to 1,m q− , and r respectively, such that

( ) *

*1*

1 2 3 2*3

( ) 0T T TX k X XX X

A A A f xγγγ

′ ′ ′=′=

⎡ ⎤⎢ ⎥⎡ ⎤ + ∇ =⎢ ⎥⎣ ⎦⎢ ⎥⎣ ⎦

(3.2)

* * *| 0, | 0, and | 0X X X X X X

F H G′ ′ ′ ′ ′ ′= = =

′ ′ ′= = = . (3.3)

Then (2.4-2.6) is satisfied by (X′) = = 0 at [γ1γ2γ3] = * * *1 2 3[ ]γ γ γ . Since ( )*D X ′ is non-singular,

(2.4-2.6) has the unique solution *X ′ and * * *1 2 3[ ]γ γ γ . It follows that ( )*Xφ ′ = 0. Conversely, if

( )*Xφ ′ = 0, then (3.2) and (3.3) are satisfied consequently, *X ′ is a constrained stationary point of *( )kf x′ .

Corollary 3.2. If the matrix

2

2 21 2 3

2

( ) [ ]T T Tk

FQ f x H

Gγ γ γ

′⎡ ⎤∇⎢ ⎥′= ∇ + ∇⎢ ⎥⎢ ⎥′∇⎣ ⎦

(3.4)


122

at the critical point *X ′ , is negative definite on the tangent plane ( )*M X ′ of the constraint

surface then *X ′ is strict local maximal point. The next Theorem concerns with the asymptotic stability on the set S, where S denote the set of point which satisfy equality constraints, i.e.

{ }1: 0, 0 0, n m rS X F H and G X + + −′ ′ ′= = = = ∈ (3.5)

Theorem 3.3. Let *X ′ be strict local maximal point. If there exist a neighbourhood U of *X ′ such

that for any X′ U ∩ S the matrix B( *X ′ ) is negative definite on the tangent plane M(X′), then ( *X ′ )

is a asymptotically stable on S. Proof. Since, S is invariant (i.e. the whole trajectory S(ξ) ⊂ S), it is sufficient to consider trajectories on S. Then for a solution X(t) ∈ U ∩ S we have:

( )

1

1

1 1

1 1

( )

0.

T

T

T T

T T

T

df X t df dXdt dX dt

Ff PB f P H

G

f PB f

fB P BPB ffB P BPB f

X BX

−

−

− −

− −

=

′⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥′= −∇ ∇ −⎢ ⎥⎢ ⎥⎢ ⎥′⎢ ⎥⎣ ⎦⎣ ⎦⎡ ⎤= −∇ ∇⎣ ⎦

= −∇ ∇

= −∇ ∇

′ ′= − >

(3.6)

We have from the assumption

( ) *( )0,

df X tX X

dt′

′ ′> ≠ (3.7)

( ) *( )0,

df X tX X

dt′

′ ′= = (3.8)

where, from the strict uniqueness of *X ′ there exists a neighbourhood V of X* such that f(X′) <

f( *X ′ ), X′ = *X ′ for any X′ V ∩ S where L(X′) = f(X′) − f( *X ′ ) satisfies L(X′) < 0, X′ = *X ′ ,


123

L(X′ ) = 0. *( ) 0,dL X X Xdt

′′< ≠ and

( ) 0dL Xdt

′= .

For X′∈ U ∩ V ∩ S. Thus the function L is a Liapunov’s function and *X ′ is asymptotic stable on S.

Clearly, a local maximal point *X ′ which satisfies the above theorem is stable, and is asymptotically

stable in the space 1n m r+ + − (See [8]). Thus, if the differential equation satisfies the assumption of Theorem 3, we get that any trajectory starting from a point within some sufficiently small

neighbourhood of the local maximal point *X ′ converges to *X ′ at t → ∞. 4 An Illustrative Example In the following we provide a numerical simple example to clarify the theory developed in the paper. (CHMOIP) Maximize ( )1 1 2 2 1 2( ) 2 , ( ) 2f x x x f x x x= + = − Subject to { }1 2 1 0.9,p x x b+ ≤ ≥


1{ } 1E b = and 21{ } 25bσ =

(MOIP) Maximize ( )1 1 2 2 1 2( ) 2 , ( ) 2f x x x f x x x= + = − Subject to 1 2 7.25x x+ ≤ ,


By using the method of constraints [1], one can write

1( )P ε Maximize 1 1 2( ) 2f x x x= + Subject to 1 2 22x x ε− ≤

1 2 7.25x x+ ≤


start with 2 7.25ε ε= = and formulate 1( )P ε with equality constraints in the form

1( )P ε Maximize 1 1 2( ) 2f x x x= + subject to

21 2 1{ : 2 7.25}x F x x x s′ ′ ′∈ = − + =

21 2 1{ : 7.25}x G x x x η′ ′ ′∈ = + + =


124

1 2 1 1, , , 0,x x s η ≥ ( 1 2,x x are integer) where x′ = (x1, x2, s1, η1). Then we obtain

[ ]1 1( ) 1 2 2 0xA F s′ ′= ∇ = −

[ ]2 1( ) 1 1 0 2xA G η′ ′=∇ = choosing B = I, where I is the identity matrix, then

21

21

5 4 11 4s

Dη

⎡ ⎤+ −= ⎢ ⎥− +⎣ ⎦

2 2 2 21 1 1 1 2 1 1

2 2 2 2 2 21 1 1 1 1 1 1 1 1 1

1 2 2 21 1 1 1 1 1 1 1 1 1

2 2 2 21 1 1 1 1 1 1 1

9 4 8 4 6 8 12 88 4 9 16 4 6 16 6 81

6 8 6 16 8 (1 2 ) 412 8 6 8 4 4 (5 4 )

s s s ss s s s s

PD s s s s s s

s s s s



⎡ ⎤+ − + + +⎢ ⎥− + + + − + +⎢ ⎥=⎢ ⎥+ − + +⎢ ⎥

+ + − +⎣ ⎦

1P I P= − and 2 21 1

21 1

1 1 12

1 1 1

3 4 2(3 2 )3 8 3 41

4 (1 2 ) 22 3 (5 4 )

sm s

PD s m s

s

ηηη

η η

⎡ ⎤+ +⎢ ⎥− − +⎢ ⎥=⎢ ⎥+⎢ ⎥

+⎣ ⎦

using the method of fourth order Rung-Kutta or Euler method with the initial point 0 0 0 01 2 1 1( , , , ) (0,0,0,0)x x s η = for solving the autonomous differential system which associated with

stochastic multi-objective integer optimization problem having chance constraints, we get the solution (7,0,0,0). References [1] Arenas, M. Bilbao, A. and Bodriguea, M. V., Solution of a multiobjective linear programming

problem, Europeam Journal of Operational Research, 119 (1999), 338-344. [2] V. Chan Kong and Y. Y. Haimes, Multiobjective Decision Making Theory and Methodology, Noth

Series in System Science and Engineering, New York, (1983). [3] D. Dubois and H. Parade, Fuzzy Sets and Systems, Theory and Applications, Academic Press, New

York, (1980). [4] Fatma M. Ali, Technique for solving multiobjective nonliner programming using differential

equation approach, J. information and optimization science, 18 (1999), 351-357. [5] Fatma M. Ali, Sensitivity analysis for parametric vector optimization problems using differential

equations approach, Int. J. math. & Math. Sci., 25:9 (2001), 621-628. [6] A. V. Fiacco, Introduction to sensitiviry and stability analysis in non-linear programming,

Academic Inc., USA, (1983). [7] Guddat, J. and Vosquez, F. G. and Tammer, K. and Wendler, K., Multiobjective and stochastic

optimization based on parametric optimization, Academic-Verleg, Berlin (1985). [8] Hiroshi Yamashira, A differential equation approach to nonlinear programming, Mathematical

Programming, 18 (1980), 155-168.


125

[9] Kelin, D. and Hanan, E., An algorithm for the multiobjective integer linear programming problem, European Journal of operational research 9 vol. 3, (1975), 378-385.

[10] J. Lasalle and S. Lefschetz, Stability theory by Liapunov’s Direct Method, Academic Press, London, (1961).

[11] W. Romisch and R. Schultz, Distribution sensitrity in stochastic programming, Mathematical Programming, 50(1991), 199-226.

[12] M. Sakawa and H. Yano, Interactive decision making for multiobjective nonlinear programming problems with fuzzy parameters, fuzzy setes and systems, 29 (1980), 315-326.

[13] M. Sakawa and H. Yano, Feasilbility and Pareto optimality for multiobjective nonlinear programming probles with fuzzy parameters, fuzzy sets and systems, 43 (1991), 1-16.

[14] Taha, H. A., Integer programming: Theory, Applications, and Computations, Academic Press, New York, (1975).

[15] R. Wets, Stochastic programming: solution techniques and approximation schemes in A. Bachem, M. Gotschel and B. Korte, Eds., Mathematical programming, (Springer Berlin 1983), 566-603.

[16] T. Yoshizawa, stability theory by Liapunov’s second method, Mathematical society of Japan, Tokyo, (1966).

[17] H. J. Zimmermann, Fuzzy sets theory and its applications, Kluwer Nijhoff, Dordrceht (1985). ical Society of Japan, Tokyo, (1966).

nonlinear science letters a mathematics, physics … science letters a: mathematics, physics and...

Documents