Research ArticleOn an Identification Problem on the Determination ofthe Parameters of the Dynamic System

Dossan Baigereyev1 Nevazi Ismailov23 Yusif Gasimov24 and Atif Namazov2

1Higher Mathematics Department D Serikbayev East Kazakhstan State Technical University Oskemen Kazakhstan2Institute of Applied Mathematics Baku State University Z Khalilov 23 AZ1148 Baku Azerbaijan3Institute of Information Technologies of ANAS B Vahabzade 9 AZ1141 Baku Azerbaijan4Institute of Mathematics and Mechanics of ANAS Baku Azerbaijan

Correspondence should be addressed to Dossan Baigereyev dbaigereyevgmailcom

Received 13 February 2015 Accepted 15 June 2015

Academic Editor Manuel Ruiz Galan

Copyright copy 2015 Dossan Baigereyev et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

An inverse problem is considered for the determination of the parameters involved in the right-hand side of the systemof nonlinearordinary differential equations by given initial and final conditions The solution of the problem is reduced to the minimizationof the quadratic functional which indeed is a deviation of the value of the solution from the given values at the end points Usingthe quasilinearization method a calculation method is proposed to the solution of the considered problem The application of thismethod is demonstrated on the example of the determination of the hydraulic resistance in the tubes

1 Introduction

As is known different classes of inverse and identificationproblems play an important role in the solition of manyapplied problems fromphysics hydrodynamics and industry[1ndash5] There exist various methods to solve these problemsas well as optimization methods [6ndash10] One of the principlesteps of these methods is a choice of suitable functionalSince many applied problems are described by the nonlinearsystems the choice of such functional and further solution ofthe corresponding optimization problem is problematic [2]These difficultiesmay be avoid for example by the iterationalquasilinearization method convergence of which is in detailstudied in [11 12]

In the present work a multidimensional identificationproblem is considered for the determination of the param-eters involving the right-hand side of the system of nonlineardifferential equations by given initial and final conditionsSolution of the problem is reduced to the optimizationproblem in which the functional under minimization isconstructed as a quadratic deviation of the solition of thesystem from the given data at the end points Since solutionof the problem in the stated nonlinear formulation presents

certain difficulties [13 14] the considered problem is reducedto the linear case with respect to phase coordinates and vectorof parameters by the help of quasilinearization method Aquadratic functional is constructed and an expression for itsgradient is derived Using Gram-Schmidt orthogonalizationmethod a calculation algorithm is proposed which allowsone to define sought parameters This algorithm is applied tothe example describing the flow in the pipes

2 Problem Statement

Let the movement of the object be described by the system ofdifferential equations

119910 (119905) = 119891 (119910 (119905) 120572) (1)

where 119910 is 119899-dimensional phase vector 119891 is 119899-dimensionaldifferentiable function continuous in the interval (0 119879) and120572 is119898-dimensional constant vector to be found

Let the following initial conditions be given

119910

119894(0) = 119910

119894

0 119894 = 1 119873 (2)

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 570475 8 pageshttpdxdoiorg1011552015570475

2 Mathematical Problems in Engineering

where 119873 119899 119898 are given natural numbers and 119905 is anindependent variable1199101198940 119894 = 1 119873 given 119899-dimensional vectorThe problem consists in the finding of the vector 120572 by whichthe solution of the Cauchy problem (1)-(2) in the point 119879satisfies the given condition

119910

119894(119879) = 119910

119894

119879 119894 = 1 119873 (3)

Such problems are often met in applications [1 3 6 15]when initial data (2) are given and final ones are statisticallymeasured In these cases it is required to find the vector120572 such that the solution of the problem by initial data (2)is maximally close to the measured data at the end pointsAs an example the problem in oil-gas production may beshown when it needs to define the coefficient of the hydraulicresistance

3 Solution Method

Since the function 119891(119910 120572) is nonlinear to solve the problem(1)ndash(3) it is expedient to use any numerical method as wellas quasilinearization method [11 13] So in the first stepwe linearize the problem (1)ndash(3) For this purpose somenominal trajectory 1199100

(119905) and parameter 1205720 are chosen and itis assumed that 119896th iteration is already hold If we linearize(1) with respect to these data we obtain

119910

119896=

120597119891 (119910

119896minus1 120572

119896minus1)

120597119910

119910

119896+

120597119891 (119910

119896minus1 120572

119896minus1)

120597120572

120572

119896

+119891 (119910

119896minus1 120572

119896minus1) minus

120597119891 (119910

119896minus1 120572

119896minus1)

120597119910

119910

119896minus1

minus

120597119891 (119910

119896minus1 120572

119896minus1)

120597120572

120572

119896minus1

(4)

After integration of the linear differential equation (4) withcondition (2) we get the representation [16]

119910

119896(119905) = Φ

119896minus1(119905 1199050) sdot 119910

119896(1199050) +Φ

119896minus11 (119905 1199050) sdot 120572

119896

+Φ

119896minus12 (119905 1199050)

(5)

where Φ

119896minus1(119905 1199050) is a fundamental matrix of the system of

homogeneous equations

119910

119896(119905) =

120597119891 (119910

119896minus1 120572

119896minus1)

120597119910

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119910=119910119896minus1

120572=120572119896minus1

sdot 119910

119896(1199050) (6)

and the matricesΦ119896minus11 (119905 1199050) Φ119896minus12 (119905 1199050) are defined as in [16]

Φ

119896minus11 (119905 1199050) = int

119905

1199050

Φ (119905 120591)

120597119891 (119910

119896minus1 120572

119896minus1)

120597120572

119889120591

Φ

119896minus12 (119905 1199050) = int

119905

1199050

Φ (119905 120591) [119891 (119910

119896minus1 120572

119896minus1)

minus

120597119891 (119910

119896minus1 120572

119896minus1)

120597119910

119910

119896minus1minus

120597119891 (119910

119896minus1 120572

119896minus1)

120597120572

120572

119896minus1]119889120591

(7)

To provide that the solution 119910

119894(119879) of the linearized

differential equation (4) with initial data (2) coincides withthe values of the measurements 119910119894

119879= 119910

119879119894in the point 119879

we construct the following quadratic functional in the 119896thiteration

119868

119896=

12

119873

sum

119894=1[119910

119896

119894(119879 120572) minus 119910

119896

119879119894]

119879

119860

119896

119894[119910

119896

119894(119879 120572) minus 119910

119896

119879119894] (8)

where the sign 119879 stands for transpore 119860119896 is a constantsymmetre 119898 times 119899-dimensional weight matrix that is chosenin each iteration considering the specifics of the concreteproblem 119910119896

119879119894is 119899times 1-dimensional vector of observation 119910119896

119894is

119899 times 1-dimensional vector defined by (5) Then the solution ofthe stated problem is reduced to the problem Find a constantvector 120572 by which the solution of (1) with conditions (2)minimizes the functional (8)

Various algorithms exist for theminimization of the func-tional (8) However in the solution of the concrete problemas well as problem arising in the oil production these algo-rithms met some difficulties [14] (eg to reach the necessaryaccuracy and speed of convergence) Therefore in [6] the useof Gram-Schmidt orthogonalization method is proposed

4 Algorithm for the Minimization ofthe Functional (8)

Here we consider the minimization of the functional (8) bythe help of the relation (4) with conditions (2) Putting 119910119896(119879)from (5) into (8) we get

119868

119896=

12

119873

sum

119894=1[Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050) +Φ

119896minus11119894 (119879 1199050) sdot 120572

+Φ

119896minus12119894 (119879 1199050) minus 119910119879119894]

119879

119860

119896

119894[Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

+Φ

119896minus11119894 (119879 1199050) sdot 120572 +Φ

119896minus12119894 (119879 1199050) minus 119910119879119894] =

16sum

119895=1119868

119895

119896

(9)

Considering the symmetricity of the matrix 119860

119896 the relation(9) may be written as

119868

1119896

def=

12

sdot

119873

sum

119894=1[Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)]119879

119860

119896

119894Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

119868

2119896

def=

12

119873

sum

119894=1[Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)]119879

119860

119896

119894Φ

119896minus11119894 (119879 1199050) sdot 120572

119868

3119896

def=

12

119873

sum

119894=1[Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)]119879

119860

119896

119894Φ

119896minus12119894 (119879 1199050)

119868

4119896

def= minus

12

119873

sum

119894=1[Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)]119879

119860

119896

119894119910

119896

119879119894

Mathematical Problems in Engineering 3

119868

5119896

def=

12

119873

sum

119894=1[Φ

119896minus11119894 (119879 1199050) sdot 120572]

119879

119860

119896

119894Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

119868

6119896

def=

12

119873

sum

119894=1[Φ

119896minus11119894 (119879 1199050) sdot 120572]

119879

119860

119896

119894Φ

119896minus11119894 (119879 1199050) sdot 120572

119868

7119896

def=

12

119873

sum

119894=1[Φ

119896minus11119894 (119879 1199050) sdot 120572]

119879

119860

119896

119894Φ

119896minus12119894 (119879 1199050)

119868

8119896

def= minus

12

119873

sum

119894=1[Φ

119896minus11119894 (119879 1199050) sdot 120572]

119879

119860

119896

119894119910

119896

119879119894

119868

9119896

def=

12

119873

sum

119894=1[Φ

119896minus12119894 (119879 1199050)]

119879

119860

119896

119894Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

119868

10119896

def=

12

119873

sum

119894=1[Φ

119896minus12119894 (119879 1199050)]

119879

119860

119896

119894Φ

119896minus11119894 (119879 1199050) sdot 120572

119868

11119896

def=

12

119873

sum

119894=1[Φ

119896minus12119894 (119879 1199050)]

119879

119860

119896

119894Φ

119896minus12119894 (119879 1199050)

119868

12119896

def= minus

12

119873

sum

119894=1[Φ

119896minus12119894 (119879 1199050)]

119879

119860

119896

119894119910

119896

119879119894

119868

13119896

def= minus

12

119873

sum

119894=1[119910

119896

119879119894]

119879

119860

119896

119894Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

119868

14119896

def= minus

12

119873

sum

119894=1[119910

119896

119879119894]

119879

119860

119896

119894Φ

119896minus11119894 (119879 1199050) sdot 120572

119868

15119896

def= minus

12

119873

sum

119894=1[119910

119896

119879119894]

119879

119860

119896

119894Φ

119896minus12119894 (119879 1199050)

119868

16119896

def= minus

12

119873

sum

119894=1[119910

119896

119879119894]

119879

119860

119896

119894119910

119896

119879119894

(10)

and gradient of the functional (9) has a form

120597119868

119896

120597120572

=

16sum

119895=1

120597119868

119895

119896

120597120572

(11)

Since the terms 1198681119896 1198683119896 1198684119896 1198689119896 11986811119896 11986812119896 11986813119896 11986815119896 11986816119896

do notdepend on the parameter 120572 we have

120597119868

1119896

120597120572

=

120597119868

3119896

120597120572

=

120597119868

4119896

120597120572

=

120597119868

9119896

120597120572

=

120597119868

11119896

120597120572

=

120597119868

12119896

120597120572

=

120597119868

13119896

120597120572

=

120597119868

15119896

120597120572

=

120597119868

16119896

120597120572

= 0

(12)

Based on the formulas

120597119909

119879119886

120597119909

=

120597119886

119879119909

120597119909

= 119886

120597119909

119879119861119909

120597119909

= (119861+119861

119879) 119909

(13)

from [17 18] for the gradients of 1198682119896 1198685119896 1198686119896 1198687119896 1198688119896 11986810119896 11986814119896 we

get the formulas

119868

2119896

120597120572

= ([Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)]119879

119860

119896

119894Φ

119896minus11119894 (119879 1199050))

119879

= Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894

119879

Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

119868

5119896

120597120572

=

120572

119879Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

120597120572

= Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

119868

6119896

120597120572

=

120572

119879Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus11119894 (119879 1199050) 120572

120597120572

= [Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus11119894 (119879 1199050)

+Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894

119879

Φ

119896minus11119894 (119879 1199050)] 120572

= 2Φ119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus11119894 (119879 1199050) 120572

119868

7119896

120597120572

=

120572

119879Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus12119894 (119879 1199050)

120597120572

= Φ

119896minus11119894119879

(119879 1199050)

sdot 119860

119896

119894Φ

119896minus12119894 (119879 1199050)

119868

8119896

120597120572

= minus

[Φ

119896minus11119894 (119879 1199050) 120572]

119879

119860

119896

119894119910

119896

119879119894

120597120572

= minus

120572

119879Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894119910

119896

119879119894

120597120572

= minusΦ

119896minus11119894119879

(119879 1199050)

sdot 119860

119896

119894119910

119896

119879119894

119868

10119896

120597120572

= Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894

119879

Φ

119896minus12119894 (119879 1199050)

119868

14119896

120597120572

= minus

[119910

119896

119879119894]

119879

119860

119896

119894Φ

119896minus11119894 (119879 1199050) 120572

120597120572

= minusΦ

119896minus11119894119879

(119879 1199050)

sdot 119860

119896

119894

119879

119910

119896

119879119894

(14)

4 Mathematical Problems in Engineering

Finally if we consider these results then the gradient ofthe functional (9) will be defined by the formula

120597119868

119896

120597120572

=

12

119873

sum

119894=1(

119868

2119896

120597120572

+

119868

5119896

120597120572

+

119868

6119896

120597120572

+

119868

7119896

120597120572

+

119868

8119896

120597120572

+

119868

10119896

120597120572

+

119868

14119896

120597120572

) =

12

sdot

119873

sum

119894=1(Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894

119879

Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

+Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

+ 2Φ119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus11119894 (119879 1199050) 120572

+Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus12119894 (119879 1199050)

minusΦ

119896minus11119894119879

(119879 1199050) 119860119896

119894119910

119896

119879119894

+Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894

119879

Φ

119896minus12119894 (119879 1199050)

minusΦ

119896minus11119894119879

(119879 1199050) 119860119896

119894

119879

119910

119896

119879119894) =

12

sdot

119873

sum

119894=1(2Φ119896minus11119894

119879

(119879 1199050) 119860119896

119894

119879

Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

+ 2Φ119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus11119894 (119879 1199050) 120572

+ 2Φ119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus12119894 (119879 1199050)

minus 2Φ119896minus11119894119879

(119879 1199050) 119860119896

119894119910

119896

119879119894) = 0

(15)

Then for the gradient of the functional 119868119896relatively to the

parameter 120572 we get the expression

120597119868

119896

120597120572

=

119873

sum

119894=1[Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus11119894 (119879 1199050)] 120572

minus

119873

sum

119894=1[Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894119910

119896

119879119894

minusΦ

119896minus11119894119879

(119879 1199050) 119860119896

119894

119879

Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

minusΦ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus12119894 (119879 1199050)]

(16)

Taking equal to zero the expression (16) we get

119873

sum

119894=1[Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus11119894 (119879 1199050)] 120572

=

119873

sum

119894=1[Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894119910

119896

119879119894

minusΦ

119896minus11119894119879

(119879 1199050) 119860119896

119894

119879

Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

minusΦ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus12119894 (119879 1199050)]

(17)

Solution of (17) with respect to 120572 gives

120572 = [

119873

sum

119894=1Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus11119894 (119879 1199050)]

minus1

sdot

119873

sum

119894=1[Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894119910

119896

119879119894

minusΦ

119896minus11119894119879

(119879 1199050) 119860119896

119894

119879

Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

minusΦ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus12119894 (119879 1199050)]

(18)

where it is assumed that [sum119873119894=1 Φ119896minus11119894119879

(119879 1199050)119860119896

119894Φ

119896minus11119894 (119879 1199050)]

minus1

existsValue of the parameter 120572 defined by (18) is a solution of

the multiparameter optimization problem for the functional(9) that gives minimum to the cost functional

Considering the above the following algorithm may beproposed to the solution of the identification problem (1) (2)

Algorithm 1 (1) Construct the function 119891(119909) from (1)initial and final data 1199101198940 and 119910

119894

119879(119894 = 1 119873) from (2) and (3)

correspondingly(2) Calculate the derivatives 120597119891(119910119896minus1 120572119896minus1)120597119910 120597119891(119910119896minus1

120572

119896minus1)120597120572 taking as initial approaches 119910

119894and 120572

119894

(3)Calculate the fundamentalmatrixΦ119896minus1(119905 1199050) from (6)reconstruct Φ119896minus11 (119905 1199050) Φ

119896minus12 (119905 1199050) from (7) and functional 119868

119896

from (8)(4) Solving the system of algebraic equations (14)

relatively 120572 find the value of the 119898-dimensional vector 120572119896 inthe 119896th iteration

(5) Check the condition1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

120597119868

119896

120597120572

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

lt 120576 (19)

where 120576 is a given small enough number If the condition(16) is satisfied the process stops otherwise go to Step (2)The convergence of this algorithm may be proved similar to[13 14]

Now we discuss the realization of this algorithm

5 Calculational Algorithm

As one can see in the realization of the above algorithmthe main step is a calculation of the fundamental matrixΦ

119896minus1(119905 119905

0) and the matricesΦ119896minus11 (119905 119905

0)Φ119896minus12 (119905 119905

0) Note that

as is mentioned in [16] construction of these matrices is anenough difficult procedure So for simplicity we try (4) andfind the corresponding derivatives by using the Eulermethod

Mathematical Problems in Engineering 5

Really 120597119891(119910(119905) 120572)120597119910 and 120597119891(119910(119905) 120572)120597120572 everywhere arereplaced by

119891(119910) = (119891(119910 + 120575 120572) minus 119891(119910 120572))120575 and

119891(120572) =

(119891(119910 120572 + 1205751) minus 119891(119910 120572))1205751 correspondingly where 120575 and 1205751are small enough numbers

To calculate the fundamental matrix Φ

119896minus1(119905 119905

0) and the

matrices Φ119896minus11 (119905 1199050) Φ119896minus12 (119905 1199050) it is proper to replace (4) by

the following discrete one

119910

119896(1199052119873)

= (

119895

prod

1198941=2119873minus1(119864+ 120575

119891 (119910

119896minus1(119905

1198941))))119910

119896(119905

119873+1)

+Φ

119896minus11 (119905 1199050) 120572 +Φ

119896minus12 (119905 1199050)

(20)

where

Φ

119896minus1119894

(1199052119873 1199050) =

119895

prod

1198941=2119873minus1(119864+ 120575

119891 (119910

119896minus1(119905

119894)))

Φ

119896minus11119894 (1199052119873 1199050)

= (

2119873minus1sum

119895=119873+2(

119895

prod

1198941=2119873minus1(119864+ 120575 sdot

119891 (119910

119896minus1(119905

119894))))

sdot 120575

119891 (120572

119896minus1119895minus1))+120575 sdot

119891 (120572

119896minus12119873minus1)

Φ

119896minus12119894 (1199052119873 1199050)

= (

2119873minus1sum

119895=119873+2(

119895

prod

1198941=2119873minus1(119864+ 120575 sdot

119891 (119910

119896minus1(119905

119894))))120575

sdot (

119891 (119910

119896minus1(119905

119895minus1) 120572119896minus1

)

minus

119891 (119910

119896minus1(119905

119895minus1)) 119910119896minus1

(119905

119895minus1) minus

119891 (120572

119896minus1) 120572

119896minus1))

+120575 sdot (

119891 (119910

119896minus1(1199052119873minus1) 120572

119896minus1) minus

119891 (119910

119896minus1(1199052119873minus1))

sdot 119910

119896minus1(1199052119873minus1) minus

119891 (120572

119896minus1) 120572

119896minus1)

(21)

119864 is unit matrix of proper dimensionThen from (20) we get that Φ119896minus1(119905 1199050) is a fundamental

matrix for the system of homogeneous equations

119910

119896(119905

119894+1) = (119864+ 120575

119891 (119910))

1003816

1003816

1003816

1003816

1003816

119910=119910119896minus1

120572=120572119896minus1

sdot 119910

119896(119905

119894) (22)

Therefore similar to the nongradient methods we pro-pose an algorithm based on the orthogonalization of gradientdirections using the Gram-Schmidt procedure

Step 1 Using Gram-Schmidt orthogonalization vectors 120596119895119894

119894 = 119895119895 + 1 119899 119895 = 2 3 1198992 are calculated and the set ofvectors

(

120597119891 (120582

1119888)

120597120582

1119888

120597119891 (120582

2119888)

120597120582

2119888

120597119891 (120582

119894

119888)

120597120582

119894

119888

120596

119895

119894+1 120596119895

119899)

(23)

is found which form an orthogonal basis in 119877

119899If we apply the orthogonalization algorithm then a

linearly independent system 1198861 1198862 119886119896 should be formedthat is orthogonal system 1198871 1198872 119887119896 and each vector 119887

119894

should be linearly expressed through 1198861 1198862 119886119894 Here 119886

119894

and 119887

119894are upper triangular matrices Thus it is possible to

ensure that the systems 119887

119894 were orthonormal where the

diagonal elements of the transition matrix are positive bythese conditions the system 119887

119894 and the transition matrix are

uniquely determinedThe algorithm considers 1198871 = 1198861 if the vectors

1198871 1198872 119887119894minus1 are constructed Then

119887

119894= 119886

119894minus

119894minus1sum

119895=1

⟨119886

119894 119887

119895⟩

⟨119887

119894 119887

119895⟩

119887

119895 (24)

where ⟨ ⟩ is the sign of the scalar product of vectors

Step 2 For the orthogonalization of the gradient directionswe compute ]119895

119894in the form

]119895119894=

119891 (120582

119888+ 120575120596

119895

119894) minus 119891 (120582

119888minus 120575120596

119895

119894)

2120575

119894 = 119895 119895 + 1 119899

(25)

Here 120575 gt 0 is any small parameter

Step 3 The orthogonal gradient directions are chosen in theform

119897

119895=

119899

sum

119894=119895

]119895119894120596

119895

119894 (26)

Replacing nabla119891(120582

(119896)

119888) by 119897

119895in (17) the nongradient iterative

minimization procedure will be

120582

(119896+1)119888

= 120582

(119896)

119888minus120594

lowast(119896)119897

119896

(27)

where 120594lowast(119896) is a scalar which is determined by golden sectionmethod

Now we apply the above proposed technique to theexample of 15 production by gas-lift method

Example 2 It is known that nonstable motion of gas in tubesand gas liquid mixture (GLM) in vertical tubes that is inthe lift pipe of the gas-lift well with constant across profile is

6 Mathematical Problems in Engineering

x = 0

x = lg

x = l

rc

rk

x

h

hst

l

lg

lj

Gas

Gas GLM

x = 2l

Gas liquid mixture

Figure 1

described by the following system of linear partial differentialequations of hyperbolic type (see Figure 1)

minus

120597119875

120597119909

=

120597 (120588120596

119888)

120597119905

+ 2120572120588120596119888

minus

120597119875

120597119905

= 119888

2 120597 (120588120596119888)

120597119909

(28)

where 120588 = 119875(119909 119905) 120596119888= 120596

119888(119909 119905) is an additional pressure on

its stationary value and averaged over across section speed ofmotion of GLM 119905 119909 time and coordinate 119888 speed of soundin gas andGLM 120588 gas oil and GLM in correspondence withcoordinates 2120572 = 119892120596

119888+120582

119888120596

1198882119863 119892 120582

119888 free fall accel 119863

interval effective diameter of the tube [19]The partial differential equation of gas and GLM motion

by are averaging over time 119905may be reduced to the followingordinary differential equation [20]

119876 =

2120572 (120582

119888) 120588119865119876

2

119888

2120588

2119865

2minus 119876

2 119876 (0) = 119906

(29)

where 119888 ≫ 120596

119888and all quantities are assumed constant 119876 =

120588120596

119888119865 and 119865 is area of across section of the pump-compressor

tubes and is constant relative to axes

It is assumed that the passing from the end of tubethrough the layer to the beginning of the lift (119909 = 119897) isdescribed by the following difference equations

119876 (119897 + 0) = 120574119876 (119897 minus 0) + 1205741 (119876 (119897 minus 0)) 119876

1205741 (119876 (119897 minus 0)) = minus 1205753 (119876 (119897 minus 0) minus 1205752)2+ 1205751

(30)

where 120574 and 1205751 1205752 1205753 are constant numbers to be found Forthe sake of simplicity we suppose that the parameters 120574 12057511205752 1205753 are known and it is required to reconstruct 120582

119888involved

in (19) due to 120572(120582119888)

Then some nominal trajectory 119876

0(119909) and parameter 1205720

are chosen assuming that 119896th iteration is already held Let uslinearize (29) among these data

119876

119896(119909) = 119860 (119876

119896minus1 120572

119896minus1) sdot 119876

119896(119909) + 119861 (119876

119896minus1 120572

119896minus1) 120572

119896

+119862 (119876

119896minus1 120572

119896minus1)

(31)

where

120597119891 (119910

119896minus1 120572

119896minus1)

120597119910

= 119860 (119876

119896minus1 120572

119896minus1)

=

4120572119896minus1119888212058831198653119876

119896minus1

(119888

2120588

2119865

2minus (119876

119896minus1)

2)

2

120597119891 (119910

119896minus1 120572

119896minus1)

120597120572

= 119861 (119876

119896minus1 120572

119896minus1)

=

2120572119896minus1120588119865119876119896minus1

119888

2120588

2119865

2minus (119876

119896minus1)

2

119891 (119910

119896minus1 120572

119896minus1) minus

120597119891 (119910

119896minus1 120572

119896minus1)

120597119910

119910

119896minus1

minus

120597119891 (119910

119896minus1 120572

119896minus1)

120597120572

120572

119896minus1= 119862 (119876

119896minus1 120572

119896minus1)

=

4120572119896minus1119888212058831198653(119876

119896minus1)

2+ 4120572119896minus1120588119865 (119876

119896minus1)

4

(119888

2120588

2119865

2minus (119876

119896minus1)

2)

2

(32)

Note that by the help of the relations (20) and (21) thematricesΦ

119896minus1(119909 0) Φ119896minus11 (119909 0) Φ119896minus12 (119909 0) are calculated as follows

Φ

119896minus1119894

(119909 0) =119895

prod

119894=2119873minus1(119864+119860 (119876

119896minus1(119909

1198941) 120572

119896minus1)) ℎ

Φ

119896minus11119894 (1199092119873 0)

= (

2119873minus1sum

119895=119873+2(

119895

prod

119894=2119873minus1(119864+119860 (119876

119896minus1(119909

1198941) 120572

119896minus1)) ℎ)

sdot 119861 (119876

119896minus1(119909

119895minus1) 120572119896minus1

) ℎ)

+119861 (119876

119896minus1(1199092119873minus1) 120572

119896minus1) ℎ

Mathematical Problems in Engineering 7

Table 1

119910

119897+0119894

55698 55732 55761 55810 55848 55852 55824119910

2119897119894

44242 44248 44254 44262 44266 44263 44251

Table 2

120575 005 01 05 054 055 06120582 01966 02141 02295 02298 02299 02302

01 02 03 04 05 06 07 08 09 10214

0216

0218

022

0222

0224

0226

0228

023

0232

Figure 2

Φ

119896minus12119894 (1199092119873 0)

= (

2119873minus1sum

119895=119873+2(

119895

prod

119894=2119873minus1(119864+119860 (119876

119896minus1(119909

1198941) 120572

119896minus1)) ℎ)

sdot 119862 (119876

119896minus1(119909

119895minus1) 120572119896minus1

) ℎ)

+119862 (119876

119896minus1(1199092119873minus1) 120572

119896minus1) ℎ

(33)

where ℎ is small enough numberLet some statistical data be givenLet us assume that some observation points for 119910119897+0

119894and

119910

2119897119894are given (see Table 1)We give in Table 2 the values of 120582 obtained by using

MATLAB by given input parametersAs we see from Table 2 by 120575 = 055 120582 gets value 02299

with error estimation 10minus3Here is the dependence of 120582 or 120575 (see Figure 2)The above algorithm reaches given accuracy after 4

iterations and gives 120582 = 02299Note that the inequality

1003816

1003816

1003816

1003816

120582

119894minus1205824

1003816

1003816

1003816

1003816

le 10minus15 (34)

holds for any 119894 gt 4 that shows the stability of the proposedquasilinearization algorithm

Conflict of Interests

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

Acknowledgment

This work is supported by the joint grant of ANAS andSOCAR 17 2013ndash2015 Baku State University ldquo50+50rdquo Grant

References

[1] A Aliev Fikret and N A Ismailov ldquoInverse problem todetermine the hydraulic resistance coefficient in the gas liftprocessrdquo Applied and Computational Mathematics vol 12 no3 pp 306ndash313 2013

[2] R Gurbanov N Nuriyev and R S Gurbanov ldquoTechnologicalcontrol and optimization problems in oil production theoryand practicerdquo Applied and Computational Mathematics vol 12no 3 pp 314ndash324 2013

[3] R N Bakhtisin and A R Latypov ldquoEstimation of the order oflinear objects by experimental informationrdquo Automation andRemote Control no 3 pp 108ndash112 1992

[4] Y S Gasimov ldquoOn a shape design problem for one spectralfunctionalrdquo Journal of Inverse and Ill-Posed Problems vol 21 no5 pp 629ndash637 2013

[5] L Lyuing Identification of the System Theory for Users NaukaMoscow Russia 1991

[6] F A AlievMMMutallimov IMAskerov and I S RaguimovldquoAsymptotic method of solution for a problem of constructionof optimal gas-lift process modesrdquo Mathematical Problems inEngineering vol 2010 Article ID 191053 10 pages 2010

[7] A S Apostolyuk and V B Larin ldquoUpdating of linear stationarydynamic systemparametersrdquoApplied andComputationalMath-ematics vol 10 no 3 pp 402ndash408 2011

[8] F Ding ldquoHierarchical multi-innovation stochastic gradientalgorithm for Hammerstein nonlinear system modelingrdquoApplied Mathematical Modelling vol 37 no 4 pp 1694ndash17042013

[9] F Ding Y Shi and T Chen ldquoAuxiliary model-based least-squares identification methods for Hammerstein output-errorsystemsrdquo Systems amp Control Letters vol 56 no 5 pp 373ndash3802007

[10] S I Kabanikhin and O I KrivorotrsquoKo ldquoA numerical methodfor determining the amplitude of a wave edge in shallow waterapproximationrdquo Applied and Computational Mathematics vol12 no 1 pp 91ndash96 2013

[11] P E Bellman and P E Kalaba Quailinearization and NonlinearBoundary Problems MirVoscow Russia 1968

[12] V E Shamansky Methods of Numerical Solution of Baun-daryProblems in PC Naukova Dumka Kiev 1966

[13] A Brayson and X Yu-shi Applied Theory of Optimal ControlMir Moscow Russia 1972

[14] DMHimmebblauAppliedNonlinear Programming Craw-HillBook Company New York NY USA 1972

[15] K R Aydazade ldquoComputatioonal problems in hydraulic net-worksrdquo Computational Mathematics and Mathematical Physicsvol 29 no 2 pp 184ndash193 1989

[16] Y N Andreev Control of the Finite Dimensional Linear ObjectsNauka Moscow Russia 1976

8 Mathematical Problems in Engineering

[17] J R Magnus and H Neudecker Matrix Differential Calculuswith Applications in Statistics and Econometrics John Wiley ampSons Chichester UK 17th edition 1988

[18] K B Petersen and M S PedersenTheMatrix Cookbook 2008httpmatrixcookbookcom

[19] D M Altshul Hidraulic Resistance Nedra Moscow Russia1970

[20] M Ghanbari S Abbasbandy and T Allahviranloo ldquoA newapproach to determine the convergence-control parameter inthe application of the homotopy analysis method to systems oflinear equationsrdquo Applied and Computational Mathematics vol12 no 3 pp 355ndash364 2013

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

where 119873 119899 119898 are given natural numbers and 119905 is anindependent variable1199101198940 119894 = 1 119873 given 119899-dimensional vectorThe problem consists in the finding of the vector 120572 by whichthe solution of the Cauchy problem (1)-(2) in the point 119879satisfies the given condition

119910

119894(119879) = 119910

119894

119879 119894 = 1 119873 (3)

Such problems are often met in applications [1 3 6 15]when initial data (2) are given and final ones are statisticallymeasured In these cases it is required to find the vector120572 such that the solution of the problem by initial data (2)is maximally close to the measured data at the end pointsAs an example the problem in oil-gas production may beshown when it needs to define the coefficient of the hydraulicresistance

3 Solution Method

Since the function 119891(119910 120572) is nonlinear to solve the problem(1)ndash(3) it is expedient to use any numerical method as wellas quasilinearization method [11 13] So in the first stepwe linearize the problem (1)ndash(3) For this purpose somenominal trajectory 1199100

(119905) and parameter 1205720 are chosen and itis assumed that 119896th iteration is already hold If we linearize(1) with respect to these data we obtain

119910

119896=

120597119891 (119910

119896minus1 120572

119896minus1)

120597119910

119910

119896+

120597119891 (119910

119896minus1 120572

119896minus1)

120597120572

120572

119896

+119891 (119910

119896minus1 120572

119896minus1) minus

120597119891 (119910

119896minus1 120572

119896minus1)

120597119910

119910

119896minus1

minus

120597119891 (119910

119896minus1 120572

119896minus1)

120597120572

120572

119896minus1

(4)

After integration of the linear differential equation (4) withcondition (2) we get the representation [16]

119910

119896(119905) = Φ

119896minus1(119905 1199050) sdot 119910

119896(1199050) +Φ

119896minus11 (119905 1199050) sdot 120572

119896

+Φ

119896minus12 (119905 1199050)

(5)

where Φ

119896minus1(119905 1199050) is a fundamental matrix of the system of

homogeneous equations

119910

119896(119905) =

120597119891 (119910

119896minus1 120572

119896minus1)

120597119910

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119910=119910119896minus1

120572=120572119896minus1

sdot 119910

119896(1199050) (6)

and the matricesΦ119896minus11 (119905 1199050) Φ119896minus12 (119905 1199050) are defined as in [16]

Φ

119896minus11 (119905 1199050) = int

119905

1199050

Φ (119905 120591)

120597119891 (119910

119896minus1 120572

119896minus1)

120597120572

119889120591

Φ

119896minus12 (119905 1199050) = int

119905

1199050

Φ (119905 120591) [119891 (119910

119896minus1 120572

119896minus1)

minus

120597119891 (119910

119896minus1 120572

119896minus1)

120597119910

119910

119896minus1minus

120597119891 (119910

119896minus1 120572

119896minus1)

120597120572

120572

119896minus1]119889120591

(7)

To provide that the solution 119910

119894(119879) of the linearized

differential equation (4) with initial data (2) coincides withthe values of the measurements 119910119894

119879= 119910

119879119894in the point 119879

we construct the following quadratic functional in the 119896thiteration

119868

119896=

12

119873

sum

119894=1[119910

119896

119894(119879 120572) minus 119910

119896

119879119894]

119879

119860

119896

119894[119910

119896

119894(119879 120572) minus 119910

119896

119879119894] (8)

where the sign 119879 stands for transpore 119860119896 is a constantsymmetre 119898 times 119899-dimensional weight matrix that is chosenin each iteration considering the specifics of the concreteproblem 119910119896

119879119894is 119899times 1-dimensional vector of observation 119910119896

119894is

119899 times 1-dimensional vector defined by (5) Then the solution ofthe stated problem is reduced to the problem Find a constantvector 120572 by which the solution of (1) with conditions (2)minimizes the functional (8)

Various algorithms exist for theminimization of the func-tional (8) However in the solution of the concrete problemas well as problem arising in the oil production these algo-rithms met some difficulties [14] (eg to reach the necessaryaccuracy and speed of convergence) Therefore in [6] the useof Gram-Schmidt orthogonalization method is proposed

4 Algorithm for the Minimization ofthe Functional (8)

Here we consider the minimization of the functional (8) bythe help of the relation (4) with conditions (2) Putting 119910119896(119879)from (5) into (8) we get

119868

119896=

12

119873

sum

119894=1[Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050) +Φ

119896minus11119894 (119879 1199050) sdot 120572

+Φ

119896minus12119894 (119879 1199050) minus 119910119879119894]

119879

119860

119896

119894[Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

+Φ

119896minus11119894 (119879 1199050) sdot 120572 +Φ

119896minus12119894 (119879 1199050) minus 119910119879119894] =

16sum

119895=1119868

119895

119896

(9)

Considering the symmetricity of the matrix 119860

119896 the relation(9) may be written as

119868

1119896

def=

12

sdot

119873

sum

119894=1[Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)]119879

119860

119896

119894Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

119868

2119896

def=

12

119873

sum

119894=1[Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)]119879

119860

119896

119894Φ

119896minus11119894 (119879 1199050) sdot 120572

119868

3119896

def=

12

119873

sum

119894=1[Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)]119879

119860

119896

119894Φ

119896minus12119894 (119879 1199050)

119868

4119896

def= minus

12

119873

sum

119894=1[Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)]119879

119860

119896

119894119910

119896

119879119894

Mathematical Problems in Engineering 3

119868

5119896

def=

12

119873

sum

119894=1[Φ

119896minus11119894 (119879 1199050) sdot 120572]

119879

119860

119896

119894Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

119868

6119896

def=

12

119873

sum

119894=1[Φ

119896minus11119894 (119879 1199050) sdot 120572]

119879

119860

119896

119894Φ

119896minus11119894 (119879 1199050) sdot 120572

119868

7119896

def=

12

119873

sum

119894=1[Φ

119896minus11119894 (119879 1199050) sdot 120572]

119879

119860

119896

119894Φ

119896minus12119894 (119879 1199050)

119868

8119896

def= minus

12

119873

sum

119894=1[Φ

119896minus11119894 (119879 1199050) sdot 120572]

119879

119860

119896

119894119910

119896

119879119894

119868

9119896

def=

12

119873

sum

119894=1[Φ

119896minus12119894 (119879 1199050)]

119879

119860

119896

119894Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

119868

10119896

def=

12

119873

sum

119894=1[Φ

119896minus12119894 (119879 1199050)]

119879

119860

119896

119894Φ

119896minus11119894 (119879 1199050) sdot 120572

119868

11119896

def=

12

119873

sum

119894=1[Φ

119896minus12119894 (119879 1199050)]

119879

119860

119896

119894Φ

119896minus12119894 (119879 1199050)

119868

12119896

def= minus

12

119873

sum

119894=1[Φ

119896minus12119894 (119879 1199050)]

119879

119860

119896

119894119910

119896

119879119894

119868

13119896

def= minus

12

119873

sum

119894=1[119910

119896

119879119894]

119879

119860

119896

119894Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

119868

14119896

def= minus

12

119873

sum

119894=1[119910

119896

119879119894]

119879

119860

119896

119894Φ

119896minus11119894 (119879 1199050) sdot 120572

119868

15119896

def= minus

12

119873

sum

119894=1[119910

119896

119879119894]

119879

119860

119896

119894Φ

119896minus12119894 (119879 1199050)

119868

16119896

def= minus

12

119873

sum

119894=1[119910

119896

119879119894]

119879

119860

119896

119894119910

119896

119879119894

(10)

and gradient of the functional (9) has a form

120597119868

119896

120597120572

=

16sum

119895=1

120597119868

119895

119896

120597120572

(11)

Since the terms 1198681119896 1198683119896 1198684119896 1198689119896 11986811119896 11986812119896 11986813119896 11986815119896 11986816119896

do notdepend on the parameter 120572 we have

120597119868

1119896

120597120572

=

120597119868

3119896

120597120572

=

120597119868

4119896

120597120572

=

120597119868

9119896

120597120572

=

120597119868

11119896

120597120572

=

120597119868

12119896

120597120572

=

120597119868

13119896

120597120572

=

120597119868

15119896

120597120572

=

120597119868

16119896

120597120572

= 0

(12)

Based on the formulas

120597119909

119879119886

120597119909

=

120597119886

119879119909

120597119909

= 119886

120597119909

119879119861119909

120597119909

= (119861+119861

119879) 119909

(13)

from [17 18] for the gradients of 1198682119896 1198685119896 1198686119896 1198687119896 1198688119896 11986810119896 11986814119896 we

get the formulas

119868

2119896

120597120572

= ([Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)]119879

119860

119896

119894Φ

119896minus11119894 (119879 1199050))

119879

= Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894

119879

Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

119868

5119896

120597120572

=

120572

119879Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

120597120572

= Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

119868

6119896

120597120572

=

120572

119879Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus11119894 (119879 1199050) 120572

120597120572

= [Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus11119894 (119879 1199050)

+Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894

119879

Φ

119896minus11119894 (119879 1199050)] 120572

= 2Φ119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus11119894 (119879 1199050) 120572

119868

7119896

120597120572

=

120572

119879Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus12119894 (119879 1199050)

120597120572

= Φ

119896minus11119894119879

(119879 1199050)

sdot 119860

119896

119894Φ

119896minus12119894 (119879 1199050)

119868

8119896

120597120572

= minus

[Φ

119896minus11119894 (119879 1199050) 120572]

119879

119860

119896

119894119910

119896

119879119894

120597120572

= minus

120572

119879Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894119910

119896

119879119894

120597120572

= minusΦ

119896minus11119894119879

(119879 1199050)

sdot 119860

119896

119894119910

119896

119879119894

119868

10119896

120597120572

= Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894

119879

Φ

119896minus12119894 (119879 1199050)

119868

14119896

120597120572

= minus

[119910

119896

119879119894]

119879

119860

119896

119894Φ

119896minus11119894 (119879 1199050) 120572

120597120572

= minusΦ

119896minus11119894119879

(119879 1199050)

sdot 119860

119896

119894

119879

119910

119896

119879119894

(14)

4 Mathematical Problems in Engineering

Finally if we consider these results then the gradient ofthe functional (9) will be defined by the formula

120597119868

119896

120597120572

=

12

119873

sum

119894=1(

119868

2119896

120597120572

+

119868

5119896

120597120572

+

119868

6119896

120597120572

+

119868

7119896

120597120572

+

119868

8119896

120597120572

+

119868

10119896

120597120572

+

119868

14119896

120597120572

) =

12

sdot

119873

sum

119894=1(Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894

119879

Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

+Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

+ 2Φ119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus11119894 (119879 1199050) 120572

+Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus12119894 (119879 1199050)

minusΦ

119896minus11119894119879

(119879 1199050) 119860119896

119894119910

119896

119879119894

+Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894

119879

Φ

119896minus12119894 (119879 1199050)

minusΦ

119896minus11119894119879

(119879 1199050) 119860119896

119894

119879

119910

119896

119879119894) =

12

sdot

119873

sum

119894=1(2Φ119896minus11119894

119879

(119879 1199050) 119860119896

119894

119879

Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

+ 2Φ119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus11119894 (119879 1199050) 120572

+ 2Φ119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus12119894 (119879 1199050)

minus 2Φ119896minus11119894119879

(119879 1199050) 119860119896

119894119910

119896

119879119894) = 0

(15)

Then for the gradient of the functional 119868119896relatively to the

parameter 120572 we get the expression

120597119868

119896

120597120572

=

119873

sum

119894=1[Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus11119894 (119879 1199050)] 120572

minus

119873

sum

119894=1[Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894119910

119896

119879119894

minusΦ

119896minus11119894119879

(119879 1199050) 119860119896

119894

119879

Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

minusΦ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus12119894 (119879 1199050)]

(16)

Taking equal to zero the expression (16) we get

119873

sum

119894=1[Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus11119894 (119879 1199050)] 120572

=

119873

sum

119894=1[Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894119910

119896

119879119894

minusΦ

119896minus11119894119879

(119879 1199050) 119860119896

119894

119879

Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

minusΦ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus12119894 (119879 1199050)]

(17)

Solution of (17) with respect to 120572 gives

120572 = [

119873

sum

119894=1Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus11119894 (119879 1199050)]

minus1

sdot

119873

sum

119894=1[Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894119910

119896

119879119894

minusΦ

119896minus11119894119879

(119879 1199050) 119860119896

119894

119879

Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

minusΦ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus12119894 (119879 1199050)]

(18)

where it is assumed that [sum119873119894=1 Φ119896minus11119894119879

(119879 1199050)119860119896

119894Φ

119896minus11119894 (119879 1199050)]

minus1

existsValue of the parameter 120572 defined by (18) is a solution of

the multiparameter optimization problem for the functional(9) that gives minimum to the cost functional

Considering the above the following algorithm may beproposed to the solution of the identification problem (1) (2)

Algorithm 1 (1) Construct the function 119891(119909) from (1)initial and final data 1199101198940 and 119910

119894

119879(119894 = 1 119873) from (2) and (3)

correspondingly(2) Calculate the derivatives 120597119891(119910119896minus1 120572119896minus1)120597119910 120597119891(119910119896minus1

120572

119896minus1)120597120572 taking as initial approaches 119910

119894and 120572

119894

(3)Calculate the fundamentalmatrixΦ119896minus1(119905 1199050) from (6)reconstruct Φ119896minus11 (119905 1199050) Φ

119896minus12 (119905 1199050) from (7) and functional 119868

119896

from (8)(4) Solving the system of algebraic equations (14)

relatively 120572 find the value of the 119898-dimensional vector 120572119896 inthe 119896th iteration

(5) Check the condition1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

120597119868

119896

120597120572

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

lt 120576 (19)

where 120576 is a given small enough number If the condition(16) is satisfied the process stops otherwise go to Step (2)The convergence of this algorithm may be proved similar to[13 14]

Now we discuss the realization of this algorithm

5 Calculational Algorithm

As one can see in the realization of the above algorithmthe main step is a calculation of the fundamental matrixΦ

119896minus1(119905 119905

0) and the matricesΦ119896minus11 (119905 119905

0)Φ119896minus12 (119905 119905

0) Note that

as is mentioned in [16] construction of these matrices is anenough difficult procedure So for simplicity we try (4) andfind the corresponding derivatives by using the Eulermethod

Mathematical Problems in Engineering 5

Really 120597119891(119910(119905) 120572)120597119910 and 120597119891(119910(119905) 120572)120597120572 everywhere arereplaced by

119891(119910) = (119891(119910 + 120575 120572) minus 119891(119910 120572))120575 and

119891(120572) =

(119891(119910 120572 + 1205751) minus 119891(119910 120572))1205751 correspondingly where 120575 and 1205751are small enough numbers

To calculate the fundamental matrix Φ

119896minus1(119905 119905

0) and the

matrices Φ119896minus11 (119905 1199050) Φ119896minus12 (119905 1199050) it is proper to replace (4) by

the following discrete one

119910

119896(1199052119873)

= (

119895

prod

1198941=2119873minus1(119864+ 120575

119891 (119910

119896minus1(119905

1198941))))119910

119896(119905

119873+1)

+Φ

119896minus11 (119905 1199050) 120572 +Φ

119896minus12 (119905 1199050)

(20)

where

Φ

119896minus1119894

(1199052119873 1199050) =

119895

prod

1198941=2119873minus1(119864+ 120575

119891 (119910

119896minus1(119905

119894)))

Φ

119896minus11119894 (1199052119873 1199050)

= (

2119873minus1sum

119895=119873+2(

119895

prod

1198941=2119873minus1(119864+ 120575 sdot

119891 (119910

119896minus1(119905

119894))))

sdot 120575

119891 (120572

119896minus1119895minus1))+120575 sdot

119891 (120572

119896minus12119873minus1)

Φ

119896minus12119894 (1199052119873 1199050)

= (

2119873minus1sum

119895=119873+2(

119895

prod

1198941=2119873minus1(119864+ 120575 sdot

119891 (119910

119896minus1(119905

119894))))120575

sdot (

119891 (119910

119896minus1(119905

119895minus1) 120572119896minus1

)

minus

119891 (119910

119896minus1(119905

119895minus1)) 119910119896minus1

(119905

119895minus1) minus

119891 (120572

119896minus1) 120572

119896minus1))

+120575 sdot (

119891 (119910

119896minus1(1199052119873minus1) 120572

119896minus1) minus

119891 (119910

119896minus1(1199052119873minus1))

sdot 119910

119896minus1(1199052119873minus1) minus

119891 (120572

119896minus1) 120572

119896minus1)

(21)

119864 is unit matrix of proper dimensionThen from (20) we get that Φ119896minus1(119905 1199050) is a fundamental

matrix for the system of homogeneous equations

119910

119896(119905

119894+1) = (119864+ 120575

119891 (119910))

1003816

1003816

1003816

1003816

1003816

119910=119910119896minus1

120572=120572119896minus1

sdot 119910

119896(119905

119894) (22)

Therefore similar to the nongradient methods we pro-pose an algorithm based on the orthogonalization of gradientdirections using the Gram-Schmidt procedure

Step 1 Using Gram-Schmidt orthogonalization vectors 120596119895119894

119894 = 119895119895 + 1 119899 119895 = 2 3 1198992 are calculated and the set ofvectors

(

120597119891 (120582

1119888)

120597120582

1119888

120597119891 (120582

2119888)

120597120582

2119888

120597119891 (120582

119894

119888)

120597120582

119894

119888

120596

119895

119894+1 120596119895

119899)

(23)

is found which form an orthogonal basis in 119877

119899If we apply the orthogonalization algorithm then a

linearly independent system 1198861 1198862 119886119896 should be formedthat is orthogonal system 1198871 1198872 119887119896 and each vector 119887

119894

should be linearly expressed through 1198861 1198862 119886119894 Here 119886

119894

and 119887

119894are upper triangular matrices Thus it is possible to

ensure that the systems 119887

119894 were orthonormal where the

diagonal elements of the transition matrix are positive bythese conditions the system 119887

119894 and the transition matrix are

uniquely determinedThe algorithm considers 1198871 = 1198861 if the vectors

1198871 1198872 119887119894minus1 are constructed Then

119887

119894= 119886

119894minus

119894minus1sum

119895=1

⟨119886

119894 119887

119895⟩

⟨119887

119894 119887

119895⟩

119887

119895 (24)

where ⟨ ⟩ is the sign of the scalar product of vectors

Step 2 For the orthogonalization of the gradient directionswe compute ]119895

119894in the form

]119895119894=

119891 (120582

119888+ 120575120596

119895

119894) minus 119891 (120582

119888minus 120575120596

119895

119894)

2120575

119894 = 119895 119895 + 1 119899

(25)

Here 120575 gt 0 is any small parameter

Step 3 The orthogonal gradient directions are chosen in theform

119897

119895=

119899

sum

119894=119895

]119895119894120596

119895

119894 (26)

Replacing nabla119891(120582

(119896)

119888) by 119897

119895in (17) the nongradient iterative

minimization procedure will be

120582

(119896+1)119888

= 120582

(119896)

119888minus120594

lowast(119896)119897

119896

(27)

where 120594lowast(119896) is a scalar which is determined by golden sectionmethod

Now we apply the above proposed technique to theexample of 15 production by gas-lift method

Example 2 It is known that nonstable motion of gas in tubesand gas liquid mixture (GLM) in vertical tubes that is inthe lift pipe of the gas-lift well with constant across profile is

6 Mathematical Problems in Engineering

x = 0

x = lg

x = l

rc

rk

x

h

hst

l

lg

lj

Gas

Gas GLM

x = 2l

Gas liquid mixture

Figure 1

described by the following system of linear partial differentialequations of hyperbolic type (see Figure 1)

minus

120597119875

120597119909

=

120597 (120588120596

119888)

120597119905

+ 2120572120588120596119888

minus

120597119875

120597119905

= 119888

2 120597 (120588120596119888)

120597119909

(28)

where 120588 = 119875(119909 119905) 120596119888= 120596

119888(119909 119905) is an additional pressure on

its stationary value and averaged over across section speed ofmotion of GLM 119905 119909 time and coordinate 119888 speed of soundin gas andGLM 120588 gas oil and GLM in correspondence withcoordinates 2120572 = 119892120596

119888+120582

119888120596

1198882119863 119892 120582

119888 free fall accel 119863

interval effective diameter of the tube [19]The partial differential equation of gas and GLM motion

by are averaging over time 119905may be reduced to the followingordinary differential equation [20]

119876 =

2120572 (120582

119888) 120588119865119876

2

119888

2120588

2119865

2minus 119876

2 119876 (0) = 119906

(29)

where 119888 ≫ 120596

119888and all quantities are assumed constant 119876 =

120588120596

119888119865 and 119865 is area of across section of the pump-compressor

tubes and is constant relative to axes

It is assumed that the passing from the end of tubethrough the layer to the beginning of the lift (119909 = 119897) isdescribed by the following difference equations

119876 (119897 + 0) = 120574119876 (119897 minus 0) + 1205741 (119876 (119897 minus 0)) 119876

1205741 (119876 (119897 minus 0)) = minus 1205753 (119876 (119897 minus 0) minus 1205752)2+ 1205751

(30)

where 120574 and 1205751 1205752 1205753 are constant numbers to be found Forthe sake of simplicity we suppose that the parameters 120574 12057511205752 1205753 are known and it is required to reconstruct 120582

119888involved

in (19) due to 120572(120582119888)

Then some nominal trajectory 119876

0(119909) and parameter 1205720

are chosen assuming that 119896th iteration is already held Let uslinearize (29) among these data

119876

119896(119909) = 119860 (119876

119896minus1 120572

119896minus1) sdot 119876

119896(119909) + 119861 (119876

119896minus1 120572

119896minus1) 120572

119896

+119862 (119876

119896minus1 120572

119896minus1)

(31)

where

120597119891 (119910

119896minus1 120572

119896minus1)

120597119910

= 119860 (119876

119896minus1 120572

119896minus1)

=

4120572119896minus1119888212058831198653119876

119896minus1

(119888

2120588

2119865

2minus (119876

119896minus1)

2)

2

120597119891 (119910

119896minus1 120572

119896minus1)

120597120572

= 119861 (119876

119896minus1 120572

119896minus1)

=

2120572119896minus1120588119865119876119896minus1

119888

2120588

2119865

2minus (119876

119896minus1)

2

119891 (119910

119896minus1 120572

119896minus1) minus

120597119891 (119910

119896minus1 120572

119896minus1)

120597119910

119910

119896minus1

minus

120597119891 (119910

119896minus1 120572

119896minus1)

120597120572

120572

119896minus1= 119862 (119876

119896minus1 120572

119896minus1)

=

4120572119896minus1119888212058831198653(119876

119896minus1)

2+ 4120572119896minus1120588119865 (119876

119896minus1)

4

(119888

2120588

2119865

2minus (119876

119896minus1)

2)

2

(32)

Note that by the help of the relations (20) and (21) thematricesΦ

119896minus1(119909 0) Φ119896minus11 (119909 0) Φ119896minus12 (119909 0) are calculated as follows

Φ

119896minus1119894

(119909 0) =119895

prod

119894=2119873minus1(119864+119860 (119876

119896minus1(119909

1198941) 120572

119896minus1)) ℎ

Φ

119896minus11119894 (1199092119873 0)

= (

2119873minus1sum

119895=119873+2(

119895

prod

119894=2119873minus1(119864+119860 (119876

119896minus1(119909

1198941) 120572

119896minus1)) ℎ)

sdot 119861 (119876

119896minus1(119909

119895minus1) 120572119896minus1

) ℎ)

+119861 (119876

119896minus1(1199092119873minus1) 120572

119896minus1) ℎ

Mathematical Problems in Engineering 7

Table 1

119910

119897+0119894

55698 55732 55761 55810 55848 55852 55824119910

2119897119894

44242 44248 44254 44262 44266 44263 44251

Table 2

120575 005 01 05 054 055 06120582 01966 02141 02295 02298 02299 02302

01 02 03 04 05 06 07 08 09 10214

0216

0218

022

0222

0224

0226

0228

023

0232

Figure 2

Φ

119896minus12119894 (1199092119873 0)

= (

2119873minus1sum

119895=119873+2(

119895

prod

119894=2119873minus1(119864+119860 (119876

119896minus1(119909

1198941) 120572

119896minus1)) ℎ)

sdot 119862 (119876

119896minus1(119909

119895minus1) 120572119896minus1

) ℎ)

+119862 (119876

119896minus1(1199092119873minus1) 120572

119896minus1) ℎ

(33)

where ℎ is small enough numberLet some statistical data be givenLet us assume that some observation points for 119910119897+0

119894and

119910

2119897119894are given (see Table 1)We give in Table 2 the values of 120582 obtained by using

MATLAB by given input parametersAs we see from Table 2 by 120575 = 055 120582 gets value 02299

with error estimation 10minus3Here is the dependence of 120582 or 120575 (see Figure 2)The above algorithm reaches given accuracy after 4

iterations and gives 120582 = 02299Note that the inequality

1003816

1003816

1003816

1003816

120582

119894minus1205824

1003816

1003816

1003816

1003816

le 10minus15 (34)

holds for any 119894 gt 4 that shows the stability of the proposedquasilinearization algorithm

Conflict of Interests

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

Acknowledgment

This work is supported by the joint grant of ANAS andSOCAR 17 2013ndash2015 Baku State University ldquo50+50rdquo Grant

References

[1] A Aliev Fikret and N A Ismailov ldquoInverse problem todetermine the hydraulic resistance coefficient in the gas liftprocessrdquo Applied and Computational Mathematics vol 12 no3 pp 306ndash313 2013

[2] R Gurbanov N Nuriyev and R S Gurbanov ldquoTechnologicalcontrol and optimization problems in oil production theoryand practicerdquo Applied and Computational Mathematics vol 12no 3 pp 314ndash324 2013

[3] R N Bakhtisin and A R Latypov ldquoEstimation of the order oflinear objects by experimental informationrdquo Automation andRemote Control no 3 pp 108ndash112 1992

[4] Y S Gasimov ldquoOn a shape design problem for one spectralfunctionalrdquo Journal of Inverse and Ill-Posed Problems vol 21 no5 pp 629ndash637 2013

[5] L Lyuing Identification of the System Theory for Users NaukaMoscow Russia 1991

[6] F A AlievMMMutallimov IMAskerov and I S RaguimovldquoAsymptotic method of solution for a problem of constructionof optimal gas-lift process modesrdquo Mathematical Problems inEngineering vol 2010 Article ID 191053 10 pages 2010

[7] A S Apostolyuk and V B Larin ldquoUpdating of linear stationarydynamic systemparametersrdquoApplied andComputationalMath-ematics vol 10 no 3 pp 402ndash408 2011

[8] F Ding ldquoHierarchical multi-innovation stochastic gradientalgorithm for Hammerstein nonlinear system modelingrdquoApplied Mathematical Modelling vol 37 no 4 pp 1694ndash17042013

[9] F Ding Y Shi and T Chen ldquoAuxiliary model-based least-squares identification methods for Hammerstein output-errorsystemsrdquo Systems amp Control Letters vol 56 no 5 pp 373ndash3802007

[10] S I Kabanikhin and O I KrivorotrsquoKo ldquoA numerical methodfor determining the amplitude of a wave edge in shallow waterapproximationrdquo Applied and Computational Mathematics vol12 no 1 pp 91ndash96 2013

[11] P E Bellman and P E Kalaba Quailinearization and NonlinearBoundary Problems MirVoscow Russia 1968

[12] V E Shamansky Methods of Numerical Solution of Baun-daryProblems in PC Naukova Dumka Kiev 1966

[13] A Brayson and X Yu-shi Applied Theory of Optimal ControlMir Moscow Russia 1972

[14] DMHimmebblauAppliedNonlinear Programming Craw-HillBook Company New York NY USA 1972

[15] K R Aydazade ldquoComputatioonal problems in hydraulic net-worksrdquo Computational Mathematics and Mathematical Physicsvol 29 no 2 pp 184ndash193 1989

[16] Y N Andreev Control of the Finite Dimensional Linear ObjectsNauka Moscow Russia 1976

8 Mathematical Problems in Engineering

[17] J R Magnus and H Neudecker Matrix Differential Calculuswith Applications in Statistics and Econometrics John Wiley ampSons Chichester UK 17th edition 1988

[18] K B Petersen and M S PedersenTheMatrix Cookbook 2008httpmatrixcookbookcom

[19] D M Altshul Hidraulic Resistance Nedra Moscow Russia1970

[20] M Ghanbari S Abbasbandy and T Allahviranloo ldquoA newapproach to determine the convergence-control parameter inthe application of the homotopy analysis method to systems oflinear equationsrdquo Applied and Computational Mathematics vol12 no 3 pp 355ndash364 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

119868

5119896

def=

12

119873

sum

119894=1[Φ

119896minus11119894 (119879 1199050) sdot 120572]

119879

119860

119896

119894Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

119868

6119896

def=

12

119873

sum

119894=1[Φ

119896minus11119894 (119879 1199050) sdot 120572]

119879

119860

119896

119894Φ

119896minus11119894 (119879 1199050) sdot 120572

119868

7119896

def=

12

119873

sum

119894=1[Φ

119896minus11119894 (119879 1199050) sdot 120572]

119879

119860

119896

119894Φ

119896minus12119894 (119879 1199050)

119868

8119896

def= minus

12

119873

sum

119894=1[Φ

119896minus11119894 (119879 1199050) sdot 120572]

119879

119860

119896

119894119910

119896

119879119894

119868

9119896

def=

12

119873

sum

119894=1[Φ

119896minus12119894 (119879 1199050)]

119879

119860

119896

119894Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

119868

10119896

def=

12

119873

sum

119894=1[Φ

119896minus12119894 (119879 1199050)]

119879

119860

119896

119894Φ

119896minus11119894 (119879 1199050) sdot 120572

119868

11119896

def=

12

119873

sum

119894=1[Φ

119896minus12119894 (119879 1199050)]

119879

119860

119896

119894Φ

119896minus12119894 (119879 1199050)

119868

12119896

def= minus

12

119873

sum

119894=1[Φ

119896minus12119894 (119879 1199050)]

119879

119860

119896

119894119910

119896

119879119894

119868

13119896

def= minus

12

119873

sum

119894=1[119910

119896

119879119894]

119879

119860

119896

119894Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

119868

14119896

def= minus

12

119873

sum

119894=1[119910

119896

119879119894]

119879

119860

119896

119894Φ

119896minus11119894 (119879 1199050) sdot 120572

119868

15119896

def= minus

12

119873

sum

119894=1[119910

119896

119879119894]

119879

119860

119896

119894Φ

119896minus12119894 (119879 1199050)

119868

16119896

def= minus

12

119873

sum

119894=1[119910

119896

119879119894]

119879

119860

119896

119894119910

119896

119879119894

(10)

and gradient of the functional (9) has a form

120597119868

119896

120597120572

=

16sum

119895=1

120597119868

119895

119896

120597120572

(11)

Since the terms 1198681119896 1198683119896 1198684119896 1198689119896 11986811119896 11986812119896 11986813119896 11986815119896 11986816119896

do notdepend on the parameter 120572 we have

120597119868

1119896

120597120572

=

120597119868

3119896

120597120572

=

120597119868

4119896

120597120572

=

120597119868

9119896

120597120572

=

120597119868

11119896

120597120572

=

120597119868

12119896

120597120572

=

120597119868

13119896

120597120572

=

120597119868

15119896

120597120572

=

120597119868

16119896

120597120572

= 0

(12)

Based on the formulas

120597119909

119879119886

120597119909

=

120597119886

119879119909

120597119909

= 119886

120597119909

119879119861119909

120597119909

= (119861+119861

119879) 119909

(13)

from [17 18] for the gradients of 1198682119896 1198685119896 1198686119896 1198687119896 1198688119896 11986810119896 11986814119896 we

get the formulas

119868

2119896

120597120572

= ([Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)]119879

119860

119896

119894Φ

119896minus11119894 (119879 1199050))

119879

= Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894

119879

Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

119868

5119896

120597120572

=

120572

119879Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

120597120572

= Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

119868

6119896

120597120572

=

120572

119879Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus11119894 (119879 1199050) 120572

120597120572

= [Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus11119894 (119879 1199050)

+Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894

119879

Φ

119896minus11119894 (119879 1199050)] 120572

= 2Φ119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus11119894 (119879 1199050) 120572

119868

7119896

120597120572

=

120572

119879Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus12119894 (119879 1199050)

120597120572

= Φ

119896minus11119894119879

(119879 1199050)

sdot 119860

119896

119894Φ

119896minus12119894 (119879 1199050)

119868

8119896

120597120572

= minus

[Φ

119896minus11119894 (119879 1199050) 120572]

119879

119860

119896

119894119910

119896

119879119894

120597120572

= minus

120572

119879Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894119910

119896

119879119894

120597120572

= minusΦ

119896minus11119894119879

(119879 1199050)

sdot 119860

119896

119894119910

119896

119879119894

119868

10119896

120597120572

= Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894

119879

Φ

119896minus12119894 (119879 1199050)

119868

14119896

120597120572

= minus

[119910

119896

119879119894]

119879

119860

119896

119894Φ

119896minus11119894 (119879 1199050) 120572

120597120572

= minusΦ

119896minus11119894119879

(119879 1199050)

sdot 119860

119896

119894

119879

119910

119896

119879119894

(14)

4 Mathematical Problems in Engineering

Finally if we consider these results then the gradient ofthe functional (9) will be defined by the formula

120597119868

119896

120597120572

=

12

119873

sum

119894=1(

119868

2119896

120597120572

+

119868

5119896

120597120572

+

119868

6119896

120597120572

+

119868

7119896

120597120572

+

119868

8119896

120597120572

+

119868

10119896

120597120572

+

119868

14119896

120597120572

) =

12

sdot

119873

sum

119894=1(Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894

119879

Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

+Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

+ 2Φ119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus11119894 (119879 1199050) 120572

+Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus12119894 (119879 1199050)

minusΦ

119896minus11119894119879

(119879 1199050) 119860119896

119894119910

119896

119879119894

+Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894

119879

Φ

119896minus12119894 (119879 1199050)

minusΦ

119896minus11119894119879

(119879 1199050) 119860119896

119894

119879

119910

119896

119879119894) =

12

sdot

119873

sum

119894=1(2Φ119896minus11119894

119879

(119879 1199050) 119860119896

119894

119879

Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

+ 2Φ119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus11119894 (119879 1199050) 120572

+ 2Φ119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus12119894 (119879 1199050)

minus 2Φ119896minus11119894119879

(119879 1199050) 119860119896

119894119910

119896

119879119894) = 0

(15)

Then for the gradient of the functional 119868119896relatively to the

parameter 120572 we get the expression

120597119868

119896

120597120572

=

119873

sum

119894=1[Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus11119894 (119879 1199050)] 120572

minus

119873

sum

119894=1[Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894119910

119896

119879119894

minusΦ

119896minus11119894119879

(119879 1199050) 119860119896

119894

119879

Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

minusΦ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus12119894 (119879 1199050)]

(16)

Taking equal to zero the expression (16) we get

119873

sum

119894=1[Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus11119894 (119879 1199050)] 120572

=

119873

sum

119894=1[Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894119910

119896

119879119894

minusΦ

119896minus11119894119879

(119879 1199050) 119860119896

119894

119879

Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

minusΦ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus12119894 (119879 1199050)]

(17)

Solution of (17) with respect to 120572 gives

120572 = [

119873

sum

119894=1Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus11119894 (119879 1199050)]

minus1

sdot

119873

sum

119894=1[Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894119910

119896

119879119894

minusΦ

119896minus11119894119879

(119879 1199050) 119860119896

119894

119879

Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

minusΦ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus12119894 (119879 1199050)]

(18)

where it is assumed that [sum119873119894=1 Φ119896minus11119894119879

(119879 1199050)119860119896

119894Φ

119896minus11119894 (119879 1199050)]

minus1

existsValue of the parameter 120572 defined by (18) is a solution of

the multiparameter optimization problem for the functional(9) that gives minimum to the cost functional

Considering the above the following algorithm may beproposed to the solution of the identification problem (1) (2)

Algorithm 1 (1) Construct the function 119891(119909) from (1)initial and final data 1199101198940 and 119910

119894

119879(119894 = 1 119873) from (2) and (3)

correspondingly(2) Calculate the derivatives 120597119891(119910119896minus1 120572119896minus1)120597119910 120597119891(119910119896minus1

120572

119896minus1)120597120572 taking as initial approaches 119910

119894and 120572

119894

(3)Calculate the fundamentalmatrixΦ119896minus1(119905 1199050) from (6)reconstruct Φ119896minus11 (119905 1199050) Φ

119896minus12 (119905 1199050) from (7) and functional 119868

119896

from (8)(4) Solving the system of algebraic equations (14)

relatively 120572 find the value of the 119898-dimensional vector 120572119896 inthe 119896th iteration

(5) Check the condition1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

120597119868

119896

120597120572

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

lt 120576 (19)

where 120576 is a given small enough number If the condition(16) is satisfied the process stops otherwise go to Step (2)The convergence of this algorithm may be proved similar to[13 14]

Now we discuss the realization of this algorithm

5 Calculational Algorithm

As one can see in the realization of the above algorithmthe main step is a calculation of the fundamental matrixΦ

119896minus1(119905 119905

0) and the matricesΦ119896minus11 (119905 119905

0)Φ119896minus12 (119905 119905

0) Note that

as is mentioned in [16] construction of these matrices is anenough difficult procedure So for simplicity we try (4) andfind the corresponding derivatives by using the Eulermethod

Mathematical Problems in Engineering 5

Really 120597119891(119910(119905) 120572)120597119910 and 120597119891(119910(119905) 120572)120597120572 everywhere arereplaced by

119891(119910) = (119891(119910 + 120575 120572) minus 119891(119910 120572))120575 and

119891(120572) =

(119891(119910 120572 + 1205751) minus 119891(119910 120572))1205751 correspondingly where 120575 and 1205751are small enough numbers

To calculate the fundamental matrix Φ

119896minus1(119905 119905

0) and the

matrices Φ119896minus11 (119905 1199050) Φ119896minus12 (119905 1199050) it is proper to replace (4) by

the following discrete one

119910

119896(1199052119873)

= (

119895

prod

1198941=2119873minus1(119864+ 120575

119891 (119910

119896minus1(119905

1198941))))119910

119896(119905

119873+1)

+Φ

119896minus11 (119905 1199050) 120572 +Φ

119896minus12 (119905 1199050)

(20)

where

Φ

119896minus1119894

(1199052119873 1199050) =

119895

prod

1198941=2119873minus1(119864+ 120575

119891 (119910

119896minus1(119905

119894)))

Φ

119896minus11119894 (1199052119873 1199050)

= (

2119873minus1sum

119895=119873+2(

119895

prod

1198941=2119873minus1(119864+ 120575 sdot

119891 (119910

119896minus1(119905

119894))))

sdot 120575

119891 (120572

119896minus1119895minus1))+120575 sdot

119891 (120572

119896minus12119873minus1)

Φ

119896minus12119894 (1199052119873 1199050)

= (

2119873minus1sum

119895=119873+2(

119895

prod

1198941=2119873minus1(119864+ 120575 sdot

119891 (119910

119896minus1(119905

119894))))120575

sdot (

119891 (119910

119896minus1(119905

119895minus1) 120572119896minus1

)

minus

119891 (119910

119896minus1(119905

119895minus1)) 119910119896minus1

(119905

119895minus1) minus

119891 (120572

119896minus1) 120572

119896minus1))

+120575 sdot (

119891 (119910

119896minus1(1199052119873minus1) 120572

119896minus1) minus

119891 (119910

119896minus1(1199052119873minus1))

sdot 119910

119896minus1(1199052119873minus1) minus

119891 (120572

119896minus1) 120572

119896minus1)

(21)

119864 is unit matrix of proper dimensionThen from (20) we get that Φ119896minus1(119905 1199050) is a fundamental

matrix for the system of homogeneous equations

119910

119896(119905

119894+1) = (119864+ 120575

119891 (119910))

1003816

1003816

1003816

1003816

1003816

119910=119910119896minus1

120572=120572119896minus1

sdot 119910

119896(119905

119894) (22)

Therefore similar to the nongradient methods we pro-pose an algorithm based on the orthogonalization of gradientdirections using the Gram-Schmidt procedure

Step 1 Using Gram-Schmidt orthogonalization vectors 120596119895119894

119894 = 119895119895 + 1 119899 119895 = 2 3 1198992 are calculated and the set ofvectors

(

120597119891 (120582

1119888)

120597120582

1119888

120597119891 (120582

2119888)

120597120582

2119888

120597119891 (120582

119894

119888)

120597120582

119894

119888

120596

119895

119894+1 120596119895

119899)

(23)

is found which form an orthogonal basis in 119877

119899If we apply the orthogonalization algorithm then a

linearly independent system 1198861 1198862 119886119896 should be formedthat is orthogonal system 1198871 1198872 119887119896 and each vector 119887

119894

should be linearly expressed through 1198861 1198862 119886119894 Here 119886

119894

and 119887

119894are upper triangular matrices Thus it is possible to

ensure that the systems 119887

119894 were orthonormal where the

diagonal elements of the transition matrix are positive bythese conditions the system 119887

119894 and the transition matrix are

uniquely determinedThe algorithm considers 1198871 = 1198861 if the vectors

1198871 1198872 119887119894minus1 are constructed Then

119887

119894= 119886

119894minus

119894minus1sum

119895=1

⟨119886

119894 119887

119895⟩

⟨119887

119894 119887

119895⟩

119887

119895 (24)

where ⟨ ⟩ is the sign of the scalar product of vectors

Step 2 For the orthogonalization of the gradient directionswe compute ]119895

119894in the form

]119895119894=

119891 (120582

119888+ 120575120596

119895

119894) minus 119891 (120582

119888minus 120575120596

119895

119894)

2120575

119894 = 119895 119895 + 1 119899

(25)

Here 120575 gt 0 is any small parameter

Step 3 The orthogonal gradient directions are chosen in theform

119897

119895=

119899

sum

119894=119895

]119895119894120596

119895

119894 (26)

Replacing nabla119891(120582

(119896)

119888) by 119897

119895in (17) the nongradient iterative

minimization procedure will be

120582

(119896+1)119888

= 120582

(119896)

119888minus120594

lowast(119896)119897

119896

(27)

where 120594lowast(119896) is a scalar which is determined by golden sectionmethod

Now we apply the above proposed technique to theexample of 15 production by gas-lift method

Example 2 It is known that nonstable motion of gas in tubesand gas liquid mixture (GLM) in vertical tubes that is inthe lift pipe of the gas-lift well with constant across profile is

6 Mathematical Problems in Engineering

x = 0

x = lg

x = l

rc

rk

x

h

hst

l

lg

lj

Gas

Gas GLM

x = 2l

Gas liquid mixture

Figure 1

described by the following system of linear partial differentialequations of hyperbolic type (see Figure 1)

minus

120597119875

120597119909

=

120597 (120588120596

119888)

120597119905

+ 2120572120588120596119888

minus

120597119875

120597119905

= 119888

2 120597 (120588120596119888)

120597119909

(28)

where 120588 = 119875(119909 119905) 120596119888= 120596

119888(119909 119905) is an additional pressure on

its stationary value and averaged over across section speed ofmotion of GLM 119905 119909 time and coordinate 119888 speed of soundin gas andGLM 120588 gas oil and GLM in correspondence withcoordinates 2120572 = 119892120596

119888+120582

119888120596

1198882119863 119892 120582

119888 free fall accel 119863

interval effective diameter of the tube [19]The partial differential equation of gas and GLM motion

by are averaging over time 119905may be reduced to the followingordinary differential equation [20]

119876 =

2120572 (120582

119888) 120588119865119876

2

119888

2120588

2119865

2minus 119876

2 119876 (0) = 119906

(29)

where 119888 ≫ 120596

119888and all quantities are assumed constant 119876 =

120588120596

119888119865 and 119865 is area of across section of the pump-compressor

tubes and is constant relative to axes

It is assumed that the passing from the end of tubethrough the layer to the beginning of the lift (119909 = 119897) isdescribed by the following difference equations

119876 (119897 + 0) = 120574119876 (119897 minus 0) + 1205741 (119876 (119897 minus 0)) 119876

1205741 (119876 (119897 minus 0)) = minus 1205753 (119876 (119897 minus 0) minus 1205752)2+ 1205751

(30)

where 120574 and 1205751 1205752 1205753 are constant numbers to be found Forthe sake of simplicity we suppose that the parameters 120574 12057511205752 1205753 are known and it is required to reconstruct 120582

119888involved

in (19) due to 120572(120582119888)

Then some nominal trajectory 119876

0(119909) and parameter 1205720

are chosen assuming that 119896th iteration is already held Let uslinearize (29) among these data

119876

119896(119909) = 119860 (119876

119896minus1 120572

119896minus1) sdot 119876

119896(119909) + 119861 (119876

119896minus1 120572

119896minus1) 120572

119896

+119862 (119876

119896minus1 120572

119896minus1)

(31)

where

120597119891 (119910

119896minus1 120572

119896minus1)

120597119910

= 119860 (119876

119896minus1 120572

119896minus1)

=

4120572119896minus1119888212058831198653119876

119896minus1

(119888

2120588

2119865

2minus (119876

119896minus1)

2)

2

120597119891 (119910

119896minus1 120572

119896minus1)

120597120572

= 119861 (119876

119896minus1 120572

119896minus1)

=

2120572119896minus1120588119865119876119896minus1

119888

2120588

2119865

2minus (119876

119896minus1)

2

119891 (119910

119896minus1 120572

119896minus1) minus

120597119891 (119910

119896minus1 120572

119896minus1)

120597119910

119910

119896minus1

minus

120597119891 (119910

119896minus1 120572

119896minus1)

120597120572

120572

119896minus1= 119862 (119876

119896minus1 120572

119896minus1)

=

4120572119896minus1119888212058831198653(119876

119896minus1)

2+ 4120572119896minus1120588119865 (119876

119896minus1)

4

(119888

2120588

2119865

2minus (119876

119896minus1)

2)

2

(32)

Note that by the help of the relations (20) and (21) thematricesΦ

119896minus1(119909 0) Φ119896minus11 (119909 0) Φ119896minus12 (119909 0) are calculated as follows

Φ

119896minus1119894

(119909 0) =119895

prod

119894=2119873minus1(119864+119860 (119876

119896minus1(119909

1198941) 120572

119896minus1)) ℎ

Φ

119896minus11119894 (1199092119873 0)

= (

2119873minus1sum

119895=119873+2(

119895

prod

119894=2119873minus1(119864+119860 (119876

119896minus1(119909

1198941) 120572

119896minus1)) ℎ)

sdot 119861 (119876

119896minus1(119909

119895minus1) 120572119896minus1

) ℎ)

+119861 (119876

119896minus1(1199092119873minus1) 120572

119896minus1) ℎ

Mathematical Problems in Engineering 7

Table 1

119910

119897+0119894

55698 55732 55761 55810 55848 55852 55824119910

2119897119894

44242 44248 44254 44262 44266 44263 44251

Table 2

120575 005 01 05 054 055 06120582 01966 02141 02295 02298 02299 02302

01 02 03 04 05 06 07 08 09 10214

0216

0218

022

0222

0224

0226

0228

023

0232

Figure 2

Φ

119896minus12119894 (1199092119873 0)

= (

2119873minus1sum

119895=119873+2(

119895

prod

119894=2119873minus1(119864+119860 (119876

119896minus1(119909

1198941) 120572

119896minus1)) ℎ)

sdot 119862 (119876

119896minus1(119909

119895minus1) 120572119896minus1

) ℎ)

+119862 (119876

119896minus1(1199092119873minus1) 120572

119896minus1) ℎ

(33)

where ℎ is small enough numberLet some statistical data be givenLet us assume that some observation points for 119910119897+0

119894and

119910

2119897119894are given (see Table 1)We give in Table 2 the values of 120582 obtained by using

MATLAB by given input parametersAs we see from Table 2 by 120575 = 055 120582 gets value 02299

with error estimation 10minus3Here is the dependence of 120582 or 120575 (see Figure 2)The above algorithm reaches given accuracy after 4

iterations and gives 120582 = 02299Note that the inequality

1003816

1003816

1003816

1003816

120582

119894minus1205824

1003816

1003816

1003816

1003816

le 10minus15 (34)

holds for any 119894 gt 4 that shows the stability of the proposedquasilinearization algorithm

Conflict of Interests

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

Acknowledgment

This work is supported by the joint grant of ANAS andSOCAR 17 2013ndash2015 Baku State University ldquo50+50rdquo Grant

References

[1] A Aliev Fikret and N A Ismailov ldquoInverse problem todetermine the hydraulic resistance coefficient in the gas liftprocessrdquo Applied and Computational Mathematics vol 12 no3 pp 306ndash313 2013

[2] R Gurbanov N Nuriyev and R S Gurbanov ldquoTechnologicalcontrol and optimization problems in oil production theoryand practicerdquo Applied and Computational Mathematics vol 12no 3 pp 314ndash324 2013

[3] R N Bakhtisin and A R Latypov ldquoEstimation of the order oflinear objects by experimental informationrdquo Automation andRemote Control no 3 pp 108ndash112 1992

[4] Y S Gasimov ldquoOn a shape design problem for one spectralfunctionalrdquo Journal of Inverse and Ill-Posed Problems vol 21 no5 pp 629ndash637 2013

[5] L Lyuing Identification of the System Theory for Users NaukaMoscow Russia 1991

[6] F A AlievMMMutallimov IMAskerov and I S RaguimovldquoAsymptotic method of solution for a problem of constructionof optimal gas-lift process modesrdquo Mathematical Problems inEngineering vol 2010 Article ID 191053 10 pages 2010

[7] A S Apostolyuk and V B Larin ldquoUpdating of linear stationarydynamic systemparametersrdquoApplied andComputationalMath-ematics vol 10 no 3 pp 402ndash408 2011

[8] F Ding ldquoHierarchical multi-innovation stochastic gradientalgorithm for Hammerstein nonlinear system modelingrdquoApplied Mathematical Modelling vol 37 no 4 pp 1694ndash17042013

[9] F Ding Y Shi and T Chen ldquoAuxiliary model-based least-squares identification methods for Hammerstein output-errorsystemsrdquo Systems amp Control Letters vol 56 no 5 pp 373ndash3802007

[10] S I Kabanikhin and O I KrivorotrsquoKo ldquoA numerical methodfor determining the amplitude of a wave edge in shallow waterapproximationrdquo Applied and Computational Mathematics vol12 no 1 pp 91ndash96 2013

[11] P E Bellman and P E Kalaba Quailinearization and NonlinearBoundary Problems MirVoscow Russia 1968

[12] V E Shamansky Methods of Numerical Solution of Baun-daryProblems in PC Naukova Dumka Kiev 1966

[13] A Brayson and X Yu-shi Applied Theory of Optimal ControlMir Moscow Russia 1972

[14] DMHimmebblauAppliedNonlinear Programming Craw-HillBook Company New York NY USA 1972

[15] K R Aydazade ldquoComputatioonal problems in hydraulic net-worksrdquo Computational Mathematics and Mathematical Physicsvol 29 no 2 pp 184ndash193 1989

[16] Y N Andreev Control of the Finite Dimensional Linear ObjectsNauka Moscow Russia 1976

8 Mathematical Problems in Engineering

[17] J R Magnus and H Neudecker Matrix Differential Calculuswith Applications in Statistics and Econometrics John Wiley ampSons Chichester UK 17th edition 1988

[18] K B Petersen and M S PedersenTheMatrix Cookbook 2008httpmatrixcookbookcom

[19] D M Altshul Hidraulic Resistance Nedra Moscow Russia1970

[20] M Ghanbari S Abbasbandy and T Allahviranloo ldquoA newapproach to determine the convergence-control parameter inthe application of the homotopy analysis method to systems oflinear equationsrdquo Applied and Computational Mathematics vol12 no 3 pp 355ndash364 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Finally if we consider these results then the gradient ofthe functional (9) will be defined by the formula

120597119868

119896

120597120572

=

12

119873

sum

119894=1(

119868

2119896

120597120572

+

119868

5119896

120597120572

+

119868

6119896

120597120572

+

119868

7119896

120597120572

+

119868

8119896

120597120572

+

119868

10119896

120597120572

+

119868

14119896

120597120572

) =

12

sdot

119873

sum

119894=1(Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894

119879

Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

+Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

+ 2Φ119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus11119894 (119879 1199050) 120572

+Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus12119894 (119879 1199050)

minusΦ

119896minus11119894119879

(119879 1199050) 119860119896

119894119910

119896

119879119894

+Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894

119879

Φ

119896minus12119894 (119879 1199050)

minusΦ

119896minus11119894119879

(119879 1199050) 119860119896

119894

119879

119910

119896

119879119894) =

12

sdot

119873

sum

119894=1(2Φ119896minus11119894

119879

(119879 1199050) 119860119896

119894

119879

Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

+ 2Φ119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus11119894 (119879 1199050) 120572

+ 2Φ119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus12119894 (119879 1199050)

minus 2Φ119896minus11119894119879

(119879 1199050) 119860119896

119894119910

119896

119879119894) = 0

(15)

Then for the gradient of the functional 119868119896relatively to the

parameter 120572 we get the expression

120597119868

119896

120597120572

=

119873

sum

119894=1[Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus11119894 (119879 1199050)] 120572

minus

119873

sum

119894=1[Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894119910

119896

119879119894

minusΦ

119896minus11119894119879

(119879 1199050) 119860119896

119894

119879

Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

minusΦ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus12119894 (119879 1199050)]

(16)

Taking equal to zero the expression (16) we get

119873

sum

119894=1[Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus11119894 (119879 1199050)] 120572

=

119873

sum

119894=1[Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894119910

119896

119879119894

minusΦ

119896minus11119894119879

(119879 1199050) 119860119896

119894

119879

Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

minusΦ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus12119894 (119879 1199050)]

(17)

Solution of (17) with respect to 120572 gives

120572 = [

119873

sum

119894=1Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus11119894 (119879 1199050)]

minus1

sdot

119873

sum

119894=1[Φ

119896minus11119894119879

(119879 1199050) 119860119896

119894119910

119896

119879119894

minusΦ

119896minus11119894119879

(119879 1199050) 119860119896

119894

119879

Φ

119896minus1119894

(119879 1199050) 119910119896

119894(1199050)

minusΦ

119896minus11119894119879

(119879 1199050) 119860119896

119894Φ

119896minus12119894 (119879 1199050)]

(18)

where it is assumed that [sum119873119894=1 Φ119896minus11119894119879

(119879 1199050)119860119896

119894Φ

119896minus11119894 (119879 1199050)]

minus1

existsValue of the parameter 120572 defined by (18) is a solution of

the multiparameter optimization problem for the functional(9) that gives minimum to the cost functional

Considering the above the following algorithm may beproposed to the solution of the identification problem (1) (2)

Algorithm 1 (1) Construct the function 119891(119909) from (1)initial and final data 1199101198940 and 119910

119894

119879(119894 = 1 119873) from (2) and (3)

correspondingly(2) Calculate the derivatives 120597119891(119910119896minus1 120572119896minus1)120597119910 120597119891(119910119896minus1

120572

119896minus1)120597120572 taking as initial approaches 119910

119894and 120572

119894

(3)Calculate the fundamentalmatrixΦ119896minus1(119905 1199050) from (6)reconstruct Φ119896minus11 (119905 1199050) Φ

119896minus12 (119905 1199050) from (7) and functional 119868

119896

from (8)(4) Solving the system of algebraic equations (14)

relatively 120572 find the value of the 119898-dimensional vector 120572119896 inthe 119896th iteration

(5) Check the condition1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

120597119868

119896

120597120572

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

lt 120576 (19)

where 120576 is a given small enough number If the condition(16) is satisfied the process stops otherwise go to Step (2)The convergence of this algorithm may be proved similar to[13 14]

Now we discuss the realization of this algorithm

5 Calculational Algorithm

As one can see in the realization of the above algorithmthe main step is a calculation of the fundamental matrixΦ

119896minus1(119905 119905

0) and the matricesΦ119896minus11 (119905 119905

0)Φ119896minus12 (119905 119905

0) Note that

as is mentioned in [16] construction of these matrices is anenough difficult procedure So for simplicity we try (4) andfind the corresponding derivatives by using the Eulermethod

Mathematical Problems in Engineering 5

Really 120597119891(119910(119905) 120572)120597119910 and 120597119891(119910(119905) 120572)120597120572 everywhere arereplaced by

119891(119910) = (119891(119910 + 120575 120572) minus 119891(119910 120572))120575 and

119891(120572) =

(119891(119910 120572 + 1205751) minus 119891(119910 120572))1205751 correspondingly where 120575 and 1205751are small enough numbers

To calculate the fundamental matrix Φ

119896minus1(119905 119905

0) and the

matrices Φ119896minus11 (119905 1199050) Φ119896minus12 (119905 1199050) it is proper to replace (4) by

the following discrete one

119910

119896(1199052119873)

= (

119895

prod

1198941=2119873minus1(119864+ 120575

119891 (119910

119896minus1(119905

1198941))))119910

119896(119905

119873+1)

+Φ

119896minus11 (119905 1199050) 120572 +Φ

119896minus12 (119905 1199050)

(20)

where

Φ

119896minus1119894

(1199052119873 1199050) =

119895

prod

1198941=2119873minus1(119864+ 120575

119891 (119910

119896minus1(119905

119894)))

Φ

119896minus11119894 (1199052119873 1199050)

= (

2119873minus1sum

119895=119873+2(

119895

prod

1198941=2119873minus1(119864+ 120575 sdot

119891 (119910

119896minus1(119905

119894))))

sdot 120575

119891 (120572

119896minus1119895minus1))+120575 sdot

119891 (120572

119896minus12119873minus1)

Φ

119896minus12119894 (1199052119873 1199050)

= (

2119873minus1sum

119895=119873+2(

119895

prod

1198941=2119873minus1(119864+ 120575 sdot

119891 (119910

119896minus1(119905

119894))))120575

sdot (

119891 (119910

119896minus1(119905

119895minus1) 120572119896minus1

)

minus

119891 (119910

119896minus1(119905

119895minus1)) 119910119896minus1

(119905

119895minus1) minus

119891 (120572

119896minus1) 120572

119896minus1))

+120575 sdot (

119891 (119910

119896minus1(1199052119873minus1) 120572

119896minus1) minus

119891 (119910

119896minus1(1199052119873minus1))

sdot 119910

119896minus1(1199052119873minus1) minus

119891 (120572

119896minus1) 120572

119896minus1)

(21)

119864 is unit matrix of proper dimensionThen from (20) we get that Φ119896minus1(119905 1199050) is a fundamental

matrix for the system of homogeneous equations

119910

119896(119905

119894+1) = (119864+ 120575

119891 (119910))

1003816

1003816

1003816

1003816

1003816

119910=119910119896minus1

120572=120572119896minus1

sdot 119910

119896(119905

119894) (22)

Therefore similar to the nongradient methods we pro-pose an algorithm based on the orthogonalization of gradientdirections using the Gram-Schmidt procedure

Step 1 Using Gram-Schmidt orthogonalization vectors 120596119895119894

119894 = 119895119895 + 1 119899 119895 = 2 3 1198992 are calculated and the set ofvectors

(

120597119891 (120582

1119888)

120597120582

1119888

120597119891 (120582

2119888)

120597120582

2119888

120597119891 (120582

119894

119888)

120597120582

119894

119888

120596

119895

119894+1 120596119895

119899)

(23)

is found which form an orthogonal basis in 119877

119899If we apply the orthogonalization algorithm then a

linearly independent system 1198861 1198862 119886119896 should be formedthat is orthogonal system 1198871 1198872 119887119896 and each vector 119887

119894

should be linearly expressed through 1198861 1198862 119886119894 Here 119886

119894

and 119887

119894are upper triangular matrices Thus it is possible to

ensure that the systems 119887

119894 were orthonormal where the

diagonal elements of the transition matrix are positive bythese conditions the system 119887

119894 and the transition matrix are

uniquely determinedThe algorithm considers 1198871 = 1198861 if the vectors

1198871 1198872 119887119894minus1 are constructed Then

119887

119894= 119886

119894minus

119894minus1sum

119895=1

⟨119886

119894 119887

119895⟩

⟨119887

119894 119887

119895⟩

119887

119895 (24)

where ⟨ ⟩ is the sign of the scalar product of vectors

Step 2 For the orthogonalization of the gradient directionswe compute ]119895

119894in the form

]119895119894=

119891 (120582

119888+ 120575120596

119895

119894) minus 119891 (120582

119888minus 120575120596

119895

119894)

2120575

119894 = 119895 119895 + 1 119899

(25)

Here 120575 gt 0 is any small parameter

Step 3 The orthogonal gradient directions are chosen in theform

119897

119895=

119899

sum

119894=119895

]119895119894120596

119895

119894 (26)

Replacing nabla119891(120582

(119896)

119888) by 119897

119895in (17) the nongradient iterative

minimization procedure will be

120582

(119896+1)119888

= 120582

(119896)

119888minus120594

lowast(119896)119897

119896

(27)

where 120594lowast(119896) is a scalar which is determined by golden sectionmethod

Now we apply the above proposed technique to theexample of 15 production by gas-lift method

Example 2 It is known that nonstable motion of gas in tubesand gas liquid mixture (GLM) in vertical tubes that is inthe lift pipe of the gas-lift well with constant across profile is

6 Mathematical Problems in Engineering

x = 0

x = lg

x = l

rc

rk

x

h

hst

l

lg

lj

Gas

Gas GLM

x = 2l

Gas liquid mixture

Figure 1

described by the following system of linear partial differentialequations of hyperbolic type (see Figure 1)

minus

120597119875

120597119909

=

120597 (120588120596

119888)

120597119905

+ 2120572120588120596119888

minus

120597119875

120597119905

= 119888

2 120597 (120588120596119888)

120597119909

(28)

where 120588 = 119875(119909 119905) 120596119888= 120596

119888(119909 119905) is an additional pressure on

its stationary value and averaged over across section speed ofmotion of GLM 119905 119909 time and coordinate 119888 speed of soundin gas andGLM 120588 gas oil and GLM in correspondence withcoordinates 2120572 = 119892120596

119888+120582

119888120596

1198882119863 119892 120582

119888 free fall accel 119863

interval effective diameter of the tube [19]The partial differential equation of gas and GLM motion

by are averaging over time 119905may be reduced to the followingordinary differential equation [20]

119876 =

2120572 (120582

119888) 120588119865119876

2

119888

2120588

2119865

2minus 119876

2 119876 (0) = 119906

(29)

where 119888 ≫ 120596

119888and all quantities are assumed constant 119876 =

120588120596

119888119865 and 119865 is area of across section of the pump-compressor

tubes and is constant relative to axes

It is assumed that the passing from the end of tubethrough the layer to the beginning of the lift (119909 = 119897) isdescribed by the following difference equations

119876 (119897 + 0) = 120574119876 (119897 minus 0) + 1205741 (119876 (119897 minus 0)) 119876

1205741 (119876 (119897 minus 0)) = minus 1205753 (119876 (119897 minus 0) minus 1205752)2+ 1205751

(30)

where 120574 and 1205751 1205752 1205753 are constant numbers to be found Forthe sake of simplicity we suppose that the parameters 120574 12057511205752 1205753 are known and it is required to reconstruct 120582

119888involved

in (19) due to 120572(120582119888)

Then some nominal trajectory 119876

0(119909) and parameter 1205720

are chosen assuming that 119896th iteration is already held Let uslinearize (29) among these data

119876

119896(119909) = 119860 (119876

119896minus1 120572

119896minus1) sdot 119876

119896(119909) + 119861 (119876

119896minus1 120572

119896minus1) 120572

119896

+119862 (119876

119896minus1 120572

119896minus1)

(31)

where

120597119891 (119910

119896minus1 120572

119896minus1)

120597119910

= 119860 (119876

119896minus1 120572

119896minus1)

=

4120572119896minus1119888212058831198653119876

119896minus1

(119888

2120588

2119865

2minus (119876

119896minus1)

2)

2

120597119891 (119910

119896minus1 120572

119896minus1)

120597120572

= 119861 (119876

119896minus1 120572

119896minus1)

=

2120572119896minus1120588119865119876119896minus1

119888

2120588

2119865

2minus (119876

119896minus1)

2

119891 (119910

119896minus1 120572

119896minus1) minus

120597119891 (119910

119896minus1 120572

119896minus1)

120597119910

119910

119896minus1

minus

120597119891 (119910

119896minus1 120572

119896minus1)

120597120572

120572

119896minus1= 119862 (119876

119896minus1 120572

119896minus1)

=

4120572119896minus1119888212058831198653(119876

119896minus1)

2+ 4120572119896minus1120588119865 (119876

119896minus1)

4

(119888

2120588

2119865

2minus (119876

119896minus1)

2)

2

(32)

Note that by the help of the relations (20) and (21) thematricesΦ

119896minus1(119909 0) Φ119896minus11 (119909 0) Φ119896minus12 (119909 0) are calculated as follows

Φ

119896minus1119894

(119909 0) =119895

prod

119894=2119873minus1(119864+119860 (119876

119896minus1(119909

1198941) 120572

119896minus1)) ℎ

Φ

119896minus11119894 (1199092119873 0)

= (

2119873minus1sum

119895=119873+2(

119895

prod

119894=2119873minus1(119864+119860 (119876

119896minus1(119909

1198941) 120572

119896minus1)) ℎ)

sdot 119861 (119876

119896minus1(119909

119895minus1) 120572119896minus1

) ℎ)

+119861 (119876

119896minus1(1199092119873minus1) 120572

119896minus1) ℎ

Mathematical Problems in Engineering 7

Table 1

119910

119897+0119894

55698 55732 55761 55810 55848 55852 55824119910

2119897119894

44242 44248 44254 44262 44266 44263 44251

Table 2

120575 005 01 05 054 055 06120582 01966 02141 02295 02298 02299 02302

01 02 03 04 05 06 07 08 09 10214

0216

0218

022

0222

0224

0226

0228

023

0232

Figure 2

Φ

119896minus12119894 (1199092119873 0)

= (

2119873minus1sum

119895=119873+2(

119895

prod

119894=2119873minus1(119864+119860 (119876

119896minus1(119909

1198941) 120572

119896minus1)) ℎ)

sdot 119862 (119876

119896minus1(119909

119895minus1) 120572119896minus1

) ℎ)

+119862 (119876

119896minus1(1199092119873minus1) 120572

119896minus1) ℎ

(33)

where ℎ is small enough numberLet some statistical data be givenLet us assume that some observation points for 119910119897+0

119894and

119910

2119897119894are given (see Table 1)We give in Table 2 the values of 120582 obtained by using

MATLAB by given input parametersAs we see from Table 2 by 120575 = 055 120582 gets value 02299

with error estimation 10minus3Here is the dependence of 120582 or 120575 (see Figure 2)The above algorithm reaches given accuracy after 4

iterations and gives 120582 = 02299Note that the inequality

1003816

1003816

1003816

1003816

120582

119894minus1205824

1003816

1003816

1003816

1003816

le 10minus15 (34)

holds for any 119894 gt 4 that shows the stability of the proposedquasilinearization algorithm

Conflict of Interests

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

Acknowledgment

This work is supported by the joint grant of ANAS andSOCAR 17 2013ndash2015 Baku State University ldquo50+50rdquo Grant

References

[1] A Aliev Fikret and N A Ismailov ldquoInverse problem todetermine the hydraulic resistance coefficient in the gas liftprocessrdquo Applied and Computational Mathematics vol 12 no3 pp 306ndash313 2013

[2] R Gurbanov N Nuriyev and R S Gurbanov ldquoTechnologicalcontrol and optimization problems in oil production theoryand practicerdquo Applied and Computational Mathematics vol 12no 3 pp 314ndash324 2013

[3] R N Bakhtisin and A R Latypov ldquoEstimation of the order oflinear objects by experimental informationrdquo Automation andRemote Control no 3 pp 108ndash112 1992

[4] Y S Gasimov ldquoOn a shape design problem for one spectralfunctionalrdquo Journal of Inverse and Ill-Posed Problems vol 21 no5 pp 629ndash637 2013

[5] L Lyuing Identification of the System Theory for Users NaukaMoscow Russia 1991

[6] F A AlievMMMutallimov IMAskerov and I S RaguimovldquoAsymptotic method of solution for a problem of constructionof optimal gas-lift process modesrdquo Mathematical Problems inEngineering vol 2010 Article ID 191053 10 pages 2010

[7] A S Apostolyuk and V B Larin ldquoUpdating of linear stationarydynamic systemparametersrdquoApplied andComputationalMath-ematics vol 10 no 3 pp 402ndash408 2011

[8] F Ding ldquoHierarchical multi-innovation stochastic gradientalgorithm for Hammerstein nonlinear system modelingrdquoApplied Mathematical Modelling vol 37 no 4 pp 1694ndash17042013

[9] F Ding Y Shi and T Chen ldquoAuxiliary model-based least-squares identification methods for Hammerstein output-errorsystemsrdquo Systems amp Control Letters vol 56 no 5 pp 373ndash3802007

[10] S I Kabanikhin and O I KrivorotrsquoKo ldquoA numerical methodfor determining the amplitude of a wave edge in shallow waterapproximationrdquo Applied and Computational Mathematics vol12 no 1 pp 91ndash96 2013

[11] P E Bellman and P E Kalaba Quailinearization and NonlinearBoundary Problems MirVoscow Russia 1968

[12] V E Shamansky Methods of Numerical Solution of Baun-daryProblems in PC Naukova Dumka Kiev 1966

[13] A Brayson and X Yu-shi Applied Theory of Optimal ControlMir Moscow Russia 1972

[14] DMHimmebblauAppliedNonlinear Programming Craw-HillBook Company New York NY USA 1972

[15] K R Aydazade ldquoComputatioonal problems in hydraulic net-worksrdquo Computational Mathematics and Mathematical Physicsvol 29 no 2 pp 184ndash193 1989

[16] Y N Andreev Control of the Finite Dimensional Linear ObjectsNauka Moscow Russia 1976

8 Mathematical Problems in Engineering

[17] J R Magnus and H Neudecker Matrix Differential Calculuswith Applications in Statistics and Econometrics John Wiley ampSons Chichester UK 17th edition 1988

[18] K B Petersen and M S PedersenTheMatrix Cookbook 2008httpmatrixcookbookcom

[19] D M Altshul Hidraulic Resistance Nedra Moscow Russia1970

[20] M Ghanbari S Abbasbandy and T Allahviranloo ldquoA newapproach to determine the convergence-control parameter inthe application of the homotopy analysis method to systems oflinear equationsrdquo Applied and Computational Mathematics vol12 no 3 pp 355ndash364 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Really 120597119891(119910(119905) 120572)120597119910 and 120597119891(119910(119905) 120572)120597120572 everywhere arereplaced by

119891(119910) = (119891(119910 + 120575 120572) minus 119891(119910 120572))120575 and

119891(120572) =

(119891(119910 120572 + 1205751) minus 119891(119910 120572))1205751 correspondingly where 120575 and 1205751are small enough numbers

To calculate the fundamental matrix Φ

119896minus1(119905 119905

0) and the

matrices Φ119896minus11 (119905 1199050) Φ119896minus12 (119905 1199050) it is proper to replace (4) by

the following discrete one

119910

119896(1199052119873)

= (

119895

prod

1198941=2119873minus1(119864+ 120575

119891 (119910

119896minus1(119905

1198941))))119910

119896(119905

119873+1)

+Φ

119896minus11 (119905 1199050) 120572 +Φ

119896minus12 (119905 1199050)

(20)

where

Φ

119896minus1119894

(1199052119873 1199050) =

119895

prod

1198941=2119873minus1(119864+ 120575

119891 (119910

119896minus1(119905

119894)))

Φ

119896minus11119894 (1199052119873 1199050)

= (

2119873minus1sum

119895=119873+2(

119895

prod

1198941=2119873minus1(119864+ 120575 sdot

119891 (119910

119896minus1(119905

119894))))

sdot 120575

119891 (120572

119896minus1119895minus1))+120575 sdot

119891 (120572

119896minus12119873minus1)

Φ

119896minus12119894 (1199052119873 1199050)

= (

2119873minus1sum

119895=119873+2(

119895

prod

1198941=2119873minus1(119864+ 120575 sdot

119891 (119910

119896minus1(119905

119894))))120575

sdot (

119891 (119910

119896minus1(119905

119895minus1) 120572119896minus1

)

minus

119891 (119910

119896minus1(119905

119895minus1)) 119910119896minus1

(119905

119895minus1) minus

119891 (120572

119896minus1) 120572

119896minus1))

+120575 sdot (

119891 (119910

119896minus1(1199052119873minus1) 120572

119896minus1) minus

119891 (119910

119896minus1(1199052119873minus1))

sdot 119910

119896minus1(1199052119873minus1) minus

119891 (120572

119896minus1) 120572

119896minus1)

(21)

119864 is unit matrix of proper dimensionThen from (20) we get that Φ119896minus1(119905 1199050) is a fundamental

matrix for the system of homogeneous equations

119910

119896(119905

119894+1) = (119864+ 120575

119891 (119910))

1003816

1003816

1003816

1003816

1003816

119910=119910119896minus1

120572=120572119896minus1

sdot 119910

119896(119905

119894) (22)

Therefore similar to the nongradient methods we pro-pose an algorithm based on the orthogonalization of gradientdirections using the Gram-Schmidt procedure

Step 1 Using Gram-Schmidt orthogonalization vectors 120596119895119894

119894 = 119895119895 + 1 119899 119895 = 2 3 1198992 are calculated and the set ofvectors

(

120597119891 (120582

1119888)

120597120582

1119888

120597119891 (120582

2119888)

120597120582

2119888

120597119891 (120582

119894

119888)

120597120582

119894

119888

120596

119895

119894+1 120596119895

119899)

(23)

is found which form an orthogonal basis in 119877

119899If we apply the orthogonalization algorithm then a

linearly independent system 1198861 1198862 119886119896 should be formedthat is orthogonal system 1198871 1198872 119887119896 and each vector 119887

119894

should be linearly expressed through 1198861 1198862 119886119894 Here 119886

119894

and 119887

119894are upper triangular matrices Thus it is possible to

ensure that the systems 119887

119894 were orthonormal where the

diagonal elements of the transition matrix are positive bythese conditions the system 119887

119894 and the transition matrix are

uniquely determinedThe algorithm considers 1198871 = 1198861 if the vectors

1198871 1198872 119887119894minus1 are constructed Then

119887

119894= 119886

119894minus

119894minus1sum

119895=1

⟨119886

119894 119887

119895⟩

⟨119887

119894 119887

119895⟩

119887

119895 (24)

where ⟨ ⟩ is the sign of the scalar product of vectors

Step 2 For the orthogonalization of the gradient directionswe compute ]119895

119894in the form

]119895119894=

119891 (120582

119888+ 120575120596

119895

119894) minus 119891 (120582

119888minus 120575120596

119895

119894)

2120575

119894 = 119895 119895 + 1 119899

(25)

Here 120575 gt 0 is any small parameter

Step 3 The orthogonal gradient directions are chosen in theform

119897

119895=

119899

sum

119894=119895

]119895119894120596

119895

119894 (26)

Replacing nabla119891(120582

(119896)

119888) by 119897

119895in (17) the nongradient iterative

minimization procedure will be

120582

(119896+1)119888

= 120582

(119896)

119888minus120594

lowast(119896)119897

119896

(27)

where 120594lowast(119896) is a scalar which is determined by golden sectionmethod

Now we apply the above proposed technique to theexample of 15 production by gas-lift method

Example 2 It is known that nonstable motion of gas in tubesand gas liquid mixture (GLM) in vertical tubes that is inthe lift pipe of the gas-lift well with constant across profile is

6 Mathematical Problems in Engineering

x = 0

x = lg

x = l

rc

rk

x

h

hst

l

lg

lj

Gas

Gas GLM

x = 2l

Gas liquid mixture

Figure 1

described by the following system of linear partial differentialequations of hyperbolic type (see Figure 1)

minus

120597119875

120597119909

=

120597 (120588120596

119888)

120597119905

+ 2120572120588120596119888

minus

120597119875

120597119905

= 119888

2 120597 (120588120596119888)

120597119909

(28)

where 120588 = 119875(119909 119905) 120596119888= 120596

119888(119909 119905) is an additional pressure on

its stationary value and averaged over across section speed ofmotion of GLM 119905 119909 time and coordinate 119888 speed of soundin gas andGLM 120588 gas oil and GLM in correspondence withcoordinates 2120572 = 119892120596

119888+120582

119888120596

1198882119863 119892 120582

119888 free fall accel 119863

interval effective diameter of the tube [19]The partial differential equation of gas and GLM motion

by are averaging over time 119905may be reduced to the followingordinary differential equation [20]

119876 =

2120572 (120582

119888) 120588119865119876

2

119888

2120588

2119865

2minus 119876

2 119876 (0) = 119906

(29)

where 119888 ≫ 120596

119888and all quantities are assumed constant 119876 =

120588120596

119888119865 and 119865 is area of across section of the pump-compressor

tubes and is constant relative to axes

It is assumed that the passing from the end of tubethrough the layer to the beginning of the lift (119909 = 119897) isdescribed by the following difference equations

119876 (119897 + 0) = 120574119876 (119897 minus 0) + 1205741 (119876 (119897 minus 0)) 119876

1205741 (119876 (119897 minus 0)) = minus 1205753 (119876 (119897 minus 0) minus 1205752)2+ 1205751

(30)

where 120574 and 1205751 1205752 1205753 are constant numbers to be found Forthe sake of simplicity we suppose that the parameters 120574 12057511205752 1205753 are known and it is required to reconstruct 120582

119888involved

in (19) due to 120572(120582119888)

Then some nominal trajectory 119876

0(119909) and parameter 1205720

are chosen assuming that 119896th iteration is already held Let uslinearize (29) among these data

119876

119896(119909) = 119860 (119876

119896minus1 120572

119896minus1) sdot 119876

119896(119909) + 119861 (119876

119896minus1 120572

119896minus1) 120572

119896

+119862 (119876

119896minus1 120572

119896minus1)

(31)

where

120597119891 (119910

119896minus1 120572

119896minus1)

120597119910

= 119860 (119876

119896minus1 120572

119896minus1)

=

4120572119896minus1119888212058831198653119876

119896minus1

(119888

2120588

2119865

2minus (119876

119896minus1)

2)

2

120597119891 (119910

119896minus1 120572

119896minus1)

120597120572

= 119861 (119876

119896minus1 120572

119896minus1)

=

2120572119896minus1120588119865119876119896minus1

119888

2120588

2119865

2minus (119876

119896minus1)

2

119891 (119910

119896minus1 120572

119896minus1) minus

120597119891 (119910

119896minus1 120572

119896minus1)

120597119910

119910

119896minus1

minus

120597119891 (119910

119896minus1 120572

119896minus1)

120597120572

120572

119896minus1= 119862 (119876

119896minus1 120572

119896minus1)

=

4120572119896minus1119888212058831198653(119876

119896minus1)

2+ 4120572119896minus1120588119865 (119876

119896minus1)

4

(119888

2120588

2119865

2minus (119876

119896minus1)

2)

2

(32)

Note that by the help of the relations (20) and (21) thematricesΦ

119896minus1(119909 0) Φ119896minus11 (119909 0) Φ119896minus12 (119909 0) are calculated as follows

Φ

119896minus1119894

(119909 0) =119895

prod

119894=2119873minus1(119864+119860 (119876

119896minus1(119909

1198941) 120572

119896minus1)) ℎ

Φ

119896minus11119894 (1199092119873 0)

= (

2119873minus1sum

119895=119873+2(

119895

prod

119894=2119873minus1(119864+119860 (119876

119896minus1(119909

1198941) 120572

119896minus1)) ℎ)

sdot 119861 (119876

119896minus1(119909

119895minus1) 120572119896minus1

) ℎ)

+119861 (119876

119896minus1(1199092119873minus1) 120572

119896minus1) ℎ

Mathematical Problems in Engineering 7

Table 1

119910

119897+0119894

55698 55732 55761 55810 55848 55852 55824119910

2119897119894

44242 44248 44254 44262 44266 44263 44251

Table 2

120575 005 01 05 054 055 06120582 01966 02141 02295 02298 02299 02302

01 02 03 04 05 06 07 08 09 10214

0216

0218

022

0222

0224

0226

0228

023

0232

Figure 2

Φ

119896minus12119894 (1199092119873 0)

= (

2119873minus1sum

119895=119873+2(

119895

prod

119894=2119873minus1(119864+119860 (119876

119896minus1(119909

1198941) 120572

119896minus1)) ℎ)

sdot 119862 (119876

119896minus1(119909

119895minus1) 120572119896minus1

) ℎ)

+119862 (119876

119896minus1(1199092119873minus1) 120572

119896minus1) ℎ

(33)

where ℎ is small enough numberLet some statistical data be givenLet us assume that some observation points for 119910119897+0

119894and

119910

2119897119894are given (see Table 1)We give in Table 2 the values of 120582 obtained by using

MATLAB by given input parametersAs we see from Table 2 by 120575 = 055 120582 gets value 02299

with error estimation 10minus3Here is the dependence of 120582 or 120575 (see Figure 2)The above algorithm reaches given accuracy after 4

iterations and gives 120582 = 02299Note that the inequality

1003816

1003816

1003816

1003816

120582

119894minus1205824

1003816

1003816

1003816

1003816

le 10minus15 (34)

holds for any 119894 gt 4 that shows the stability of the proposedquasilinearization algorithm

Conflict of Interests

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

Acknowledgment

This work is supported by the joint grant of ANAS andSOCAR 17 2013ndash2015 Baku State University ldquo50+50rdquo Grant

References

[1] A Aliev Fikret and N A Ismailov ldquoInverse problem todetermine the hydraulic resistance coefficient in the gas liftprocessrdquo Applied and Computational Mathematics vol 12 no3 pp 306ndash313 2013

[2] R Gurbanov N Nuriyev and R S Gurbanov ldquoTechnologicalcontrol and optimization problems in oil production theoryand practicerdquo Applied and Computational Mathematics vol 12no 3 pp 314ndash324 2013

[3] R N Bakhtisin and A R Latypov ldquoEstimation of the order oflinear objects by experimental informationrdquo Automation andRemote Control no 3 pp 108ndash112 1992

[4] Y S Gasimov ldquoOn a shape design problem for one spectralfunctionalrdquo Journal of Inverse and Ill-Posed Problems vol 21 no5 pp 629ndash637 2013

[5] L Lyuing Identification of the System Theory for Users NaukaMoscow Russia 1991

[6] F A AlievMMMutallimov IMAskerov and I S RaguimovldquoAsymptotic method of solution for a problem of constructionof optimal gas-lift process modesrdquo Mathematical Problems inEngineering vol 2010 Article ID 191053 10 pages 2010

[7] A S Apostolyuk and V B Larin ldquoUpdating of linear stationarydynamic systemparametersrdquoApplied andComputationalMath-ematics vol 10 no 3 pp 402ndash408 2011

[8] F Ding ldquoHierarchical multi-innovation stochastic gradientalgorithm for Hammerstein nonlinear system modelingrdquoApplied Mathematical Modelling vol 37 no 4 pp 1694ndash17042013

[9] F Ding Y Shi and T Chen ldquoAuxiliary model-based least-squares identification methods for Hammerstein output-errorsystemsrdquo Systems amp Control Letters vol 56 no 5 pp 373ndash3802007

[10] S I Kabanikhin and O I KrivorotrsquoKo ldquoA numerical methodfor determining the amplitude of a wave edge in shallow waterapproximationrdquo Applied and Computational Mathematics vol12 no 1 pp 91ndash96 2013

[11] P E Bellman and P E Kalaba Quailinearization and NonlinearBoundary Problems MirVoscow Russia 1968

[12] V E Shamansky Methods of Numerical Solution of Baun-daryProblems in PC Naukova Dumka Kiev 1966

[13] A Brayson and X Yu-shi Applied Theory of Optimal ControlMir Moscow Russia 1972

[14] DMHimmebblauAppliedNonlinear Programming Craw-HillBook Company New York NY USA 1972

[15] K R Aydazade ldquoComputatioonal problems in hydraulic net-worksrdquo Computational Mathematics and Mathematical Physicsvol 29 no 2 pp 184ndash193 1989

[16] Y N Andreev Control of the Finite Dimensional Linear ObjectsNauka Moscow Russia 1976

8 Mathematical Problems in Engineering

[17] J R Magnus and H Neudecker Matrix Differential Calculuswith Applications in Statistics and Econometrics John Wiley ampSons Chichester UK 17th edition 1988

[18] K B Petersen and M S PedersenTheMatrix Cookbook 2008httpmatrixcookbookcom

[19] D M Altshul Hidraulic Resistance Nedra Moscow Russia1970

[20] M Ghanbari S Abbasbandy and T Allahviranloo ldquoA newapproach to determine the convergence-control parameter inthe application of the homotopy analysis method to systems oflinear equationsrdquo Applied and Computational Mathematics vol12 no 3 pp 355ndash364 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

x = 0

x = lg

x = l

rc

rk

x

h

hst

l

lg

lj

Gas

Gas GLM

x = 2l

Gas liquid mixture

Figure 1

described by the following system of linear partial differentialequations of hyperbolic type (see Figure 1)

minus

120597119875

120597119909

=

120597 (120588120596

119888)

120597119905

+ 2120572120588120596119888

minus

120597119875

120597119905

= 119888

2 120597 (120588120596119888)

120597119909

(28)

where 120588 = 119875(119909 119905) 120596119888= 120596

119888(119909 119905) is an additional pressure on

its stationary value and averaged over across section speed ofmotion of GLM 119905 119909 time and coordinate 119888 speed of soundin gas andGLM 120588 gas oil and GLM in correspondence withcoordinates 2120572 = 119892120596

119888+120582

119888120596

1198882119863 119892 120582

119888 free fall accel 119863

interval effective diameter of the tube [19]The partial differential equation of gas and GLM motion

by are averaging over time 119905may be reduced to the followingordinary differential equation [20]

119876 =

2120572 (120582

119888) 120588119865119876

2

119888

2120588

2119865

2minus 119876

2 119876 (0) = 119906

(29)

where 119888 ≫ 120596

119888and all quantities are assumed constant 119876 =

120588120596

119888119865 and 119865 is area of across section of the pump-compressor

tubes and is constant relative to axes

It is assumed that the passing from the end of tubethrough the layer to the beginning of the lift (119909 = 119897) isdescribed by the following difference equations

119876 (119897 + 0) = 120574119876 (119897 minus 0) + 1205741 (119876 (119897 minus 0)) 119876

1205741 (119876 (119897 minus 0)) = minus 1205753 (119876 (119897 minus 0) minus 1205752)2+ 1205751

(30)

where 120574 and 1205751 1205752 1205753 are constant numbers to be found Forthe sake of simplicity we suppose that the parameters 120574 12057511205752 1205753 are known and it is required to reconstruct 120582

119888involved

in (19) due to 120572(120582119888)

Then some nominal trajectory 119876

0(119909) and parameter 1205720

are chosen assuming that 119896th iteration is already held Let uslinearize (29) among these data

119876

119896(119909) = 119860 (119876

119896minus1 120572

119896minus1) sdot 119876

119896(119909) + 119861 (119876

119896minus1 120572

119896minus1) 120572

119896

+119862 (119876

119896minus1 120572

119896minus1)

(31)

where

120597119891 (119910

119896minus1 120572

119896minus1)

120597119910

= 119860 (119876

119896minus1 120572

119896minus1)

=

4120572119896minus1119888212058831198653119876

119896minus1

(119888

2120588

2119865

2minus (119876

119896minus1)

2)

2

120597119891 (119910

119896minus1 120572

119896minus1)

120597120572

= 119861 (119876

119896minus1 120572

119896minus1)

=

2120572119896minus1120588119865119876119896minus1

119888

2120588

2119865

2minus (119876

119896minus1)

2

119891 (119910

119896minus1 120572

119896minus1) minus

120597119891 (119910

119896minus1 120572

119896minus1)

120597119910

119910

119896minus1

minus

120597119891 (119910

119896minus1 120572

119896minus1)

120597120572

120572

119896minus1= 119862 (119876

119896minus1 120572

119896minus1)

=

4120572119896minus1119888212058831198653(119876

119896minus1)

2+ 4120572119896minus1120588119865 (119876

119896minus1)

4

(119888

2120588

2119865

2minus (119876

119896minus1)

2)

2

(32)

Note that by the help of the relations (20) and (21) thematricesΦ

119896minus1(119909 0) Φ119896minus11 (119909 0) Φ119896minus12 (119909 0) are calculated as follows

Φ

119896minus1119894

(119909 0) =119895

prod

119894=2119873minus1(119864+119860 (119876

119896minus1(119909

1198941) 120572

119896minus1)) ℎ

Φ

119896minus11119894 (1199092119873 0)

= (

2119873minus1sum

119895=119873+2(

119895

prod

119894=2119873minus1(119864+119860 (119876

119896minus1(119909

1198941) 120572

119896minus1)) ℎ)

sdot 119861 (119876

119896minus1(119909

119895minus1) 120572119896minus1

) ℎ)

+119861 (119876

119896minus1(1199092119873minus1) 120572

119896minus1) ℎ

Mathematical Problems in Engineering 7

Table 1

119910

119897+0119894

55698 55732 55761 55810 55848 55852 55824119910

2119897119894

44242 44248 44254 44262 44266 44263 44251

Table 2

120575 005 01 05 054 055 06120582 01966 02141 02295 02298 02299 02302

01 02 03 04 05 06 07 08 09 10214

0216

0218

022

0222

0224

0226

0228

023

0232

Figure 2

Φ

119896minus12119894 (1199092119873 0)

= (

2119873minus1sum

119895=119873+2(

119895

prod

119894=2119873minus1(119864+119860 (119876

119896minus1(119909

1198941) 120572

119896minus1)) ℎ)

sdot 119862 (119876

119896minus1(119909

119895minus1) 120572119896minus1

) ℎ)

+119862 (119876

119896minus1(1199092119873minus1) 120572

119896minus1) ℎ

(33)

where ℎ is small enough numberLet some statistical data be givenLet us assume that some observation points for 119910119897+0

119894and

119910

2119897119894are given (see Table 1)We give in Table 2 the values of 120582 obtained by using

MATLAB by given input parametersAs we see from Table 2 by 120575 = 055 120582 gets value 02299

with error estimation 10minus3Here is the dependence of 120582 or 120575 (see Figure 2)The above algorithm reaches given accuracy after 4

iterations and gives 120582 = 02299Note that the inequality

1003816

1003816

1003816

1003816

120582

119894minus1205824

1003816

1003816

1003816

1003816

le 10minus15 (34)

holds for any 119894 gt 4 that shows the stability of the proposedquasilinearization algorithm

Conflict of Interests

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

Acknowledgment

This work is supported by the joint grant of ANAS andSOCAR 17 2013ndash2015 Baku State University ldquo50+50rdquo Grant

References

[1] A Aliev Fikret and N A Ismailov ldquoInverse problem todetermine the hydraulic resistance coefficient in the gas liftprocessrdquo Applied and Computational Mathematics vol 12 no3 pp 306ndash313 2013

[2] R Gurbanov N Nuriyev and R S Gurbanov ldquoTechnologicalcontrol and optimization problems in oil production theoryand practicerdquo Applied and Computational Mathematics vol 12no 3 pp 314ndash324 2013

[3] R N Bakhtisin and A R Latypov ldquoEstimation of the order oflinear objects by experimental informationrdquo Automation andRemote Control no 3 pp 108ndash112 1992

[4] Y S Gasimov ldquoOn a shape design problem for one spectralfunctionalrdquo Journal of Inverse and Ill-Posed Problems vol 21 no5 pp 629ndash637 2013

[5] L Lyuing Identification of the System Theory for Users NaukaMoscow Russia 1991

[6] F A AlievMMMutallimov IMAskerov and I S RaguimovldquoAsymptotic method of solution for a problem of constructionof optimal gas-lift process modesrdquo Mathematical Problems inEngineering vol 2010 Article ID 191053 10 pages 2010

[7] A S Apostolyuk and V B Larin ldquoUpdating of linear stationarydynamic systemparametersrdquoApplied andComputationalMath-ematics vol 10 no 3 pp 402ndash408 2011

[8] F Ding ldquoHierarchical multi-innovation stochastic gradientalgorithm for Hammerstein nonlinear system modelingrdquoApplied Mathematical Modelling vol 37 no 4 pp 1694ndash17042013

[9] F Ding Y Shi and T Chen ldquoAuxiliary model-based least-squares identification methods for Hammerstein output-errorsystemsrdquo Systems amp Control Letters vol 56 no 5 pp 373ndash3802007

[10] S I Kabanikhin and O I KrivorotrsquoKo ldquoA numerical methodfor determining the amplitude of a wave edge in shallow waterapproximationrdquo Applied and Computational Mathematics vol12 no 1 pp 91ndash96 2013

[11] P E Bellman and P E Kalaba Quailinearization and NonlinearBoundary Problems MirVoscow Russia 1968

[12] V E Shamansky Methods of Numerical Solution of Baun-daryProblems in PC Naukova Dumka Kiev 1966

[13] A Brayson and X Yu-shi Applied Theory of Optimal ControlMir Moscow Russia 1972

[14] DMHimmebblauAppliedNonlinear Programming Craw-HillBook Company New York NY USA 1972

[15] K R Aydazade ldquoComputatioonal problems in hydraulic net-worksrdquo Computational Mathematics and Mathematical Physicsvol 29 no 2 pp 184ndash193 1989

[16] Y N Andreev Control of the Finite Dimensional Linear ObjectsNauka Moscow Russia 1976

8 Mathematical Problems in Engineering

[17] J R Magnus and H Neudecker Matrix Differential Calculuswith Applications in Statistics and Econometrics John Wiley ampSons Chichester UK 17th edition 1988

[18] K B Petersen and M S PedersenTheMatrix Cookbook 2008httpmatrixcookbookcom

[19] D M Altshul Hidraulic Resistance Nedra Moscow Russia1970

[20] M Ghanbari S Abbasbandy and T Allahviranloo ldquoA newapproach to determine the convergence-control parameter inthe application of the homotopy analysis method to systems oflinear equationsrdquo Applied and Computational Mathematics vol12 no 3 pp 355ndash364 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Table 1

119910

119897+0119894

55698 55732 55761 55810 55848 55852 55824119910

2119897119894

44242 44248 44254 44262 44266 44263 44251

Table 2

120575 005 01 05 054 055 06120582 01966 02141 02295 02298 02299 02302

01 02 03 04 05 06 07 08 09 10214

0216

0218

022

0222

0224

0226

0228

023

0232

Figure 2

Φ

119896minus12119894 (1199092119873 0)

= (

2119873minus1sum

119895=119873+2(

119895

prod

119894=2119873minus1(119864+119860 (119876

119896minus1(119909

1198941) 120572

119896minus1)) ℎ)

sdot 119862 (119876

119896minus1(119909

119895minus1) 120572119896minus1

) ℎ)

+119862 (119876

119896minus1(1199092119873minus1) 120572

119896minus1) ℎ

(33)

where ℎ is small enough numberLet some statistical data be givenLet us assume that some observation points for 119910119897+0

119894and

119910

2119897119894are given (see Table 1)We give in Table 2 the values of 120582 obtained by using

MATLAB by given input parametersAs we see from Table 2 by 120575 = 055 120582 gets value 02299

with error estimation 10minus3Here is the dependence of 120582 or 120575 (see Figure 2)The above algorithm reaches given accuracy after 4

iterations and gives 120582 = 02299Note that the inequality

1003816

1003816

1003816

1003816

120582

119894minus1205824

1003816

1003816

1003816

1003816

le 10minus15 (34)

holds for any 119894 gt 4 that shows the stability of the proposedquasilinearization algorithm

Conflict of Interests

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

Acknowledgment

This work is supported by the joint grant of ANAS andSOCAR 17 2013ndash2015 Baku State University ldquo50+50rdquo Grant

References

[1] A Aliev Fikret and N A Ismailov ldquoInverse problem todetermine the hydraulic resistance coefficient in the gas liftprocessrdquo Applied and Computational Mathematics vol 12 no3 pp 306ndash313 2013

[2] R Gurbanov N Nuriyev and R S Gurbanov ldquoTechnologicalcontrol and optimization problems in oil production theoryand practicerdquo Applied and Computational Mathematics vol 12no 3 pp 314ndash324 2013

[3] R N Bakhtisin and A R Latypov ldquoEstimation of the order oflinear objects by experimental informationrdquo Automation andRemote Control no 3 pp 108ndash112 1992

[4] Y S Gasimov ldquoOn a shape design problem for one spectralfunctionalrdquo Journal of Inverse and Ill-Posed Problems vol 21 no5 pp 629ndash637 2013

[5] L Lyuing Identification of the System Theory for Users NaukaMoscow Russia 1991

[6] F A AlievMMMutallimov IMAskerov and I S RaguimovldquoAsymptotic method of solution for a problem of constructionof optimal gas-lift process modesrdquo Mathematical Problems inEngineering vol 2010 Article ID 191053 10 pages 2010

[7] A S Apostolyuk and V B Larin ldquoUpdating of linear stationarydynamic systemparametersrdquoApplied andComputationalMath-ematics vol 10 no 3 pp 402ndash408 2011

[8] F Ding ldquoHierarchical multi-innovation stochastic gradientalgorithm for Hammerstein nonlinear system modelingrdquoApplied Mathematical Modelling vol 37 no 4 pp 1694ndash17042013

[9] F Ding Y Shi and T Chen ldquoAuxiliary model-based least-squares identification methods for Hammerstein output-errorsystemsrdquo Systems amp Control Letters vol 56 no 5 pp 373ndash3802007

[10] S I Kabanikhin and O I KrivorotrsquoKo ldquoA numerical methodfor determining the amplitude of a wave edge in shallow waterapproximationrdquo Applied and Computational Mathematics vol12 no 1 pp 91ndash96 2013

[11] P E Bellman and P E Kalaba Quailinearization and NonlinearBoundary Problems MirVoscow Russia 1968

[12] V E Shamansky Methods of Numerical Solution of Baun-daryProblems in PC Naukova Dumka Kiev 1966

[13] A Brayson and X Yu-shi Applied Theory of Optimal ControlMir Moscow Russia 1972

[14] DMHimmebblauAppliedNonlinear Programming Craw-HillBook Company New York NY USA 1972

[15] K R Aydazade ldquoComputatioonal problems in hydraulic net-worksrdquo Computational Mathematics and Mathematical Physicsvol 29 no 2 pp 184ndash193 1989

[16] Y N Andreev Control of the Finite Dimensional Linear ObjectsNauka Moscow Russia 1976

8 Mathematical Problems in Engineering

[17] J R Magnus and H Neudecker Matrix Differential Calculuswith Applications in Statistics and Econometrics John Wiley ampSons Chichester UK 17th edition 1988

[18] K B Petersen and M S PedersenTheMatrix Cookbook 2008httpmatrixcookbookcom

[19] D M Altshul Hidraulic Resistance Nedra Moscow Russia1970

[20] M Ghanbari S Abbasbandy and T Allahviranloo ldquoA newapproach to determine the convergence-control parameter inthe application of the homotopy analysis method to systems oflinear equationsrdquo Applied and Computational Mathematics vol12 no 3 pp 355ndash364 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

[17] J R Magnus and H Neudecker Matrix Differential Calculuswith Applications in Statistics and Econometrics John Wiley ampSons Chichester UK 17th edition 1988

[18] K B Petersen and M S PedersenTheMatrix Cookbook 2008httpmatrixcookbookcom

[19] D M Altshul Hidraulic Resistance Nedra Moscow Russia1970

[20] M Ghanbari S Abbasbandy and T Allahviranloo ldquoA newapproach to determine the convergence-control parameter inthe application of the homotopy analysis method to systems oflinear equationsrdquo Applied and Computational Mathematics vol12 no 3 pp 355ndash364 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of