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International Journal of

Mathematical Modelling& ComputationsWinter 2015

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Mathematical Modelling& Computations

Vol. 05, No. 01, Winter 2015, 1- 14

Non-Polynomial Spline for the Numerical Solution of Problems in

Calculus of Variations

R. Jalilian a,, J. Rashidinia b, K. Farajyan c and H. Jalilian b

aDepartment of Mathematics, Razi University Tagh Bostan, Kermanshah P.O. Box

6714967346 Iran;bSchool of Mathematics, Iran University of Science and Technology Narmak, Tehran

16844, Iran;cDepartment of Mathematics, College of Science, Bonab Branch, Islamic Azad University

of Bonab, Iran.

Abstract.A Class of new methods based on a septic non-polynomial spline function for thenumerical solution of problems in calculus of variations is presented. The local truncationerrors and the methods of order 2th, 4th, 6th, 8th, 10th, and 12th. are obtained. The inverseof some band matrixes are obtained which are required in proving the convergence analysis ofthe presented method. Convergence analysis of these methods is discussed. Numerical resultsare given to illustrate the efficiency of methods and compared with the methods in [23,32-34].

Received: 1 March 2014, Revised: 14 April 2014, Accepted: 21 June 2014.

Keywords:Two-point boundary value problem, Non-polynomial spline, Convergenceanalysis, Calculus of variation.

Index to information contained in this paper

1 Introduction

2 Description of the Method and Development of Boundary Conditions

3 Convergence Analysis

4 Numerical Illustrations

5 Conclusion

1. Introduction

In some problems arising in analysis, mechanics, geometry, etc., it is necessaryto determine the maximal and minimal of a certain functional. Such problemsare called variational problems. Many authors obtained analytical and numericalmethods for the solution of the calculus of variations. The direct method of Ritzmethods, Walsh series method, Bernstein direct method, Haar wavelet, orthogonal

Corresponding author. Email: [email protected],[email protected]


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2 R. Jalilian et al./ IJM2C, 05 - 01 (2015) 1-14.

polynomials, Legendre wavelets, Adomian decomposition, and Galerkin method insolving variational problems has been of considerable concern and is well covered inmany textbooks and papers see [2,6-14,17,26 ] and references therein. The simplestform of a variational problem can be considered as:

J[u1(t), u2(t),...,un(t)] =

ba

H(t, u1(t),...,un(n), u1(t),...,u

n(n))dt, (1)

with the given boundary conditions:u1(a) =1, u2(a) =2,...,un(a) =n,u1(b) =1, u2(b) =2,...,un(b) =n.

(2)

In this paper we consider special form of the variational problem in the followingform:

J[u(t)] =

ba

H(t, u(t), u(t))dt, (3)

with boundary conditions

u(a) =1, u(b) =2, (4)

We known that the function solution should satisfy in the following equation (Euler-Lagrange equation):

Hu ddt

Hu = 0, (5)

with same boundary conditions. Some authors obtained numerical methods forthe solution of boundary value problems using finite difference, spline and non-

polynomial spline for example see [1,3-5,15,16,18-25,27-34]. The basic motivationof this paper is discussed convergence analysis of the non-polynomial spline forsolutions of calculus of variations problems. The paper is organized in four sections.We use the consistency relation of non-polynomial septic spline for approximatethe solution of (3)-(4). Section 2 is devoted to the description of the method anddevelopment of boundary conditions and also we obtain the methods of order2th, 4th, 6th, 8th, 10th, and 12th. The new approach for convergence analysisis discussed in section 3. Finally, in section 4, numerical evidences are included toshow the practical applicability and superiority of our method and compare withthe other methods.

2. Description of the Method and Development of Boundary Conditions

Let us consider a mesh with nodal points xi = a + ih on [a, b] such that:

:a = x0 < x1< x2

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R. Jalilian et al./ IJM2C, 05 - 01 (2015) 1-14. 3

+i(x xi) 2 + gi(x xi) + ri, i= 0,...,n, (6)

where ai, bi, ci, di, ei, i, gi, and ri are constants and also k is the frequency ofthe trigonometric part of the spline functions which can be real or pure imaginaryand which will be used to raise the accuracy of the method. Let u(x) be the exactsolution, and let Si be an approximation to ui obtained by the segment Qi(x)passing through the points (xi, Si) and (xi+1, Si+1). The non-polynomial spline is

defined by the following relations:S(x) =Qi(x), x[xi, xi+1], i= 0,...,n 1,S(x)C[a, b].

To derive the coefficients ai, bi, ci, di, ei, i, gi, and ri, we first define:Qi(xi) =Si, Q

i(xi) =mi, Q

(2)i (xi) =Mi, Q

(6)i (xi) =Li,

Qi(xi+1) =Si+1, Qi+1(xi+1) =mi+1, Q

(2)i (xi+1) =Mi+1, Q

(6)i (xi+1) =Li+1.

By algebraic manipulation we get the following expression where = kh:

ai =Lik6

, bi =cot()Li+ csc()L1+ik6

, i =Li k4Mi

2k4 , ri =

Li

k6+ ui,

gi =cot()Li csc()L1+i k5mi

k5 ,

ci= 1

20h5k6(

120Li 10h2

k

2

Li 60hkcot()Li 60hkcos()cot()Li60hksin()Li 120L1+i+ 10h2k2L1+i+ 60hkcot()L1+i+ 60hkcsc()L1+i+60hk6mi+ 60hk

6m1+i+ 10h2k6Mi 10h2k6M1+i+ 120k6ui 120k6u1+i

),

di = 14h4k6

(60Li+ 6h2k2Li+ 32hkcot()Li+ 28hkcos()cot()Li+28hksin()Li+ 60L1+i 4h2k2L1+i 28hkcot()L1+i 32hkcsc()L1+i

32hk6mi

28hk6m1+i

6h2k6Mi+ 4h

2k6M1+i

60k6ui+ 60k

6u1+i) ,

ei = 12h3k6

(20Li 3h2k2Li 12hkcot()Li 8hkcos()cot()Li

8hksin()Li 20L1+i+ h2k2L1+i+ 8hkcot()L1+i+ 12hkcsc()L1+i+12hk6mi+ 8hk

6m1+i+ 3h2k6Mi h2k6M1+i+ 20k6ui 20k6u1+i

).

Using the continuity of the third, fourth and fifth derivatives, that is Q()i1(x) =

Q()i (x), = 3, 4,and 5,we obtain the following useful consistency relation in terms

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of second derivative of spline Mi and ui.

1(Mi3+ Mi+3) + 2(Mi2+ Mi+2) + 3(Mi1+ Mi+1) + 4Mi = [(ui+3+ ui3)

+1(ui+2+ ui2) + 2(ui+1+ ui1) + 3ui]1

h2, i= 3,...,n 3, (7)

where1=

(120203+5120sin())202(6+3+6sin()) ,

2= 2(240+203135+(120202+4)cos()360sin())

202(6+3+6sin()) ,

3= (840+1003+675+(960+803525)cos()1800sin())

202(6+3+6sin()) ,

4=4(240+203135+3(120+202+114)cos[]600sin())

202(6+3+6sin()) ,

1=402((12+2)+(6+2)cos()+18sin())

202(6+3+6sin()) ,

2=202(42+53+4(12+2)cos()90sin())

202(6+3+6sin()) ,

3= 802(12+3+3(6+2)cos()30sin[])

202(6+3+6sin()) ,

By expanding (7) in Taylor series about xi, we obtain the following local trun-cation error ti:

ti =(2 + 21+ 22+ 3) ui+ h2u(2)i (9 + 21+ 22+ 23+ 4 41 2)

+1

12h4u

(4)i (81 + 1081+ 482+ 123 161 2)

+ 1

360

h6u(6)i (

729 + 24301+ 4802+ 303

641

2)

+ 1

20160h8u

(8)i (6561 + 408241+ 35842+ 563 2561 2)

+ h10

1814400u

(10)i (59049 + 5904901+ 230402+ 903 10241 2)

+ h12

239500800u

(12)

i

(

531441 + 77944681+ 1351682+ 1323

40961)

+ h14u

(14)i

43589145600(4782969 + 967222621+ 7454722+ 1823 163841 2)

+O(h15). (8)

By using the above truncation error to eliminate the coefficients of various powersh we can obtain classes of the methods. For different choices of parameters

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1, 2, 3, 4, 1, 2 and 3, we get the class of methods such as:

(I) Second-order method. For 1 = 50, 2 = 10, 3 = 4 = 0, 1 = 24, 2 = 15 and3=80, we have

ti = 450h4u

(4)i + O(h

5).

(II) Fourth-order method. For 1 = 50, 2 = 10, 3 = 450, 4 = 900, 1 =24, 2= 15, and 3=80, we have

ti = 307h6u

(6)i + O(h

7).

(III) Sixth-order method. For 1 = 142 , 2 =

207, 3 =

39714, 4 =

120821 , 1 = 24, 2 =

15, and 3=80, we have

ti = h8u

(8)i

252 + O(h9).

(IV) Eighth-order method. For 1 = 3374927 , 2 = 3971244343 , 3 = 1628544343 , 4 = 0, 1 =33536

14781 , 2= 1875514781 , and 3= 0, we have

ti = 65629h10u

(10)i

27936090 + O(h11).

(V) Tenth-order method. For1= 286762559 , 2=

3418020853 , 3=

15833920853, 4=

52503262559 , 1 =

6272993, 2=7265993, and 3= 0, we have

ti = 3319h12u

(12)i

35390520

+ O(h13).

(VI) Twelve-order method. For 1 = 1857

49483 , 2 = 110322

49483 , 3 = 989739

49483, 4 =2175924

49483 , 1= 112266

7069 , 2 = 112995

7069 , and 3 =4646607069 , we have

ti = 114669h14u

(14)i

19812993200 + O(h15).

we assume that

ui =f(xi, ui) =fif(xi, u(xi)), (9)

whereui is the approximation of the exact valueu(xi) andSi(x) is non-polynomialseptic spline function [20]. By substituting (9) in the spline relation (7), we obtainthe nonlinear equations in the following form.

1(fi3+ fi+3) + 2(fi2+ fi+2) + 3(fi1+ fi+1) + 4fi

1h2

((ui+3+ ui3) + 1(ui+2+ ui2) + 2(ui+1+ ui1) + 3ui) = 0,

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i= 3,...,n 3. (10)

To obtain unique solution for the nonlinear system (10) we need four more equa-tions. We define the following identities:

4k=0

kuk+ h2 12k=1

kuk+ t1h

14u(14)0 = 0, i= 1,

5k=0

kuk+ h2

12k=1

kuk+ t2h

14u(14)0 = 0, i= 2,

5k=0

kunk+ h2

12k=1

kunk+ tn2h

14u(14)0 = 0, i= n 2,

4k=0

kunk+ h2

12k=1

kunk+ tn1h

14u(14)0 = 0, i= n 1,

(11)

In order that t1, t2, tn2and tn1 are O(h14),we find that the unknown coefficientsin (11) as follows:

(0, 1, 2, 3, 4) = (65, 104, 14, 24, 1), t1= tn1= ( 1210210269217326918592000

),

(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12) =

(2248215317

19958400 ,

4539179

17600 ,6055918291

6652800 ,

9918918899

4989600 ,892246279

285120 ,

18019157507

4989600 ,

306500223179979200

,2380569353

1247400 ,

1685171232119958400

,503717713

1995840 ,130447723

2851200 ,

18976637

4989600),

(0, 1, 2, 3, 4, 5, ) = (26, 14, 80, 15, 24, 1), t2 = tn2= ( 521085679991373621248000

),

(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12) =

(7712745923

159667200 ,

259885933

5702400 ,19222747601

53222400 ,

8351791919

11404800 ,3712133107

3193344 ,

95883047

71280 ,

26143703812280960

,28466192407

39916800 ,

720403331322809600

,16761737

177408 ,43435489

2534400 ,

113795873

79833600).

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3. Convergence Analysis

In this section, we investigate the convergence analysis of the tenth-order methodand also in the same way we can prove the convergence analysis for any of the othermethods. The equations (10) along with boundary condition (11) yields nonlinearsystem of equations, and may be written in a matrix form as

A0U

(1)

+ h

2

Bf(1)

(U

(1)

) =R

(1)

, (12)

(where f(1)(U(1)) = (f1,...,fn1)t,

the matricesA0and B are an (n1)(n1)-dimensional which have the followingforms

A0 =(Pn1(1, 2, 1))3 + 30(Pn1(1, 2, 1))2 120Pn1(1, 2, 1), (13)

where the matrix Pn1(x, z , y) has the following form:

Pn1(x, z , y) =

z yx z y

. . . . . .

. . .

x z yx z

, (14)

B =

1 2 3 4 5 6 7 8 9 10 11 121 2 3 4 5 6 7 8 9 1011122 3 4 3 2 11 2 3 4 3 2 1

. . . . . . . . . . . . . . . . . . . . .. . .

. . . . . .

. . . . . .

. . . . . .

1 2 3 4 3 2 11 2 3 4 3 2

1211 10 9 8 7 6 5 4 3 2 112 11 10 9 8 7 6 5 4 3 2 1

. (15)

R(1) =

65u0,26u0,

u0 1,0...0

un 1,26u0,65u0,

. (16)

We assume that

A0U(1)

+ h2Bf(1)(U(1)

) =R(1) + t(1), (17)

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where the vector U(1)

=u(xi), (i= 1, 2,...,n 1), is the exact solution and t(1) =[t1, t2,...,tn1]

T, is the vector of local truncation error.By using (12) and (17) we get

AE(1) = [A0+ h2BFk(U

(1))]E(1) =t(1), (18)

where

E(1) =U(1) U(1),

f(1)(U(1)

) f(1)(U(1)) =Fk(U(1))E(1), (19)

and Fk(U(1)) = diag{ fi

ui}, (i= 1, 2,...,n 1), is a diagonal matrix of order n 1.

To prove the existence of A1, since A = A0 +h4BFk(U

(1)), we have to showA0=P3n1(1, 2, 1) + 30P2n1(1, 2, 1) 120Pn1(1, 2, 1), is nonsingular.

By using Henrici [15] we have

(Pn1(1, 2, 1))1 (b a)2

8h2 . (20)

It is clear that the matrix A0 is nonsingular and alsoA10 < where is apositive number (. is the L norm).

Theorem 3.1 If Y < 1h2BA1

0

, then the matrix A given by (18) is monotone

(Y =max| fiui |, i= 1, 2,...,n 1).Proof From (18) we have

A= A0+ h2BFk(U

(1)),

hence AA10 =I+ h2BFk(U

(1))A10 , so that

A0A1 = (I+ h2BFk(U

(1))A10 )1 =

=I (h2BFk(U(1))A10 ) + (h2BFk(U(1))A10 )2 (h2BFk(U(1))A10 )3 + ....

= [I (h2BFk(U(1))A10 )][I+ (h2BFk(U(1))A10 )2 + (h2BFk(U(1))A10 )4 + ....].

Also if(h2BFk(U(1))A10 )< 1 then, the two infinite series convergence.

LetFk(U(1)) Y= max| fiui |, i= 1, 2,...,n 1, then

A1 =

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[A10 A10 h2BFk(U(1))A10 ][I+(h2BFk(U(1))A10 )2+(h2BFk(U(1))A10 )4+....],

where the infinite series is nonnegative. Hence to show that A is monotone, itsufficient to show that [A10 A10 h2BFk(U(1))A10 ]> 0. Here we have

I > A10 h2BFk(U

(1)),

h2A10 BFk(U(1)) h2A10 BFk(U(1))< 1. (21)

Then

Y < 1

h2BA10 .

Theorem 3.2 Letu(x)be the exact solution of the boundary value problem (4)-(5)and assumeu

i, i= 1, 2,...,n

1, be the numerical solution obtained by solving the

system (12). Then we have

E O(h10),

providedY < 17325279548636h2 , where

1= 2867

62559, 2 =

34180

20853, 3=

158339

20853 , 4=

525032

62559 , 1=

6272

993 , 2=7265

993 , 3 = 0,

ProofWe can write the error equation (18) in the following form

E(1) = (A0+ h2BFk(U

(1)))1t(1) = (I+ h2A10 BFk(U(1)))1A10 t

(1),

E(1) (I+ h2A10 BFk(U(1)))1A10 t(1),

it follows that

E(1) A10 t(1)

1

h2

A1

0 B

F

k(U(1))

, (22)

provided that h2A10 BFk(U(1))< 1. Following [22] we have

t(1) 3319h12M12

35390520 , (23)

where M12= max|u(12)()|, a b.From inequalities (22), (23),A10 < ,Fk(U(1)) Y (Y = max| fiui |, i =1, 2,...,n 1,) andB 27954863617325 we obtain

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E 17325h12M12

252(1 279548636h2Y) O(h10), (24)

provided that

Y 0 such that

T xT y Lxy, x, yC, (1.1)

(2) pseudocontractive[4] if for any x, yC, there exists j (x y)J(x y) suchthat

T xT y , j(x y) xy2, (1.2)

and it is well known that condition (1.2) is equivalent to the following:

xy xy+s[(IT x)(IT y)], s >0, x , yC, (1.3)

(3) strongly pseudocontractive[4] if there exists a constant k(0, 1) andj(xy)J(xy) such that for any x, yC,

T x

T y , j(x

y)

k

x

y

2, (1.4)

(4) -strictly pseudocontractive[4] in the terminology of Browder and Petryshyn(-strictly pseudocontractive, for short) if there exists >0 and

j(xy)J(xp)such that for any x, yC,

T xT y , j(xy) xy2 (IT)x(IT)y2, (1.5)

(5) -demicontractive[4] ifF(T)= and there exists a constant >0 andj(xp)J(xp) such that for any xC, pF(T),

T xp, j(xp) xp2 x T x2. (1.6)

(6) nonexpansive[32] if

T xT y xy, x, yC. (1.7)

(7)asymptotically nonexpansive[32] if there exists a sequence{kn} [1, )with kn1as n such that

TnxTny knxy, n 1, x , yC. (1.8)

(8) asymptotically nonexpansive in the intermediate sense[24] ifT is continuousand the following inequality holds:

limn

sup supx,yC

(TnxTny xy) 0. (1.9)

Observe that if we define

n = max

0, sup

x,yC

(TnxTny xy)

, (1.10)

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In real Hilbert spaces, we observe that (1.21) is equivalent to

TnxTny2 (2kn1)xy2 +(ITn)x(ITn)y2 +2n, n 1, x , yC.(1.22)

Definition 1.2. [4] A Banach space E is said to satisfy the Opial conditionif forany sequence{xn} Ewith xn x, the following inequality holds:

limn

sup xnx< limn

sup xny

for any yEwith y=x.

Definition 1.3. [28] Let H be a real Hilbert space with inner product., .and norm., respectively and let C be a closed convex subset of H. For everyxH, there exists a unique nearest point in C, denoted by PCx,such that

x

PCx

x

y

y

C.

PC is called the metric projectionofH onto C.Goebel and Kirk [7] introduced the class of asymptotically nonexpansive map-

pings as a generalization of the class of nonexpansive mappings. They establishedthat ifCis a nonempty closed convex and bounded subset of a real uniformly con-vex Banach space and T is an asymptotically nonexpansive mapping on C, thenThas a fixed point. The class of asymptotically nonexpansive mapping in the inter-mediate sense was introduced by Bruck et al.[3] in 1993. In 1974, Kirk [11] provedthat ifC is a nonempty close convex subset of a uniformly convex Banach spaceE and T is asymptotically nonexpansive in the intermediate sense, then T has afixed point. We remark that the class of mappings which are asymptotically non-expansive in the intermediate sense contains properly the class of asymptoticallynonexpansive mappings. The class of strict pseudocontractive maps was introducedby Browder and Petryshyn [2]. Marino and Xu [13] established that the fixed pointset of strict set of strict pseudocontractions is closed convex, and they obtained aweak convergence theorem for strictly pseudocontractive mappings by Mann iter-ative process.

The class of asymptotically strict pseudocontractive mappings was introducedby Liu [12]. Sahu et al. [28], introduced the class of asymptotically strict pseudo-contractive mappings in the intermediate sense in 2009. The class of asymptoti-cally nonexpansive mapping was introduced by Schu [29]. Rhoades [27] producedan example to show that the class of asymptotically pseudocontractive mappings

contains properly the class of asymptotically nonexpansive mappings. The classof asymptotically pseudocontractive mappings in the intermediate sense was in-troduced by Qin et al. [24]. They obtained some convergence results of Ishikawaiterative processes for the class of mappings which are asymptotically pseudocon-tractive mappings in the intermediate sense. Olaleru and Okeke [21] introduced theclass of asymptotically demicontractive mappings in the intermediate sense and theclass of asymptotically hemicontractive mappings in the intermediate sense. We es-tablished some interesting fixed points results for this class of nonlinear mappings(see, [21]).

Noor et al. [18] gave the following three-step iteration process for solving non-linear operator equations in real Banach spaces. Let T :CCbe a mapping. For

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G. A. Okeke& J. O. Olaleru/ IJM2C, 05 - 01 (2015) 15-28. 19

an arbitrary x0C, the sequence{xn}n=0C defined by

xn+1= (1n)xn+nT ynyn = (1n)xn+nT znzn = (1n)xn+nT xn, n 0,

(1.23)

where{n}

n=0,{

n}

n=0 and{n}

n=0,are three sequences satisfying

n

, n

, n[0, 1] for each n.

It was established by Bnouhachem et al. [1] that three-step method performsbetter than two-step and one-step methods for solving variational inequalities.Glowinski and P. Le Tallec [6] applied three-step iterative sequences to finding theapproximate solutions of the elastoviscoplasticity problem, eigenvalue problems andin the liquid crystal theory. Moreover, three-step schemes are natural generalizationof the splitting methods to solve partial differential equations. What this meansis that Noor three-step methods are robust and more efficient than the Mann(one-step) and Ishikawa (two-step) type schemes for solving problems in pure andapplied sciences.

The following results will be useful to us in this study.

Lemma 1.4. [24]. Let {rn}, {sn}, and {tn} be three nonnegative sequencessatisfying the following condition:

rn+1 (1 +sn)rn+tn, n n0, (1.24)

where n0 is some nonnegative integer. If

n=1sn

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generated by the modified Mann iteration process:

xn+1= (1n)xn+nTnxn nN. (1.26)

Then{xn} converges weakly to an element ofF(T).Qin et al. [24] proved the following theorem:

Theorem QCK. Let C be a nonempty closed convex bounded subset of areal Hilbert space H and T : C Ca uniformly L-Lipschitz and asymptoticallypseudocontractive mapping in the intermediate sense with sequences{kn} [1, )and{n} [0, ) defined as in (1.21). Assume that F(T) is nonempty. Let{xn}be a sequence generated in the following manner:

x1C,yn = nT

nxn+ (1n)xn,xn+1 = nT

nyn+ (1n)xn, n 1,()

where{n}and{n}are sequences in (0, 1).Assume that the following restrictionsare satisfied:

(a)

n=1n

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where{n}n=0,{n}n=0 and{n}n=0, are three sequences satisfying n, n, n[0, 1] for each n. Assume that the following restrictions are satisfied:

(i)

n=1n

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Tnynp2 qnynp2 +ynTnyn2 + 2n qn{q2nxnp2 nn(1nnqn2nL2)Tnxnxn2+

n(1n)xnTnzn2 + 2n(1 +qn)}+(1n)xnTnyn2 +3nL2(1n)xnTnzn23nL

2n(1n2nL2)Tnxnxn2n(1n)Tnznxn2 + 2n

=q3nxnp2 nnqn(1nnqn2nL2)Tnxnxn2+nqn(1n)xnT

n

zn2

+ 2qnn(1 +qn)+(1n)xnTnyn2 +3nL2(1n)xnTnzn23nL

2n(1n2nL2)Tnxnxn2n(1n)Tnznxn2 + 2n

q3nxnp2 nnqn(1nnqn2nL2)Tnxnxn2+(1n)xnTnyn2 + 2n(1 +qn+q2n). (2.6)

Using Lemma 1.5, (1.22) and (2.6), we obtain:xn+1p2 =(1n)(xnp) +n(Tnynp)2

= (1n)xnp2 +nTnynp2 n(1n)Tnynxn2 (1n)xnp2 +n{q3nxnp2

nnqn(1nnqn2nL2)Tnxnxn2+(1

n)

xn

Tnyn

2 + 2n(1 +qn+q

2n)

}n(1n)Tnynxn2 (2.7) q3nxnp2 nnnqn(1nnqn2nL2)Tnxnxn2+

n(1n)xnTnyn2 + 2n(1 +qn+q2n)n(1n)Tnynxn2

q3nxnp2 nnnqn(1nnqn2nL2)Tnxnxn2+2n(1 +qn+q

2n). (2.8)

From condition (b), we observe that there exists n0 N

nnnqn(1nnqn2nL2) 12bL2b2

2 >0, n n0. (2.9)

We note that

xn+1p2

1 + (q3n1) xnp2 + 2n(1 +qn+q2n), n n0. (2.10)

Using Lemma 1.4, we see that limn xn p exists. For each n n0, weobserve thata2(12bL2b2)

2 Tnxnxn2 (q3np)xnp2 +xnp2xn+1p2 + 2n(1 +qn+q2n), (2.11)Hence,

limn

Tnxn

xn

2 = 0. (2.12)

Note thatxn+1xn nTnynxn n(TnynTnxn +Tnxnxn)

n(Lynxn+Tnxnxn) (1 +nL)Tnxnxn. (2.13)

Hence, from (2.12) we obtain

limn

Tnxnxn2 = 0. (2.14)

By triangle inequality, we have

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xnT xn xnxn+1+xn+1Tn+1xn+1+Tn+1xn+1Tn+1xn+Tn+1xnT xn

(1 +L)xnxn+1+xn+1Tn+1xn+1+LTnxnxn. (2.15)

From (2.12) and (2.14), we have

limn

Tnxnxn2 = 0. (2.16)

But{xn}is bounded, hence we observe that there exists a susequence{xni} {xn}such that xni x

. From Lemma 1.7, we have that x F(T).We now prove that{xn} converges weakly to x. Next, we prove that x is

unique. Suppose that there exists some subsequence{xnj} {xn}such that{xnj}converges weakly to x C and x= x. From Lemma 1.7, we can show thatx F(T). Put d = limn xnx. Since H satisfies Opial property, we seethatd= limniinfxnix< limniinfxnix

= limnjinfxnjx< limnjinfxnjx= limniinfxnix= d. (2.17)

Which is a contradiction. It follows that x = x. The proof of the theorem is

complete.

Next, we establish the hybrid Noor algorithm for L-Lipschitzian asymptoticallypseudocontractive mappings in the intermediate sense to obtain a strong conver-gence theorem without any compact assumption.

Theorem 2.2. Let C be a nonempty closed convex bounded subset of areal Hilbert space H, PC the metric projection from H onto C, and T : C Ca uniformly L-Lipschitz and asymptotically pseudocontractive mapping in theintermediate sensewith sequences{kn} [1, ) and n [0, ) defined as in(1.21). Let qn = 2kn 1 for each n 1. Assume that F(T)=.Let{xn}n=0 be asequence in Cgenerated by the following hybrid Noor algorithm:

yn = (1n)xn+nTnzn

zn = (1n)xn+nTnsn

sn = (1n)xn+nTnxn, n 0, (2.18)

Cn ={uC:ynu2 xnu2 +nn+nnnqn(n+nqn+

2nL

2 1)Tnxnxn2}

Qn ={uC :x1xn, xnu 0},xn+1= PCnQnx1,

where {n}n=0, {n}n=0 and {n}n=0, are three sequences satisfyingn, n, n[0, 1] for each n and n = qn([1+ n(qn 1)] 1)M+ 2n(1 + qn+ q2n)for each n 1.Assume that the control sequences{n}, {n}and{n}are chosensuch thata n n n bfor somea >0 and someb(0, L2[

1 +L21]).

Then the sequence generated by (2.18) converges strongly to a fixed point ofT.

Proof. The proof is divided into seven steps.

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Step 1. We must show that CnQn is closed and convex for all n 1.By definition of Qn, it is clear that it is closed and convex and Cn is closed foreach n 1. We, therefore, only need to show that Cn is convex for each n 1.Observe thatCn ={uC:ynu2 xnu2 +nn+

nnnqn(n+nqn+2nL

2 1)Tnxnxn2} (2.19)is equivalent toCn =

{u

C: 2

xn

yn, u

xn

2

yn

2 +nn+

nnnqn(n+nqn+2nL2 1)Tnxnxn2} (2.20)Clearly, we see that Cn is convex for each n 1. This implies that CnQn isclosed and convex for each n 1.This completes Step 1.Step 2. We must show that F(T)CnQn for each n 1.

Let pF(T).From Lemma 1.3 and the algorithm (2.18), we observe thatynp2 =(1n)(xnp) +n(Tnznp)2

= (1n)xnp2 +nTnznp2 n(1n)Tnznxn2 (1n)xnp2 +n(qnznp2 +znTnzn2 + 2n)

n(1n)Tnznxn2= (1n)xnp2 +nqnznp2 +nznTnzn2+

2nnn(1n)Tn

znxn2

. (2.21)snp2 =(1n)(xnp) +n(Tnxnp)2= (1n)xnp2 +nTnxnp2 n(1n)Tnxnxn2 (1n)xnp2 +n(qnxnp2 +xnTnxn2 + 2n)

n(1n)Tnxnxn2= (1n)xnp2 +nqnxnp2 +nxnTnxn2+

2nnn(1n)Tnxnxn2. (2.22)znp2 =(1n)(xnp) +n(Tnsnp)2

= (1n)xnp2 +nTnsnp2 n(1n)Tnsnxn2 (1n)xnp2 +n(qnsnp2 +snTnsn2 + 2n)

n(1n)Tnsnxn2

= (1n)

xn

p

2

+n

qn

sn

p

2

+n

sn

Tnsn

2

+2nnn(1n)Tnsnxn2. (2.23)snTnsn2 =(1n)(xnTnsn) +n(TnxnTnsn)2

= (1n)xnTnsn2 +nTnxnTnsn2n(1n)Tnxnxn2

(1n)xnTnsn2 +nL2xnsn2n(1n)Tnxnxn2. (2.24)

znTnzn2 =(1n)(xnTnzn) +n(TnsnTnzn)2= (1n)xnTnzn2 +nTnsnTnzn2

n(1n)Tnsnxn2 (1n)xnTnzn2 +nL2snzn2

n(1

n)

Tnsn

xn

2. (2.25)

Using (2.22)-(2.25), we obtain:

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ynp2 (1n)xnp2 +nqnznp2 +nznTnzn2+2nnn(1n)Tnznxn2

(1n)xnp2 +nqn{(1n)xnp2 +nqnsnp2+nsnTnsn2 + 2nnn(1n)Tnsnxn2}+n{(1n)xnTnzn2 +nL2snzn2n(1n)Tnsnxn2}+ 2nnn(1n)Tnznxn2

= (1n)xnp2 +nqn(1n)xnp2+nnq

2

nsnp2

+nnqnsnTn

sn2

+ 2nqnnnnqnn(1n)Tnsnxn2 +n(1n)xnTnzn2+nnL

2snzn2 nn(1n)Tnsnxn2 + 2nnn(1n)Tnznxn2

((1 n) +nqn(1n))xnp2 +nnq2n{(1n)xnp2+nqnxnp2 +nxnTnxn2 + 2nnn(1n)Tnxnxn2} +nnqn{(1n)xnTnsn2+nL

2xnsn2 n(1n)Tnxnxn2} + 2nqnnnnqnn(1n)Tnsnxn2 +n(1n)xnTnzn2+nnL

2snzn2 nn(1n)Tnsnxn2 + 2nnn(1n)Tnznxn2

{(1

n) +nqn(1

n) +nnq

2n(1

n) +nnq

3n

}xn

p

2+

{nnnq2nnnq2nn(1n)nnqnn(1n)}Tnxnxn2+2nn(1 +nqn+nnq

2n)

xnp2 +nnnqn(n+nqn+2nL2 1)Tnxnxn2+nn. (2.26)

where n = qn([1+ n(qn 1)] 1)M+ 2(qn+ 1)n for each n 1.It follows thatpCn for all n 1.This shows that F(T)Cn for all n 1.

We now show that F(T) Qn for all n 1. We prove this by inductions.Clearly, F(T)Q1 =C. Suppose that F(T)Qk for some k >1. Since xk+1 isthe projection ofx1 onto CkQk, we see that

x1

xk+1, xk+1

y

0,

x

Ck

Qk.

By induction, we know that F(T)CkQk. Hence, for each y F(T)C, weobtain:

x1xk+1, xk+1y 0, (2.27)

this implies that yQk+1. Hence, F(T)Ck+1. This shows that F(T)Qn forall n 1. Hence, F(T)CnQn for each n 1.This completes step 2.Step 3. We must show that limn xnx1 exists.From (2.18), we observe that xn = PQnx1 and xn+1Qn which shos that

x1xn x1xn+1. (2.28)

Hence the sequencexnx1 is nondecreasing. Recall that C is bounded. Thisimplies that limn xnx1 exists. This completes step 3.Step 4. We must show that xn+1xn0 as n .Observe that xn = PQnx1 and xn+1= PCnQn Qn. This implies that

xn+1xn, x1xn 0, (2.29)

from which it follows that

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xn+1xn2 =(xn+1x1) + (x1xn)2=xn+1x12 +x1xn2 + 2xn+1x1, x1xn xn+1x12 x1xn2. (2.30)

Hence, we obtain xn+1xn0 as n .This completes step 4.Step 5. We must show that Tnxnxn0 and Tnsnxn0 as n .In view ofxn+1Cn,we obtainynxn+12 xnxn+12 +nn (2.31)

+[n

3n

L2n(

qn

n+

L22n1)]

Tnxn

xn

2

+(nqn2nnnqn2n+n3nL2 +n2nnn)Tnsnxn2.

On the other hand, we see that

ynxn+12 =ynxn+xnxn+12 =ynxn2+xnxn+12+2ynxn, xnxn+1.(2.32)

Combining (2.31) and (2.32) and recalling that yn = (1n)xn + nTnzn, weobtain:nTnznxn2 + 2Tnznxn, xnxn+1

n+ [n3nL

2n(qnn+L22n1)]Tnxnxn2

+(nqn2nnnqn2n+n3nL2

+n2nnn)Tnsnxn2. (2.33)

From the asumption, we observe that there exists n0 such that

1qnnL22n 12b L2b2

2 >0, n n0. (2.34)

For any n n0, it follows from (2.33) that

a(12b L2b2)2

Tnxnxn2 n+ 2Tnznxnxnxn+1. (2.35)

Similarly,

a(12b L2b2)2

Tnsnxn2 n+ 2Tnznxnxnxn+1. (2.36)

Note that n0 as n . Hence, by Step 4, we have:

limn

Tnxnxn= 0. (2.37)

and

limn

Tnsnxn= 0. (2.38)

This completes Step 5.Step 6. We must show that T xnxn0 and T snxn0 as n .xnT xn xnxn+1+xn+1Tn+1xn+1

+Tn+1xn+1Tn+1xn+Tn+1xnT xn (1 +L)xnxn+1+xn+1Tn+1xn+1

+LTnxnxn. (2.39)Similarly,xnT sn xnxn+1+xn+1Tn+1sn+1

+Tn+1sn+1Tn+1sn+Tn+1snT sn (1 +L)xnxn+1+xn+1Tn+1sn+1

+LTnsnxn. (2.40)

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The conclusion follows from Step 5. This completes Step 6.Step 7. We must show that xnq,where q= PF(T)x1 as n .

From Lemma 1.5, we have that w(xn)F(T). From xn = PQnx1 and F(T)Qn,we observe that

x1xn x1q. (2.41)

From Lemma 1.5 of Yanes and Xu [31], we obtain Step 7. The proof of Theorem2.2 is complete.

Remark 2.3. The results of Theorem 2.2 is more general and it is an im-provement of X. Qin et al. [24], in the sense that ifn = 0n 1,then, we obtainthe results of Qin et al. [24]. Ifn = 0,n 1, then we obtain the results of Kimand Xu [10], Marino and Xu [13], Qin et al. [24], Sahu et al. [28] and Zhou [33].

3. Conclusion

The fixed points results established in this paper improves, generalizes and extendsseveral other fixed point results in literature including Schu [29], Qin et al. [24]among others.

Acknowledgements

The authors wish to thank the referees for their comments and observations whichlead to the improvement of the manuscript.

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[5] S.-S. Chang, J. K. Kim, H. W. J. Lee and C. K. Chan On the convergence theorems for a count-able family of lipschitzian pseudocontraction mappings in banach spaces, Fixed Point Theory andApplications, vol. 2011, Article ID 680930, 11 pages, (2011).

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[7] K. Goebel and W. Kirk,A fixed point theorem for asymptotically nonexpansive mappings, Proceed-ings of the American Mathematical Society, 35 (1972) 171-174.

[8] M. G. Kamalvand,On coeigenvalues of a complex square matrix, International Journal of Mathe-matical Modelling & Computations, 3 (3) (2013) 253-258.

[9] J. K. Kim and Y. M. Nam, Modified Ishikawa iterative sequences with errors for asymptotically set-valued pseudocontractive mappings in Banach spaces, Bulletin of the Korean Mathematical Society,43 (4) (2006) 847-860.

[10] T.-H. Kim and H.-K. Xu, Convergence of the modified Manns iteration method for asymptoticallystrict pseudo-contractions in Hilbert spaces, Nonlinear Analysis: Theory, Methods & Applications,68 (9) (2008) 2828-2836.

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[13] G. Marino and H.-K. Xu, Weak and strong convergence theorems for strict pseudo-contractions inHilbert spaces, Journal of Mathematical Analysis and Applications, 329 (1) (2007) 336-346.

[14] A. A. Mogbademu and J. O. Olaleru, Modified Noor iterative methods for a family of stronglypseudocontractive maps, Bulletin of Mathematical Analysis and Applications.3 (4) (2011) 132-139.

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[16] M. A. Noor, Three-step iterative algorithms for multivalued quasi variational inclusions, J. Math.Anal. Appl. 255 (2001) 589-604.

[17] M. A. Noor,Some developments in general variational inequalities, Appl. Math. Comput.152(2004)199-277.

[18] M. A. Noor, T. M. Kassias and Z. Huang, Three-step iterations for nonlinear accretive operatorequations, J. Math. Anal. Appl. 274 (2001) 59-68.

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[24] X. Qin, S. Y. Cho and J. K. Kim, Convergence theorems on asymptotically pseudocontractive map-pings in the intermediate sense, Fixed Point Theory and Applications, vol. 2010, Article ID 186874,

14 pages, (2010).[25] X. Qin, Y. J. Cho, S. M. Kang and M. Shang, A hybrid iterative scheme for asymptoticallyk-strictpseudo-contractions in Hilbert spaces, Nonlinear Analysis: Theory, Methods & Applications, 70 (5)(2009) 1902-1911.

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Vol. 05, No. 01, Winter 2015, 29- 40

Analysis of Discrete-Time Machine Repair Problem with two

Removable Servers under Triadic Policy

V. Goswamia, and P. Vijaya Laxmib

aSchool of Computer Application, KIIT University, Bhubaneswar 751024, India;bDepartment of Applied Mathematics, Andhra University, Visakhapatnam - 530003,

India.

Abstract.This paper analyzes a controllable discrete-time machine repair problem with Loperating machines and two technicians. The number of working servers can be adjusteddepending on the number of failed machines in the system one at a time at machines failure orat service completion epochs. Analytical closed-form solutions of the stationary probabilities ofthe number of failed machines in the system are obtained. We develop the total expected costfunction per machine per unit time and obtain the optimal operating policy and the optimalservice rate at minimum cost using quadratic fit search method and simulated annealingmethod. Various performance measures along with numerical results to illustrate the influenceof various parameters on the buffer behavior are also presented.

Received: 7 Janurary 2014, Revised: 2 August 2014, Accepted: 17 August 2014.

Keywords:Discrete-time, Triadic policy, Machine-repair, Cost, Quadratic fit searchmethod, Queue.


1 Introduction

2 Model Description

3 System Characteristics

4 Cost Analysis and Optimization Investigation

5 Numerical Results

6 Conclusion

1. Introduction

During the last few decades, discrete-time queueing systems have been consider-ably investigated and have been extensively employed to various fields, such astelecommunication systems, communication networks, production systems, man-ufacturing systems, etc. These queueing systems are more accurate and effective

Corresponding author. Email: veena [email protected]


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30 V. Goswami& P. Vijaya Laxmi/ IJM2C, 05 - 01 (2015) 29-40.

than their continuous-time counterparts to study and design slotted digital trans-mitting communication systems. Extensive study of discrete-time queueing modelshave been reported in [1-2].

Many researchers have studied queueing systems with threshold policy in whichthe working servers can be adjusted one at a time at any arrival or service comple-tion epochs depending on the number of customers present in the system. Tadj andChoudhury [15] reported a survey of continuous-time queue with single threshold.

The M/M/1 queue with Npolicy has been first studied by Yadin and Naor [21],where the server turns on whenever N or more customers are present in the sys-tem, and turns the server off when the system becomes empty. When the server isturned off, the server can not work till Ncustomers are present in the system.

The service resources can be better utilized with the two service rate control toenhance the operational economy. The optimal operation of anM/M/2 queue withtwo removable servers has been investigated by Bell [1]. Rhee and Sivazlian [13]presented the busy period distribution in the controllable M/M/2 queue operatingunder the triadic (0, K , N , M ) policy. The M/M/2 queue, the N policy M/M/1queue, and the M/M/1 queue are the special cases of the controllable M/M/2queue operating under the triadic (0, Q , N , M ) policy. The optimal control of anM/M/2 queue with finite capacity L operating machines under the triadic pol-

icy has been examined by Wang and Wang [16]. The optimal operating policyfor a controllable queueing model in which cost elements, arrival rate and servicerate are all fuzzy numbers have been discussed by Lin and Ke [8]. Discrete-timeGI/D-MSP/1/K queue with N threshold policy has been studied by Goswamiand Vijaya Laxmi [3]. In continuous-time, the analysis of optimal thresholds of aninfinite buffer two severs system with triadic policy has been investigated by Linand Ke [9]. An infinite buffer discrete-time two severs system with triadic policyhas been studied by Goswami and Mund [2].

In many manufacturing systems, machine interference is a significant problem.A machine may fail due to some unpredicted fault and thus requires a repair fa-cility, after which it can again begin functioning properly. Extensive research of a

machine repair problems have been reported in [5, 14]. Wang [17] presented thesteady-state analytic solutions of an M/M/1 machine repair problem with a sin-gle service station subject to breakdowns. The machine repair problem with Runreliable service stations has been developed by Wang [18]. The M/Ek/1 ma-chine repair problem with an unreliable server was considered by Wang and Kuo[19]. The reliability characteristics of a repairable system with warm standbys andserver breakdowns was investigated by Wang et al. [20]. Sensitivity analysis of ma-chine repair problems in manufacturing systems with service interruptions can befound in Ke and Lin [6]. Lv et al. [11] discussed unreliable multi-server machinerepairable system with variable breakdown rates. Ke et al. [7] studied a machine re-pair problem with warm standbys, imperfect coverage, service pressure coefficient,and unreliable multi-technicians.

Many practical and highly automated manufacturing processes, production ormachines facilities are subject to failures leading to significant loss of productionoutput which in succession affects the companys revenue. The theory of machinerepairable systems with spares has many applications, such as power stations, hos-pitals, manufacturing systems and industrial systems, where standby equipmentsare needed. Generally, there is a trade-off between the magnitude of machine inter-ference and the operator staffing level. For a practical view, machine interferenceproblems cannot simply take the relationship between machines and the operatorbut also impacts upon many other components such as the part arrival rate, exis-tence of external operations and machine failures.

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V. Goswami& P. Vijaya Laxmi/ IJM2C, 05 - 01 (2015) 29-40. 31

In many real life congestion setting of machine repair problem, the system ef-ficiency can be enhanced by rendering sufficient spare part back-up in case ofmachine failure. In a multi-processor computer with a shared memory, the proces-sors work for a time before they need data from the memory and enter the memoryqueue for servicing. As soon as the shared memory responds, they resume opera-tion. Liou et al. [10] studied the controllableM/M/2 machine repair problem withL operating machines operating under the triadic policy. Many continuous-time

queueing systems with machine repair problems have been studied, but in the lit-erature their discrete-time counterparts have got very little attention. In computerand digital telecommunication systems, discrete-time queueing systems are moresuitable than their continuous-time counterparts to determine system performancemeasures, due to clock-driven procedure of those systems.

This paper analyzes an optimal thresholds of a discrete-time machine repairproblem with two removable servers operating under the triadic policy. In the tri-adic policy, whenever there are no failed machines in the system, both servers aretemporarily inactive until certain specified conditions arise. First of all, we assumethat both the servers are turned off. By using the recursive method the station-ary probabilities and some performance measures with numerical results have beenpresented. The results found in this paper are appropriate for implementation in

software packages developed for various management control systems where an au-tomatic threshold control limits alert can be employed as a cautionary mechanism.

The rest of the paper is organized as follows. Section 2 presents model descrip-tion and analysis of the queueing model for the stationary probabilities at arbitraryepoch. System characteristics have been discussed in Section 3. The cost analysisand optimization investigation is carried out in Section 4. Numerical results in theform of tables and graphs to study the parameter effect on the system performanceare presented in Section 5. Section 6 concludes the paper.

2. Model Description

We consider a discrete-time machine repair model with L identical operating ma-chines that are served by two removable servers which are subject to breakdownsunder the triadic policy. In the triadic policy, initially, we assume that both theservers remain inactive until certain specified conditions develop. When the num-ber of failed machines waiting for service reaches a specific number, say N, one ofthe servers becomes active instantly. When the number of failed machines waitingfor service increases to another specified level, say M(M > N), then the left serveralso turns active instantly. If both the servers are active and the number of failedmachines in the system decreases to Q, where 3 Q < N, the server just finishinga service becomes inactive at that time. In addition, if the number of failed ma-chines drops down to zero while one server is active, the server becomes inactive.

In discrete-time queues, number of failed machines and their onward service mayoccur simultaneously around slot boundaries. Their occurrence order may be takencare of by either early arrival system (EAS) or late arrival system with delayed ac-cess (LAS-DA), which are also known as departure-first (DF) or arrival-first (AF)policies. Here we discuss the model with LAS-DA and therefore, a potential ma-chine fails in (t, t) and a potential service occurs in (t, t+), fort = 0, 1, 2, . . .. Formore details on this topic, see [4? ]. The inter-arrival times of failed machines areindependent and geometrically distributed with probability mass function (p.m.f.)an =

n1, 0 < < 1, n 1. When an operating machine fails, it is instantlysent to a server and is served in order of its breakdown and the service timesof both the servers are assumed to be independent and geometrically distributed

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with p.m.f. sn = n1, 0 <

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where () is equal to 1 when the expression is satisfied; otherwise, its value is0. In the steady-state, above equations reduce to

0P0,0= 1P1,1, (1)

nPn,0= n1Pn1,0, 1 n N 1, (2)

(n + n)Pn,1= n+1Pn+1,1+ (2 n Q 2)n1Pn1,1,

1 n Q 2, (3)

(n + n)Pn,1= n+1Pn+1,1+ (Q 1 n Q)n+22Pn+2,2

+n1Pn1,1+ (n= Q)(

2n+1 + 2)

Pn+1,2,

Q 1 n N 1, (4)

(N + N)PN,1= N+1PN+1,1+ N1PN1,1+ N1PN1,0, (5)

(n + n)Pn,1= (N+ 1 n M 2)n+1Pn+1,1+ n1Pn1,1,

N+ 1 n M 1, (6)

nPn,2=(

2n+1 + n+12)

Pn+1,2+ (n= M)n1Pn1,1

+n+22Pn+2,2+ (Q + 2 n L 3)n1

2Pn1,2,

Q + 1 n L 3, (7)

L2PL2,2=(

2L1 + L12)

PL1,2+ L32PL3,2

+2PL,2, (8)

L1PL1,2= L22PL2,2+ 2PL,2, (9)

(1 2

)PL,2= L1

2PL1,2, (10)

where n =(

1 n2 2n)

, 1 n L 1. The steady-state probabilitiesPn,0 (0 n N 1), Pn,1 (1 n M 1) and Pn,2, (Q+ 1 n L) arecomputed by solving the equations (1) - (10). From (2), we get

Pn,0=

L

L n

P0,0, 1 n N 1. (11)

Using (1) and (3) recursively, we obtain

Pn,1= hnP0,0, 1 n Q 1, (12)

where

h1= 01

, h2=

1 + 1

2

h1 and

hn=

(n1 + n1

)hn1 (n2) hn2

n , 1 n Q 1.

From (6),

Pn,1= hnPM1,1, n= M 1, M 2, . . . , N , (13)

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where

hM1 = 1, hM2=M1 + M1

M2 and

hn=

(n+1 + n+1

)hn+1

(n+2

)hn+2

n , n= M 3, . . . , N .

From (4), using (13) and simplifying, we obtain

Pn,1= hnPM1,1+ fnP0,0, n= N 1, N 2, . . . , Q , (14)

where fN1= L

(LN+1) , fN2 = (N1+N1)fN1

N2 and

fn=(n+1 + n+1) fn+1 (n+2)fn+2

n , n= N 3, N 4, . . . , Q ,

hn=

(n+1 + n+1

)hn+1 (n+2)hn+2

n , n= N 1, N 2, . . . , Q .

Using (7) - (10), we obtain

Pn,2= dnPL,2, n= L, L 1, . . . , M , (15)

where

dL= 1, dL1=2 2

L12,

dL2= L1(2

2)

(L12)(L22)

2

L22,

dL3=

L2dL2 (2L1 + L12)dL1

2dLL32 ,

dn=n+1dn+1 (2n+2 + n+2

2)dn+2 n+32dn+3

n2 ,

n= L 4, . . . , Q .

From (6), we obtain

Pn,2= dnPL,2+ enPM1,1, n= M 1, . . . , Q + 1, (16)

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where

eM1=1

, eM2=

M1eM1M22

,

eM3= M2eM2

(2M1 + M1

2)

eM1

M32 ,

en= n+1en+1

(2n+2 + n+22

)en+2 n+32en+3

n2 ,

n= M 4, . . . , Q .

Setting n = Q in (5), we get after simplification

PL,2=

eQdQ

PM1,1. (17)

Putting n = Q 1 in (4), we get

P0,0= K1PM1,1, (18)

where

K1=

QhQ+Q+1

2 (eQ+1dQ eQdQ+1)

dQ

(Q1 + Q1)hQ1 Q2hQ2 QfQ

.

Now using the normalization condition, we obtain

PM1,1=

K1 N

n=0

L

L n

+ K1

N1n=Q

fn+L

n=Q+1

dn+ K1

Q1n=1

hn

+M1n=Q

hn+M1n=Q+1

en

1

. (19)

3. System Characteristics

There are various system features of the controllable discrete-time machine repairqueueing system operating under the triadic (0, Q , N , M ) policy.Let the expected number of failed machines in the system when all the servers areturned off (L0), the expected number of failed machines in the system when oneof the servers is turned on and working (L1) and the expected number of failedmachines when both the servers are turned on and working (L2) are given by

L0=N1n=0

nPn,0, L1=M1n=1

nPn,1 L2=L

n=Q+1

nPn,2.

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The expected number of failed machines in the system (Ls) is given by

Ls =N1n=0

nPn,0+M1n=1

nPn,1+L

n=Q+1

nPn,2.

The expected number of operating machines in the system (E[O]) is given by

E[O] =L

Ls.Let the expected number of idle servers in the system (E[I]), the expected numberof one busy server (E[B1]), the expected number of two busy servers (E[B2]) andthe expected number of busy servers (E[B]) in the system, can be obtained as

E[I] =N1n=0

2Pn,0+M1n=1

Pn,1,

E[B1] =M1n=1

Pn,1, E[B2] =L

n=Q+1

2Pn,2, E[B] =E[B1] + E[B2].

The fraction of total time the machines are working, that is, machine availability(MA) is given by

M A= E[O]

L .

The fraction of the busy servers, that is, operative utilization (OU) is given by

OU=E[B1] + E[B2]

2 .

4. Cost Analysis and Optimization Investigation

The performance measures derived can now be used to optimize the performanceof the system. We develop the total expected cost function per unit time for thediscussed queueing system, assuming the decision variables as (Q ,N ,M,). Ourobjective is to determine the optimum values (Q ,N ,M,), say (Q, N, M, ),so that the expected cost function is minimized. LetCh cost per unit time per failed machine in the system;C1 cost incurred per unit time for keeping one busy server;C2 cost incurred per unit time for keeping two busy server;C3 fixed cost for every repair rate of the working server.

LetF(Q ,N ,M,) be the expected cost per unit time. Using the definitions of eachcost element and its corresponding system characteristics, we have

F(Q ,N ,M,) =C0Ls+ C1E[B1] + C2E[B2] + C3E[I] + C4

L . (20)

It is indeed a difficult job to find the optimal values (Q, N, M, ) analyticallyfrom (20), as the expected cost function is complex and non-linear. A direct searchoptimization algorithm is used over a grid whose boundaries for decision variables(Q ,N ,M) are selected in order to guarantee the global optimum in the interiorregion. To search the optimal values (Q, N, M, ), it is necessary to have bounds

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for decision variables.

F(Q, N, M, ) = min3Q

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Table 1. Optimal values of for various values of.

F Ls E[O] E[B1] E[B2] M A OU 0.09 0.2692 13.4650 5.3095 4.6905 0.4302 1.1378 0.4690 0.78400.08 0.2612 13.2227 5.1277 4.8723 0.5026 0.9896 0.4872 0.74610.07 0.2545 12.9530 4.9028 5.0971 0.5840 0.8179 0.5097 0.70100.06 0.2471 12.6499 4.6383 5.3617 0.6635 0.6383 0.5362 0.65090.05 0.2329 12.3108 4.3739 5.6261 0.7193 0.4882 0.5626 0.6038

Table 2. Optimal values of for various values of from QFSM and SA.

F (QFSM) F (SA) Ls E[O] MA OU 0.09 0.2692 13.4650 13.4650 5.3095 4.6905 0.4690 0.78400.08 0.2612 13.2227 13.2227 5.1277 4.8723 0.4872 0.74610.07 0.2545 12.9530 12.9530 4.9029 5.0971 0.5097 0.70100.06 0.2471 12.6499 12.6499 4.6383 5.3617 0.5362 0.65090.05 0.2329 12.3108 12.3108 4.3739 5.6261 0.5626 0.6038

Table 3. Sensitivity analysis of expected cost for various values of.

M= 11 M= 12 M= 13 (Q, N) F (Q, N) F (Q, N) F

0.021 (3,4) 6.0199 (3,4) 5.8314 (3,4) 5.75700.023 (3,4) 6.6266 (3,4) 6.2963 (3,4) 6.14550.025 (6,7) 7.2669 (3,4) 6.8510 (3,4) 6.57740.027 (7,8) 7.7844 (8,9) 7.3680 (3,4) 7.05540.029 (7,8) 8.3960 (8,9) 7.7863 (8,9) 7.4888

service rates, the expectation of both the servers being busy is almost zero.

0.1 0.2 0.3 0.4 0.5 0.613.4

13.6

13.8

14

14.2

14.4

14.6

14.8

15

Expected

cost

C0= 10; C

1= 20; C

2= 30; C

3= 40; C

4= 80,

=0.09,

Minimum cost = 13.465

at = 0.26922

0.1 0.2 0.3 0.4 0.5 0.6 0.70

1

2

3

4

5

6

7

8

E[O]

E[I]

E[B1

]

E[B2

]

Figure 1. vs. cost Figure 2. vs. E[O], E[I],(q= 3, N = 5, M= 8, L= 10) E[B1] and E[B2]

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3 4 5 6 75.2

5.6

6

6.4

6.8

Q

Ls

= 0.5

= 0.45

= 0.4

=0.05, l = 15, N=8, M=11

4

6

8

10

11

12

13

14

5.6

5.7

5.8

5.9

6

6.1

NM

Ls

Figure 3. Effect ofQ on Ls Figure 4. Effect ofN and M on Ls

6

8

10

12

14

3

4

5

6

6

6.05

6.1

6.15

6.2

6.25

NQ

Ls

4

6

8

10

11

12

13

14

0.59

0.6

0.61

0.62

0.63

NM

Machineava

ilability

Figure 5. Effect ofQ and N on Ls Figure 6. Effect ofN and Mon MA

Figure 3 shows that the increasing trend of (Ls) with Q for different values. Fora fixed Q, however, as decreases, there is a considerable increase in Ls. Figure

4 provides the effect of M and Non the expected number of failed machines inthe system (Ls) for L = 15, Q = 3, = 0.055, = 0.5. The graph showing theincreasing trend of Ls with both M and N and more prominently so in case ofMand becomes steady at higher values. A similar observation is made in case ofFigure 5 where the effect ofQand N on Ls is shown.

For the same parameters as used for Figure 4, Figure 6 shows machine availabilityas a function ofN andM. We can observe that for a fixedM, asN increases thereis a decrease in MA and in case ofM increasing, MA slightly decreases first andthen remains steady.

6. Conclusion

In this paper, we consider a discrete-time machine repair model with L identicaloperating machines that are subject to breakdowns and are served by two remov-able servers under the triadic policy. The inter-arrival times of failed machines areindependent and geometrically distributed and the service times of both the serversare assumed to be independent and geometrically distributed. Using the recursivemethod, we have obtained the queue length distributions and further the variousperformance measures and cost analysis are considered. The parameter effect onthe performance of the system is analyzed using some numerical computations.The method of analysis used in this paper can be applied to bulk-arrival and bulk

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service Geo/Geo/1 models under different constraints of service patterns. Thesetopics are left for future investigations.

References

[1] Bell. C. E., Optimal operation of anM/M/2 queue with removable servers, Oper. Res., 28 (1980)11891204.

[2] Goswami. V and Mund. G. B., Optimal thresholds of an infinite buffer discrete-time two-serversystem with triadic policy, Int. J. Strategic Decision Sci. 2 (2011) 7689.[3] Goswami. V and Vijaya Laxmi. P.,Discrete-timeGI/DMSP/1/KQueue withNthreshold Policy,

International Journal of Mathematical Modelling & Computations, 3 (2013) 8394.[4] Gravey. A. and Hebuterne. G., Simultaneity in discrete time single server queues with Bernoulli

inputs, Perf. Eval., 14 (1992) 123131.[5] Haque. L and Armstrong. M. J., A survey of the machine interference problem, Eur. J. Oper. Res.,

179 (2007) 469482.[7] Hunter. J. J., Mathematical Techniques of Applied Probability, Volume II, Discrete-Time Models:

Techniques and Applications, Academic Press, New York, (1983).[6] Ke. J. C and Lin. C. H., Sensitivity analysis of machine repair problems in manufacturing systems

with service interruptions, App. Math. Modell., 32 (2008) 20872105.[7] Ke. J. C, Hsu. Y. L., Liu. T. H and Zhang. Z. G.,Computational analysis of machine repair problem

with unreliable multi-repairmen, Comput. Oper. Res., 40 (2013) 848855.[8] Lin. C. H and Ke. J. C.,Optimal operating policy for a controllable queueing model with a fuzzy

environment, J. Zhejiang Univ. Sci. Ed.. 10 (2009) 311318.[9] Lin. C. H and Ke. J. C.,Genetic algorithm for optimal thresholds of an infinite capacity multi-server

system with triadic policy, Expert Syst. Appl., 37 (2010) 42764282.[10] Liou. C. D, Wang. K. H and Liou. M. W.,Genetic algorithm to the machine repair problem with two

removable servers operating under the triadic (0,Q,N,M) policy, App. Math. Modell., 37 (2013)84198430.

[11] Lv. S. L, Li. J. B and Yue. D. Q., Unreliable multi-server machine repairable system with variablebreakdown rates, J. Chinese Institute Ind. Eng., 28 (2011) 400409.

[12] Rardin. R. L.,Optimization in Operations Research, Prentice Hall, New Jersey, (1997).[13] Rhee. H. K and Sivazlian. B. D., Distribution of the busy period in a controllableM/M/2 queue

operating under the triadic(0 ,K,N,M) policy, J. App. Prob., 27 (1990) 425432.[14] Stecke. K. E. and Aronson. J. E., Review of operator / machine interference models, Int. J. Prod.

Res.,23 (1985) 129151.[15] Tadj. L and Choudhury. G., Optimal design and control of queues, TOP 13 (2005) 359414.[16] Wang. K. H and Wang. Y. L.,Optimal control of anM/M/2 queueing system with finite capacity

operating under the triadic(0 ,Q,N,M) policy, Math. Method. Oper. Res., 55 (2002) 447460.[17] Wang. K. H.,Profit analysis of the machine repair problem with a single service station subject to

breakdowns, J. Oper. Res. Soc. 41 (1990) 11531160.[18] Wang. K. H., Cost analysis of the machine-repair problem with R non-reliable service stations,

Microelectron. Reliab., 35 (1995) 923934.[19] Wang. K. H and Kuo. M. Y., Profit analysis of theM/Ek/1 machine repair problem with a non-

reliable service station, Comput. Ind. Eng., 32 (1997) 587594.[20] Wang. K. H., Lai. Y. J and Ke. J. B., Reliability and sensitivity analysis of a system with warm

standbys and a repairable service station, Int. J. Oper. Res., 1 (2004) 6170.[21] Yadin. M. and Naor. P., Queueing system with a removable service station, Oper. Res., 14 (1963)

393405.

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Vol. 05, No. 01, Winter 2015, 41- 48

A Modified Steffensens Method with Memory for Nonlinear

Equations

F. Khaksar Haghani

Department of Mathematics, Shahrekord Branch, Islamic Azad University, Shahrekord,

Iran.

Abstract.In this work, we propose a modification of Steffensens method with some freeparameters. These parameters are then be used for further acceleration via the concept ofwith memorization. In this way, we derive a fast Steffensen-type method with memory forsolving nonlinear equations. Numerical results are also given to support the underlying theoryof the article.

Received: 9 July 2014, Revised: 9 August 2014, Accepted: 25 August 2014.

Keywords: Steffensens method, Iterative methods, Root, Order, With memory.


1 Introduction

2 Modified Steffensens Method

3 Modified Steffensens Method with Memory

4 Numerical Computations

5 Conclusion

1. Introduction

Finding the root of a nonlinear equation

f(x) = 0, (1)

where fis a sufficiently differentiable function in a neighborhood of a simple zero is a classical problem in scientific computing, [10].

A derivative-free family of methods proposed by Steffensen in [9] for solving (1)as follows (SM)

xk+1= xk f(xk)f[xk, wk]

, R\{0}, k= 0, 1, 2, , (2)

Corresponding author. Email: [email protected]


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42 F. khaksar Haghani/ IJM2C, 05 - 01 (2015) 41-48.

where wk = xk+ f(xk). Considering this iterative expression, many higher ordermethods with higher computational efficiency indices have been developed in theliterature (see e.g. [3], [4] and [7]).

The aim of this short communication is to state a one-step method with memoryof very high computational efficiency. We start from a modification of the one-steptwo-point method of Steffensen (2) without memory with order 2, and increasethe convergence order to 3.56155 without any additional function evaluation which

provides a high computational efficiency index.In general, the free nonzero parameter in Steffensen-like methods such (2) isof great importance for accelerating the R-order of convergence without additionalcalculations.

In this manner, we would like to obtain new methods for finding simple roots ofnonlinear equations, whose computational efficiency is higher than the efficiencyof existing methods known in literature in the class of one-step methods and evenhigher than the efficiency indices of optimal three- and four-step methods of orderseight and sixteen, respectively.

Recently, Dzunic in [2] designed an efficient one-step bi-parametric iterativemethod with memory (DM) possessing 12 (3 +

17) R-order of convergence as fol-

lows:

wk =xk+kf(xk),

k = 1N2(xk)

, pk =N

3 (wk)

2N3(wk), k 1,

xk+1= xk f(xk)f[xk, wk] +pkf(wk)

, k 0,

(3)

whereNj(l) stands for the Newtons interpolatory polynomial ofj-th order passingthroughj +1 nodes at the pointl. For example, we can defineN3(t) as the Newtonsinterpolation polynomial of third degree, set through four available approximations

xk, wk, wk1, xk1.It should be remarked that the notation of divided difference is used frequently

in this study.Here, we aim at presenting a simple quadra-parametric modification of Stef-

fensens method without memory and make it with memory. Higher order of con-vergence is attained without additional function evaluations, making the derivedmethod very efficient. Numerical examples are also given to demonstrate excellentconvergence features of the presented method with memory.

The rest of this paper is organized as follows. In Section 2, a modification of (2)without memory is given possessing quadratic convergence. The main goal of thispaper is presented in Section 3 by contributing an iterative method with memory.

The proposed scheme is an extension over (2) and has a simple structure with anincreased computational efficiency. Its efficiency index is the same to (3). In Section4, numerical reports are stated. Some discussions will be given in Section 5 to endthe paper.

2. Modified Steffensens Method

In order to modify (2) and have as much as possible of free parameters, we firstapply the backward finite difference approximation in the denominator of (2) and

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F. khaksar Haghani/ IJM2C, 05 - 01 (2015) 41-48. 43

then introduce a weight function in what follows (MSM):

wk = xkf(xk), R\{0}, k 0,xk+1= xk f(xk)

f[xk, wk] +qf(xk) +pf(wk)

1 + f(wk)

f[xk,wk]

, q, p, R. (4)

Theorem 2.1 Let the function f(x) be sufficiently differentiable in a neighbor-hood of its simple zero . If an initial approximation x0 is sufficiently close to .Then, the order of convergence of the method (4) without memory is two with fourparameters.

Proof Introducing the notations ci = 1i!fi()f()

, df a = f(), e1 = xk+1, e =xk, ew = wk , and t = f(wk)/f[xk, wk], we can provide the final errorequation of (4) by applying the following piece of code written in the symbolicpackage of Mathematica:

ClearAll["Global*"]

f[e_] := dfa (e^1 + c2 e^2 + c3 e^3 + c4 e^4)fe = f[e]; f1e = f[e];

ew = e - \[Beta] fe; fw = f[ew];

FD1 = (fe - fw)/(e - ew); t = fw/FD1;

e1 = e - Series[fe/( FD1 + q fe + p fw),

{e, 0, 2}] (1 + \[Zeta] t) //FullSimplify

This gives the following error equation

ek+1 = (c2+p+qc2f()+f()(p+))e2k+O(e3k). (5)

The relation (5) shows the quadratic convergence of (4). Now, the proof is complete.

In the next section, we aim at accelerating convergence without imposing furtherfunctional evaluations per cycle. This means that using two function evaluations,we must increase the R-order of convergence more than two [5]. This is possible byapproximating the free parameters involved in (4). The best method for our recentgoal would be the following bi-parametric case of (4)

wk = xkf(xk), R\{0}, k 0,xk+1= xk f(xk)

f[xk, wk]

1 + f(wk)f[xk,wk]

, R, (6)

where it reads

ek+1=(1 +f())(c2)e2k+O(e3k). (7)

Note that the effect of the other two parameters p and q could be ignored byconsidering the other two parameters.

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3. Modified Steffensens Method with Memory

Now we try to construct a method with memory consists of the calculation ofparameters

= k, (8)

and

= k, (9)

as the iteration proceeds by the formulas k = 1

f() and k = c2, fork = 1, 2, ,

where f() and c2 are approximations to f() and c2, respectively.As a matter of fact, in this way we minimize the factors1 + f()k andc2 k

that appear in (7). Hence, we present the following scheme (PM)

wk = xkkf(xk),

k =

1

N2(xk) , k =

N3 (wk)

2N3(wk) , k 1,

xk+1= xk f(xk)f[xk, wk]

1 +k

f(wk)f[xk,wk]

, k 0.

(10)

The two self-accelerating parameters, i.e., k and k are calculated using infor-mation available from the current and previous iterations. Moreover, it is assumedthat initial estimates0 and 0 should be chosen before starting the iterative pro-cess.

Theorem 3.1 Let the function f(x) be sufficiently differentiable in a neighbor-hood of its simple zero . If an initial approximation x0 is sufficiently close to .Then, the R-order of convergence of the one-step method (10) with memory is at

least 12 (3 + 17).Proof. Let{xk} be a sequence of approximations generated by (10). We first

must find the asymptotic error constants for the two self-accelerator parameters.

Following the substitutionsk = 1

N2(xk) andk =

N3 (wk)

2N3(wk), and similar notations

as in Theorem 2.1, we could write the following two pieces of Mathematica codes:

ClearAll["Global*"]

A[t_]:= InterpolatingPolynomial[{{e, fx}, {ew, fw}, {e1, fx1}}, t]

Approximation = 1/A[e1] // Simplify;

fx = f1a*(e + c2*e^2 + c3*e^3 + c4*e^4);

fw = f1a*(ew + c2*ew^2 + c3*ew^3 + c4*ew^4);

fx1 = f1a*(e1 + c2*e1^2 + c3*e1^3 + c4*e1^4);

b = Series[Approximation, {e, 0, 2}, {ew, 0, 2},

{e1, 0, 0}] //Simplify;

Collect[Series[-1 + b*f1a, {e, 0, 1}, {ew, 0, 1},

{e1, 0, 0}], {e, ew, e1}, Simplify]

which results in

1 +kf()c3ek1ek1,w, (11)

and also

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ClearAll["Global*"]

A[t_] :=InterpolatingPolynomial[{{e, fx}, {ew, fw},

{e1, fx1}, {ew1, fw1}}, t]

Approximation = A[ew1]/(2 A[ew1]) // Simplify;

fx = f1a*(e + c2*e^2 + c3*e^3 + c4*e^4);

fw = f1a*(ew + c2*ew^2 + c3*ew^3 + c4*ew^4);

fx1 = f1a*(e1 + c2*e1^2 + c3*e1^3 + c4*e1^4);

fw1 = f1a*(ew1 + c2*ew1^2 + c3*ew1^3 + c4*ew1^4);h = Series[Approximation, {e, 0, 2}, {ew, 0, 2},

{e1, 0, 0}, {ew1, 0, 0}] //Simplify;

Collect[Series[c2 - h, {e, 0, 1}, {ew, 0, 1}, {e1, 0, 0},

{ew1, 0, 0}], {e, ew, e1, ew1}, Simplify]

which results in

c2kc4ek1ek1,w, (12)

wherein ew = ek1,w, e = ek1, e1 =ek. Therefore, one may obtain

ek+1 (c3ek1ek1,w)(c4ek1ek1,w)e2k. (13)

We also have ek1,w(1 +k1f())ek1. So, we attain

ek+1 (c3c4e2k1)((1 +k1f())ek1)2e2k. (14)

and

ek+1 c3c4(1 +k1f())2e4k1e2k. (15)

Note that in general we know that the error equation should read ek+1

Aepk,

whereA and p are to be determined. Hence, one has ekAepk1, and subsequently

ek1A1/pe1/pk . (16)

Hence, we firstly have

ek+1 c3c4(1 +k1f())2

ek(1 +k1f())(c2k1)

e2k1e

2k. (17)

Thus, it is easy to obtain

epkA2/pCe3+2

p

k , (18)

wherein

C=c3c4(1 +k1f())2

1

(1 +k1f())(c2k1)

. (19)

This results in the equation p = 3 + 2p , with two solutions{12(

3 17),12

(3 +

17)}. Clearly the value for p = 12 (3 + 17) 3.56155 is acceptable and

would be the convergence R-order of the method (10) with memory. The proof is

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complete.

We emphasize that the increase of the convergence order is obtained without anyadditional function evaluations, which points to very high computational efficiency.

The computational efficiency index of (10) is 3.561551

2 1.8872, which is sameto (3), and is clearly much higher than 2

1

2 1.4142 of (2) and (4), 8 14 1.6817of optimal eighth-order method [6] and 16

1

5 1.7411 of optimal sixteenth-orderschemes.

4. Numerical Computations

In this section we compare the behavior of different methods for solving an examplein programming package Mathematica [11]. Numerical experiments have been per-formed with 1500 precision digits, being large enough to minimize round-off errorsas well as to clearly observe the computed asymptotic error constants requiringsmall number divisions. In fact, it clearly shows the high computational R-order ofthe proposed method.

We compare the methods SM using = 0.1, MSM using = 0.1, p = q= 1/4,= 0, DM and PM. All experiments have been carried out on a personal computerequipped with an AMD 3.1 GHz dual-core processor and Windows 32-bit XPoperating system.

In the meantime, the computational order of convergence (coc) has been com-puted by

= ln |f(xk)/f(xk1)|ln |f(xk1)/f(xk2)| . (20)

Here the stop termination is|f(xk)| 10250. We have used 0 = 0 = 0.1whenever required.

Example 4.1 We consider the following nonlinear test function

f(x) = (x2 tan(x))(x3 8), (21)

where = 2. The results are provided in Tables 1-2 for two different initial ap-proximations.

The values of initial guess x0 were selected close to to guarantee convergenceof iterative methods.

Tables 1-2 evidently show that proposed scheme (10) exhibits 3.56 convergence.Iterative scheme (10) is evidently believed to be more favorable than other listed

methods due to its fast speed and acceptable accuracy.We remark that convergence behavior was verified for additional test functions

and we attained similar superiority for our proposed Steffensen-like method withmemory.

5. Conclusion

For the first time, iterative methods with memory for solving nonlinear equations,that use information from the current and previous iterate, were considered in [10].After Traubs research this class of methods was studied seldom in the literature

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Table 1. Results of comparisons for Example 4.1 by x0 = 1.7.

Methods |f(x3)| |f(x4)| |f(x5)| |f(x6)| SM 4.1583 3.0743 1.4436 0.25430 2.MSM 23.499 18.452 12.275 0.60559 2.DZ 0.13132 2.0026107 1.01811027 5.17311099 3.57002PM 1.8921106 4.58641024 1.05691088 7.526910318 3.54512

Table 2. Results of comparisons for Example 4.1 by x0 = 1.92.

Methods |f(x3)| |f(x4)| |f(x5)| |f(x6)| SM 0.032743 0.00010819 1.1761109 1.38981019 2.MSM 0.0018889 2.9274107 7.02851015 4.05161030 2.DZ 0.041691 5.5105108 8.44571032 5.217710115 3.57209PM 1.44251015 1.37311057 1.632210207 2.484810741 3.56056

in spite of its capability to reach high computational efficiency. Recent results pub-

lished in [2] and [3] showed considerably high computational efficiency of methodswith memory using new accelerating techniques based on varying free parameterscalculated by interpolating polynomials in each iteration.

For these reasons, in this paper we have constructed a family of Steffensen-type methods without memory possessing quadratic convergence with four freeparameters. Then, by using suitable accelerators we have constructed a methodwith memory possessing a high computational efficiency index.

Numerical results have also been provided to support the theoretics given inSections 2-3. Observing the analytical and numerical results, one may concludethat the proposed variant of Steffensens method with memory is an efficient toolfor solving nonlinear equations.

The application of the developed method for matrix problems (such as the onesin [8]) and its dynamical studies (see e.g. [1]), will be pursued in future works.

Acknowledgements

The author sincerely thanks Islamic Azad University, Shahrekord Branch,Shahrekord, Iran for financial support.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publicationof this paper.

References

[1] A. Cordero, F. Soleymani and J. R. Torregrosa, Dynamical analysis of iterative methods for nonlinearsystems or how to deal with the dimension?, Applied Mathematics and Computation, 244 (2014)398-412.

[2] J. Dzunic, On efficient two-parameter methods for solving nonlinear equations, Numer. Algor. 63(2013) 549-569.

[3] T. Lotfi and E. Tavakoli, On a new efficient Steffensen-like iterative class by applying a suitableself-accelerator parameter, The Scientific World Journal, Article ID 769758, 9 pages.(2014).

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[4] F. Mirzaee and A. Hamzeh, A sixth order method for solving nonlinear equations, InternationalJournal of Mathematical Modelling & Computations,04 (2014) 55-60.

[5] J. M. Ortega and W. C. Rheinbolt, Iterative Solution of Nonlinear Equations in Several Variables,Academic Press, New York, (1970).

[6] F. Soleymani, An optimally convergent three-step class of derivative-free methods, World AppliedSciences Journal, 13 (2011) 2515-2521.

[7] F. Soleymani, M. Sharifi, S. Shateyi and F. Khaksar Haghani, A class of Steffensen-type iterativemethods for nonlinear systems, Journal of Applied Mathematics,9 Pages, (2014).

[8] F. Soleymani, M. Sharifi, S. Karimi Vanani, F. Khaksar Haghani and A. Kilicman, An inversion-freemethod for finding positive definite solution of a rational matrix equation, The Scientific WorldJournal, Article ID 560931, 5 pages,(2014).

[9] J. F. Steffensen, Remarks on iteration, Skand. Aktuarietidskr, 16 (1933) 64-72.[10] J. F. Traub, Iterative Methods for the Solution of Equations, Prentice Hall, New York, (1964).[11] S. Wagon, Mathematica in Action, Third edition, Springer, New York, NY, USA, (2010).

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Vol. 05, No. 01, Winter 2015, 49- 67

A Non-Markovian Batch Arrival Queue with Service Interruption

& Extended Server Vacation

K. Sathiya a, and G. Ayyappanb

aDepartment of Mathematics, Rajiv Gandhi Arts and Science College, Pondicherry,

India;bDepartment of Mathematics, Pondicherry Engineering College, Pondicherry, India.

Abstract. A single server provides service to all arriving customers with service time followinggeneral distribution. After every service completion the server has the option to leave forphase one vacation of random length with probability p or continue to stay in the systemwith probability 1p. As soon as the completion of phase one vacation, the server may takephase two vacation with probability q or to remain in the system with probability 1 q,after phase two vacation again the server has the option to take phase three vacation withprobabilityr or to remain in the system with probability 1r. The vacation times are assumedto be general. The server is interrupted at random and the duration of attending interruptionfollows exponential distribution. Also we assume, the customer whose service is interruptedgoes back to the head of the queue where the arrivals are Poisson. The time dependentprobability generating functions have been obtained in terms of their Laplace transforms andthe corresponding steady state results have been obtained explicitly. Also the mean numberof customers in the queue and system and the waiting time in the queue and system are alsoderived. Particular cases and numerical results are discussed.

Received: 14 February 2014, Revised: 3 July 2014, Accepted: 17 August 2014.

Keywords:Batch arrival queue, General service, Three phases of vacation, Serviceinterruption, Mean queue size, Mean waiting time.


1 Introduction

2 Mathematical Description of the Model

3 Definitions and Equations Governing the System

4 Generating Functions of the Queue Length: The Time-Dependent Solution5 The Steady State Results

6 The Average Queue Size

7 The Average Waiting Time

8 Particular Cases

9 Numerical Results

10 Conclusion

Corresponding author. [email protected]


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50 K. Sathiya& G. Ayyappan/ IJM2C, 05 - 01 (2015) 49-67.

1. Introduction

Queueing system are powerful tool for modeling communication networks, trans-portation networks, production lines, operating systems, etc. In recent years, com-puter networks and data communication systems are the fastest growing technolo-gies, which lead to glorious development in many applications. For example, theswift advance in internet, a

ijm2c 2015 - vol 5, no 1, sn 17

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