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A New Linear Programming Method for Weights Generation and Group Decision Making in the Analytic Hierarchy Process

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This paper proposes a new linear programming method entitled by LP-GW-AHP for weights generation in analytic hierarchy process (AHP) which employs concepts from data envelopment analysis. We propose four specially constructed linear programming (LP)
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   International Journal of Astronomy and Astrophysics , 2011, 1, 66-71 doi:10.4236/ijaa.2011.12010 Published Online June 2011 (http://www.scirp.org/journal/ijaa)Copyright © 2011 SciRes.  IJAA   An Approximation Algorithm for the Solution of Astrophysics Equations Using Rational ScaledGeneralized Laguerre Function CollocationMethod Based on Transformed Hermite-Gauss Nodes Ali Pirkhedri 1* , Parisa Daneshjoo 1 , Hamid Hai Seied Javadi 2 , Hamid Navidi 2 , Salem Khodamoradi 3 ,Kamal Ghaderi 3   1  Department of Computer Engineering ,  Islamic Azad University , Science and Research Branch , Tehran ,  Iran 2  Department of Applied Mathematics And Computer Sciences , Shahed University , Tehran ,  Iran 3  Department of Science ,  Islamic Azad University ,  Branch of Marivan ,  Marivan ,  Iran E-mail : alipirkhedri @ gmail.com Received March 12, 2011; revised April 28, 2011; accepted May 5, 2011   Abstract In this paper we propose a collocation method for solving Lane-Emden type equation which is nonlinear or-dinary differential equation on the semi-infinite domain. This equation is categorized as singular initial valueproblems. We solve this equation by the generalized Laguerre polynomial collocation method based onHermite-Gauss nodes. This method solves the problem on the semi-infinite domain without truncating it to afinite domain and transforming domain of the problem to a finite domain. In addition, this method reducessolution of the problem to solution of a system of algebraic equations. Keywords: Lane-Emden Equation, Generalized Laguerre Functions, Collocation Method, Hermite-Gauss Nodes,Nonlinear ODE, Semi-Infinite 1. Introduction There are many problems in science and engineeringarising in unbounded domains.Spectral methods are famous ways to solve these kindsof problems. The most common approach on spectralmethods, that is used in this paper too, is through the useof functions that are orthogonal over unbounded domains,such as the Hermite and the Laguerre functions [1-8].The second approach is reformulating the srcinalproblem in the semi-infinite domain to a singular prob-lem in a bounded domain by variable transformation andthen using the Jacobi polynomials to approximate theresulting singular problem [9-11]. A third approach of spectral method is based on rational orthogonal functions,for example, Christov [12] and Boyd [13,14] developedsome spectral methods on unbounded intervals by usingmutually orthogonal systems of rational functions. Boyd[14] defined a new spectral basis, named rational Che-byshev functions on the semi-infinite interval, by map-ping it to the Chebyshev polynomials. Guo et al . [15]proposed and analyzed a set of Legendre rational func-tions which are mutually orthogonal in ( ) 2 0,  L χ ∞ witha non-uniform weight function ( ) ( ) 2 =1  xx χ − + . A forthapproach is replacing the semi-infinite domain with [0,]  L interval by choosing L, sufficiently large, thismethod is named as the domain truncation [16].In this paper, we investigate the Generalized Laguerre-collocation method based on Hermite-Gauss Nodeswhich is another approach for solving ODEs on the half line. In [7] proposed spectral methods using Laguerrefunctions and analyzed for model elliptic equations onregular unbounded domains. It is shown that spec-tral-Galerkin approximations based on Laguerre func-tions are stable and convergent with spectral accuracy inthe Sobolev spaces. Siyyam [8] applied two numericalmethods for solving initial value problem differentialequations using the Laguerre Tau method. He generatedlinear systems and solved them. Maday, et al. [6] pro-posed a Laguerre type spectral method for solving partialdifferential equations. They introduced a general presen-tation of the method and a description of the derivationdiscretization matrices and then determined the optimumestimations in the adapted Hilbert norms.  A. PIRKHEDRI  ET AL .   Copyright © 2011 SciRes.  IJAA  67 2. The Lane-Emden Equation This equation is one of the basic equations in the theoryof stellar structure and has been the focus of many stud-ies [17-21]. This equation describes the temperaturevariation of a spherical gas cloud under the mutual at-traction of its molecules and subject to the laws of clas-sical thermodynamics. The polytropic theory of starsessentially follows out of thermodynamic considerations,that deal with the issue of energy transport, through thetransfer of material between different levels of the star.We simply begin with the Poisson equation and the con-dition for hydrostatic equilibrium: ( ) 2 d=,d GMr Pr r  − (1) ( ) 2 d=4,d  Mr r r  πρ (2)Where G is the gravitational constant, P is thepressure, ( )  Mr  is the mass of a star at a certain radius r  , and ρ is the density, at a distance r  from the cen-ter of a spherical star. Combination of these equationsyields the following equation, which as should be noted,is an equivalent form of the Poisson Equation. 22 1dd=4 π .dd rPGrr r  ρρ   −     (3)From these equations one can obtain the Lane-Emdenequation through the simple supposition that the densityis simply related to the density, while remaining inde-pendent of the temperature. We already know that in thecase of a degenerate electron gas that the pressure anddensity are 35 P ρ : , assuming that such a relation existsfor other states of the star we are led to consider a rela-tion of the following form: 11 =, m PK  ρ + (4)Where K  and m are constants, at this point it is impor-tant to note that m is the polytropic index which is relatedto the ratio of specific heats of the gas comprising thestar. Based upon these assumptions we can insert thisrelation into our first equation for the hydrostatic equi-librium condition and from this rewrite equation to: ( ) 1122 11dd=,4dd mm Km yryGrr r  λπ −  +  −       (5)Where the additional alteration to the expression fordensity has been inserted with λ representing the cen-tral density of the star and  y that of a related dimen-sionless quantity that are both related to ρ through thefollowing relation=. m  y ρ λ (6)Additionally, if place this result into the Poisson equa-tion, we obtain a differential equation for the mass, witha dependance upon the polytropic index m . Though thedifferential equation is seemingly difficult to solve, thisproblem can be partially alleviated by the introduction of an additional dimensionless variable  x , given by the fol-lowing:=, rax (7) ( ) 1121 1=.4 m KmaG λπ −  +   (8)Inserting these relations into our previous relations weobtain the famous form of the Lane-Emden equation,given below: 22 1dd=.dd m  y xy xx x   −     (9)Taking this simple relation we will have the Lane-Emden equation:2=0,>0. m  yyyx x ′′ ′+ + (10)At this point it is also important to introduce theboundary conditions, which are based upon the followingboundary conditions for hydrostatic equilibrium, andnormalization consideration of the newly introducedquantities  x and  y . What follows for0 r  = is: ( ) =0=0,=0=1 rxy ρ λ → → (11)As a result an additional condition must be introducedin order to maintain the condition of Equation (11) si-multaneously: ( ) 0=0.  y ′ (12)In other words, the boundary conditions are as follow: ( ) ( ) 0=1,0=0.  yy ′ (13)Physically interesting value of  m lie in the interval [0,5]. Exact soloution for Equation (??) are known only for=0,1 m and 5. For other value of  m the Lane-Emdenequation is to be integrated numerically. In this paper, wesolve it for =1.5,2,2.5,3 m and 4.This paper is arranged as follows: in Section 3, we ex-plain the formulation of rational scaled generalized La-guerre polynomials and Hermite functions required forour subsequent development. In Section 4, we summa-rize the application of the method for solvingLane-Emden equation and compare it with the existingmethods in the literature. Finally we give a brief conclu-sion in the last section.  A. PIRKHEDRI  ET AL .Copyright © 2011 SciRes.  IJAA  68 3. Rational Scaled Generalized LaguerrePolynomials and Hermite FunctionsProperties This section is devoted to the introduction of the basicnotions and working tools concerning orthogonal rationalscaled generalized Laguerre polynomials and later wepresent some properties of Hermite function and Her-mite-Gauss nodes. 3.1. Properties of Rational Scaled GeneralizedLaguerre Polynomials   ( ) n  Lx α (generalized Laguerre polynomial) is the n theigenfunction of the Sturm-Liouville problem [2,22,23]: ( ) ( ) ( ) ( ) 22 1=0,dd nnn dd  xLxxLxnLx x x α α α α + + − +   [ ) 0,,=0,1,2,...  xn ∈ ∞ (14)The generalized Laguerre polynomials are definedwith the following recurrence formula: ( ) ( ) ( ) 01 =1,=1,  LxLxx α α α + −   ( ) ( ) ( ) ( ) ( ) 12 =211, nnn nLxnxLxnLx α α α α α − − − + − − + −   2,>1 n α ≥ − (15)With the normalizing condition: ( ) 0=. n n Ln α α +       (16)Let () wx denotes a non-negative, integrable, real-valued function over the interval ( ) 0, ∞ , we define ( ) { } 2 =:0,v is measurable and<, ww  LvRv ∞ → ∞   (17)Where ( ) ( ) () 1220 =||d, w vvxwxx ∞ ∫  (18)Is the norm induced by the inner product of the space 2 w  L , ( ) ( ) ( ) 0 <,>=d. w uvuxvxwxx ∞ ∫  (19)These are orthogonal polynomials for the weight func-tion ( ) =  x wxxe αα − where nm δ is the Kronecker deltafunction.   ( ) ( ) ( )( ) 0 1=.! nmnm na LxLxwxn α αα δ ∞ Γ + +       ∫   Let1  N  ≥ be an integer and we define ,  jN   x α ,  j =0, … ,  N  - 1 to be zeroes of   N  d  Ldx α and the point  x = 0. Itcan be shown that , >0  jN   x α ,  j = 0, … ,  N  - 1 and the cor-responding weights are: ( ) ( )( )( ) 20, 111!=1  N   N w N  α α αα + Γ + −Γ + +   ( ) ( ) ( ) 1,,1, =,!=1,2,...,1.  jNNjNNjN   N d wLxLx Ndx jN  α α α α α α −− Γ +      −  The following quadrature formula is known: ( ) ( ) ( ) 1,,0=0 d=  N  NjNjN  j  fxwxxfxw α α −+∞ ∑∫    ( ) ( )( ) 21 (1),0<<!2!  N   N  f  NN  αξ ξ −   Γ + ++ ∞       (20)In particular, the second term on the right hand sidevanishes when  f  is a polynomial of degree at most22  N  − . For convenience, we shall set , =  jNj  xx α and , =  jNj ww α .Pseudospectral approximations in unbounded domainsby Laguerre polynomials lead to ill-conditioned algo-rithms and [24] introduced a scaling function and appro-priate numerical procedures in order to limit these un-pleasant phenomena.We define scaled Laguerre functions { } n l as fol-lows: ( ) ( ) ( ) ( ) 0 ;=1,;=//,=1,2,..... nnn  xkxkSxkLxk n α α α l l (21)Where>0 k  is a constant and ( ) 1 n  Lx is generalizedLaguerre polynomials for=1 α and ( ) n Sx is definedas follows: ( ) ( ) ( ) ( ) 10=1 =1,=11/4,=1,2,.... nnt  SxSxnxt nn −   + +       ∏ (22)We denote scaled laguerre functions with (SLF) .Boyd [26] offered guidelines for optimizing the mapparameter k  where>0 k  is the scaling parameter .From Equations. (21), (22) for=1 α we obtain the fol-lowing formula: ( ) ( )( )( ) 1101 42;=1,;=,24 kx xkxk  xk  −+ l l   ( )( )( )( ) ( ) 111 4;=[2/;14/  nn n xknxkxk nnxk  − −+ + l l   ( )( ) 212 41;],2,4/4 n n xknnxk  − −− ≥+ − l (23)This system is an orthogonal basis with weight func-  A. PIRKHEDRI  ET AL .   Copyright © 2011 SciRes.  IJAA  69 tion ( )( )  / 2 = /   xk  N   xewxkSxk  − [24].Some of the relations of scaled Laguerre functions andgeneralized Laguerre-Gauss-type interpolation were in-troduced by [24,25]. It is convenient to define theweights of  ( ) ; n  xk  l as follows: ( )( )( ) 00,2 22==!0  N  N   N ww NN S − Γ +   ( ) ,2 ==  j jN  Nj wwSx −   ( )( ) ( ) 11112=1 4(2)d1,d44(!)  N  j NjNjm j Nj  Nx N  xx xmx NN  x −−− −   +Γ ++    +   ∑ l ll =1,2,...,1.  jN  − (24)Where we noted that, for2 n ≥   ( ) ( ) [ ) 11111=1 d1[]=[],=0,d4  N  NN m j SxSxxI  xmx −− −− − ∈ ∞+ ∑   3.2. Properties of Hermite Functions The Hermite function is defined for allR  x ∈ and canbe written in recursive formula as follows [27-29]: ( ) ( ) ( ) 11 2=,111 nnn n HxxHxHxnnn + − − ≥+ +   ( ) ( ) 222201 =,=2  xx  HxeHxxe − − (25) { } n  H  is an orthogonal system in ( ) 2 R  L , i.e., ( ) ( ) , =, nmnm  HxHxdx πδ +∞−∞ ∫  (26)Where , nm δ is the Kronecker delta function. Usingthe recurrence relation of Hermite polynomials and theabove formula leads to ( ) ( ) ( ) 1 =2 nnn  HxnHxxHx − ′−   ( ) ( ) 11 122 nn nn HxHx − + +−  Let us define 22 =:=,  x NN  PuuevvP −   ∀ ∈    (27)Where  N  P is the set of all Hermite polynomials of degree at most  N  . We now introduce the Gauss quad-rature associated with the Hermite functions approach.Let { } =0  N  j j  x be the Hermite-Gauss nodes which can be  N  + 1 roots of  1  N   H  + and define the weights ( ) ( ) 2 =,0.1  jnj wjN nHx π ≤ ≤+ (28)Then we have: ( ) ( ) 21=0 =,.  N  jjN  j  pxdxpxwpP + ∀ ∈ ∑∫  R   4. Solving the Lane-Emden Equation To apply rational scaled generalized Laguerre colloca-tion method to the standard Lane-Emden Equation in-troduced in Equation (10) with boundary conditions Eq-uation (13), we define ( ) ( ) =0 =;.  N  Njj j  yxcxk  α ζ ∑ l (29)And we construct the residual function by substituting ( )  yx by ( )  N   yx ζ in the Lane-Emden Equation (10): ( ) ( ) ( ) ( ) ( ) 22 =2. m NNN  dd  Resxxyxyxxyxdxdx ζ ζ ζ + +  (30)To find the unknown coefficients { } =0  N  j j c 's, we equ-alize ( )  Resx to zero at1  N  − Hermite-Gauss Nodes { } 2=0  N  j j  x − and we equalize boundary conditions in Equa-tion (13) too, therefore we have: ( ) =0,=02  j  ResxjN  − K   ( ) ( ) 0=1,0=0.  NN  d  yydx ζ ζ (31)But as mentioned before Lane-Emden equations aredefined on the interval ( ) 0, ∞ and we know propertiesof Hermite functions are derived in the infinite domain ( ) , −∞ ∞ . Also we know approximations can be con-structed for infinite, semi-infinite and finite intervals. Oneof the approaches to construct Hermite-Gauss nodes onthe interval ( ) 0, ∞ which is used in the current paper, isto use a mapping, that is a change of variable of the form: ( ) 2 ==ln1  x x j jjj  zxee ϕ   + +     (32)Where is the inverse map of  ( ) ( ) ( ) 1 ==lnsinh  jjj  xzz ϕ − (33)Finally to find the unknown coefficients { } =0  N  j j c 's,we have: ( ) =0,=0..2  j  ReszjN  − (34) ( ) ( ) 0=1,0=0  NN  d  yydx ζ ζ (35)Where  z  j is transformed root of  ( ) 1  N   Hx − . Equations.  A. PIRKHEDRI  ET AL .Copyright © 2011 SciRes.  IJAA  70 (34), (35) gives  N  + 1 nonlinear algebraic equationswhich can be solved for the unknown coefficients  j c byusing the well known Newton's method by Maple pro-gramming and we use=0,=01  j cjN  + K  as starting points to obtain convergence of the method,consequently, ()  yx given in Equation. (10) can be cal-culated.The resulting graph of Lane-Emden equation obtainedby present method for =1.5,2,2.5,3 m and 4 is shownin Figure 1 . Tables 1 shows the comparison of the first zero of  ( )  N   yx ζ , between Pad é approximation used by [17] andthe present method for=1 α and =1.5,2,2.5,3 m and4 respectively. Table 2 and 3 show the approximations of  ( )  N   yx ζ  for standard Lane-Emden with =2.5,3 m obtained bythe rational scaled generalized Laguerre collocation me-thod for=1 α and those obtained by Horedt [20]. Table 4 shows the coefficients of rational scaled gen-eralized Laguerre functions obtained by present methodfor the Lane-Emden equation with various m . 5. Summary and Conclusions   A set of rational scaled generalized laguerre orthogonalfunctions are proposed to solve Lane-Emden equationwhich is defined in the semi-infinite interval and has sin-gularity at  x = 0, by collocation method. But to approxi-mate unknown function in Lane-Emden equation by col-location method we must equalize ( )  Resx to zero atsuitable points in ( ) 0, ∞ interval. Since Hermite func-tions are derived in the infinite domain ( ) , −∞ ∞ , we can'tapply Hermite-Gauss nodes for equalizing ( )  Resx tozero, therefore we transform this nodes from (,) −∞ ∞ to ( ) 0, ∞ interval by a mapping ( ) ( ) 2 =ln1  xx  xee ϕ + + . Figure 1. Lane-Emden equation graph obtained by presentmethod for various  m.  Table 1. Comparison the first zero of   y (  x ), between Pad é  approximation used by [17] and the rational scaled gener-alized Laguerre collocation method for various  m and  a = 1. m N k  Present method Bender[17] Exact value [20]1.5 4 0.83200 3.653710928 - 3.653753742 4 0.52990 4.352761605 4.3603 4.352874602.5 4 0.39101 5.355236655 - 5.355275463 6 0.29354 6.896951110 7.0521 6.896848624 5 0.10022 1497174961 17.967 14.9715463 Table 2. Approximation of   y (  x ) obtained by present methodand solutions of Horedt [20] for  m = 2.5 and  a = 1,  N  = 4.  x Present method Solutions of Horedt [20]0.000 1.000000 1.0000000.100 0.998749 0.9983350.500 0.960372 0.9599781.000 0.851467 0.8519445.000 0.024267 0.0290195.355 0.000005 0.000021 Table 3. Approximation of   y (  x ) obtained by present meth-od and solutions of Horedt [20] for  m = 3 and  a = 1,  N  = 6.  x Present method Solutions of Horedt [20]0.000 1.000000 1.0000000.100 0.998293 0.9983360.500 0.959837 0.9598391.000 0.854839 0.8550585.000 0.110415 0.1108206.000 0.048132 0.0437386.800 0.004952 0.0041686.896 0.000048 0.000036 Table 4. Coefficients of the rational scaled generalized Lag-uerre functions of the Lane-Emden equation for various  m . c i  I m = 1.5 m = 2 m = 2.5 m = 3 m = 40 0.5552687 0.7618023 0.8553158 0.9108075 0.97863091 0.7098163 0.5800179 0.3986864 0.2657800 0.04839112 0.5796016 0.2517954 0.2748482 0.2308814 0.05250973 1.4222221 0.2455873 0.0468485 0.0164809 0.04559874 0.5775354 0.1555624 0.0676945 0.0448774 0.01521425 - - - 0.0032424 0.00489686 - - - - 0.0103069 Through the comparisons among the exact solutions of Horedt and the approximate solutions of Bender and thecurrent work, it has been shown that the present work hasprovided more exact solutions for Lane-Emden equations. 6. References [1]   D. Funaro and O. Kavian, “ Approximation of Some Dif-fusion Evolution Equations in Unbounded Domains byHermite Functions, ”    Mathematics of Computation , Vol.57, No. 196, 1991, pp. 597-619.doi:10.1090/S0025-5718-1991-1094949-X [2]   O. Coulaud, D. Funaro and O. Kavian, “ Laguerre Spec-tral Approximation of Elliptic Problems in Exterior Do-mains, ”   Computer Methods in Applied Mechanics and 
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