Numerical Approximation Of Generalised Riccati Equations By Iterative Decomposition Method by ijcsiseditor

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									                                                     (IJCSIS) International Journal of Computer Science and Information Security,
                                                     Vol. 9, No. 6, June 2011




           Numerical Approxima
           N         A                            Riccati Equ
                             ation of Generalised R         uations by Iterative
                                Decomp position Meethod
                                                 Kadiri, K.O.2, Odetunde, O.S.3
                                  Adekun Y.A.1*, K
                                       nle,
    1
           artment of Com
        Depa                         e           atics, Babcock University, Ilis
                        mputer Science and Mathema                                        un           ria
                                                                               san-Remo Ogu State, Niger
            dekunleya@g
           ad         gmail.com
    2
            partment Electr
        . Dep                            cs                         technic, Offa, K
                          rical/Electronic Engineering, Federal Polyt                           Nigeria
                                                                                   Kwara State, N
            adiritoyin2007@yahoo.c
           ka                    com
    3
            partment of Ma
        . Dep                          ience, Olabisi O
                         athematical Sci                          versity, Ago-Iw
                                                      Onabanjo Univ                          ate,Nigeria
                                                                                woye, Ogun Sta
            odetunde@ya
           to         ahoo.co.uk


                            I              position Method
Abstract-In this paper, the Iterative Decomp             d          appli in [7] to s
                                                                        ied         solve one-dime            rmonic
                                                                                                 ensional Bihar
           to            neralized Riccat equations. We
is applied t solve the gen              ti            e                 tions. In [8] t method w applied to solve
                                                                    equat             the      was
                                                          s
considered equations with variable coefficients, as well as             ational problem while in [9 delay differ
                                                                    varia             ms          9]           rential
                       fficients. The p
those with constant coeff                          d
                                      present method
                                                                        tion were appr
                                                                    equat                         ng         method.
                                                                                     roximated usin the same m
                          ly                          t
presents solutions as easil computable, fast convergent
                                                                         l             ers,                      ws
                                                                    In all of these pape the results obtained show very
            ies, requiring no discretization Examples are
infinite seri              n               n.           e
                                                                         ble
                                                                    reliab      accuracy     when        mpared
                                                                                                       com            with     exact
           o                            efficiency of the
presented to establish the accuracy and e               e
method.                                                                 tions, and with results obtaine using other k
                                                                    solut                             ed            known
keyword: G
         Generalised Riccati Equations, I
                                        Iterative                      hods.
                                                                    meth
         tion Method.
Decomposit                                                                  The aim of this paper is to continu the
                                                                                                              ue
                                                                        ication of the It
                                                                    appli                            mposition on G
                                                                                        terative Decom            General
                    I.   INTRODUCTION
                          N
                                                                        ati
                                                                    Ricca Equations.
        ccati equations find applicatio in Random
  The Ric                             ons       m
                                                                        layout of the re of the paper is as follows:
                                                                    The l              est          r
                                               d
processes, optimal control, network analysis and
                                                                    In    section   2,     the
                                                                                           t      analysis    of    the      erative
                                                                                                                           Ite
          problem, as well as in Finance mathematics.
diffusion p                            e
                                                                       omposition Me
                                                                    Deco                                      ions is
                                                                                   ethod IDM for Riccati equati
          ons          e            e             d
The solutio of Riccati equations have been obtained
                                                                        ented. We nex present the application o the
                                                                    prese           xt          e             of
         e              umerical metho such as the
using some traditional nu            ods         e
                                                                       hod                                  ples
                                                                    meth developed in section 2 to some examp in
         hod, and the Runge-Kutta m
Euler meth            R                       r
                                  method. Newer
                                                                         on         lusion is drawn on the meth
                                                                    sectio 3. A concl             n           hod, in
                     e            d           e
numerical schemes have been applied by [3]. The
                                                                         on
                                                                    sectio 4.
         domian Decom
popular Ad                       hod     s
                    mposition Meth ADM has
         ied          o             nlinear Riccati
been appli in [2, 6] to solve the non             i
equations. In [10],the Homotopy An
                       H         nalysis Method
                                              d                     II. So
                                                                         olution Techni
                                                                                      ique By Iterat Decompos
                                                                                                   tive     sition

                                    tion of Riccati
was used to obtain the analytic solut             i                    hod
                                                                    Meth
equations.                                                             nsider
                                                                     Con             the         General        Riccati        Diffe

            he                          work is the need
           Th basic motivation for this w              d
                                                                         al
                                                                    rentia Equation.
          tion technique which can be applied with
for a solut            e            e            h
          ase, and requi
relative ea                                     l
                       iring minimal mathematical
           thout any loss of accuracy or e
details, wit              o                            e
                                         efficiency. The            Wher re                                               are scalar
          Decomposition Method (ID
Iterative D           n          DM) has been
                                            n                           tions
                                                                    funct
                                                                    Equaation (i) can b written in the form
                                                                                      be



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                                                                                                  ISSN 1947-5500
                                                   (IJCSIS) International Journal of Computer Science and Information Security,
                                                   Vol. 9, No. 6, June 2011




        he
 Where th differential operator L is gi
                       o              iven by                     III   Numerical Examples
                                                                        Example 3.1
                                                                  Conssider the R Riccati equation             with      onstant
                                                                                                                        co
The Unive erse operator is thus the int
                        i             tegral operator
                                                    r                 ficients
                                                                  coeff
         y
defined by

                                                                      exact solution of the problem is
                                                                  The e                           m
         the         perator (4) on (2), it follows
Applying t universe op                            s
that


                                                                      Iterative Decom
                                                                  The I                         thod gives
                                                                                    mposition Met



The iterativ Decomposit
            ve             tion method asssumes that thee
unknown f  function                      ssed in terms of
                             can be expres              f
an infinite series of the fo
                           orm

where                                               e
                                          so that the
component   t         b
                 can be determined iteratively
To convey the idea of the method, a well as for
           y                       as           r
                                                                     thus have an ap
                                                                  We t             pproximation
            ess      hnique [7] we c see that (8)
completene of the tech             can          )
           orm
is of the fo

where k is a constant and
                        d           nonlinear term.
                             is the n
We can decompose the nonlinear term a
                        n            as                               e
                                                                  Table 1
                                                                   t      Exact Solution          Approximate             rror
                                                                                                                         Er
                                                                                                solution b IDM
                                                                                                         by
                                                                  0.0
                                                                  0.1    0.099667994                   81
                                                                                              0.09966798                   -8
                                                                                                                      1.3 E-
                                                                  0.2    0.197375320                   04
                                                                                              0.19737370              1.616 E-6
From (8) a (10), equati (9) is equiv
         and          ion          valent to                      0.3    0.291312612                   272
                                                                                              0.29128522              2.738 E-5
                                                                  0.4    0.379948962                   26
                                                                                              0.37974602              2.029 E-4
                                                                  0.5    0.462117157                   031
                                                                                              0.46116290              9.543 E-4
                                                                  0.6    0.537049366                   643
                                                                                              0.53368786              3.362 E-3
                                                                  0.7    0.604367777                   936
                                                                                              0.59467219              9.696 E-3
                                                                  0.8    0.664036770                   089
                                                                                              0.63989270              2.414 E-2
                                                                  0.9    0.716297870                   771
                                                                                              0.66254907              5.375 E-2
                                                                  1.0    0.761594155                   824
                                                                                              0.65196588              1.096 E-1

                                                                                       erical comparison o results for Exam 3.1
                                                                  Table 1 shows the nume                 of               mple
                                                                       ned
                                                                  obtain from our present scheme and the exact solutio at aon
                                                                  numbe of points in the interval [0, 1]. Error is def
                                                                        er                                                fined as



          ,…                                                         mple
                                                                  Exam 3.2
Thus,                                                                sider the Riccat equation with variable coef
                                                                  Cons              ti            h             fficient



                                                                      exact solution of the problem is
                                                                  The e                           m
and
                                                                  Appl              ative Decompo
                                                                      lying the Itera                       od,
                                                                                                osition Metho we
                                                                  have



                                                            311                                http://sites.google.com/site/ijcsis/
                                                                                               ISSN 1947-5500
                                                         (IJCSIS) International Journal of Computer Science and Information Security,
                                                         Vol. 9, No. 6, June 2011




                                                                             quation. Int. M
                                                                            Eq                            6):
                                                                                           Math. For. 2 (56 2759-2770
                                                                             007
                                                                            20
Then,                                                                    [4] H. He, Varia                n
                                                                                         ational iteration method for
                                                                             utonomous ord
                                                                            au                           tial
                                                                                         dinary different systems. Appl.
                                                                            Math. Comput. 1
                                                                            M             114:115-123. 2 2000

                                                                         [5] L. Junfeng, Va                ation method fo
                                                                                            ariational itera              or
                                                                             solving two-poi boundary v
                                                                             s               int           value problems. J.
                                                                             Comput. Appl. 2006
                                                                             C
                                                                             T.Ramesh Rao, The use of Ad
                                                                         [6] T                             domian
                                                                            D
                                                                            Decomposition M  Method for Sol lving Generalissed
We thus ha an approximation for y (t as
         ave                       t)                                       Riiccati Differential Equations. Proc. 6th. IM
                                                                                                            .,           MT-GT
                                                                              onf.           t.
                                                                            Co Math. Stat Appl.: 935-9 2010941.
                                                                             O.
                                                                         [7] O Taiwo, S, Od                Y.
                                                                                             detunde, and Y Adekunle,
Table 2:                                                                     N
                                                                             Numerical App  proximation of One Dimensio  onal
t      Exact Solution    Appproximate            Error                       Biharmonic Eq
                                                                             B              quations by an IIterative
                         Soluution by IDM                                    decomposition Method, Int. J. Math. Sc. 20 (1):
                                                                             d
0.0     0.0              0.0                     0.0                         3
                                                                             34-44. 2009
0.1     0.1                 0000000000
                         0.10                    1.000E-26
                                                                             O.              S.
                                                                         [8] O Taiwo, and S Odetunde, N    Numerical
0.2     0.2                 0000000000
                         0.20                    1.000E-26
0.3     0.3              0.30
                            0000000000           1.257E-26                   A                              l
                                                                             Approximation of Variational Problems by a    an
0.4     0.4              0.40
                            0000000000           3.954E-22                   I
                                                                             Iterative Decommposition Meth  hod, Maejo J. M
                                                                                                                          Math.
0.5     0.5                 0000000000
                         0.50                    1.218E-18                                  433
                                                                             3 (03): 426-4 2009
0.6     0.6                 0000000000
                         0.60                    8.636E-16                   O.              S.
                                                                         [9] O Taiwo, and S Odetunde, Ap   Approximation of
0.7     0.7                 0000000000
                         0.70                    2.220E-13                   D               tial
                                                                             Delay Different Equations by a Decompo       osition
0.8     0.8                 0000000000
                         0.80                    2.717E-11
0.9     0.9                 0000000001
                         0.90                    1.886E-9
                                                                             M                              3:
                                                                             Method, Asian J. Math. Stat. 3 1-7. 2010
1.0     1.0                 0000000837
                         1.00                    8.372E-8
                                                                         [10] Y. Tan, and S. Abbasbandy, Homotopy ana   alysis
            ws                                                 2
Table 2 show the comparison of the exact solution of Example 3.2             m
                                                                             method for quaadratic Riccati differential
             tion obtained by th Iterative Decom
with the solut                 he              mposition Method.
                                                                             equations. Com
                                                                             e             mmun. Nonlin. Sci. Numer. Simul.
                                                                             d             cnsns.2006
                                                                             doi:10.1016/j.c
IV. Conclusion
     From the given exam               revious section,
                         mples in the pr
           t             cheme is very e
we see that our current sc                             e
                                       efficient for the
          f
classes of Riccati equa   ations. The aapproximations  s
          are            y
obtained a reasonably accurate, wh      hen compared   d
          xact           e              ew
with the ex solutions, even for very fe terms of the   e
approxima ating infinite series. Th    he level of     f
                         on
mathematical computatio required is very minimal,
with extremmely low com                 st.
                        mputational cos The present    t
           n              s            nt
method, on the whole, is very efficien and accurate    e
for the Ri               ns            ant
          iccati equation with consta coefficients     s
           with          c
and those w variable coefficients.

REFEREN  NCES
[1] G. Doomian, Solving Frontier problems in
         cs:
    Physic The Decomp                od.
                        position Metho Kluwer
          mic          s,
    Academ Publishers Dordrecht. 19  994

         hnasawi, M. El-
[2] A. Bah               -Tawil, and A. Abdel-Naby,
                                       ns
   Solving Riccati Differential Equation using
         an              ion          Appl.
   Adomia Decompositi Method., Ap Math.
          t.             4
   Comput 157: 503-514 2004

[3] B. Ba               ani,
         atiha, M. Noora I. Hashim, Application of   f
          onal Iteration Method to a Ge
   Variatio              M            eneral Riccati




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