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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
310 http://sites.google.com/site/ijcsis/
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
312 http://sites.google.com/site/ijcsis/
ISSN 1947-5500
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