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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