MODIFIED ADOMIAN DECOMPOSITION METHOD FOR SOLVING RICCATI DIFFERENTIAL EQUATIONS

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1 Review of he Air Force Academy No 3 (3) 15 ODIFIED ADOIAN DECOPOSIION EHOD FOR SOLVING RICCAI DIFFERENIAL EQUAIONS 1. INRODUCION Adomia decomposiio mehod was foud by George Adomia ad has recely become a very well - kow mehod i applied scieces. he mehod does o eed ay liearizaio or smalless assumpios o solve he differeial equaios ad his makes he mehod very effecive amog he oher mehods. ay works have bee examied i various differe areas such as hea or mass rasfer, oliear opics, icompressible fluid ad gas dyamics pheomea ec [1-5]. Noliear differeial equaios arise from may impora applicaios i physics, egieerig, applied sciece such as dampig laws, diffusio processes, rasmissio lie pheomea, ec. hey have bee solved by may differe echiques [6-9]. Riccai differeial equaio is oe of he mos sigifica oliear differeial equaios. hey are geerally used i he sudies of opimal corol problems. Obaiig he exac soluio of his equaio is o always possible. So, umerical mehods are eeded o ge he approximae soluio. radiioal Adomia decomposiio mehod is also used o solve Riccai differeial equaio i may differe papers [1-1]. Necde BİLDİK, Sia DENİZ Celal Bayar Uiversiy, aisa/urkey DOI: 1.196/ Absrac: I his sudy, we solve Riccai differeial equaios by modified Adomia decomposiio mehod which is cosruced by differe orhogoal polyomials. Here, Chebyshev polyomials are used isead of aylor polyomials o expad he source fucio. We see he beefis of usig hese expasios o ge beer resuls. Keywords: Chebyshev polyomials, Adomia decomposiio mehod, oliear differeial equaios.. ADOIAN DECOPOSIION EHOD I his secio we give some brief ad basic iformaio abou Adomia decomposiio mehod. For much more iformaio, we refer o [5,13,14]. Cosider he differeial equaio Ly + Ry + Ny = g( ) (1) where L is he highes-order derivaive which is assumed o be iverible, R is a liear differeial operaor of less order ha L, N is he oliear operaor ad g is he source 1 erm. If we apply he operaor L - which is he iverse of he L o he equaio (1), we ge L ( Ly) = y = L ( g) - L ( Ry) - L ( Ny). () Le us suppose he soluio of he Eq.(1): y( ) = y ( ). (3) Besides ha he oliear erms is obaied by Ny = A (4) where A are Adomia polyomials which ca be calculaed from: 1

2 odified Adomia Decomposiio ehod for Solvig Riccai Differeial Equaios 1 d i A = N i,,1,, λ! dλ = i= K (5) λ= Usig he equaios ()-(5), we ge y = f - L R y - L A. (6) where f is calculaed from he source erm ad he give codiio(s) which are assumed o be prescribed. We ow cosruc he recursive relaio as : -1 y = f = Ψ + L ( g( )) y1 = -L R ( y ) - L A ( ) ( ( )) ( ) y = -L R y - L A, k k + 1 k k I ca be easily said ha he soluio is. (7) m y = lim y m (8) provided ha he series coverges suiably. 3.ODIFIED ADOIAN DECOPOSIION EHOD I his secio, we give he cosrucio of modified Adomia decomposiio mehod by usig Chebyshev polyomials. Normally, we use aylor polyomials i calculaios for Adomia decomposiio mehod. However, as we will see ha usig Chebyshev polyomials yields beer resuls ha aylor polyomials. Geerally, he source erm is usually wrie as m g () g( ). (9)! Hosseii [15] used Chebyshev polyomials o modify he AD by expadig: where a are coefficies ad are Chebyshev polyomials [16,17]. I fac, here are may orhogoal polyomials as Laguerre, Legedre ec. ha we ca use isead of aylor polyomials. Bu, ie [18] ad ahmoudi [19] showed ha hese modificaios are o good eough as much as Chebyshev polyomials. 4. NUERICAL EXAPLES I his secio, we solve wo Riccai equaios o illusrae he pheomea. hese problems are brad ew ad cao be foud i he lieraure. Example 1) Cosider he Riccai differeial equaio 1 y - y + y = e, y () = 1 (11) which have he exac soluio y = e. Soluio: We proceed accordig o secio. We have d -y L =, R( y) =, Ny = F( y) = y d ad g( ) = e. (1) Cosrucig Adomia polyomials accordig o (5), we obai: A = F( y) = y A1 = y1f ( y) = y y1 1 A = y F ( y ) + y F ( y ) = y y + y! Wriig he source fucio i aylor series form for oly 4 erms:. (13) 3 g( ) = e (14) 6 he we form he recursive relaio as i (7): m g( ) a ( ) (1)

3 Review of he Air Force Academy No 3 (3) y = f = y() + L ( ) 6 y1 = -L R ( y ) - L ( A ) yields y = -L R y - L A, k ( ( )) ( ) k + 1 k k 3 4 y = y y = L = L (15) (16) Afer doig much more calculaios, we ge he soluio as: yc ( ) = (1) For larger m more accurae resuls we ge. Figure 1 ad Figure displays he errors for oly m = 4. able 1 shows he comparisos of hese errors for m = 8. Example ) Cosider he Riccai differeial equaio - + =, () 1 y y y e which have he exac soluio y = () y = e. m y ( ) = y = y + y + y + L y (17) 1 m where y ( ) deoes he approximae soluio compued by usig aylor series expasios. For m = 4 we obai 3 y ( ) = 1+,5 +,15 +, 833 (18) 4 5 +, 64 +, 5 Now, we make he same calculaios by usig Chebyshev expasio which ca be calculaed as i [17,18]: g( ) e Agai, we have ow he recursive relaio: 3 = (19) Figure 1. he errors Example 1 y - y for 3 y = 1+, , , L 3 4 y1 = -.5 -, L 3 4 y = L () Figure. he errors y - y for Example 1 C By proceedig for m = 4 we ge 3

4 odified Adomia Decomposiio ehod for Solvig Riccai Differeial Equaios able 1: Absolue errors for m = 8 for Example 1. y( ) y y C - y - y E E E E-6 Soluio: We here have d - L, R( y) y, Ny F( y) y d E E E E- = = = = ad g( ) = e. (3) Figure 3 ad Figure 4 displays he absolue errors for oly m = 4. We ca see he differece eve for small m. able shows he comparisos of hese errors for m = 8. CONCLUSIONS I his sudy, we show ha usig Chebyshev polyomials is good idea o improve he effeciveess of he Adomia decomposiio mehod. We use Chebyshev expasios of he source erm o obai more accurae resuls. Figures eable us o see ha he differece bewee he usig boh wo mehods by graphically. ables are also give o show he variaio of he absolue errors for larger approximaio, amely for larger m. aple 18 is used for calculaios ad skechig graphs. We ca easily compue he aylor ad Chebyshev series expasios of he source erms we eed: g ( ) = e ad 4 6 (4) g ( ) = e C (5) Followig he same procedure as i he previous example 1, we ge he approximae soluios: Figure 3: he errors y - y for Example. y ( ) = ad (6) yc ( ) =

5 Review of he Air Force Academy No 3 (3) 15 able : Absolue errors for m = 8 for Example. y( ) y y C - y - y E E E E E E E E-5 Figure 4: he errors y - y for Example. C BIBLIOGRAPHY 1. Bildik, Necde, ad Sia Deiz. Implemeaio of aylor collocaio ad adomia decomposiio mehod for sysems of ordiary differeial equaios. Proceedigs of he Ieraioal Coferece o Numerical Aalysis ad Applied ahemaics 14 (ICNAA-14). Vol AIP Publishig, 15.. Wazwaz, A.. Cosrucio of soliary wave soluios ad raioal soluios for he KdV equaio by Adomia decomposiio mehod. Chaos, Solios & Fracals 1.1 (1): Evas, David J., ad Hasa Bulu. A ew approach o he gas dyamics equaio: A applicaio of he decomposiio mehod. Ieraioal joural of compuer mahemaics 79.7 (): Bulu, Hasa, e al. Numerical soluio of a viscous icompressible flow problem hrough a orifice by Adomia decomposiio mehod. Applied mahemaics ad compuaio (4): Bildik, Necde, ad Ali Kouralp. he use of variaioal ieraio mehod, differeial rasform mehod ad Adomia decomposiio mehod for solvig differe ypes of oliear parial differeial equaios. Ieraioal Joural of Noliear Scieces ad Numerical Simulaio 7.1 (6): Aagaa, Abdo, ad Adem Kılıçma. he use of Sumudu rasform for solvig cerai oliear fracioal hea-like equaios. Absrac ad Applied Aalysis. Vol. 13. Hidawi Publishig Corporaio, Poalagusamy, R., ad S. Sehilkumar. A ew fourh order embedded RKAHe (4, 4) mehod wih error corol o mulilayer raser cellular eural ework. Sigal, image ad video processig 3.1 (9): Elbeleze, Asma Ali, Adem Kılıçma, ad Bachok. aib. Homoopy perurbaio mehod for fracioal Black-Scholes Europea opio pricig equaios usig Sumudu rasform. ahemaical problems i egieerig13 (13). 5

6 odified Adomia Decomposiio ehod for Solvig Riccai Differeial Equaios 9. Fu, Xiaolig, e al. A Asymmeric Proximal Decomposiio ehod for Covex Programmig wih Liearly Couplig Cosrais. Advaces i Operaios Research 1 (1). 1. El-awil, agdy A., Ahmed A. Bahasawi, ad Ahmed Abdel-Naby. Solvig Riccai differeial equaio usig Adomia s decomposiio mehod. Applied ahemaics ad Compuaio 157. (4): Abbasbady, Saeid. Homoopy perurbaio mehod for quadraic Riccai differeial equaio ad compariso wih Adomia s decomposiio mehod. Applied ahemaics ad Compuaio 17.1 (6): Bulu, Hasa, ad David J. Evas. O he soluio of he Riccai equaio by he decomposiio mehod. Ieraioal joural of compuer mahemaics 79.1 (): Öziş, urgu, ad Ahme Yıldırım. Compariso bewee Adomia s mehod ad He s homoopy perurbaio mehod. Compuers & ahemaics wih Applicaios 56.5 (8): Deiz, Sia, ad Necde Bildik. Compariso of Adomia Decomposiio ehod ad aylor arix ehod i Solvig Differe Kids of Parial Differeial Equaios. Ieraioal Joural of odelig ad Opimizaio 4.4 (14): Hosseii, ohammad ahdi. Adomia decomposiio mehod wih Chebyshev polyomials. Applied ahemaics ad Compuaio 175. (6): Fox, Leslie, ad Ia Bax Parker. Chebyshev polyomials i umerical aalysis. (1968). 17. aso, Joh C., ad David C. Hadscomb. Chebyshev polyomials. CRC Press,. 18. ie, Wei-Chug. Adomia decomposiio mehod by Legedre polyomials. Chaos, Solios & Fracals 39.5 (9): ahmoudi, Y., e al. Adomia Decomposiio ehod wih Laguerre Polyomials for Solvig Ordiary Differeial Equaio. (1). 6