REDUCED DIFFERENTIAL TRANSFORM METHOD FOR GENERALIZED KDV EQUATIONS. Yıldıray Keskin and Galip Oturanç

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1 Mahemaical ad Compuaioal Applicaios, Vol. 15, No. 3, pp , 1. Associaio for Scieific Research REDUCED DIFFERENTIAL TRANSFORM METHOD FOR GENERALIZED KDV EQUATIONS Yıldıray Kesi ad Galip Ouraç Deparme of Mahemaics, Sciece Faculy Selçu Uiversiy, Koya 43, Turey ad Absrac- I his paper, a geeral framewor of he reduced differeial rasform mehod is preseed for solvig he geeralized Koreweg de Vries equaios. This echique does require ay discreizaio, liearizaio or small perurbaios ad herefore i reduces sigificaly he umerical compuaio. Comparig he mehodology wih some ow echiques shows ha he prese approach is effecive ad powerful. I addiio, hree es problems of mahemaical physics are discussed o illusrae he effeciveess ad he performace of he reduced differeial rasform mehod. Key Words- Reduced differeial rasform mehod, Adomia decomposiio mehod, Variaioal ieraio mehod, Homoopy perurbaio mehod, Koreweg de Vries equaios 1.INTRODUCTION Parial differeial equaios (PDEs) have umerous esseial applicaios i various fields of sciece ad egieerig such as fluid mechaic, hermodyamic, hea rasfer, physics [1]. Oe of he mos aracive ad surprisig wave pheomeo is he creaio of soliary waves or solios. Mos of hese equaios are oliear parial differeial equaios. I is difficul o hadle oliear par of hese equaios. Alhough mos of scieiss applied umerical mehods o fid he soluio of hese equaios, solvig such equaios aalyically is of fudameal imporace sice he eise umerical mehods which approimae he soluio of parial differeial equaios do resul i such a eac ad aalyical soluio which is obaied by aalyical mehods. Hiroa s biliear mehod [], he balace mehod [3], iverse scaerig mehod [4], sie cosie mehod [5], he homoopy aalysis mehod [6], he homoopy perurbaio mehod (HPM) [7-8], he differeial rasform mehod (DTM) [9-11], he variaioal ieraio mehod (VIM) [1,14,15], he Adomia s decomposiio mehod (ADM) [16-19] are some eamples of aalyical mehods. I was approimaely hudred years ago ha a adequae heory for soliary waves was developed, i he form of a modified wave equaio ow as he Koreweg de Vries equaio (KdV) [17]. I his paper we will apply he reduced differeial rasform mehod (RDTM) [-] o he geeralized Koreweg de Vries equaio defied i [3] as follows: p u + ( p+ 1)( p+ ) u u + u = g(, ) (1)

2 Y. Kesi ad G. Ouraç 383 where g(, ) is a give fucio ad p= 1,,... wih u, u, u as. If p=, p= 1 ad p=, Equaio (1) becomes liearized KdV, oliear KdV, ad modified KdV equaio, respecively [17,4].. ANALYSIS OF THE METHOD The basic defiiios of reduced differeial rasform mehod are iroduced as follows: u, is aalyic ad differeiaed coiuously Defiiio.1. If fucio ( ) wih respec o ime ad space i he domai of ieres, he le 1 U ( ) = u(, )! = () where he -dimesioal specrum fucio U ( ) is he rasformed fucio. I his paper, he lowercase u(, ) represe he origial fucio while he uppercase U ( ) sad for he rasformed fucio. The differeial iverse rasform of U ( ) is defied as follows: u(, ) = U ( ) =. (3) The combiig equaio () ad (3) we wrie 1 u(, ) = u(, )! = =. (4) From he above defiiios, i ca be foud ha he cocep of he reduced differeial rasform is derived from he power series epasio. For illusraio of he mehodology of he proposed mehod, we wrie he geeralized Koreweg de Vries equaio i he sadard operaor form wih iiial codiio ( ) ( ) ( ) L u(, ) + R u(, ) + N u(, ) = g(, ) (5) u(,) = f ( ) (6)

3 384 Reduced Differeial Trasform Mehod for Geeralized KDV Equaios 3 where L= ad R =, are liear operaors which have parial derivaives, 3 N( u(, ) ) = ( p+ 1)( p+ ) u p u is a oliear erm ad g(, ) is a ihomogeeous erm. Table 1. Reduced differeial rasformaio [-] Fucioal Form Trasformed Form 1 u(, ) U ( ) = u(, )! = (, ) (, ) (, ) w = u ± v W ( ) = U ( ) ± V ( ) (, ) αu(, ) w = W ( ) = αu ( ) (α is a cosa) (, ) w y = m m 1, = W ( ) = δ ( ), δ ( ) =, ( ), m m w y = u(, ) W ( ) = U ( ) (, ) = u(, ) v(, ) w W ( ) = V ( ) U ( ) = U ( ) V ( ) r r r r r= r= r ( + r)! w(, ) = u(, ) W ( ) ( 1)...( ) r = + + r U + 1( ) = U + r ( )! w(, ) = u(, ) W ( ) = U ( )

4 Y. Kesi ad G. Ouraç 385 Maple Code for Noliear Fucio Nu(, ) resar; NF:=Nu(,):#Noliear Fucio m:=5: # Order u[]:=sum(u[b]*^b,b=..m): NF[]:=subs(Nu(,)=u[],NF): s:=epad(nf[],): d:=uapply(s,): for i from o m do [i]:=((d@@i)(d)()/i!): pri(n[i],[i]); #Trasform Fucio od: Accordig o he RDTM ad Table 1, we ca cosruc he followig ieraio formula: ( ) ( ) ( + 1) U ( ) = G ( ) R U ( ) N U ( ) (7) where ( ( )), ( ( )) R( u(, )), ( (, )) + 1 R U N U ad G ( ) are he rasformaios of he fucios N u ad g(, ) respecively. For he easy o follow of he reader, we ca give he firs few oliear erm are p N = ( p+ 1)( p+ ) U ( ) U( ) p 1 p N1 = ( p+ 1)( p+ ) pu ( ) U1( ) U( ) + U ( ) U1( ) p 1 p( p 1) p pu ( ) U ( ) U( ) + U ( ) U1 ( ) U1( ) + N = ( p+ 1)( p+ ) p 1 p pu ( ) U1( ) U1( ) U ( ) U ( ) + From iiial codiio (6), we wrie U ( ) = f ( ) (8) Subsiuig (8) io (7) ad by a sraigh forward ieraive calculaios, we ge he followig U ( ) values. The he iverse rasformaio of he se of values { U ( ) } = gives approimaio soluio as,

5 386 Reduced Differeial Trasform Mehod for Geeralized KDV Equaios u% (, ) U ( ) (9) = = where is order of approimaio soluio. Therefore, he eac soluio of problem is give by u(, ) = lim u% (, ). (1) 3.APPLICATIONS To illusrae he effeciveess of he prese mehod, several es eamples are cosidered i his secio. The accuracy of he mehod is assessed by compariso wih he eac soluios. Eample 1: We cosider he ihomogeeous KdV equaios [17] u uu u e e + = (11) wih iiial codiio u(,) = 1 (1) Taig differeial rasform of (11) ad he iiial codiio (1) respecively, we obai ( + 1) U ( ) = U ( ) U ( ) U ( ) ( ) e ( ) e r r 3 δ δ r= (13) where he -dimesioal specrum fucio U ( ) From he iiial codiio (1) we wrie are he rasformed fucio. U( ) = 1 (14). successively Now, subsiuig (14) io (13), we obai he followig U ( ) values ( ) ( ) U = e, U =, =,3,... 1 Fially he differeial iverse rasform of U ( ) gives

6 Y. Kesi ad G. Ouraç 387 u(, ) = U ( ) = 1 e = which is eacly he same as ha obaied by ADM [17]. Eample. Cosider he KdV equaio which aes he form [13,17]: u + 6uu + u =, R, (15) ad iiial codiio 1 u(,) sech = (16) where u u(, ) = is a fucio of he variables ad. The, by usig he basic properies of he reduced differeial rasformaio, we ca fid he rasformed form of equaio (15) as 3 ( + 1) U ( ) = 6 U ( ) U ( ) U ( ), for 1,,... 1 = (17) r r 3 + r= where he -dimesioal specrum fucio U ( ) From he iiial codiio (16) we wrie is he rasformed fucio. 1 U( ) = sech (18) Subsiuig (18) io (17), we obai he followig U ( ) values successively

7 388 Reduced Differeial Trasform Mehod for Geeralized KDV Equaios sih cosh U1( ) =, U 3 ( ) = 4 8 cosh cosh sih cosh 3 1 U3( ) =, 5 1 cosh 4 cosh cosh 1 U4( ) = 6 96 cosh M 1 1 U ( ) = sech! = ad so o. The, he iverse rasformaio of he se of values { U } = approimaio soluio as ( ) gives -erms sih u% (, ) = U ( ) = + 3 = cosh cosh 4 cosh sech 96! cosh = (19) Therefore, he eac soluio of problem is give by u(, y) = lim u% (, y). The approimae soluio u% (, ) U ( ) is coverge o he eac soluio as i [17] ad i is also aalogous o he approimae soluio obaied by variaioal ieraio mehod i [5]. (see Figure 1). Therefore he soluio is obaied as = =

8 Y. Kesi ad G. Ouraç u(, ) sech = () which is he eac soluios of (14) (15). Figure 1. The umerical resuls for u (, ) 4 y : (a) i compariso wih he aalyical soluios Eq.(), (b) for he soliary wave soluio wih he iiial codiio of Eample. Figure 1 shows he compariso of he approimae soluio by RDTM of order four, he (a) eac soluio Eq.() he solid lie represes he soluio by he reduced differeial rasform mehod, while he circle represes he eac soluio. From he figure 1, i is clearly see ha he RDTM approimaio ad he eac soluio are i good agreeme. Eample 3. Lasly, Cosider he mkdv equaio [13] u + u u + u = (1) 6 ad iiial codiios u(,) = sech( ) () Taig differeial rasform of (1) ad he iiial codiio () respecively, we obai 3 ( + 1) U+ 1( ) = N ( ) U ( ) 3 (3) ad he rasformed iiial codiio

9 39 Reduced Differeial Trasform Mehod for Geeralized KDV Equaios ( ) U = sech( ) (4). successively Now, subsiuig (4) io (5), we obai he followig U ( ) values ( ) ( ) ( ) sih( ) 1 cosh( ) U =, U =, 3 1 cosh( ) cosh( ) + U ( ) = 3 6 cosh( ) 1 sih( )( 6 cosh( ) ) + U ( ) = 4 4 cosh( ) 4 1 cosh( ) cosh( ) 4 5, + U ( ) = cosh( ) M U sih( )(cosh( ) 6 cosh( ) 1) 1 =! sech( ) = ad so o. The, he iverse rasformaio of he se of values { U } = approimae soluio as ( ) gives -erms = 1 sih( ) cosh( ) 3 u% (, ) = U ( ) = + + cosh( ) cosh( ) cosh( ) sih( )( 6+ cosh( ) ) sech( ) 4 6cosh( )! Therefore, he eac soluio of problem is give by u(, ) = lim u% (, ). This soluio is coverge o he eac soluio as i [5] ad he same as approimae soluio of he variaioal ieraio mehod [5]. (see Table ) u(, ) = sech( ) which is he eac soluios of (1) () =

10 Y. Kesi ad G. Ouraç 391 Table. Compariso of he approimae soluios wih eac soluio u(, ) = sech( ) Absolue error: Absolue error: Eac RDTM VIM eac soluio eac soluio ad RDTM ad VIM (hree (hree ieraio) ieraio) CONCLUSION The mai cocer of his aricle is o cosruc a approimae aalyical soluio for he geeralized Koreweg de Vries equaio. We have achieved his goal by applyig reduced differeial rasform mehod. The mai advaage of he mehod is he fac ha i provides is user wih a aalyical approimaio, i may cases a eac soluio, i a rapidly coverge sequece wih elegaly compued erms. Aalyical soluios eable researchers o sudy he effec of differe variables or parameers o he fucio uder sudy easily. Is small size of compuaio i compariso wih he compuaioal size required i oher umerical mehods, ad is rapid covergece show ha he mehod is reliable ad iroduces a sigifica improveme i solvig he geeralized Koreweg de Vries equaio over eisig mehods. As he mehod is usually edious o use by had, we have o use he Maple Pacage o calculae he series obaied from he RDTM. 5. REFERENCES [1] L. Debah, Noliear Parial Differeial Equaios for Scieis ad Egieers, Birhauser, Boso, [] X.-B. Hu, Y.-T. Wu, Applicaio of he Hiroa biliear formalism o a ew iegrable differeial- differece equaio, Physics Leers A 46 (6), 53 59, [3] M. Wag, Y. Zhou, Z. Li, Applicaio of a homogeeous balace mehod o eac soluios of oliear equaios i mahemaical physics, Physics Leers A 16, 67 75, [4] V.O. Vaheo, E.J. Pares, A.J. Morriso, A Bäclud rasformaio ad he iverse scaerig rasform mehod for he geeralized Vaheo equaio, Chaos Solios Fracals 17 (4), , 3.

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