Poceedigs of he Wold Cogess o Egieeig Vol I WCE, July 6-8,, Lodo, U.K. Nueical Soluio of Sie-Godo Equaio by Reduced Diffeeial Tasfo Mehod Yıldıay Kesi, İbahi Çağla ad Ayşe Beül Koç Absac Reduced diffeeial asfo ehod (RDTM), which does o eed sall paaee i he equaio is ipleeed fo solvig he sie-godo equaio. The appoiae aalyical soluio of he equaio is calculaed i he fo of a seies wih easily copuable copoes. Copaig he ehodology wih soe ohe ow echiques shows ha he pese appoach is effecive ad poweful. Thee es odelig pobles fo aheaical physics, boh oliea ad coupled ae discussed o illusae he effeciveess ad he pefoace of he poposed ehod. Ide Tes Reduced diffeeial asfo ehod, sie- Godo equaios Vaiaioal ieaio ehod. I. INTRODUCTION Oe of he os ipoa of all paial diffeeial equaios occuig i applied aheaics is ha associaed wih he ae of sie-godo. The sie-godo equaio plays a ipoa ole i he popagaio of fluos i Josephso jucios [-3] bewee wo supecoducos, he i ay scieific fields such as he oio of a igid pedula aached o a seched wie [], solid sae physics, oliea opics, sabiliy of fluid oios. We coside he sie-godo equaio u u si( u) () subjec o iiial codiios u (,) f( ), u (,) g ( ) () whee u is a fucio of ad, f() ad g() ae a ow aalyic fucio. May ueical ehods wee developed fo his ype of oliea paial diffeeial equaios such as he Adoia Decoposiio Mehod (ADM) [5-9], he EXP fucio ehod [], he Hooopy Peubaio Mehod (HPM) [-3], he Hooopy Aalysis Mehod (HAM) [], he vaiable sepaaed ODE ehod [,5] ad Vaiaioal Ieaio Mehod (VIM) [6-7]. I his pape, we solve soe sie-godo equaios by he educed diffeeial asfo ehod [8-] which is peseed o ovecoe he deei of cople calculaio of diffeeial asfo ehod (DTM) []. The ai advaage of he ehod is he fac ha i povides is use wih a aalyical appoiaio, i ay cases a eac soluio, i a apidly covege sequece wih elegaly copued es. Mauscip eceived Febuay,. Y. Kesi is wih he Selcu Uivesiy, Depae of Maheaics, Koya, 3 Tuey (coespodig auho o povide phoe: + 9 33-3 39 66; fa:+9-33- 99; e-ail: yildiayesi@ gail.co). A. B. Koc is wih wih he Selcu Uivesiy, Depae of Maheaics, Koya, 3 Tuey e-ail: aysebeuloc@ yahoo.co. The sucue of his pape is ogaized as follows. I secio, we begi wih soe basic defiiios ad he use of he poposed ehod. I secio 3, we apply he educed diffeeial asfo ehod o solve hee es eaples i ode o show is abiliy ad efficiecy. II. TRADITIONAL DIFFERENTIAL TRANSFORM METHOD A. Oe Diesioal Diffeeial Tasfo Mehod The diffeeial asfo of he fucio w is defied as follows: whee W w! w is he oigial fucio ad asfoed fucio. Hee d d (3) W is he eas he he deivaive wih espec o. The diffeeial ivese asfo of W is defied as w W. () Cobiig (3) ad () we obai d w w (5)! d Fo above defiiios i is easy o see ha he cocep of diffeeial asfo is deived fo Taylo seies epasio. Wih he aid of (3) ad () he basic aheaical opeaios ae eadily be obaied ad give i Table. Table Oe-diesioal diffeeial asfoaio Fucioal Fo Tasfoed Fo u v U V cu d u d cu! U! u v W UV w B. Two Diesioal Diffeeial Tasfo Mehod Siilaly, he wo diesioal diffeeial asfo of he fucio w, ca be defied as follows: h W, h w, h h!! whee w, is he oigial fucio ad W, h is he asfoed fucio. The diffeeial ivese asfo of W, h is (,) (6) h, W, h. (7) w h The cobiig equaio (6) ad (7) we wie ISBN: 978-988-8-6-5 ISSN: 78-958 (Pi); ISSN: 78-966 (Olie) WCE
Poceedigs of he Wold Cogess o Egieeig Vol I WCE, July 6-8,, Lodo, U.K. h w w h!! h,, h. (8) h (,) Theefoe we ca obai basic aheaical opeaios of wo-diesioal diffeeial asfo as follows i Table. Table. Two diesioal diffeeial asfoaio Fucioal Tasfoed Fo Fo u, v, U, h V, h cu, cu, h u, U, h u, s u, s u, v, h U, h hs!! U, h s! s! h s Uh (, sv ) ( s,) Now we ca sae ou ai esuls i he e secio. III. ANALYSIS OF THE REDUCED DIFFERENTIAL TRANSFORM METHOD The basic defiiios of educed diffeeial asfo ehod ae ioduced as follows: Defiiio. If fucio u, is aalyic ad diffeeiaed coiuously wih espec o ie ad space i he doai of iees, he le U ( ) u,! whee he -diesioal specu fucio U is he asfoed fucio. I his pape, he lowecase u, epese he oigial fucio while he uppecase U sad fo he asfoed fucio. The diffeeial ivese asfo of is defied as follows: U (9),. () u U The cobiig equaio (9) ad () we wie u, u,. ()! Fo he above defiiios, i ca be foud ha he cocep of he educed diffeeial asfo is deived fo he powe seies epasio. Fo he pupose of illusaio of he ehodology o he poposed ehod, we wie he gas dyaic equaio i he sadad opeao fo Lu (, ) Ru (, ) Nu (, ) () wih iiial codiio u (,) f( ) (3) whee L((,)) u u (,) is a liea opeao which has paial deivaives, R((,)) u u (,), Nu(, ) si u(, ) is a oliea e. Accodig o he RDTM ad Table 3, we ca cosuc he followig ieaio foula: ( )! U( ) U ( ) NU( )! () whee NU ( ) is he asfoaios of he fucios Nu(, ) especively. Table 3. Reduced diffeeial asfoaio [8-] Fucioal Fo Tasfoed Fo u (, ) U ( ) u,!, v, u U ( ) V ( ) u, U ( ) ( ) ( is a cosa) u (, ) U( ) u, v, u (, ) u (, ) Nu(, ) U ( ) V ( ) ( )! U! U ( ) ( ) Maple Code fo Noliea Fucio esa; NF:=Nu(,):#Noliea Fucio :=5: # Ode u[]:=su(u[b]*^b,b=..): NF[]:=subs(Nu(,)=u[],NF): s:=epad(nf[],): d:=uapply(s,): fo i fo o do [i]:=((d@@i)(d)()/i!): pi(n[i],[i]); #Tasfo Fucio od: Fo he easy o follow of he eade, we ca give he fis few oliea e ae N si U( ) N cos U ( ) U ( ) N cos U( ) U( ) si U( ) U ( ) Fo iiial codiio (), we wie U ( ) f( ) (5) Subsiuig (5) io () ad by a saigh fowad ieaive calculaios, we ge he followig U ( ) values. The he ivese asfoaio of he se of values U ( ) gives appoiaio soluio as, u (, ) U( ) (6) whee is ode of appoiaio soluio. Theefoe, he eac soluio of poble is give by u (, ) li u (, ). (7) IV. APPLICATIONS To show he efficiecy of he ew ehod descibed i he pevious pa, we pese soe eaples. ISBN: 978-988-8-6-5 ISSN: 78-958 (Pi); ISSN: 78-966 (Olie) WCE
Poceedigs of he Wold Cogess o Egieeig Vol I WCE, July 6-8,, Lodo, U.K. A. Eaple We fis coside he hoogeeous sie-godo equaio [,6,,7] u u si( u) (8) wih iiial codiios: u (,), u (,) sech( ) (9) whee u u, is a fucio of he vaiables ad. Now if we use he VIM, based o he coecio fucioal give [7,] u u u u ( ) si( u ) d he we will fid i oo difficul o evaluae he soluio copoes because we should evaluae he iegal ( )si( u ) d, which is o easily copued. So, he RDTM will be oe efficie fo his eaple. The, by usig he basic popeies of he educed diffeeial asfoaio, we ca fid he asfoed fo of equaio (8) as U ( )! U( ) U ( ) ( ) N!. () Usig he iiial codiios (9), we have U( ), U( ) sech( ) () Now, subsiuig () io (), we obai he followig ( ) values successively U ( ), U3( ), U 3 ( ), 3cosh ( ) U5( ), U 5 6( ), U7( ) 7 5cosh ( ) 5cosh ( ) ( ), fo is odd U ( ) cosh ( ), fo is eve Fially he diffeeial ivese asfo of U ( ) gives (),3,...,3,... u ( ), U ( ) cosh ( ) Hece he closed fo of () is u, aca sec( ) which is he eac soluios of (8) (9) (see Figue ad ). Figue. The ueical esuls fo u 7 (, y ): (a) i copaiso wih he aalyical soluios u (, ) aca sec( ), (b) fo he soliay wave soluio wih he iiial codiio of Eaple. Figue. The ueical esuls fo u 7 (, y )(show i +) (a) i copaiso wih he aalyical soluios u (, ) aca sec( ) (show i ) ad VIM soluio (show i --) fo, (b). B. Eaple Now, we will fid he appoiae aalyical soluio of he sie-godo equaio u u si( u) (3) wih he iiial codiios u (,) cos( ), u(,) () whee ad is a cosa. Taig diffeeial asfo of (3) ad he iiial codiios () especively, we obai ( )! U( ) U ( ) ( ) N! si u,. (5) whee N is asfoed fo of The asfoed iiial codiios U( ) cos( ), U( ) (6) Subsiuig (6) io (5), we obai he followig U values successively U ( ) cos( ) si cos( ), U ( ) 3 U( ) cos( ) si cos( ) cos ( )si cos( ) cos( )cos cos( ) si cos( ) cos cos( ) U ( ) 5 T The, he ivese asfoaio of he se of values U ( ) 5 gives five-e appoiaio soluio as u (, y) cos( ) si cos( ) cos( ) 5 cos( ) si cos( ) cos ( )si cos( ) cos( )cos cos( ) si cos( ) cos cos( ) (7) To deosae he ueical sabiliy of he RDTM, we ae fou γ values (γ=., γ=.5, γ=. ad γ=.), hese values have peviously bee used by Kaya [6], ad soe of γ values (γ=.5,.) have bee used by Ablowiz e al. []. I he pese ueical epeie, (3.) has bee used o daw he gaphs as show i Figue 3-. The ueical soluios of eaple have bee show i Figue 3- usig VIM. I he pese ueical copuaio we have assued γ=., γ=.5, γ=. ad γ= especively. Figue 3. The copaiso of he RDTM (lie) appoiaio ad he VIM soluio (cicle) fo (a) γ=., (b) γ=.5 ISBN: 978-988-8-6-5 ISSN: 78-958 (Pi); ISSN: 78-966 (Olie) WCE
Poceedigs of he Wold Cogess o Egieeig Vol I WCE, July 6-8,, Lodo, U.K. Figue. The copaiso of he RDTM (lie) appoiaio ad he VIM soluio (cicle) fo (a) γ=. ad (b) γ= si( Acos( )) v 5(, y) c si( Acos( )) A c si( Acos( ))cos ( ) A c cos( Acos( )) A cos( ) cos( Acos( )) A cos( ) cos( Acos( ))si( Acos( )) cos( Acos( ))si( Acos( )) This soluio is covege o he adoia decoposiio ehod soluio [8,3] ad he sae as appoiae soluio of he vaiaioal ieaio ehod [6]. (see Figue 5) Figue 5. The copaiso of he RDTM (lie) appoiaio ad he VIM soluio (poi). C. Eaple 3 We ow coside a syse of coupled sie-godo equaios [8,3,6] u u u v (,) (,) si((,) (,)) v c v u v (, ) (, ) si( (, ) (, )) wih iiial codiios u (,) Acos( ), u(,) v (,), v(,) Taig he diffeeial asfo of (8), i ca be obaied ha ( )! U U N! ( )! V ( ) c V ( ) ( ) N! ( ) ( ) ( ) (8) (9) (3) whee he -diesioal specu fucio U ad V ae he asfoed fucio ad N is asfoed fo of si u, v,. Fo he iiial codiio (.) we wie U( ) Acos( ), U( ) (3) V ( ), V ( ) Subsiuig (3) io (3), we obai he followig U ad V values successively. The, he ivese asfoaio of he se of values U ( ) 5 ad V ( ) 5 gives five e appoiaio soluio as Acos( ) si( Acos( )) u 5(, y) Acos( ) Acos( ) si( Acos( )) A si( Acos( )) A cos ( ) cos( Acos( )) A cos( ) cos( Acos( ))si( Acos( )) cos( A cos( ))s i( Acos( )) Figue 5 (a-b) shows he copaiso of he RDTM appoiaio soluio of ode five ad he VIM soluio u (, )(Figue a) ad v (, ) (Figue b), he solid lie epeses he soluio by he RDTM (show i ed), while he cicle epeses he VIM (show i blue). V. CONCLUSIONS The sie-godo equaios have bee aalyzed usig he educed diffeeial asfo ehod. All he eaples show ha he educed diffeeial asfo ehod is a poweful aheaical ool o solvig sie-godo equaio. I is also a poisig ehod o solve ohe oliea equaios. This ehod solves he poble wihou ay eed o disceizaio of he vaiables, heefoe, i is o affeced by copuaio oud off eos ad oe does o face he eed of lage copue eoy ad ie. I ou wo, we ade use of he Maple Pacage o calculae he seies obaied fo he educed diffeeial asfo ehod. ACKNOWLEDGEMENT This sudy was suppoed by he Coodiaoship of Selcu Uivesiy s Scieific Reseach Pojecs (BAP). REFERENCES [] M. J. Ablowiz, B. M. Hebs; Schobe, Cosace O he ueical soluio of he sie-godo equaio. I. Iegable disceizaios ad hoocliic aifolds. J. Copu. Phys. 6 (996), o., 99--3. [] M. J. Ablowiz, B. M. Hebs, C. M. Schobe ; O he ueical soluio of he sie-godo equaio. II. Pefoace of ueical schees J. Copu. Phys. 3 (997), o., 35--367. [3] A. D. Polyai, V. F. Zaisev, Hadboo of Noliea Paial Diffeeial Equaios, Chapa & Hall/CRC Pess, Boca Rao. [] A. M. Wazwaz, A vaiable sepaaed ODE ehod fo solvig he iple sie-godo ad he iple sih-godo equaios Chaos Solios & Facals, 7, 33: 73-7. ISBN: 978-988-8-6-5 ISSN: 78-958 (Pi); ISSN: 78-966 (Olie) WCE
Poceedigs of he Wold Cogess o Egieeig Vol I WCE, July 6-8,, Lodo, U.K. [5] S. M. El-Sayed, The decoposiio ehod fo sudyig he Klei- Godo equaio Chaos Solios & Facals, 3, 8: 5-3. [6] D. Kaya, A ueical soluio of he sie-godo equaio usig he odified decoposiio ehod Applied Maheaics ad Copuaio, 3, 3: 39-37. [7] D. Kaya. A applicaio of he odified decoposiio ehod fo wo diesioal sie-godo equaio, Applied Maheaics ad Copuaio,, 59: -9. [8] S. S. Ray, A ueical soluio of he coupled sie-godo equaio usig he odified decoposiio ehod, Applied Maheaics ad Copuaio, 6, 75: 6-5. [9] Q. Wag, A applicaio of he odified Adoia decoposiio ehod fo (N+)-diesioal sie-godo field, Applied Maheaics ad Copuaio, 6, 8: 7-5. [] A. Bei, ad A. Boz, Eac soluios fo a class of oliea paial diffeeial equaios usig ep-fucio ehod, Ieaioal Joual of Noliea Scieces ad Nueical Siulaio, 7,8: 55-5. [] A. Aslaov, The Hooopy-Peubaio Mehod fo Solvig Klei-Godo-Type Equaios wih Ubouded Righ-Had Side, Zeischif Fu Naufoschug Secio A-A Joual of Physical Scieces, 9, 6: 9-5. [] M. S. H. Chowdhuy ad I. Hashi, Applicaio of hooopypeubaio ehod o Klei-Godo ad sie-godo equaios, Chaos Solios & Facals, 9, 39: 98-935. [3] A. Sadighi, D. D. Gaji, e al., Tavelig Wave Soluios of he Sie-Godo ad he Coupled Sie-Godo Equaios Usig he Hooopy-Peubaio Mehod, Scieia Iaica Tasacio B- Mechaical Egieeig, 9, 6: 89-95. [] U. Yucel, Hooopy aalysis ehod fo he sie-godo equaio wih iiial codiios, Applied Maheaics ad Copuaio, 8, 3: 387-395. [5] N. H. Kuo, ad C. D. Hu, A sudy of he soluios of he cobied sie-cosie-godo equaio, Applied Maheaics ad Copuaio, 9, 5: 5-9. [6] B. Baiha,, M. S. M. Nooai, e al., Appoiae aalyical soluio of he coupled sie-godo equaio usig he vaiaioal ieaio ehod, Physica Scipa, 7, 76: 5-8. [7] B. Baiha, M. S. M. Nooai, e al., Nueical soluio of sie- Godo equaio by vaiaioal ieaio ehod, Physics Lees A, 7, 37: 37-. [8] Y. Kesi, G. Ouac, Reduced Diffeeial Tasfo Mehod fo Paial Diffeeial Equaios, Ieaioal Joual of Noliea Scieces ad Nueical Siulaio, 9, (6), 7-79. [9] Y. Kesi, Ouaç, G., Reduced diffeeial asfo ehod fo geealized KdV Equaios, Maheaical ad Copuaioal Applicaios,, 5 (3), 38-393. [] Y. Kesi, G. Ouac, Reduced Diffeeial Tasfo Mehod fo facioal paial diffeeial equaios, Noliea Sciece Lees A,, () 6-7. [] N. Bildi, A. Koualp, The use of vaiaioal ieaio ehod, diffeeial asfo ehod ad Adoia decoposiio ehod fo solvig diffee ypes of oliea paial diffeeial equaios, Ieaioal Joual of Noliea Scieces ad Nueical Siulaio, 6, 7 (), 65 7 [] Z. M. Odiba, Reliable appoaches of vaiaioal ieaio ehod fo oliea opeaos, Maheaical ad Copue Modellig, 8, 8: -3. ISBN: 978-988-8-6-5 ISSN: 78-958 (Pi); ISSN: 78-966 (Olie) WCE