Numerical approximatons for solving partial differentıal equations with variable coefficients

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1 Appled ad Copuaoal Maheacs ; () : 9- Publshed ole Februar (hp:// do:.648/j.ac.. Nuercal approaos for solvg paral dffereıal equaos wh varable coeffces Ves TURUT Depare of Maheacs Facul of Ars ad Sceces Baa Uvers BaaTurke Eal address: vesuru@gal.co (V. TURUT) To ce hs arcle: Ves TURUT. Nuerıcal Approıaos for Solvıg Parıal Dıffereıal Equaıos wıh Varıable Coeffıcıes Appled ad Copuaoal Maheacs. Vol. No. pp. 9-. do:.648/j.ac.. Absrac: I hs paper varaoal erao ehod (VIM) ad ulvarae padé approao (MPA) were copared. Frs paral dffereal eqauo has bee solved ad covered o power seres b varaoal erao ehod (VIM) he he uercal soluo of paral dffereal eqauao was pu o ulvarae padé seres. Thus he uercal soluos of he paral dffereal eqauos were obaed. Nuercal soluos of wo eaples were calculaed ad resuls were preseed ables ad fgures. Kewords: Varaoal Ierao Mehod (VIM) Mulvarae Padé Approao (MPA) Paral Dffereal Equao (PDE). Iroduco Ma powerful uercal ad aalcal ehods have bee preseed. Aog he he Adoa decoposo ehod (ADM) [-4] he varaoal erao ehod (VIM) [ 8] dffereal rasfor ehod (DTM) ad ulvarae padé approao (MPA) [9-] are relavel ew approaches provdg a aalcal ad uercal approao o lear ad olear probles. The varaoal eraoal ehod (VIM) was frs proposed b He [67] ad has bee succesfull appled o auooous dffereal equaos o-lear paral dffereal equaos o-lear polcrsalle solds ad oher felds. Mulvarae padé approao (MPA) has bee successfull appled o solve paral dffereal equaos. Ma defos ad heores have bee developed for Mulvarae Padé Approaos (MPA) (see [8] for a surve o Mulvarae Padé approao).. The Varaoal Ierao Mehod The basc coceps ad prcples varaoal erao ehod ca be see [9-]. Al ad Rasla [] obaed he followg erao forula for geeral PDE equao () b usg he basc coceps ad prcples of varaoal erao ehod: Lu L u L u L u Nu g( ). () L ad where L L L are lear operaors of ad respecvel ad N s a o-lear operaor. Accordg o VIM he followg correco fucoal ca be epressed ad -drecos respecvel as follows [] : u ( ) u( ) { Lu s ɶ ( L L L N) u g ds u ( ) u ( ) { s ɶ L u ( L L L N ) u g ds u ( ) u( ) 4 { Lu ɶ ( L L L N ) u g ds s u ( ) u( ) { Lu s ɶ ( L L L N) u g ds. where ad 4 are geeral Lagrage ulplers [] whch ca be defed opall va he varaoal heor [ ] ad uɶ s a resrced varao whch eas () () (4) ()

2 Ves TURUT: Nuercal approaos for solvıg paral dffereal equaos wh varable coeffces ɶ. B hs ehod frs he Lagrage ulplers δ u are deered ( 4) whch wll be defed opall. The succesve approaos u of he soluo u wll be readl obaed b suable choce of ral fuco u []. Cosequel he correco fucoal wll gve several approaos. The oe of hs approaos was copared wh ulvarae padé approao b pug o ulvarae padé seres.. Mulvarae Padé Approao Cosder he bvarae fuco f ( ) wh Talor seres develope f ( ) (6) j cj j aroud he org[4]. We kow ha a soluo of uvarae Padé approao proble for s gve b ad f ( ) c (7) p( ) c c c q( ) Le us ow ulpl j h row p( ) ad q( ) b j ( j... ) ad aferwards dvde j h colu p( ) ad q( ) j b j. Ths resuls a (... ) ulplcao of ueraor ad deoaor b Havg doe so we ge c c c c c c p( ) c c c q( ) c c c c c c. f ( D ded ). Ths quoe of deeras ca also edael be (8) (9) () wre dow for a bvarae fuco f ( ). The su k c shall be replaced k h paral su of he Talor seres develope of f ( ) ad he epresso ck k b a epresso ha coas all he ers of degree k j f ( ). Here a bvarae er cj s sad o be of degree j. If we defe p ( ) ad q ( ) j j j cj cj cj j j j c c c j j j j j j j j j c c c j j j j j j j j j c c c j j j j j j j j j c c c j j j j j j j j j The s eas o see ha p( ) ad q( ) are of he for p ( ) q ( ) j a j j j b j j We kow ha p( ) ad q( ) are called Padé equaos[4]. So he ulvarae Padé approa of order ( ) for f ( ) s defed as r () () () p ( ) ( ) (4) q ( ) 4. Applcaos ad Resuls I hs seco he wo ehods VIM ad MPA wl be llusraed b wo eaples. All he resuls are calculaed b usg sofware apple. Eaple 4.. Cosder he oe-desoal hea equao wh varable coeffces u ( ) u ( ) () ad he al codo u( ). The varaoal eraoal schea of equao () has he for[] u ( ) u ( ) ( u ( s)) s ( u ( s)) ds ɶ (6) where ad u ( ). Ths elds he saoar codos

3 Appled ad Copuaoal Maheacs () : 9- s Hece he Lagrage ulpler s ' ( s). (7). (8) Subsug hs value of he Lagrage ulpler o he fucoal (6) gves he erao forula[] u ( ) u( ) ( u( s)) s ( u( s)) ds. (9) Al ad Rasla obaed [] he folowg succesve approaos sarg wh a al approao: ad usg he erao forula (9) u ( ) ( ) u! ( ) ( ) u!! ( ) ( ) u!! 4! 4 ( ) ( ) 4 u!! 4!! 4 ( ) ( ) u ( ) u( ) 4 6 ( ) ( ) u. 6 ()!! 4!! 6! The eac soluo s gve as u( ) e [8]. Now le us calculae he approae soluo of Eq.() for 6 ad b usg Mulvarae Padé approao. To oba Mulvarae Padé equaos of Eq.() for 6 ad we use Eqs.() ad (). B usg Eqs.() ad () We oba p( ) ( 7 4 6) 68 ad q( ) ( ) So he Mulvarae Padé approao of order (6 ) for eq.() ha s ( 7 4 6) r6 ( ) ( ) 4 () Eaple 4.. Cosder he oe-desoal wave equao wh varable coeffces u ( ) u ( ) () wh al codos u( ) u ( ). The correco fucoal for equao () s gve as[] u u ( ) ( ) ( u ( s ) ) ss ( uɶ ( s ) ) d s where ad codos are u ( ). The he saoar () ( ). (4) Ths ur gves s s s s () Subsug hs value of he Lagrage ulpler o fucoal () gves he erao forula [] u ( ) u ( ) ( s ) ( u ( s)) ss ( u ( s)) ds. (6) Selecg he al approao: u ( ) wh he rao Forula (6) Al ad Rasla obaed [] he followg succesve approaos ( ) ( ) u!!!!! 7!!! 7! 9! ( ) ( ) u 7 ( ) ( ) u 7 9 4( ) ( ) u u!! 7! 9!! 7 9 ( ) ( ) (7) The eac soluo s gve asu( ) sh [8]. Now le us calculae he approae soluo of Eq.() for ad b usg Mulvarae Padé approao. To oba Mulvarae Padé equaos of Eq.() for ad we use Eqs.() ad (). B usg Eqs.() ad () We oba

4 Ves TURUT: Nuercal approaos for solvıg paral dffereal equaos wh varable coeffces p ( ) ( ad ) q ( ) ( ) So he Mulvarae Padé approao of order () for eq.(7) ha s r ( ( ) ) (844( )) (8) Table. Coparso of VIM ad MPA for eaple. Eac soluo Approae Absolue error of soluo whmpa MPA Table. Coparso of VIM ad MPA for eaple. Eac soluo Approae soluo wh- MPA Absolue error of MPA Fgure. Eac soluo of paral dffereal equao eaple Fgure. Mulvarae Padé approao for VIM soluo of paral dffereal equao Eaple. Fgure. Eac soluo of paral dffereal equao eaple.

5 Appled ad Copuaoal Maheacs () : [9] F. Aa Soluos of he Sse of Dffereal Equaos b Dffereal Trasfor Mehod Appled Maheacs ad Copuao (4) 47: [] N. Bldk A. Kouralp F. Bek S. Kucukarsla Soluo of dffere pe of he paral dffereal equao b dffereal rasfor ehod ad Adoa s decoposo ehod Appl. Mah. Copu. (6) 7: 67. [] V. Turu E. Çelk M. Yğder Mulvarae padé approao for solvg paral dffereal equaos (PDE) Ieraoal Joural For Nuercal Mehods I Fluds () 66 (9): 9 7. Fgure 4. Mulvarae Padé approao for VIM soluo of paral dffereal equao Eaple.. Cocluso The fgure whch s obaed usg MPA ad he fgure of he eac soluo hree-desoal are show fgure (-) ad fgure (-4). As ca be see able able ad fgure (-) fgure (-4) he approao soluos wh MPA are que close o eac soluos. I s also observed ha MPA s robus ad applcable o varous pes of paral dffereal equaos. Refereces [] G. Adoa A revew of he decoposo ehod appled aheacs J Mah Aal Appl (988) : 44. [] A. M. Wawa A relable odfcao of Adoa decoposo ehod Appl Mah Copu (999) : [] I. H. Abdel-Hal Hassa Coparso dffereal rasforao echque wh Adoa decoposo ehod for lear ad olear al value probles Chaos Solos Fracals (8) 6: 6. [4] N. Bldk H. Baraoglu The soluo of wo desoal olear dffereal equao b he Adoa decoposo ehod Appled Maheacs ad Copuao () 6: 9 4. [] J. H. He ad X. H.Wu Varaoal erao ehod: ew develope ad applcaos Copu Mah Appl (7) 4: [6] J. H. He Varaoal erao ehod soe rece resuls ad ew erpreaos J Copu Appl Mah (7) 7: 7. [7] S. Moa S. Abuasad ad Z. Odba Varaoal erao ehod for solvg olear boudar value probles Appl Mah Copu (6) 8: 8. [8] A. Yıldırı ad T. Öş Soluos of Sgular IVPs of Lae-Ede pe b he varaoal erao ehod Nolear Aalss Ser A: Theor Mehods Appl (9) 7: [] V. Turu N. Güel Mulvarae padé approao for solvg paral dffereal equaos of fracoal order Absrac ad Appled Aalss () press. [] V. Turu N. Güel Coparg Nuercal Mehods for Solvg Te-Fracoal Reaco-Dffuso Equaos ISRN Maheacal Aalss () do:.4//776. [4] V. Turu Applcao of Mulvarae padé approao for paral dffereal equaos Baa Uvers Joural of Lfe Sceces () (acceped). [] Ph. Gullaue A. Huard Mulvarae Padé Approas Joural of Copuaoal ad Appled Maheacs () :97-9. [6] J.H. He A ew approach o olear paral dffereal equaos. Coucaos Nolear Scece ad Nuercal Sulao (997) :. [7] J. H. He Approae aalcal soluo for seepage flow wh fracoal dervaves porous eda. Copu Mehod Appl Mech Eg (998) 67: [8] Ph. Gullaue Nesed Mulvarae Padé Approas J. Copu. Appl. Mah. (997) 8:49-8. [9] J. H. He Varaoal erao ehod a kd of olear aalcal echque: soe eaples Iera J Nolear Mech (999) 4: [] J. H. He Varaoal erao ehod for auooous ordar dffereal sses Appl Mah Copu () 4 :. [] J. H. He Soe aspoc ehods for srogl olear equaos. I J Mod Phs B (6) : [] J. H. He Wu XH Cosruco of solar soluo ad copac o-lke soluo b varaoal erao ehod. Chaos Solos & Fracals (6) 9: 8. [] J.H. He Varaoal prcples for soe olear paral dffereal equaos wh varable coeffces. Chaos Solos & Fracals (4) 9: 847. [4] A. Cu L. Wuack Nolear Mehods Nuercal Aalss Elsever Scece Publshers B.V. (987) Aserda. [] A.H.A. Al K.R. Rasla Varaoal eraso ehod for solvg paral dffereal eqauaos wh varable coeffces chaos Solos ad Fracals (9) 4:-9.