Exact Solutions for Some Nonlinear Partial Differential Equations in Mathematical Physics
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1 ISSN Englnd K Journl of Informion nd Compuing Sin Vol. 6 No. pp. 9- E Soluions for Som Nonlinr Pril Diffrnil Equions in Mhmil Phsis A.R. Shh + E.M.E.Zd. * nd K.A.prl Mhmis Dprmn Ful of Sin Tif nirsi El-Tif El- Hih P.O.Bo Kingdom of Sudi Ari Mhmis DprmnFul of Sin El-Mini nirsi El-Mini Egp. Mhmis Dprmn Ful of Sin Zgzig nirsi Zgzig Egp. Rid Dmr pd Dmr Asr. In his ril inroduing n gnrl nsz h improd / - pnsion - mhod is proposd o onsru soluions of som nonlinr pril diffrnil quions in mhmil phsis i h gnrlizd Zhro quions h oupld Mris quions h +- dimnsionl Wu-Zhng quions nd h + dimnsionl Fornrg Whihm quion in rms of h hproli funions rigonomri funions nd rionl funion hr sisfis sond ordr linr ordinr diffrnil quion. Whn h prmrs r n spil lus h solir r drid from h rling s. This mhod is rlil simpl nd gis mn n soluions. / Kords: Th improd - pnsion mhod Trling soluions Th gnrlizd Zhro quions Th oupld Mris quions Th + dimnsionl Fornrg Whihm quion Th +-dimnsionl Wu-Zhng quions.. Inroduion Nonlinr pril diffrnil quions r non o dsri id ri of phnomn no onl in phsis hr ppliions nd or mgno fluid dnmis r surf gri s lromgni rdiion rions nd ion ousi s in plsm jus o nm f u lso in iolog nd hmisr nd srl ohr filds. I is on of h imporn ss in h sud of h nonlinr pril diffrnil quions o s nd plii soluions. In h ps srl dds oh mhmiins nd phsiiss h md mn mps in his dirion. Vrious mhods for oining soluions o nonlinr pril diffrnil quions hd n proposd. Among hs r h inrs sring mhod [] Hiro s ilinr mhod [] Blund rnsformion [] Pinlépnsion [5] sin osin mhod [6] homognous ln mhod [7] homoop prurion mhod [ ] riion mhod [] Adomin domposiion mhod [5] nh - funion mhod [6 ] Joi llipi funion pnsion mhod [9 ] F-pnsion mhod [ 5] nd Ep-funion mhod [6 ]. Wng l [9] proposd n mhod lld h / pnsion mhod o loo for h rling soluions for nonlinr pril diffrnil quions NPDEs. B using h / pnsion mhod Zd l [] nd h modifid / pnsion mhod Shh [] h sussfull oind mor rling soluions for som imporn NPDEs. Rnl uo l [] hd dlopd h / pnsion mhod for soling h NPDEs. In his ppr us h impromn / pnsion mhod o find h rling soluions for h gnrlizd Zhro quions h oupld Mris quions h +-dimnsionl Wu-Zhng quions nd h + dimnsionl Fornrg Whihm quion.. Dsripion of h impromn / pnsion mhod for NPDEs + Corrsponding uhor. E-mil ddrss: shh@hoo.om * E-mil ddrss: gprl@hoo.om Pulishd World Admi Prss World Admi nion
2 A.R. Shh l: E Soluions for Som Nonlinr Pril Diffrnil Equions in Mhmil Phsis In his sion gi h dild dsripion of our mhod. Suppos h nonlinr oluion quion s in o indpndn rils nd is gin P u u u u u u.. hr u u is n unnon funion P is polnomil in u u nd is rious pril driis in hih h highs ordr driis nd nonlinr rms r inold. To drmin u pliil h folloing fi sps []: Sp : W us h folloing rlling rnsformion: u hr is onsn o drmind lr. Th NPDE is rdud o nonlinr ordinr diffrnil quion NODE in : Sp. W suppos h folloing sris pnsion s soluion of Eq. : P.... i n m i n i im n n hr i i... m r onsns o drmind lr m is posii ingr nd sisfis sond ordr linr ordinr diffrnil quion 5 hr is rl onsns. Th gnrl soluions of Eq. 5 n lisd s follos. Whn oin h hproli funion soluion of Eq.5 C osh C sinh. 6 Whn oin h rigonomri funion soluion of Eq.5 C sin C os. 7 Whn oin h rionl funion soluion of Eq.5 hr C nd C r rirr onsns. C C. Sp. Drmin h posii ingr m lning h highs ordr nonlinr rms nd h highs ordr drii in Eqs. or. Sp. Susiuing Eq. long ih 5 ino lning h dnominor nd hn sing ll h offiins of / i i.. o zro ild s of lgri quions for hih h onsns i... m nd. i Sp 5. Assuming h h onsns i i... m nd n oind soling h lgri quions in Sp hn susiuing hs onsns nd h non gnrl soluions of Eq. 5 ino n oin h plii soluions of Eq. immdil.. Appliions of h improd / pnsion mhod for NPDEs In his sion ppl h improd / - pnsion mhod o onsru h rling JIC mil for onriuion: dior@ji.org.u
3 Journl of Informion nd Compuing Sin Vol. 6 No. pp 9- JIC mil for susripion: pulishing@wa.org.u soluions for som nonlinr PDEs i h gnrlizd Zhro quions h oupld Mris quions h +-dimnsionl Wu-Zhng quions nd h + dimnsionl Fornrg Whihm quion hih r r imporn in h mhmil phsis nd h n pid nion mn rsrhrs... Empl. Th gnrlizd- Zhro quions In his sion h gnrlizd- Zhro quions for h ompl nlop [] rds: i 9 hr is nonzro onsn. L us ssum h rling soluion of Eqs 9 in h form: V i hr V r rl funions nd r onsns o drmind lr. Susiuing ino Eqs.9 h:. V V B lning h highs ordr drii rms nd nonlinr rms in Eqs. suppos h Eqs. on h soluions in h folloing forms: V hr sisfis Eq.5 nd r onsns o drmind lr. Susiuing Eqs. long ih 5 ino Eqs. nd lning h dnominor nd olling ll rms ih h sm ordr of / oghr h lf hnd sid of Eqs. r onrd ino polnomils in /. Sing h offiin of hs polnomils o zro dri s of lgri quions for nd. Soling h s of lgri quions using Mpl or Mhmi h Cs. hr nd r rirr onsns.
4 A.R. Shh l: E Soluions for Som Nonlinr Pril Diffrnil Equions in Mhmil Phsis Cs. hr nd r rirr onsns. No h hr r ohr ss hih r omid hr. W jus lis som soluions orrsponding o ss o illusr h ffinss of h improd pnsion mhod. / sing s nd h gnrl soluions of Eq.5 n find h folloing rling soluions of h gnrlizd- Zhro quions 9. Whn oin h hproli funion soluions of Eq.9 i C osh C sinh C sinh C osh nd V [ C {osh [ C {osh sinh sinh [ C [ C sinh } C sinh } C C C {sinh {sinh osh osh ] ] osh }] osh }] 5 hr. In priulr sing C C h folloing solir soluions of gnrlizd- Zhro quions 9 r disord i nh nd V nh. 6 Sing gin C C C hn h solir soluions of gnrlizd- Zhro quions 9 h folloing form: i oh[ ] JIC mil for onriuion: dior@ji.org.u
5 Journl of Informion nd Compuing Sin Vol. 6 No. pp 9- nd V oh [ ] 7 C hr nh. I is s o s h if C C C nd r n s ohr spil lus in propr mor solir soluions of Eq. 9 n oind hr omi hm for simplii. Whn g h rigonomri funion soluions of Eq.9 sin os i C C C os C sin nd V [ C {sin os [ C } C os {os C sin }] sin ] [ C{sin os } C{os sin }] [ C os C sin ] In priulr sing C C h folloing solir soluions of gnrlizd- Zhro quions 9 r disord i o nd V o 9 Sing gin C C C hn h solir soluions of gnrlizd- Zhro quions 9 h folloing form: i n nd V n C hr n. C Whn g h soluions of Eq.9 h folloing form: i C C C JIC mil for susripion: pulishing@wa.org.u
6 A.R. Shh l: E Soluions for Som Nonlinr Pril Diffrnil Equions in Mhmil Phsis nd C C V. C In h s nd h gnrl soluions of Eq.5 n find h folloing rling soluions of h gnrlizd- Zhro quions 9. Whn oin h hproli funion soluions of Eq.9 [ C sinh C osh ] [ C{osh sinh } C{sinh i[ ] osh }] nd V [ C {osh [ C {osh sinh sinh [ C sinh [ C sinh C } C C } C osh {sinh osh ] {sinh ] osh }] osh }] Whn g h rigonomri funion soluions of Eq.9 [ C os C sin ] [ C {sin os } C {os i[ ] sin }] nd V [ C {sin os [ C {sin os [ C os [ C os C } C C } C sin ] {os sin ] {os sin sin }] }]. hr. I is s o s h if C C nd r n s ohr spil lus in propr mor solir soluions of Eq. 9 n oind hr omi hm for simplii... Empl. Th oupld Mris quions In his susion sud h oupld Mris quions []. iq Q QR R R Q. In ordr o s h soluions of Eqs. suppos JIC mil for onriuion: dior@ji.org.u
7 Journl of Informion nd Compuing Sin Vol. 6 No. pp 9-5 Q u [ i l] 5 hr nd l r onsns o drmind lr. Susiuing Eq.5 ino Eqs. h : i u R u R u uu. u ur 6 W us h folloing rling rnsformions u R V 7 hr nd r onsns o drmind lr is n onsn Eqs. 6 om h folloing NODEs: V V B lning h highs ordr drii rms nd nonlinr rms in Eqs. suppos h Eqs. on h soluions. Susiuing Eqs. long ih 5 ino Eqs. nd lning h dnominor nd olling ll rms ih h sm ordr of / oghr h lf hnd sid of Eqs. r onrd ino polnomils in /. Sing h offiin of hs polnomils o zro dri s of lgri quions for l nd. Soling h s of lgri quions using Mpl or Mhmi h Cs. 9 hr nd r rirr onsns. Cs. hr nd r rirr onsns. sing s 9 nd h gnrl soluions of Eq.5 n find h folloing rling soluions of h oupld Mris quions. Whn oin h hproli funion soluions of Eq. JIC mil for susripion: pulishing@wa.org.u
8 6 A.R. Shh l: E Soluions for Som Nonlinr Pril Diffrnil Equions in Mhmil Phsis Q [ C {osh [ i l] [ C sinh C osh ] sinh } C{sinh osh }] nd R [ C {osh [ C{osh [ C sinh C osh ] sinh } C {sinh osh [ C ] sinh C osh sinh } C{sinh osh }] }] hr [ ]. In priulr sing C C h folloing solir soluions of h oupld Mris quions r disord [oh ] Q [ oh }] [ i l] nd R oh oh oh [ oh ] Whn oin h hproli funion soluions of Eq. Q [ C{sin [ C os C sin ] os } C{os sin [ i l] }] 5 nd R [ C os C sin ] [ C {sin os } C {os sin [ C ] os C sin [ C{sin os } C{os sin }] }] 6 In priulr sing C C h folloing solir soluions of h oupld Mris quions r disord n ] Q [ i l] 7 [ n ] JIC mil for onriuion: dior@ji.org.u
9 Journl of Informion nd Compuing Sin Vol. 6 No. pp 9- JIC mil for susripion: pulishing@wa.org.u 7 nd ] n [ n ] n [ n R hr ] [... Empl. Th +-dimnsionl Wu-Zhng quions In his susion sud h +-dimnsionl Wu-Zhng quions [56].. u u u u u u uu u 9 L us ssum h rling soluions of Eqs 9 in h folloing forms: W V u hr is n rirr onsn. Susiuing ino Eqs. 9 h: L V W V W W VV V V W V hr L is h ingrion onsn. B lning h highs ordr drii rms nd nonlinr rms in Eqs. suppos h Eqs. on h soluions in h folloing forms:. W V hr nd r onsns o drmind lr. Susiuing Eqs. long ih 5 ino Eqs. nd lning h dnominor nd olling ll rms ih h sm ordr of / oghr h lf hnd sid of Eqs. r onrd ino polnomils in /. Sing h offiin of hs polnomils o zro dri s of lgri quions for L nd. Soling h s of lgri quions using Mpl or Mhmi h
10 Cs. A.R. Shh l: E Soluions for Som Nonlinr Pril Diffrnil Equions in Mhmil Phsis L hr nd r rirr onsns. Cs L sing s nd h gnrl soluions of Eq.5 n find h folloing rling soluions of h +-dimnsionl Wu-Zhng quions 9. Whn oin h hproli funion soluions of Eqs.9 [ C sinh C osh ] [ C{osh sinh } C{sinh osh }] 5 [ C{osh sinh } C{sinh osh }] [ C sinh C osh ] V 6 [ C{osh [ C sinh C osh ] [ C{osh sinh } C{sinh osh sinh } C{sinh [ C sinh C osh osh }] ] }] 6 JIC mil for onriuion: dior@ji.org.u
11 Journl of Informion nd Compuing Sin Vol. 6 No. pp 9-9 nd 6 W [ ][ C sinh C osh ] [ C{osh sinh } C{sinh osh }] [ C sinh C osh ] [ C{osh sinh } C{sinh osh }] [ C{osh sinh } C{sinh osh }] [ C sinh C osh ] 7 [ C{osh sinh } C{sinh osh }] [ C sinh C osh ] hr. In priulr sing C C h folloing solir soluions of h +-dimnsionl Wu-Zhng quions r disord oh ] [nh ] oh oh V 6 [nh ] 9 [ oh }] nd 6 W [ ] oh [ oh }] oh [nh [ oh }] ] [nh ] 5 Whn g h rigonomri funion soluions of Eqs.9 [ C{sin os [ C{sin os [ C os [ C os } C{os } C{os C sin ] C sin ] sin sin }] }] 5 V 6 [ C{sin os [ C{sin os } C{os } C{os [ C os C sin ] [ C os C sin ] sin }] sin }] 5 nd JIC mil for susripion: pulishing@wa.org.u
12 A.R. Shh l: E Soluions for Som Nonlinr Pril Diffrnil Equions in Mhmil Phsis 6 W [ ][ C os C sin ] [ C{sin os } C{os sin }] [ C os C sin ] [ C{sin os } C{os sin }] [ ][ C{sin os } C{os sin }] [ C os C sin ] [ C{sin os } C{os sin }]. [ C os C sin ] 5 In priulr sing C C h folloing solir soluions of h +-dimnsionl Wu- Zhng quions r disord n ] [o ] 5 [ n }] n V 6 [o ] 55 [ n }] nd 6 W [ ]n [ n ] n [ [ n ] ][o [o ]. B h similr mnnr n oin h soluions for h s. W omid h rsuls of s. for onnin... Empl Th + dimnsionl Fornrg Whihm quion In his susion sud h + dimnsionl Fornrg Whihm quion [7]. u u u uu uu u u. 57 Th rling rnsformion prmis us onring Eq.57 o h folloing ODE:. 5 B lning h highs ordr drii rms nd nonlinr rms in Eqs. 5 g 59 On susiuing Eq.59 long ih 5 ino Eqs. 5 nd lning h dnominor nd olling ll rms ih h sm ordr of / oghr h lf hnd sid of Eq. 5 r onrd ino ] 56 JIC mil for onriuion: dior@ji.org.u
13 Journl of Informion nd Compuing Sin Vol. 6 No. pp 9- polnomil in /. Sing h offiin of his polnomil o zro dri s of lgri quions for nd. Soling h s of lgri quions using Mpl or Mhmi h 6 6 hr nd r rirr onsns. Consqunl h rling soluion s h folloing form: [ ]{ Asinh B osh } 6 B A osh A B sinh hr A B r rirr onsns nd.. Conlusion Th proposd mhod in his ppr is mor gnrl hn h nsäz in '/-pnsion mhod [] nd modifid '/- pnsion mhod []. If s h prmrs in. nd.5 o spil lus h o o mhods n rord our proposd mhod. Thrfor h n mhod is mor porful hn h '/-pnsion mhod nd modifid '/- pnsion mhod nd som n ps of rlling soluions nd solir soluions ould pd for som PDEs. This mhod is onis ffi nd n pplid o ohr nonlinr oluion quions in mhmil phsis. 5. Anoldgmn This or is prill suppord Tif nirsi Sudi Ari undr h rn No Th uhors ish o hn Tif nirsi for his suppor. 6. Rfrns [] M.J. Aloiz nd P.A. Clrson. Solions nonlinr Eoluion Equions nd Inrs Sring Trnsform. Cmridg: Cmridg ni. Prss 99. [] R.Hiro. E soluion of h KdV quion for mulipl ollisions of soluions. Phs. R. Lrs. 97 7: 9-9. [] M.R.Miur. Blund Trnsformion. Brlin: Springr-Vrlg 97. [] C.Rogrs nd W.F.Shdi. Blund Trnsformions. N Yor: Admi Prss 9. [5] J.Wiss M.Tor nd.rnll. Th Pinl propr for pril diffrnil quions. J.Mh.Phs. 9 : 5-56 [6] D.S.Wng Y.J.Rn nd H.Q.Zhng. Furhr ndd sinh-osh nd sin-os mhods nd n non rling soluions of h +-dimnsionl disprsi long quions. Appl. Mh.E-Nos. 5 5: [7] M.L. Wng. E soluions for ompound KdV-Burgrs quion. Phs. L. 996 A : [] J.H. H. Th homoop prurion mhod for nonlinr osillors ih disoninuiis. Appl. Mh. Compu. 5: 7-9. [9] J.H. H. Comprison of homoop prurion mhod nd homoop nlsis mhod. Appl. Mh. Compu. 56: [] J.H. H. Homoop prurion mhod for ifurion of nonlinr quions. In. J. Nonlinr Si. Numr. Simul. 5 6: 7-. [] E.M.E.Zd T.A. Nofl nd K.A.prl. Th homoop prurion mhod for soling nonlinr Burgrs nd n oupld MKdV quions. Zishrif fur Nurforshung. 6: [] H.M. Liu. nrlizd riionl prinipls for ion ousi plsm s H s smi-inrs mhod. Chos Solions & Frls. 5 : [] H.M. Liu. Vriionl pproh o nonlinr Elrohmil ssm. In. J. Nonlinr Si. Numr. Simul. 5: [] E. Bolin J. Bizr nd A.R.Vhidi. A n ompuionl mhod for Lpl rnsforms domposiion mhod. Appl. Mh. Compu. 5: -6. JIC mil for susripion: pulishing@wa.org.u
14 A.R. Shh l: E Soluions for Som Nonlinr Pril Diffrnil Equions in Mhmil Phsis [5] E.Bolin J.Bizr nd A.R.Vhidi. Soluion of ssm of nonlinr quions Adomin domposiion mhod. Appl. Mh. Compu. 5: 7-5. [6] H.A. Aduslm. On n improd ompl nh -funion mhod. In. J. Nonlinr Si.Numr. Simul. 5 6: [7] E.M.E.Zd Hssn A.Zdn nd Khld A. prl. roup nlsis nd modifid ndd Tnh- funion o find h inrin soluions nd solion soluions for nonlinr Eulr quions. In. J. Nonlinr Si. Numr. Simul. 5: -. [] L. D-Shng. Fng nd Z. Hong-Qing. Soling h + dimnsionl highr ordr Bror-Kup ssm i rnsformion nd nh- funion mhod. Chos Solions & Frls. : -5. [9] Y. Chn nd Q. Wng. Endd Joi llipi funion rionl pnsion mhod nd undn fmilis of Joi llipi funions soluions o + dimnsionl disprsi long quion. Chos Solions nd Frls. 5 : [] S.Liu Z. Fu S.D. Liu nd Q. Zho. Joi llipi funion pnsion mhod nd priodi soluions of nonlinr quions. Phs. Lrs A. 9: [] D. Lu. Joi llipi funion soluions for o rin Boussinsq quions. Chos Solions nd Frls. 5 : 7-5. [] E.M.E. Zd H.A. Zdn nd K.A. prl. On h solir soluions for nonlinr Hiro Ssum oupld KdV of quions. Chos Solions Frls. : 5. [] M.A. Adou. Th ndd F-pnsion mhod nd is ppliions for lss of nonlinr oluion quion. Chos Solions nd Frls. 7 : [] M.Wng nd X. Li. Appliions of F-pnsion o priodi soluions for n Hmilonin mpliud quion. Chos Solions nd Frls. 5 : [5] S. Zhng nd T.C. Xi. A gnrlizd F-pnsion mhod nd n soluions of Konoplhno- Duros quions. Appl. Mh. Compu. 6 : 9-. [6] J.H. H nd X.H. Wu. Ep-funion mhod for nonlinr quions. Chos Solions nd Frls. 6 : 7-7. [7] S. Zhng. Appliion of Ep-funion mhod o highr dimnsionl nonlinr oluion quion. Chos Solions nd Frls. : [] S. Zhng. Appliion of Ep-funion mhod o Rii quion nd n soluions ih hr rirr funions of Bror- Kup- Kuprshmid quions. Phs. Lrs A. 7: 7-. [9] M.Wng X.Li nd J.Zhng. Th / - pnsion mhod nd rling soluions of nonlinr oluion quions in mhmil phsis. Phs.Lrs A. 7: 7-. [] E.M.E.Zd nd K.A.prl. Th / - pnsion mhod for finding rling soluions of nonlinr PDEs in mhmil phsis. J. Mh. Phs. 9 5: 5-5. [] E.M.E.Zd nd Khld A.prl. Som ppliions of h / pnsion mhod o non-linr pril diffrnil quions. Appl. Mh. nd Compu. 9 : -. [] A.R. Shh. Th rling soluions of h prurd nonlinr Shrödingr quion nd h ui quini inzurg Lndu quion using h modifid /-pnsion mhod. Applid Mh. Compuion. 7: -. [] S. uo Y. Zhou nd C. Zho. Th improd '/- pnsion mhod nd is ppliions o h Bror Kup quions nd pproim long r quions. Applid Mh. Compuion. 6: [] M. A. Adou. An ndd Rii quion rionl pnsion mhod nd is ppliion. In. Journl Nonlinr Sin. 9 7: [5] Z.Y.M. Homoop prurion mhod for h Wu-Zhng quion in fluid dnmis. J. Phs.: Conf. Sr. 96: -. [6] X. Ji C. Chn J.E. Zhng nd Y. Li. Li smmr nlsis nd som n soluions of h Wu-Zhng quion. Journl Mh. Phs. pp.-6. [7] L. Tin nd Y. o. Th glol ror of h isous Fornrg - Whihm quion. Nonlinr Anlsis. 9 7: JIC mil for onriuion: dior@ji.org.u
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