Solutions for Nonlinear Partial Differential Equations By Tan-Cot Method

Transcription

1 IOSR Journl of Mhemics (IOSR-JM) e-issn: Volume 5, Issue 3 (Jn. - Feb. 13), PP 6-11 Soluions for Nonliner Pril Differenil Equions By Tn-Co Mehod Mhmood Jwd Abdul Rsool Abu Al-Sheer Al -Rfidin Universiy College, Bghdd, Irq Absrc: In his pper, we esblished rveling wve soluion by using he proposed Tn-Co funcion lgorihm for nonliner pril differenil equions. The mehod is used o obin new soliry wve soluions for vrious ype of nonliner pril differenil equions such s, he (1+1)-dimensionl Io equion, Pochhmmer-Chree (PC) equion, MIKP equion, Konopelcheno nd Dubrovsy (KD) sysem of equions which re he imporn Solion equions. Proposed mehod hs been successfully implemened o esblish new soliry wve soluions for he nonliner PDEs. Keywords: Nonliner PDEs, Ec Soluions, Tn-Co funcion mehod. I. INTRODUCTION The ec soluions of nonliner pril differenil equions (NLPDEs) ply n imporn role in he sudy of mny physicl phenomen. Wih he help of ec soluions, when hey eis, he mechnism of compliced physicl phenomen nd dynmicl processes modeled by hese NLPDEs cn be beer undersood. They cn lso help o nlyze he sbiliy of hese soluions nd o chec numericl nlysis for hese NLPDEs. Lrge vrieies of physicl, chemicl, nd biologicl phenomen re governed by nonliner pril differenil equions. One of he mos eciing dvnces of nonliner science nd heoreicl physics hs been he developmen of mehods o loo for ec soluions of nonliner pril differenil equions [1]. Ec soluions o nonliner pril differenil equions ply n imporn role in nonliner science, especilly in nonliner physicl science since hey cn provide much physicl informion nd more insigh ino he physicl specs of he problem nd hus led o furher pplicions. Nonliner wve phenomen of dispersion, dissipion, diffusion, recion nd convecion re very imporn in nonliner wve equions. In recen yers, quie few mehods for obining eplici rveling nd soliry wve soluions of nonliner evoluion equions hve been proposed. A vriey of powerful mehods, such s, nh-sech mehod [], eended nh mehod [3], hyperbolic funcion mehod [4], Jcobi ellipic funcion epnsion mehod [5], F-epnsion mehod [6], nd he Firs Inegrl mehod [7]. The sine-cosine mehod [8] hs been used o solve differen ypes of nonliner sysems of PDEs. In his pper, we pplied he Tn-Co mehod o solve he (1+1)-dimensionl Io equion, Pochhmmer- Chree (PC) equion, MIKP equion, Konopelcheno nd Dubrovsy KD) sysem of equions, given respecively by: u + u + 3 u u + uu + 3u u d = (1) u u ( u b u 3 ) = () u + u u + u u + u + b u yy = (3) u 6 b u u u + 3 u u 3 v y + 3 u v =, u y = v (4) II. THE TAN-COT FUNCTION METHOD Consider he nonliner pril differenil equion in he form [9] F u, u, u, u y, u, u, u y, u yy, = (5) where u(, y, ) is rveling wve soluion of nonliner pril differenil equion Eq. (5). We use he rnsformions, u, y, = f ξ (6) where ξ = + δy λ This enbles us o use he following chnges:. = λ d.,. = d.,. = δ d. (7) dξ dξ y dξ Using Eq. (7) o rnsfer he nonliner pril differenil equion Eq. (5) o nonliner ordinry differenil equion Q f, f, f, f,. = (8) The ordinry differenil equion (8) is hen inegred s long s ll erms conin derivives, where we neglec he inegrion consns. The soluions of mny nonliner equions cn be epressed in he form [9]: 6 Pge

2 Soluions for Nonliner Pril Differenil Equions by Tn-Co Mehod f ξ = α n β μξ, ξ π μ or in he form (9) f ξ = α co β μξ, ξ π μ Where α, μ, nd β re prmeers o be deermined, We use f ξ = α n β μξ f = α β μ [ n β 1 μξ + n β + 1 μξ ] (1) f = α βμ [(β 1) n β μξ + β n β μξ + (β + 1) n β + μξ ] f = βμ 3 α[ β 1 β n β 3 μξ + 3β 3β + n β 1 μξ + β + 1 β + n β μξ + β n β + 1 μξ + (β + 1)(β + ) n β + μξ ] nd heir derivive. Or use f ξ = α co β μξ f = α β μ [ co β 1 μξ + co β + 1 μξ ] f = α βμ [ β 1 co β μξ + β co β μξ + (β + 1) co β + μξ ] (11) nd so on. We subsiue Eq.(1) or Eq.(11) ino he reduced equion (8), blnce he erms of he n funcions when Eq. (1) re used, or blnce he erms of he co funcions when Eq. (11) re used, nd solve he resuling sysem of lgebric equions by using compuerized symbolic pcges. We ne collec ll erms wih he sme power in n μξ or co μξ nd se o zero heir coefficiens o ge sysem of lgebric equions mong he unnown's α, μ nd β, nd solve he subsequen sysem. III. Applicions In his secion we pply he Tn-Co mehod o differen nonliner pril differenil equions: 1. The (1+1)-dimensionl Io equion Consider The (1+1)-dimensionl Io equion [1]: u + u + 3 u u + uu + 3u u d = (1) Assume: u, = v (, ) (13) Then Eq.(1) cn be wrien s: v + v + 3 v v + v v + 3v v = (14) we inroduce he rnsformions ξ = λ (15) where, nd λ re rel consns. Equion (14) becomes λ v 3 v (5) 3 [v ] = (16) Inegring Eq. (16) wice wih zero consns, we ge λ v 3 v 3 v = (17) Le w(ξ) = v (ξ) (18) Eq.(18) becomes λ w 3 w 3 w = (19) Applying he n funcion mehod s in Eq.(1), hen Eq.(19) becomes λ α n β μξ 3 α βμ [(β 1) n β μξ + β n β μξ + (β + 1) n β + μξ ] 3 α n β μξ = () Then Eq.() cn wrien s λ n β μξ 3 βμ [(β 1) n β μξ + β n β μξ + (β + 1) n β + μξ ] 3 α n β μξ = (1) Blncing he eponens β + nd β hen β + = β nd we ge β = Subsiue β = in Eq. (1) o ge he vlue of : λ = 8 3 μ, α = μ Then : w(ξ) = μ n μξ () Inegre Eq.() for ξ o ge: v(, ) = μ [n μ λ μ( λ)] (3) From Eq.(3) we ge: u(, ) = v = μ n μ 8 μ (4) 7 Pge

3 u(,) Soluions for Nonliner Pril Differenil Equions by Tn-Co Mehod For μ = = 1, Eq.(4) becomes: u(, ) = n 8 u, in (5) is represened in Figure (1) for 1 1 nd 1. (5) Figure (1) represen u, in (5) for 1 1 nd 1.. Pochhmmer-Chree (PC) equion Consider he Pochhmmer-Chree (PC) equion [11] u u ( u b u 3 ) = (6) We inroduce he rnsformion ξ = ( λ), where, nd λ re rel consns. Equion (6) rnsforms o he ODE: λ u λ u (4) ( u b u 3 ) = (7) Inegring Eq.(7) wice wih zero consn o ge he following ordinry differenil equion: λ u λ u ( u b u 3 ) = (8) Seeing he soluion in Eq.(11) λ α co β μξ λ α βμ [ β 1 co β μξ + β co β μξ + (β + 1) co β + μξ ] α co β μξ + b α 3 co 3β μξ = (9) Equing he eponens nd he coefficiens of ech pir of he co funcions we find he following lgebric sysem: 3β = β + β = 1 (3) Subsiuing Eq. (3) ino Eq. (9) o ge: λ co μξ λ μ [ co μξ + co 3 μξ ] co μξ + b α co 3 μξ = (31) Equing he eponens nd he coefficiens of ech pir of he co funcion, we obin sysem of lgebric equions: co μξ λ λ μ = co 3 μξ λ μ + b α = (3) By solving he lgebric sysem (3), we ge, λ = 1 μ, α = b 1 μ μ (33) Then by subsiuing Eq.(33) ino Eq.(11), he ec solion soluion of equion (6) cn be wrien in he form u, = b 1 μ μ co μ ( For, μ = = ε = 1, hen (34) becomes: u, = b ), < μ ( 1 μ 1 μ ) < π (34) co ( ), <, b > (35) For, = 1, b = 1, hen (35) becomes: u, = co( ) (36) u, in (36) is represened in Figure () for 1 1 nd 1. 8 Pge

4 u(,) Soluions for Nonliner Pril Differenil Equions by Tn-Co Mehod Figure (). Represens u, in (36) for 1 nd MIKP equion Consider he MIKP equion [1] u + u u + u u + u + b u yy = (37) Le us now solve Eq.(37) by he proposed mehod. We inroduce he rnsformion u, y, = U(ξ), ξ = + ly λ (38), where, nd λ re rel consns. Equion (38) rnsforms o he ODE: λ u + [ u u + u u ] + 4 u + b l u = (39) Eq.(39) cn be wrien s λ u + 3 [ u3 ] + 4 u + b l u = (4) Inegring (4) once wih zero consn o ge he following ordinry differenil equion: (b l λ ) u + 3 u3 + 4 u = (41) Seeing he soluion in (11) b l λ α co β μξ + 3 α3 co 3β μξ + 4 α βμ [ β 1 co β μξ + β co β μξ + β + 1 co β + μξ ] = (4) Equing he eponens nd he coefficiens of ech pir of he co funcions we find he following lgebric sysem: 3β = β + β = 1 b l λ co μξ + 3 α co 3 μξ + 4 μ [ co μξ + co 3 μξ ] = (43) co 3 μξ 3 α + 4 μ = co 1 μξ b l λ + 4 μ = (44) By solving he lgebric sysem (44), we ge, λ b l α = i 3 λ b l, μ = (45) 4 Then by subsiuing Eq. (45) ino Eq. (11), he ec solion soluion of equion (37) cn be wrien in he form u, = i 3 λ b l λ b l co + ly λ, < λ b l ly λ < π (46) For λ = = l = 1, = b = 1/ Eq.(46) becomes u, = i 3 co y (47) u, in (47) is represened in figure (3) for 1 nd y = 1, nd Pge

5 u(,) Soluions for Nonliner Pril Differenil Equions by Tn-Co Mehod Figure (3). Represens u, in (46) for 1 nd Konopelcheno-Dubrovsy (KD) equion Konopelcheno nd Dubrovsy (1984) presened he Konopelcheno-Dubrovsy (KD) equion[13] u 6 b u u u + 3 u u 3 v y + 3 u v = (48) u y = v (49) where nd b re rel prmeers. Equions (48), nd (49) is new nonliner inegrble evoluion equion on wo spil dimensions nd one emporl. This equion ws invesiged by he inverse scering rnsform mehod. The F-epnsion mehod is lso used in Wng nd Zhng [14] o invesige he KD equion. To solve Eqs.(48), nd (49) by he proposed mehod. We inroduce he rnsformion ξ = ( + ly λ), where, l, nd λ re rel consns. Equion (48) rnsforms o he ODE: λ u 6 b u u 3 u + 3 u u 3 l v + 3 u v = (5) And Eq.(49) rnsforms o: v = l u Inegring Eq.(51) wih zero consn v = l u (5) Subsiue Eqs.(51) nd (5) in Eq.(5) o ge λ 3 l u 3 u 3 b l [ u ] + 1 u 3 = (53) Inegring Eq(53) once wih zero consn o ge λ + 3 l u 3 u 3 b l u + 1 u 3 = (54) Seeing he soluion in (1) λ + 3 l α n β μξ 3 α βμ [(β 1) n β μξ + β n β μξ + (β + 1) n β + μξ ] 3 b l α n β μξ + 1 α 3 n 3β μξ = (55) From (55), equing eponens β + nd 3β yield β + = 3β, so h β = 1 (56) hen Eq.(56) becomes: λ + 3 l n μξ 3 μ [n μξ + n 3 μξ ] 3 b l α n μξ α n 3 μξ = (57) Blncing he sme eponens o give: n μξ : λ + 3 l 3 μ = n μξ : 3 b l α = n 3 μξ : 3 μ + 1 α = (58) By solving he lgebric sysem (58), we ge, λ = 6 b + μ b, l =, α = μ (59) Then by subsiuing Eq. (59) ino Eq. (1), he ec solion soluion of equion (54) cn be wrien in he form u, = μ n [ μ( + b y + 6 b + μ ) ] (6) 5 1 (51) 1 Pge

6 u(,) Soluions for Nonliner Pril Differenil Equions by Tn-Co Mehod bμ v, = 4 n [ μ( + b y + 6 b + μ ) ] (61) for μ = = = b = 1 u, = n [ ( + y + 14 ) ] (6) v, = 4 n [ ( + y + 14 ) ] (63) Figures (4) represens u, in (6), for 1, y = 1, nd Figure (4) represens u, in (61) for 1 nd.1 1. IV. Conclusion In his pper, he Tn-Co funcion mehod hs been implemened o esblish new soliry wve soluions for vrious ypes of nonliner PDEs. We cn sy h he new mehod cn be eended o solve he problems of nonliner pril differenil equions which rising in he heory of solions nd oher res. References [1] Mrwn Alqurn, Kmel Al-Khled, Hsn Annbeh (11). New Solion Soluions for Sysems of Nonliner Evoluion Equions by he Rionl Sine-Cosine Mehod. Sudies in Mhemicl Sciences, Vol. 3, No. 1, pp [] Mlflie, W. (199). Soliry wve soluions of nonliner wve equions. Am. J. Phys, Vol. 6, No. 7, pp [3] El-Wil, S.A, Abdou, M.A. (7). New ec rvelling wve soluions using modified eended nh-funcion mehod, Chos Solions Frcls, Vol. 31, No. 4, pp [4] Xi, T.C., Li, B. nd Zhng, H.Q. (1). New eplici nd ec soluions for he Nizhni- Noviov-Vesselov equion, Appl. Mh. E-Noes, Vol. 1, pp [5] Inc, M., Ergu, M. (5). Periodic wve soluions for he generlized shllow wer wve equion by he improved Jcobi ellipic funcion mehod, Appl. Mh. E-Noes, Vol. 5, pp [6] Zhng, Sheng. (6). The periodic wve soluions for he (+1) -dimensionl Konopelcheno Dubrovsy equions, Chos Solions Frcls, Vol. 3, pp [7] Feng, Z.S. (). The firs ineger mehod o sudy he Burgers-Koreweg-de Vries equion, J Phys. A. Mh. Gen, Vol. 35, No., pp [8] Michell A. R. nd D. F. Griffihs (198), The Finie Difference Mehod in Pril Differenil Equions, John Wiley & Sons. [9] Anwr J. M. Jwd, (1), New Ec Soluions of Nonliner Pril Differenil Equions Using Tn-Co Funcion Mehod, Sudies in Mhemicl Sciences Vol. 5, No., pp [1] A.M. Wzwz, Muliple-solion soluions for he generlized (1+1)-dimensionl nd he generlized (+1)-dimensionl Io equions, Appl. Mh. Compu. (8): [11] Bio Li, Yong Chen, nd Hongqing Zhng, Trvelling Wve Soluions for Generlized Pochhmmer-Chree Equions, Verlg der Zeischrif fu r Nurforschung, Tu bingen " [1] N. Tghizdeh nd S.R. Moosvi Noori, Two Relible Mehods for Solving he Modified Improved Kdomsev-Pevishvili Equion, Applicions nd Applied Mhemics Vol. 7, Issue (December 1), pp [13] N. Tghizdeh nd M. Mirzzdeh, Modificion of Trunced Epnsion Mehod for Solving Some Imporn Nonliner Pril Differenil Equions, Applicions nd Applied Mhemics Vol. 7, Issue (December 1), pp [14] Wng, D., Zhng, H.-Q. (5). Furher improved F-epnsion mehod nd new ec soluions of Konopelcheno-Dubrovsy equion, Chos, Solions nd Frcls, Vol. 5, pp Pge