Solitons Solutions to Nonlinear Partial Differential Equations by the Tanh Method

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1 IOSR Journal of Mahemaics (IOSR-JM) e-issn: 7-7,p-ISSN: 319-7X, Volume, Issue (Sep. - Oc. 13), PP 1-19 Solions Soluions o Nonlinear Parial Differenial Equaions by he Tanh Mehod YusurSuhail Ali Compuer Science Deparmen, Al-Rafidain Universiy College, Iraq Absrac:The Tanh mehod is implemened for he eac soluions of some differen kinds of nonlinear parial differenial equaions. New soluions for nonlinear equaions such as Benjamin-Bona-Mahony (BBM) equaion, Gardner equaion,cassama-holm equaion, and wo componen Kdv evoluionary sysem are obained. Keywords:Tanh mehod, Benjamin-Bona-Mahony (BBM) equaion, Gardner equaion, Cassama-Holm equaion I. INTRODUCTION Non-linear evoluion equaions (NLEEs) had come a long way hrough. These NLEEs appear in various areas of Physics, Engineering, Biological Sciences, Geological Sciences and many oher places [1 1]. These equaions arise of necessiy. Subsequenly hey are sudied in hese various scienific cones. There are various aspecs of hese NLEEs ha are sudied by various scieniss and engineers as he need arises. Some of he commonly sudied aspecs are inegrabiliy, conservaion laws,numerical soluions and many oher aspecs. II. OUTLINE OFTHE TANH METHOD: The mehod is applied o find ou eac soluions of a coupled sysem of nonlinear differenial equaion wih unknown: P u, u, u, u,.. = (1) where P is a polynomial of he variable u and is derivaives. If we consider u(, ) = u(ξ), and ξ = k + λ, so ha u(, ) = U ξ, we can use he following changes: = k d, d dξ = k, 3 d3 dξ 3 = k3 = λ d, dξ dξ 3() and so on, hen Eq. (1) becomes an ordinary differenial equaion Q U, U, U, U,.. = (3) wih Q being anoher polynomial form, which will be called he reduced ordinary differenial equaion. Inegraing Eq.(3) as long as all erms conain derivaives, he inegraion consans are considered o be zeros in view of he localized soluions. However, he nonzero consans can be used and handled as well. Now finding he raveling wave soluion o Eq.(1) is equivalen o obaining he soluion o he reduced ordinary differenial equaion (3). The anh mehod is inroduced by Malflie and Wazwaz [1,3,and]. I is based on a priori assumpion ha he raveling wave soluions can be epressed in erms of he anh funcion o solve he coupled KdV equaions. For he anh mehod, we inroduce he new independen variable Y, = anh(ξ)() ha leads o he change of variables: d dξ = (1 Y ) d dy d d = Y 1 Y dξ dy + 1 Y dy d 3 = 1 dξ 3 Y 3Y 1 d d 3 d3 Y 1 Y dy dy + 1 Y dy 3() Then he soluion is epressed in he form u, = U ξ = m i= a i Y i () where he parameer m can be found by balancing he highes-order linear erm wih he nonlinear erms in Eq.(3), and k, λ, a, a 1,, a m are o be deermined. Subsiuing Eq.() or Eq.() ino Eq.(3) will yield a se of algebraic equaions for k, λ, a, a 1,, a m because all coefficiens of Y i have o vanish for i=,1,,3... From hese relaions k, λ, a, a 1,, a m can be obained. Having deermined hese parameers, we can obain he analyic soluion u(, ) in a closed form. These mehods seem o be powerful ool in dealing wih coupled nonlinear physical models. d 1 Page

2 u(,) Solions Soluions o Nonlinear Parial Differenial Equaions by he Tanh Mehod III. APPLICATIONS: 1. Benjamin-Bona-Mahony (BBM) equaion: Consider he following BBM equaion [11], u = u u uu (7) Le u(, ) = u(ξ)and by using he wave variable ξ = k + λ,equaion (7) urns o be he following ordinary differenial equaion: λu λk u + ku + kuu = () by inegraing equaion () once wih zero consan, we have λ + k u λk u + k u = (9) Applying Tanh Mehod, eq.(9) becomes λ + k u λk Y 1 Y du + 1 dy Y d u + k dy u = (1) Where he soluion can be formed as follows: u, = m i= a i Y i (11) Where he parameer m can be deermined by balancing he highes order of linear and non-linear erms in equaion (3), which gives m=, hen equaion (11) will become: u, = u = a + a 1 Y + a Y, a (1) Where a, a1 and a are consan parameers ha mus be deermined; ando calculae hose parameers subsiue equaion u and u from equaion (11) in equaion (1), and equaing he erms wih idenical power of he parameer Y, hen we obain he following sysem of equaions: i= a i Y i Y : λ + k a λk a + k a = Y 1 : λ + k a 1 + ka a 1 = Y : λ + k a + k (a a + a 1 ) = Y 3 : ka 1 a = Y : k a = (13) Solving sysem of equaions (13), we ge: a = (λ+k), a k 1 =, a = (λ+k) λk 3 And by subsiuing hose parameers in equaion (3.), we ge he soluion: u(, ) = λ+k k Now, for λ = k = 1, we have: 1 + λ+k λk anh (k + λ) (1) u(, ) = anh ( + ) (1) Figure (1) shows he behavior of he soluion u(,) in (1) for BBM equaion Fig.(1) soluion u(,) in (1) for BBM equaion.. Gardner equaion: Consider he following Gardner equaion[11], u = u + uu (1) Le u(, ) = u(ξ) and by using he wave variable ξ= k + λ, equaion (1) urns o be he following ordinary differenial equaion: λu k 3 u 3k u = (17) And by inegraing equaion (.1) once wih zero consan, we have 1 1 Page

3 u(,) Solions Soluions o Nonlinear Parial Differenial Equaions by he Tanh Mehod λu k 3 u 3ku = (1) Applying Tanh Mehod, equaion (1) becomes: λu k 3 Y 1 Y du + 1 dy Y d u + dy 3ku = (19) Where he soluion can be formed as follows: u, = m i= a i Y i () Where he parameer m can be deermined by balancing he highes order of linear and non-linear erms in equaion (19), which gives m=, hen equaion () will becomes: u = a + a 1 Y + a Y, a (1) Where a, a1 and a are consan parameers ha mus be deermined; and o calculae hose parameers subsiue equaion uand u from equaion (1) in equaion (19), and equaing he erms wih idenical power of he parameer Y, hen we obain: Y : λa k 3 a 3ka = Y 1 : λa 1 ka a 1 = Y : λa ka a 3ka 1 = Y 3 : ka 1 a = Y : 3ka = () Solving sysem of equaions (), we ge: a = λ, a k 1 =, a = λ k by subsiuing hose parameers in equaion (3), we ge he soluion: u(, ) = λ 1 + λ k k 3 anh (k + λ) (3) Now, for λ = k = 1, we have: u(, ) = anh ( + ) () Figure () shows he behavior of he soluion u(,) in () for Gardner equaion. Fig.() he soluion u(,) in () for Gardner equaion. 3. Cassama-Holm equaion: Consider he following Cassama-Holm equaion [11], u + μu u + 3uu u u uu = () Le u(, ) = u(ξ) and by using he wave variable ξ = k + λ, equaion () urns o be he following ordinary differenial equaion: λ + μk u + 3 k u λk u k 3 1 u + uu = () by inegraing equaion () once wih zero consan, we have λ + μk u + 3 ku λk u k 3 1 u k 3 uu = (7) Applying Tanh Mehod, equaion (7) becomes: λ + μk u + 3 ku λk Y 1 Y du + 1 dy Y d u k3 dy YdudY+1 YdudY=() Y du dy k 3 u Y 1 Where he soluion can be formed as follows: u, = m i= a i Y i (9) Where he parameer m can be deermined by balancing he highes order of linear and non-linear erms in equaion (), which gives m=, hen equaion (9) will becomes: u = a + a 1 Y + a Y, a (3) 1 1 Page

4 u(,) Solions Soluions o Nonlinear Parial Differenial Equaions by he Tanh Mehod Where a, a1 and a are consan parameers ha mus be deermined; and o calculae hose parameers subsiue equaion (3), u and u from equaion (3) in equaion (), and equaing he erms wih idenical power of he parameer Y, hen we obain: Y : λ + μk a + 3 ka λk a k3 a 1 k 3 a a = Y 1 : λ + μk a 1 + 3ka a 1 k 3 a 1 a = Y : 3 ka = (31) Solving sysem of equaions (31), we ge: a = λ +λμ k, a k(μk λ) 1 =, a = λ+μk, k 3 (μk λ) Y : λ + μk a + 3ka a + 3 ka 1 k 3 a = Y 3 : 3ka 1 a = μk > λ by subsiuing hose parameers in equaion (3), we ge he soluion: u(, ) = λ+μk λ + λ+μk anh (k + λ) (3) k(μk λ) k Now, for λ= k = 1 and μ = 3, we have: u(, ) = 1 + anh ( + ) (33) Figure (3) shows he behavior of he soluion u(,) in (33) for Cassama-Holm equaion. 7 Fig.(3) he soluion u(,) in (33) for Cassama-Holm equaion.. Two componen Kdv evoluionary sysem [11]: Consider he following wo componen evoluionary sysemof homogeneous Kdv equaions of order, u = 3v (3) v = u + u (3) Le u, = u ξ and v, = v ξ and by using he wave variable ξ = k + λ, equaions in (3), and (3) urn o be he following ordinary differenial sysem: λu = 3k v (3) λv = k u + u (37) by inegraing equaion (3) once wih zero consan, we have λu = 3k v (3) ha leads o: v = λ 3k u(39) 1 Now subsiue in equaion (39) in (37), we ge: λ 3k u k u u = () Applying Tanh Mehod, equaion (), becomes: 1 λ u 3k k Y 1 Y du + 1 dy Y d u dy u = (1) Where he soluion can be formed as follows: u, = m i= a i Y i () Where he parameer m can be deermined by balancing he highes order of linear and non-linear erms in equaion (1), which gives m=, hen : u = a + a 1 Y + a Y, a (3) 1 17 Page

5 v(,) u(,) Solions Soluions o Nonlinear Parial Differenial Equaions by he Tanh Mehod Where a, a1 and a are consan parameers ha mus be deermined; and o calculae hose parameers subsiue equaion (3)and u from equaion (3) in equaion (1), and equaing he erms wih idenical power of he parameer Y, hen we obain: Y : λ 3k a k a a = Y 1 : λ 3k a 1 a a 1 = Y : λ 3k a a a a 1 = Y 3 : a 1 a = Y : a = () Solving sysem of equaions (), we ge: a = λ, a k 1 =, a = λ k subsiuinghose parameers in equaion (3), we ge he soluion: u β = λ 1 + λ k 1k anh (β) () subsiuing equaion () in equaion (39) and inegraing once wih respec o ξ, we have: v ξ = λ 3 ξ + λ 7k 1k (ξ anh ξ) +c () Then he soluions of u(,) and v(,) will become: u, = λ 1 + λ k 1k anh (k + λ) (7) And v, = λ 3 7k (k + λ) + λ Now, for λ = k = 1, we have: u(, ) = anh ( + ) (9) 1k k + λ anh (k + λ) +c () v, = 1 ( + ) + 1 ( + ) anh ( + ) +c () 7 1 Figures () and () show he behavior of he soluion of u(,) and v(,) in (9) and () respecively for he Kdv sysem Fig.() he soluion of u(,) in (9) for he Kdv sysem Fig.() he soluion of v(,) in () for he Kdv sysem. 1 1 Page

6 Solions Soluions o Nonlinear Parial Differenial Equaions by he Tanh Mehod IV. CONCLUSION In his paper, he anh mehod has been successfully applied o find he soluion for four nonlinear parial differenial equaions such as he Benjamin-Bona-Mahony (BBM) equaion, Gardner equaion,cassama-holm equaion, and wo componen Kdv evoluionary sysem of equaions. The anh mehod is used o find a new comple ravelling wave soluions. The resuls show ha he anh mehod is a powerful mahemaical ool o solve he nonlinear PDEs. V. ACKNOWLEDGEMENTS The auhor graefully acknowledges he suppor from Dr. Anwar Ja afar Mohamad Jawad, Al-Rafidain Universiy College Baghdad-Iraq. This suppor is genuinely and sincerely appreciaed. REFERENCES [1] W. Malflie, The anh mehod: a ool for solving cerain classes of nonlinear evoluion and wave equaions, Journal of Compuaional and Applied Mahemaics 1 1,, 9 1. [] W. Malflie, Soliary wave soluions of nonlinear wave equaions, American Journal of Physics, 199,. [3] Abdul-Majid Wazwaz, Parial Differenial Equaions and Soliary Waves Theory, Springer Verlag, New York, NY. USA, 9. [] Abdul-Majid Wazwaz, The Cole-Hopf ransformaion and muliple solion soluions for he inegrable sih-order Drinfeld Sokolov Sasuma Hiroa equaion, Applied Mahemaics and Compuaion. 7, (1), 9,. [] Anwar Ja afar Mohamad Jawad, Marko D. Pekovic, and Anjan Biswas, Solion soluions of Burgers equaions and perurbed Burgers equaion, Applied Mahemaics and Compuaion 1, (11), 1, [] D.D. Ganji, M. Nourollahi, M. Rosamian, A comparison of variaional ieraion mehod wih Adomian s decomposiion mehod in some highly nonlinear equaions, Inernaional Journal of Science & Technology, (), 7, [7] Anwar Ja afar Mohamad Jawad, Marko D. Pekovic, and Anjan Biswas, Solion soluions of a few nonlinear wave equaions, Applied Mahemaics and Compuaion 1, (9), 1, 9. [] Y.C. Hon, E.G. Fan, A series of new soluions for a comple coupled KdV sysem, Chaos Solions Fracals 19,, 1. [9] Y. Ugurlu, D. Kaya, Eac and numerical soluions of generalized Drinfeld Sokolov equaions, Physics Leers A 37, (1),, [1] L. Wu, S. Chen, C. Pang, Traveling wave soluions for generalized Drinfeld Sokolov equaions, Applied Mahemaical Modelling33, (11), 9, [11] Marwan T. Alquran, Solions and Periodic Soluions o Nonlinear Parial Differenial Equaions by he Sine-Cosine Mehod, Appl. Mah. Inf. Sci., (1), 1, Page