Enhanced (G /G)-Expansion Method to Find the Exact Solutions of Nonlinear Evolution Equations in Mathematical Physics

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1 Inernaional Journal of Parial Differenial Equaions and Applicaions 013 Vol. 1 No Available online a hp://pubs.sciepub.com/ijpdea/1/1/ Science and Educaion Publishing DOI: /ijpdea-1-1- Enhanced ( /)-Expansion Mehod o Find he Exac Soluions of Nonlinear Evoluion Equaions in Mahemaical Physics Md. Ekramul Islam 1 Kamruzzaman Khan 1* M. Ali Akbar Rafiqul Islam 1 1 Deparmen of Mahemaics Pabna Universiy of Science and Technology Pabna Bangladesh Deparmen of Applied Mahemaics Universiy of Rajshahi Rajshahi Bangladesh *Corresponding auhor: k.khanru@gmail.com Received July 5 013; Revised November ; Acceped November Absrac In he presen paper we consruc he raveling wave soluions involving parameers for he (+1)- dimensional cubic Klein-ordon equaion (ck) via Enhanced ( /)-expansion mehod. The efficiency of his mehod for finding hese exac soluions has been demonsraed. As a resul a se of soliary wave soluions are derived which are expressed by he combinaions of raional hyperbolic and rigonomeric funcions involving several parameers. I is shown ha he mehod is effecive and can be used for many oher nonlinear evoluion equaions (NLEEs) in mahemaical physics. Keywords: enhanced ( /)-expansion mehod ck equaion soliary wave raveling wave Cie This Aricle: Md. Ekramul Islam Kamruzzaman Khan M. Ali Akbar and Rafiqul Islam Enhanced ( /)-Expansion Mehod o Find he Exac Soluions of Nonlinear Evoluion Equaions in Mahemaical Physics. Inernaional Journal of Parial Differenial Equaions and Applicaions 1 no. 1 (013): 6-1. doi: /ijpdea Inroducion Nowadays NLEEs have been he subjec of allembracing sudies in various branches of nonlinear sciences. A special class of analyical soluions named raveling wave soluions for NLEEs have a lo of imporance because mos of he phenomena ha arise in mahemaical physics and engineering fields can be described by NLEEs. NLEEs are frequenly used o describe many problems of proein chemisry chemically reacive maerials in ecology mos populaion models in physics he hea flow and he wave propagaion phenomena quanum mechanics fluid mechanics plasma physics propagaion of shallow waer waves opical fibers biology solid sae physics chemical kinemaics geochemisry meeorology elecriciy ec. Therefore invesigaion raveling wave soluions is becoming more and more aracive in nonlinear sciences day by day. However no all equaions posed of hese models are solvable. As a resul many new echniques have been successfully developed by diverse groups of mahemaicians and physiciss such as he Hiroa s bilinear ransformaion mehod [1] he anh-funcion mehod [34] he exended anh-mehod [56] he Expfuncion mehod [7-14] he Adomian decomposiion mehod [15] he F-expansion mehod [16] he auxiliary equaion mehod [17] he Jacobi ellipic funcion mehod [18] Modified Exp-funcion mehod [19] he expansion mehod [0-9] Weiersrass ellipic funcion mehod [30] he homoopy perurbaion mehod [ ] he homogeneous balance mehod [36 37] he Modified simple equaion mehod [38-43] He's polynomial [44] asympoic mehods and nanomechanics [45] he variaional ieraion mehod [4647] casoraion formulaion [48] frobenius inegrable decomposiion [49] he exended muliple Riccai equaions expansion mehod [5051] Enhanced ('/)-expansion Mehod [5] and so on. Among hose approaches an enhanced - expansion mehod is a ool o reveal he solions and periodic wave soluions of NLEEs in mahemaical physics and engineering. The main ideas of he enhanced -expansion mehod are ha he raveling wave soluions of NLEEs can be expressed as he combinaion of raional and irraional funcions of = ( ξ ) saisfies he second order linear ordinary differenial equaion + µ. The objecive of his aricle is o apply he enhanced -expansion mehod o consruc he exac soluions for nonlinear evoluion equaions in mahemaical physics via he ck equaion. The ck equaion is compleely inegrable and has N-solion soluions. The aricle is prepared as follows: In secion an enhanced -expansion mehod is discussed. In secion 3 we apply his mehod o he nonlinear evoluion equaions poined ou above ; in secion 4 physical explanaions and in secion 5 conclusions are given.

2 Inernaional Journal of Parial Differenial Equaions and Applicaions 7. Mehodology In his secion we describe enhanced - expansion mehod for finding raveling wave soluions of nonlinear evoluion equaions. Suppose ha a nonlinear evoluion equaion say in wo independen variables x and is given by Ruu ( ux u uxx u x...) (.1) u( ξ ) = ux () is an unknown funcion R is a polynomial of ux () and is parial derivaives in which he highes order derivaives and nonlinear erms are involved. In he following we give he main seps of his mehod [5]: Sep 1. Combining he independen variables x and ino one variable ξ we suppose ha ( R { 0} ) ω. u( ξ) = ux ( ) ξ = x± ω (.) The raveling wave ransformaion Eq. (.) permis us o reduce Eq. (.1) o he following ODE: Ruu ( u ). (.3) Where R is a polynomial in and is derivaives while du d u u ( ξ) = u ( ξ) = and so on. dξ dξ Sep.We suppose ha Eq.(.3) has he formal soluion i ai n i (1 + λ ) u( ξ ) = i= n i 1 + bi σ 1+ µ (.4) = ( ξ ) saisfy he equaion + µ (.5) in which ai bi( n i nn ; N) and λ are consans o be deermined laer and σ = ± 1 µ 0. Sep 3. We deermine he posiive ineger n in Eq. (.4) by considering he homogeneous balance beween he highes order derivaives and he nonlinear erms in Eq. (.3). Sep 4. We subsiue Eq. (.4) ino Eq.(.3) using Eq. (.5) and hen collec all erms of same powers of j and j σ 1+ ogeher hen µ se each coefficien of hem o zero o yield a overdeermined sysem of algebraic equaions solve his sysem for ai bi( n i nn ; N) and λω. Sep 5. The soluion of Eq.(.5) can be wrien as follows: When µ < 0 we ge and = µ anh + µξ (.6) and = µ coh + µξ (.7) Again when µ > 0 he soluions are = µ an µξ (.8) = µ co + µξ (.9) A is an arbirary consan. Finally subsiuing ai bi( n i nn ; N) λω and Eqs.(.6)- (.9) ino Eq. (.4) we obain raveling wave soluions of Eq. (.1). 3. Applicaion In his secion we will exer enhanced - expansion mehod o solve he ck equaion in he form u + u u + u+ u (3.1) xx yy are posiive consans. The raveling wave ransformaion equaion u( ξ) = uxy ( ) ξ = x+ y ω reduces Eq. (3.1) o he following ordinary differenial equaion: 3 ( ω ) u u u. (3.) Now aking he homogeneous balance beween he highes order derivaive u and he nonlinear erm 3 u from Eq.(3.) we ge n = 1. Hence for n = 1 Eq. (.4) reduces o ( / ) ( ) 1 + b0 + b λ ( ) ( ) a1 a 1 / u( ξ ) = a λ / / 1 µ + b1 σ 1 + µ µ (3.3) = ( ξ ) saisfies Eq. (.5). Subsiue Eq. (3.3) along wih Eq. (.5) ino Eq. (3.). As a resul of his subsiuion we ge a polynomial of j j and From his µ polynomial we equae he coefficiens of j j and and seing µ hem o zero we ge a over-deermined sysem ha

3 8 Inernaional Journal of Parial Differenial Equaions and Applicaions consiss of weny-five algebraic equaions. Solving hese over deermined sysem of equaions we obain he following valid ses. se1: Se: ω ω = ± + λ a1 = ± a0 µ µ a 1 b1 b0 b 1 1 8µ = ± λ a1 = a0 µ µ µ a 1 = ± b1 b0 b 1 Se3: ω 1 (16 µ + ) 1 = ± λ a1 = 4 µ µ µ 1 a0 a 1 = ± b1 b0 b 1 Se4: ω ( µ + ) = ± λ a1 = ± a0 µ µ σ a 1 b1 = ± b0 b 1 Se5: (4 µ + ) 4 + ω = ± λ = ± a1 = µ µ µ a0 = a 1 b1 b0 b 1 Se 6: (4 µ + ) µ = ± = a1 a0 = µ ω λ λ λ µ a 1 = ± b1 b0 b 1 Se7: ( µ ) = ± = a1 a0 a 1 µ ω λ λ b1 = ± b0 b 1 Se8: σ ( µ + ) µ = ± = a1 a0 = µ ω λ λ λ µ µ σ a 1 = ± b1 b0 = ± b 1 Now for µ < 0 subsiuing he values of ω a 1 a0 a1 b 1 b0 b1 ino Eq. (3.3) from he above Se-1 o Se-8 we ge he following hyperbolic funcion soluions of ck equaion. Family -1: Family-: Family-3: u 1 ( ξ ) = ± anh( A + µξ) u ( ξ ) = ± coh( A + µξ) x y ξ = + +. µ u3 = ± csc h(( A+ µξ)) x y 1 8µ µ 1 u4 = (coh( A+ µξ ) + anh( A+ µξ)) Family-4: Family-5: x y 1 (16 µ + ) 4 µ anh( A + µξ) u5 = ± sec ha ( µξ) + + coh( A + µξ) u6 = ± + csc ha ( + µξ) ( µ + ). µ (4 + ) µ anh( A + µξ) µ u7 = (1 ± µ anh( A + µξ)) µ (4 + ) µ coh( A + µξ) µ u8 = (1 ± µ coh( A + µξ)) µ

4 Inernaional Journal of Parial Differenial Equaions and Applicaions 9 Family-6: Family-7: Family-8: x y (4 µ + ) µ u9 = anh( A+ µξ) u10 = coh( A+ µξ) x y (4 µ + ) µ u 11 = ± sec ha ( + µξ ) u 1 = ± csc ha ( + µξ ) ( µ ). µ u13 = ± (1 + sec ha ( + µξ)) coh( A + µξ) u14 = ± (1 + Icsc ha ( + µξ)) anh( A + µξ) ( µ + ). µ Similarly for µ > 0; we ge he following periodic soluions of ck equaion. Family -9: Family-10: u 15 ( ξ ) = ± an( A µξ) u 16 ( ξ ) = ± co( A + µξ) x y ξ = + +. µ Family-11: u 17 = csc(( A µξ )) u 18 = csc(( A + µξ )) x y 1 8µ µ 1 u19 = an( A ) co( A ) 1 u0 = co( A+ ) an( A+ ) x y 1 (16 µ + ) 4 µ Family-1: u1 = ± an( A ) + sec( A ) u = ± co( A+ ) + csc( A+ ) Family-13: ( µ + ). µ (4 + ) µ an( A µξ) µ u3 = (1 ± µ an( A µξ)) µ (4 + ) µ co( A + µξ) µ u4 = (1 ± µ co( A + µξ)) Family-14: Where µ x y (4 µ + ) µ u5 = ± co( A µξ) u6 = ± an( A+ µξ)

5 10 Inernaional Journal of Parial Differenial Equaions and Applicaions Family-15: x y (4 µ + ) µ u 7 = sec( A µξ ) u 8 = csc( A + µξ )) ( µ ). µ Family-16: u9 = ± (1 + sec( A µξ ))co( A µξ) Figure. (Family 7) Shape of µ 11=(ξ) for µ=-1 =1 =1 A=0 y=0 in inerval -5 x 5 u30 = ± (1 + csc( A µξ )) an( A µξ) ( µ + ) µ 4. Resuls and Discussion In his secion we will discuss he physical explanaions of obained soluions of ck equaion. I is ineresing o poin ou ha he delicae balance beween he nonlineariy effec of u 3 and he dissipaive effec of uxx u yy and u gives rise o solions ha afer a fully ineracion wih ohers he solions come back reaining heir ideniies wih he same speed and shape. The ck equaion has soliary wave soluions ha have exponenially decaying wings. If wo solions of he ck equaion collide he solions jus pass hrough each oher and emerge unchanged. We make graphs of obained soluions so ha hey can represen he imporance of each obained soluion and physically inerpre he consequence of parameers as well. Some of our obained raveling wave soluions are represened in Figure 1-Figure 4 wih he aid of Maple sofware: Figure 3. (Family 1) Shape of µ 1=(ξ) for µ=1 =1 =1 A=1 y=0 in inerval -5 x 5 Figure 4. (Family 15) Shape of µ 7=(ξ) for µ=1 =1 =3 A=0 y=0 in inerval -10 x Conclusion Figure 1. (Family 3) Shape of µ 4=(ξ) for µ=-1 = =1 A=1 y=0 in inerval -3 x 3 - In shor we have illusraed he Enhanced ( expansion mehod and uilized i o find he exac soluions of nonlinear equaions wih he help of Maple 13. We have successfully obained some solions singular solions and plane periodic soluions of he ck equaion

6 Inernaional Journal of Parial Differenial Equaions and Applicaions 11 involving parameers. When he parameers are aken as special values he soliary wave soluions and he periodic wave soluions are obained. Taken as a whole i is worhwhile o menion ha his mehod is effecive for solving oher nonlinear evoluion equaions in mahemaical physics. References [1] R. Hiroa Exac envelope solion soluions of a nonlinear wave equaion. J. Mah. Phy. 14 (1973) [] R. Hiroa J. Sasuma Solion soluions of a coupled KDV equaion. Phy. Le. A. 85 (1981) [3] M. Malflie Soliary wave soluions of nonlinear wave equaions. Am. J. Phys. 60 (199) [4] H.A.Nassar M.A. Abdel-Razek A.K. Seddeek Expanding he anh-funcion mehod for solving nonlinear equaions Appl. Mah. (011) [5] E.. Fan Exended anh-mehod and is applicaions o nonlinear equaions. Phy. Le. A. 77 (000) [6] M.A. Abdou The exended anh-mehod and is applicaions for solving nonlinear physical models. App. Mah. Compu. 190(007) [7] J.H. He X.H. 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7 1 Inernaional Journal of Parial Differenial Equaions and Applicaions [45] J. H. He An elemenary inroducion o recenly developed asympoic mehods and nanomechanics in exile engineering In. J. Mod. Phys. B (1) (008) [46] S. T. Mohyud-Din M. A. Noor and K. I. Noor Some relaively new echniques for nonlinear problems Mahemaical Problems in Engineering Hindawi 5 pages. [47] S. T. Mohyud-Din M. A. Noor K. I. Noor and M. M. Hosseini Soluion of singular equaions by He's variaional ieraion mehod Inernaional Journal of Nonlinear Sciences and Numerical Simulaion 11 () (010) [48] W. X. Ma and Y. You Raional soluions of he Toda laice equaion in Casoraian form Chaos Solions & Fracals (004) [49] W. X. Ma H. Y. Wu and J. S. He Parial differenial equaions possessing Frobenius inegrable decomposiions Phys. Le. A 364 (007) 9-3. [50] K. A. epreel Exac Complexion Solion Soluions for Nonlinear Parial Differenial Equaions Inernaional Mahemaical Forum Vol. 6( 011) no [51] K. A. epreel A. R. Shehaa Exac complexion solion soluions for nonlinear parial differenial equaions in mahemaical physics Scienific Research and Essays Vol. 7() pp [5] K. Khan and M. Ali Akbar. Traveling Wave Soluions of Nonlinear Evoluion Equaions via he Enhanced ('/)-expansion Mehod. Journal of he Egypian Mahemaical Sociey 013JOEMS-D (Acceped for publicaion).