IMPROVED HYPERBOLIC FUNCTION METHOD AND EXACT SOLUTIONS FOR VARIABLE COEFFICIENT BENJAMIN-BONA-MAHONY-BURGERS EQUATION

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1 THERMAL SCIENCE, Year 015, Vol. 19, No. 4, pp IMPROVED HYPERBOLIC FUNCTION METHOD AND EXACT SOLUTIONS FOR VARIABLE COEFFICIENT BENJAMIN-BONA-MAHONY-BURGERS EQUATION by Hong-Cai MA a,b*, Xiao-Fang PENG a, and Dan-Dan YAO a a Deparmen of Applied Mahemaics, Donghua Universiy, Shanghai, China b Deparmen of Mahemaics and Saisics, Universiy of Souh Florida, Tampa, Fla., USA Original scienific paper DOI: 10.98/TSCI M By using he improved hyperbolic funcion mehod, we invesigae he variable coefficien Benjamin-Bona-Mahony-Burgers equaion which is very imporan in fluid mechanics. Some exac soluions are obained. Under some condiions, he periodic wave leads o he kink-like wave. Key words: Benjamin-Bona-Mahony-Burgers equaion, exac soluions, hyperbolic funcion mehod Inroducion Nowadays, non-linear parial differenial equaions (NLPDE) are becoming more and more imporan, especially heir broad applicaions on modeling many physical phenomena. For insance, solid sae physics, fluid mechanics, and so on [1]. I is very valuable o do some research on how o solve NLPDE exac soluions and invesigae he propery of he soluions. The Benjamin-Bona-Mahony-Burgers (BBMB) equaion: u u αu uu u = 0 xx xx x x is proposed as a model o sudy he unidirecional long waves of small ampliudes in waer, which is an alernaive o he KdV equaion []. I is an imporan model in fluid mechanics. In his paper we will sudy he variable coefficien BBMB equaion: u λ() u α() u β() uu γ() u = 0 (1) xx xx x x λ(), α(), β(), and γ () are arbirary ime-dependen coefficiens. If α() = 0, λ() = 1, and γ() = 1, eq. (1) is an alernaive o he regularized long wave equaion proposed by Benjamin e al. [3] and Peregrine [4]. The variable coefficien BBMB eq. (1) [5] is proposed o describe long waves of small ampliudes broadcasing in non-linear dispersive media. This model has been inroduced [6], which plays a very imporan role in mahemaics and fluid. During he pas few years, numerous mehods in solving BBMB equaions have been discovered by many auhors, such as G'/G mehod, homoopy analysis mehod, He' mehod, ec. [7, 8]. * Corresponding auhor; hongcaima@dhu.edu.cn

2 1184 THERMAL SCIENCE, Year 015, Vol. 19, No. 4, pp Hyperbolic funcion mehod was firs proposed in [9], which is based on he facha mos soliary wave soluions have form of hyperbolic funcion. This mehod has been used o find he exac soluions of many non-linear wave equaions, such as Burgers' equaion, KdV equaion, ec. [9, 10]. Improved hyperbolic funcion mehod For parial differenial equaion (PDE) of wo independen variables x, : P(, x, ux, u, u xx,...) = 0 () we consider is raveling wave soluion of he form: ux (, ) = ϕ( ξ), ξ = kx ( ) c() l (3) k(x) and c() are funcions of single variable. By using his raveling wave ransacion, we can change non-linear PDE eq. () o a variable coefficien differenial equaion (VCDE). Supposing hahe soluion of his VCDE is: m m i i i= 0 j= 1 j 1 φξ ( ) = af bf g (4) j f 1 sinhξ =, g = cosh ξ r cosh ξ r (5) and a i, b j, and r are coefficiens o be deermined. m can be deermined by balancing [11] he highes order derivaives and highes order non-linear erms appearing in VCDE, wha's more: f = fg, g = 1 g rf, g = 1 rf ( r 1) f (6) ξ ξ By subsiuing eq. (4) ino eq. (), and using eq. (5) o simplify he VCDE unil i only involves exponenial erm of f and g (he degree of g is less han 1). Then combining he similar erms of f and g, we will esablish a se of equaion. By solving his se of equaion, he exac soluion of eq. () will be obained. Obain exac soluion of BBMB equaion by improved hyperbolic funcion mehod Firsly, we consider he following ransformaions: ux (, ) = ϕ( ξ), ξ = kx ( ) c ( ) l (7) kx ( ), c () are arbirary funcions. Subsiuing eq. (7) ino eq. (1) we have he equaion: [ c () α() k ( x) γ()] ϕ ( ξ) [ λ() c () k ( x) α() k ( x)] ϕ ( ξ) β( k ) ( x) ϕ( ξ)( ξ) λ( c ) ( k ) ( x) ϕ ( ξ) = 0 (8) By using eq. (4) and balancing he highes order derivaives and highes order nonlinear erms appearing in eq. (8), we ge m = 1. Thus we have:

3 THERMAL SCIENCE, Year 015, Vol. 19, No. 4, pp φξ ( ) = a0 a1f g (9) By subsiuing eq. (9) ino eq. (8), simplifying he eq. (8) unil i only involves exponenial erm of f and g (he degree of g is less han 1), by supposing heir coefficien equal o zero, we obain he sysem: (1) ( r 1)( r 1) λ( ) c ( ) k ( x) = 0 () ( r 1)( r 1){ a1λ() c () k "( x) ab 1 1() k ( x) [ a1α() 1 br 1 λ() c ()] k ( x)} = 0 (3) [3 ar 1 λ( ) c ( ) (1 r ) α( )] k ( x) [(3 ar 1 a0 ar 0 ) β( ) (1 r ) γ( )] k ( x) [3 ar 1 α( ) (7r 4) λ() c ()] k ( x) (1 r ) c () = 0 (4) [ brc 1 () br 1 α() a1λ() c ()] k ( x) [ a0rβ() ab 1 1β() br 1 γ()] k ( x) [ a1α( ) br 1 λ( ) c ( )] k ( x) brc 1 ( ) = 0 (5) [ br 1 λ() c () a1α()] k ( x) [( r a0a1) β() a 1 γ ()] k ( x) [ a1γ( ) c ( ) br 1 α( )] k ( x) a1c ( ) = 0 (6) ( r 1) λ( ) c ( ) k ( x) ( r a1 ) β( ) k ( x) [ ( r 1) α( ) 6 a1rλ( ) c ( )] k ( x) = 0 (7) a1 ( r 1)( r 1) λ( ) c ( ) k ( x) = 0 Solving his sysem by Maple gives he following resuls: (1) When λ () = 0, we obain: β () k ( x) = (10) α() 1 ( 1) 1 a =± r b (11) γ() β() a0β () () = (1) c α() Equaion (10) indicaes ha β ( )/ α ( ) mus be a consan: β() = cα(), k ( x) = bc, c () = bc [ γ() a β()] We obain he soluion of variable coefficien BBMB eq. (1): 1 1 b r 1 b sinh[ k( x) c( ) l] ux (, ) = a0 ± r cosh[ k( x) c( ) l] r cosh[ k( x) c( ) l] (13) and a 0, b 1, and c 1 are consans. kx ( ) = bcx, c () = bc [ γ() aβ()]d

4 1186 THERMAL SCIENCE, Year 015, Vol. 19, No. 4, pp Figure 1 shows a special soluion of eq. (1), when we suppose ha: a0 = 1, = c1 = r =, l = 0, γ() = cos, β() = sin From he figure we can see hahe wave is a kind of kink-like wave and he ampliude is consan in he process of ransmission. Since γ() and β() are ime dependen periodic funcion, wave velociy (boh size and direcion) periodically changes over ime. () When a 1 = r = 0, we have: Figure 1. a ( ) () 0 = 1, b 1 = c 1 = r =, l = 0, γ() = cos, β k x = (14) and β() = sin α( ) γ() β() a0β () c () = (15) α( ) Equaion (14) indicaes ha β()/α() mus be a consan. Then, we have: kx ( ) = cx, c () = c [ γ() a0β()]d The soluion of variable coefficien BBMB equaion eq. (1) is: sinh[ k( x) c( ) l] ux (, ) = a0 (16) r cosh[ k( x) c( ) l] Figure. a 0 = b 1 = 1, c =, l = 0, γ() = cos, and β() = sin kx ( ) = cx c () = c [ γ() a0β()]d and a 0, b 1, and c are consans. Figure shows a special soluion when we suppose ha: a0 = = 1, c =, l = 0, γ() = cos, β() = sin From he figure we can see hahe ampliude of he wave end o be a consan over ime in he process of ransmission. Conclusions In his aricle, he improved hyperbolic funcion mehod is uilized o find he soluions of he variable coefficien BBMB equaion. I is very imporan in fluid mechanics. As we

5 THERMAL SCIENCE, Year 015, Vol. 19, No. 4, pp know, hyperbolic funcion mehod is an effecive mehod in solving non-linear parial differenial equaions. We also indicae ha hyperbolic funcion mehod is a srong mehod no only in solving consan coefficien non-linear parial differenial equaions bu also in solving variable coefficien non-linear parial differenial equaions. Acknowledgmens The work is in par suppored by he Naional Naural Science Foundaion of China (projec No ), he Fund of Science and Technology Commission of Shanghai Municipaliy (projec No.ZX ) and he Fundamenal Research Funds for he Cenral Universiies. References [1] Zhang, P., Lin, F. H., Lecures on he Analysis of Non-Linear Parial Differenial Equaions, Higher Educaion Press, Beijing, China, 013 [] Bona, J. L., Smih, R., The Iniial-Value Problem for he Koreweg-de Vries Equaion, Philosophical Transacions of he Royal Sociey of London Series A, 78 (1975), 187, pp [3] Benjamin, T. B., e al., Model Equaions for Long Waves in Non-Linear Dispersive Sysems, Philosophical Transacions of he Royal Sociey of London Series A, 7 (197), 10, pp [4] Peregrine, D. H., Calculaions of he Developmen of an Under Bore, Journal of Fluid Mechanics, 5 (1966),, pp [5] Mei, M., Large-Time Behavior of Soluion for Generalized Benjamin-Bona-Mahony-Burgers Equaions, Non-Linear Analysis, 33 (1998), 7, pp [6] Kumar, V., e al., Painleve Analysis, Lie Symmeries and Exac Soluions for Variable Coefficiens Benjamin-Bona-Mahony-Burger (BBMB) Equaion, Communicaions in Theoreical Physics, 60 (013),, pp [7] Tari, H., Ganji, D. D., Approximae Explici Soluions of Non-Linear BBMB Equaion by He's Mehods and Comparison wih he Exac Soluion, Physics Leers A, 367 (007), 1, pp [8] Kadri, T., e al., Mehods for he Numerical Soluion of he Benjamin-Bona-Mahony-Burgers Equaion, Numer. Mah. Parial. Diff. Eq., 4 (008), 6, pp [9] Shi, Y., e al., Exac Soluions of Generalized Burgers' Equaion wih Variable Coefficiens, Eas China Normal Universiy Transacion, 5 (006), pp [10] Shi, Y. R., e al., Exac Soluions of Variable Coefficiens Burgers' Equaions, Journal of Lanzhou Universiy (Naural Sciences), 41 (005),, pp [11] Wang, M., e al., Applicaion of a Homogeneous Balance Mehod o Exac Soluions of Non-Linear Equaions in Mahemaical Physics, Physics Leers A, 16 (1996), 1, pp Paper submied: January 5, 015 Paper revised: February 7, 015 Paper acceped: March 31, 015