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International Journal of Science and Research (IJSR) ISSN: 2319-7064 ResearchGate Impact Factor (2018): 0.28 | SJIF (2018): 7.426 Hopf-Cole Transformation with Effect of the Dissipation Coefficient on Nonlinearity in the Benjamin-Bona-Mahony (BBM) Generalized Pseudo-Parabolic Equation R. Gilles Bokolo University of Kinshasa, Department of Mathematics and Computer Sciences, BP. 190 Kinshasa XI, Kinshasa, Democratic Republic of the Congo Abstract: In this paper, we consider the general form of the Benjamin-Bona-Mahony's pseudo-parabolic nonlinear equation (BBM) in which we use the Hopf-Cole transformation [10]. We also set up a conjecture to get rid of several terms throughout our development. Finally, we present a comparative analysis of solutions found either by the aforementioned transform or by the functional variable method formulated by A. Zerarka [11]-[12]. Hence, we show that the result obtained by the approach of Hopf- Cole transformation has some significant features. Keywords: pseudo-parabolic nonlinear equation, generalized Benjamin-Bona-Mahony equation (BBM), functional variable method, Hopf-Cole transformation 1. Introduction The Benjamin-Bona-Mahony equation (BBM) [1] has been studied for the first time as a regularized version of the Korteweg De Vries (KdV) equation for modeling low amplitude waves moving over a long surface under the action of gravity; hence the name of regularized long wave equation (RLW). The term "regularized" refers to the fact that, from the point of view of such properties as existence, uniqueness and stability, the BBM equation offers considerable technical advantages over the KdV equation. As the authors H. Zhang, GM Wei and Y.T. Gao [2] point out, unlike the KdV equation, the BBM equation is not an equation of evolution in the strict sense of the term, following the appearance of u xxt . The latter makes this equation pseudo-parabolic [9] since the term of the highest order of this equation has mixed derivatives (i.e. derivatives dependent on both time and space). Investigations on exact solutions of traveling waves of nonlinear evolution equations play an important role in the study of non-linear physical phenomena. Solitons are the most important types of solutions among all traveling wave solutions. The existence of multi-soliton, in particular two-soliton type solutions, is crucial in information technologies: it makes possible the simultaneous undisturbed propagation of several pulsations in both directions [13]. In this paper, we attempt to linearize the following generalized BBM equation: u t  u x  uu x  u xxt  0 1 with the coefficients  ,  ,   R , which respectively correspond to the transport, nonlinearity and dispersion coefficients. This equation describes the unidirectional propagation of long waves.   u : the amplitude or the velocity (or the speed of the flow) x : is proportional to the distance in the direction of fluid  flow t : the time of fluid flow The classical form of Hopf-Cole transformation is applied in the search for solution of the nonlinear diffusion equation. It should be noted that Eq. (1) incorporates non-linear dispersive and dissipative effects. Although there are several ineluctably efficient methods by which the multi-solution solutions of Eq. (1) can be derived, the main aim of this paper is to present the remarkable strength of the Hopf-Cole transformation [10]. We perform a comparative analysis of our solution with a solution obtained by the functional variable method formulated by A. Zerarka [11]-[12]. 2. Hopf-Cole Transformation 2.1 Description Below are some essential steps to apply to the cited transformation. Step 1: Consider the differential equation given by Px, t , u, u t , u x , u xxt ,...  0 where P is a given function, ut , x  is the independent variable. Step 2: Find a smooth solutions to the form    u  Step 3: Substitute the expression resulting from step 2 into the PDE of step 1. Step 4: Use the notation equation having    u   as variable. to get a polynomial Volume 8 Issue 6, June 2019 www.ijsr.net Licensed Under Creative Commons Attribution CC BY Paper ID: ART20198788 10.21275/ART20198788 1707 International Journal of Science and Research (IJSR) ISSN: 2319-7064 ResearchGate Impact Factor (2018): 0.28 | SJIF (2018): 7.426 Step 5: By equaling the coefficients of most of the powers of  u  to zero, we obtain an algebraic equation of unknowns  ,  and  . Step 6: Find solutions of the linearized equation to the form  x, t   1  exp Ax  Bt  C  Where    xt Then   u u t u x   u u xt  0 In the other hand, to satisfy 3 , we choose  so that  u   0 . We solve this PDE by establishing respect to u . So we see that if Step 7: By solving this equation and interchanging, we finally get the solutions of our PDE. 2.2 Application Suppose that u is a smooth solution of Eq. 1 . We define    u  :RR where  such as  solve a linear equation. We should have the following special format   x, t      u x, t  Where m and n are integers  2 determined as follows: t   u u t 2  x   u u x ,  xt   u u t u x   u u xt  xx   u u x2   u u xx and n t  xxt   u u t u x2   u 2u x u xt   u u t u xx   u t u xxt and therefore, 2  implies t   u  u x  uu x  u xxt   t   x   u uu x      u u t u x2   u u t u xx  2 u u x u xt   xxt u solve 1 , then “Hopf-Cole transformation” Solve this IVP for the linearized form of the BBM equation:  t   x   xxt  0 dans R n  0,    sur R n  t  0  K Considering the following transformation    xt (4) becomes  xt t   xxt   xxxtt 4 0 3 whose trail solution has the form  x, t   1  exp Ax  Bt  C  A, B and C are arbitrary constants, with A and B satisfying the following relations 2 : A  0, B  0 and B 1  A 2  A  0 Where   Therefore,      t  C1  if   0 1  exp A x  2    1  A     x, t    with 1  A 2  0 1      1  exp   2 x  Bt  C 2  otherwise      Simultaneously, Eq. 4  and 3 are automatically satisfied  with the substitution of the above special solution for both equations. Putting all the term together, we have   with is a smooth function, not yet specified. We will try to choose m x K   K constant  Explaining the cancelling effects of different mechanisms that act to change waveforms to result in a soliton wave, Authors (2) pointed out the fact that these mechanisms included dispersion, dissipation and nonlinearity either separately or in various combinations. We now make the conjecture that the cancelling act concentrate on terms mixing up coefficient of dispersion  2 acting on such nonlinear terms as u t u x , u t u xx and u x u xt in As we had   x, t  , which is thus a   u u t u x   u u xt  0 Finally, the set of exact solutions of the general form of Eq. 1 are clearly  ux, t     x t    xt  3A2 1     sec h 2  A x  t  C1 if   0  2 2  1  A  2     1  A 2  with 1  A  0 5 u  x, t    1   1 2    3 B sec h 2  1     2 x  Bt  C  otherwise 2     2        Eq. (3) In one hand, Conservation laws clearly shows that      u u t u x2   u u t u xx  2 u u x u xt  0 Which is equivalent to   u u t u x2   u u x u xt   u u x u xt  0 That implies   u u t u x   u u xt  0 for   0 2 Which are the solitary waves having the sec h profiles. A and B are parameters specifying the amplitude and speed of the wave. 3. Functional Variable Method Volume 8 Issue 6, June 2019 www.ijsr.net Licensed Under Creative Commons Attribution CC BY Paper ID: ART20198788 10.21275/ART20198788 1708 International Journal of Science and Research (IJSR) ISSN: 2319-7064 ResearchGate Impact Factor (2018): 0.28 | SJIF (2018): 7.426 3.1 Description the possible solutions of the original equation of step 1. Step 1: Consider the nonlinear differential equation given by 3.2 Application Px, t , u, u t , u x , u xxt ,...  0   where P is a given function, u t , x is the independent variable or the functional variable to be determined.. Consider the Eq. 1 . We use the following wave variable   x  ct The different derivatives are carried out via the chain rule, Eq.(1) is transformed as Step 2: Define a new wave variable such that m      ci xi cU   U   i 0 Where  and c i are free parameters (with c i which variables that represents the wave vector ux0 , x1 ,...  U   The integration of 8 gives Step 4: introduce the following derivation rule Step 5: transformation from Partial Differential Equation (PDE) to Ordinary Differential Equation (ODE). Using the transformations from step 4, our equation from step 1 converts to the following ODE Thus, we get the functional F U   and the solution U is  U   cU 2 2 9 F U  , c   U 2c 1  2 U 3c    dU 2 1 U 3c    c     2 Arc tanh U by posing  2  1  U  3c      1 1 2 U   tanh 3c    2 1 2 U  sec h 2  3c    2 We deduce Step 8: Using these transformations, our equation of step 5 takes the form RU , G U , G U , G U G U , G U ,...  0 The interest of the obtained form is that it admits analytical solutions for a large class of nonlinear equations. If Step 10: the use of the formula of step 6 allow to construct 2c 12  c     14  2c  U which is written: c    > 0 , If 2c  1 c     3c    U    sec h 2    15 2 2c  2 c    < 0 , 2c U    1 3c    sec 2  2 2 The solutions in terms of c     16  2c x and t are : Volume 8 Issue 6, June 2019 www.ijsr.net Licensed Under Creative Commons Attribution CC BY Paper ID: ART20198788 11 c     13  Relationship 13  provides the following expression Step 9: the integration of the expression obtained in step 8 provides the expression of F through G . 10 We obtain the following relation after integration of 11 it comes then G  F2. 1 1 U   G U , U   G U  G , 2 2 1 U   G U G  G U G U ,... 2 7  0 2  c    2  3  U  U  c  2 6  2c 2c Step 7: We give some successive derivatives of  8 c     U   F U  F is a function of U alone. 6  cU   0 U is obtained by integration QU , U  , U  , U  U  , U  ,...  0 Step 6: transformation in which the unknown function a function-dependent variable  2  c   U   U 2   c  2  G U   G 2 2  .  ci d U .,  .  ci c j d 2 U .,... x i d x i x j d 2 G U  , Eq. 7  transforms as We use transformation U   Step 3: Introduce the following transformation for wave solutions of the main equation U  2 cU  U  . In this case x 0  x and x1  t .  A first integration of Eq. 6  provides represents the pulsation of the wave). x i are independent Where ux, t   U   . ,and we choose 10.21275/ART20198788 1709  International Journal of Science and Research (IJSR) ISSN: 2319-7064 ResearchGate Impact Factor (2018): 0.28 | SJIF (2018): 7.426 Case 1 : If c    > 0 , 2c U  x, t   Case 2 : If [6] 1 3c    sec h 2  2 2 c    < 0 , 2c 1 3c    U  x, t   sec 2  2 2 c    x  ct  17  2c  c    x  ct  18  2c  It is clear that in case1, the set of exact solution of the general form of BBM are solitary waves having the [7] [8] [9] 2 profile sech . Hence, we have soliton-type solutions. On the other hand, case2 gives us solutions of the type [10] 2 compactons with the profile sec . Which clearly are not solitons. [11] 4. Conclusion As a conclusion, we used the Hopf-Cole transformation [10] to linearize Benjamin-Bona-Mahony's pseudo-parabolic nonlinear generalized equation, and we obtained a suitable form for determining a solution that was difficult to get in the past. In addition, it is appropriate to note the ease of the method, which offers a solution with the soliton profile. We have also presented a comparative analysis of the results obtained using the functional variable method [11]-[12], and we note that solutions are visibly very close. Even if, we have used two different methods, results are fascinating. We would like to thank the Ordinary Professor Walo Omana Rebecca for her considerable contribution throughout the development of this text. References [2] [3] [4] [5] [13] Author Profile 5. Acknowledgement [1] [12] A.El Achab, A.Bekir , “travelling wave solutions to generalized Benjamin-Bona-Mahony (BBM)”, international journal of nonlinear science, vol.19 NO.1,pp.40-46, 2015 A.M.Wazwaz , “ nonlinear variants to the BBM equation with compact and non-compact physical structure”, Chaos, solitons and fractals, 26 : pp.767776,2005 Ben Muatjetjeja and Massod Khalique , “BenjaminBona-Mahony equation with variable coefficients : conservation laws”, symmetry,6, pp.1026-1036, 2014 Gözükizil and Akçagil , “ Exact solutions of BenjaminBona-Mahony-Burgers-Type non linear pseudoparabolic equations”, Boundary Value Problems 2012 pp. 144, 2012 Lawrence C.Evans, “ Partial Differential Equations, Graduate studies in Mathematics”, volume 19, American Mathematical society, pp.194-195, 1997 A. Zerarka, S.Ouamane, A.Attaf , “On the functional variable method for finding exacts solutions to a class of wave equations”, applied mathematics and computation 217 , pp. 2897-2904, 2010 A. Zerarka, S.Ouamane , “Application of functional variable method to a class of nonlinear wave equations”, world journal of modelling and simulation vol.6 NO.2,pp.150-160, 2010 O.Asayyed, H.M.Jaradat, M.M.M. jaradat, Zead Mustafa, Feras Shatat , “Multi-Solutions of BBM equation arisen in shallow water”, Journal Of Nonlinear Science And Application, 9,pp 1807-1814, 2016 T. B. Benjamin, J.L. Bona, and J. J. Mahony, “Model equations for long waves in nonlinear dispersive systems”, Philosophical Transactions Royal Society of London. Series A, vol. 272, NO 1220, pp 47-78, 1972 H.Zhang,G.M.Wei and Y.T.Gao , “on the general form of the Benjamin-Bona-Mahony equation in fluid mechanics”, czechoslovak journal of physics, vol.51,NO.3,pp. 373-377, 2001 Jacek Dziubanski and Grzegirz Karch, Wroclan , “nonlinear scattering for some dispersive equations generalizing Benjamin-Bona-Mahony”, Mh. Math.122, pp. 35-43, 1996 H.Gündogdu and O.F. Gözükizil , “solving BenjaminBona-Mahony equation by using the sn-sn method and the tanh-coth methode”, mathematica MORAVICA, vol.21, NO.1, pp. 95-103, 2017 Muhammmad Ikram, Abbas Muhammad, Atiq Ur Rahm , “ Analytic solution to Benjamin-Bona-Mahony equation by using Laplace Adomian decomposition method”, Matrix Science Mathematic, 3(1) :pp. 01-04, 2019 R.Gilles Bokolo is born in Laxou (France). He received the B.S. and M.S. degrees in Applied Mathematics from the University of Kinshasa in 2010 and 2019, respectively. During 2010-2019, he worked as Relationship Manager Support at Citigroup DRC and in the Credit Department of Standardbank. Alumni of the International Visitor Leadership Program (IVLP) of the US State Department, he now keep working as Teaching Assistant in the Department of Mathematics and Computer Sciences of the University of Kinshasa, to supervise Analysis 2, complex analysis and Ordinary Differential Equations. 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