This paper obtains the solitary wave solutions of two different forms of Boussinesq equations that model the study of shallow water waves in lakes and ocean beaches. The decomposition method using He’s polynomials is applied to solve the governing equations. The travelling wave hypothesis is also utilized to solve the generalized case of coupled Boussinesq equations, and, thus, an exact soliton solution is obtained. The results are also supported by numerical simulations. Keywords: Decomposition Method, He’s polynomials, cubic Boussinesq equation, Coupled Boussinesq equation

Ahmad, Jamshad

Mushtaq, Mariyam

Naeem, Muhammad

English

International Institute for Science, Technology and Education (IISTE): E-Journals

Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.6, No.9, 2016  7  Decomposition Method for Kdv Boussinesq and Coupled Kdv Boussinesq Equations  Muhammad Naeem1, Mariyam Mushtaq2, Jamshad Ahmad3  1Department of Mathematics, Faculty of Sciences, UET Lahore, Pakistan 2Department of Mathematics, GC Women University Sialkot, Pakistan 3Department of Mathematics, Faculty of Sciences, University of Gujrat, Pakistan  Abstract: This paper obtains the solitary wave solutions of two different forms of Boussinesq equations that model the study of shallow water waves in lakes and ocean beaches. The decomposition method using He’s polynomials is applied to solve the governing equations. The travelling wave hypothesis is also utilized to solve the generalized case of coupled Boussinesq equations, and, thus, an exact soliton solution is obtained. The results are also supported by numerical simulations. Keywords: Decomposition Method, He’s polynomials, cubic Boussinesq equation, Coupled Boussinesq equations  1. Introduction In recent years, remarkable developments have taken place in the study of nonlinear evolutionary partial differential equations. It is realized that many such equations possess special solutions in the form of pulses which retain their shapes and velocities after interacting with each other. Such solutions are called solitons. Many equations admitting soliton solutions are as follows: sine Gordon and double sine Gordon equations, Schrodinger equation, and KDV, MKDV, and complex modified KDV equations many research works have been done on these equations. Most of the current research is directed to solve coupled nonlinear systems analytically and numerically. Solitons are of great interest in many physical Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.6, No.9, 2016  8  areas, as, for example, in dislocation theory of crystals, plasma and fluid dynamics, magneto hydrodynamics, laser and fiber optics, and the study of the water waves. Many research works on Boussinesq equation have been developed analytical solution of this equation was studied by many authors [1-32], such as the construction of 𝑁-soliton solutions using the bilinear form [4], multiple soliton solutions for the GB equation using a simplified version of Hirota method [27] and decomposition method [28]. Construction of soliton solutions and periodic solution of Boussinesq equation by modified decomposition method are given in [29, 30]. A variational iteration method is developed for GB equation [31]. A solitary wave solution of the Boussinesq equation with power law nonlinearity is derived in [32]. Many numerical methods have been developed for solving the Boussinesq equation, suchas Petrov-Galerkin method [19].  Mohebbi and Asgari [26] also have solved the GB equation using a fourth order time stepping schemes with combination of discrete Fourier transform. Split step Fourier method is also used to solve Boussinesq-type equations.  2. Decomposition Method Using He’s Polynomials To illustrate the basic concept of modified decomposition method, we consider the following general differential equation 𝐿(𝑢) + 𝑁(𝑢) = 𝑔(𝑥)                                                                                                (1) Where L is the liner operator and N is the nonlinear operator and g(x) is the homogeneous term. According to the ADM we construct the  𝑢𝑛+1 = 𝑢𝑛 − 𝐿𝑡−1{𝑁(𝑢) + 𝑔(𝑥)},                                                                           (2) Where𝐿𝑡−1 = ∫ 𝑑𝑡,𝑡0. The embedding parameter p ∈(0, 1] can be considered as an Expanding parameters. The homotopy perturbation method uses the homotopy parameter p as an expanding parameter to obtain 𝑢 = ∑ 𝑝𝑛𝑢𝑛 = 𝑢0 + 𝑝𝑢1 + 𝑝2𝑢2 + 𝑝3𝑢3 + ⋯∞𝑛=0 ,                                                   (3) Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.6, No.9, 2016  9  If𝑝 → 1, the approximate solution of the form, 𝑓 = lim𝑝→1 𝑢 = ∑ 𝑢𝑛,∞𝑛=0                                                                                             (4) It is well known that series (4) is convergent for most of the cases and also the rate of convergence is dependent on L (u). We assume that (4) has a unique solution. The comparisons of like powers of p give solutions of various orders. In sum, according to, He’s considers the solution u(x), of the homotopy equation in a series of p as follows: 𝑢(𝑥) = ∑ 𝑝𝑛𝑢𝑛 = 𝑢0 + 𝑝𝑢1 + 𝑝2𝑢2 + 𝑝3𝑢3 + ⋯∞𝑛=0, And the method consider the nonlinear term N(u), as 𝑁(𝑢) = ∑ 𝑝𝑛𝐻𝑛 = 𝐻0 + 𝑝𝐻1 + 𝑝2𝐻2 + 𝑝3𝐻3 + ⋯∞𝑛=0, Where 𝐻𝑛 are so called He’s polynomials which can be calculated by using the formula 𝐻𝑛(𝑢0, 𝑢1, 𝑢2, … ) =1𝑛!𝜕𝑛𝜕𝑝𝑛[𝑁(∑ 𝑝𝑖𝑢𝑖)]∞𝑖=0 𝑝=0,           𝑛 = 0,1,2,3, …,                                    (5) The successive approximation 𝑢𝑛+1 , 𝑛 ≥ 0 of the solution of u will be obtained by selective function𝑢0, Consequently the solution is given by 𝑢 = lim𝑛→∞ 𝑢𝑛. 3. Numerical Applications In this section, we apply the decomposition method using He’s polynomials for solving Boussinesq and coupled Boussinesq equations. Example 3.1 Consider the cubic KDV Boussinesq equation 𝑢𝑡𝑡 − 𝑢𝑥𝑥 + 2(𝑢3)𝑥𝑥 − 𝑢𝑥𝑥𝑥𝑥 = 0                                                                         (6) With subject to initial condition u(x, 0) =1𝑥,      𝑢𝑡(𝑥, 0) = −1𝑥2                                                                             (7) According to the above procedure Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.6, No.9, 2016  10  𝑢(𝑥, 𝑡) =1𝑥−1𝑥2𝑡 + 𝑝𝐿𝑡−1𝐿𝑡−1{∑ 𝑝𝑛(𝑢𝑛𝑥𝑥)∞𝑛=0 − 2 ∑ 𝑝𝑛(𝑢3)𝑛𝑥𝑥∞𝑛=0 + ∑ 𝑝𝑛(𝑢𝑛𝑥𝑥𝑥𝑥)∞𝑛=0 },  (8) 𝑢(𝑥, 𝑡) =1𝑥−1𝑥2𝑡 + 𝑝𝐿𝑡−1𝐿𝑡−1{∑ 𝑝𝑛(𝑢𝑛𝑥𝑥)∞𝑛=0 − 2 ∑ 𝑝𝑛(𝐻𝑛(𝑢))∞𝑛=0 + ∑ 𝑝𝑛(𝑢𝑛𝑥𝑥𝑥𝑥)∞𝑛=0 }, (9) Equating the like power components of p, we get 𝑝0: 𝑢0 =1𝑥−1𝑥2𝑡, 𝑝1: 𝑢1 = 𝐿𝑡−1𝐿𝑡−1{𝑢0𝑥𝑥 − 2𝐻0 + 𝑢0𝑥𝑥𝑥𝑥}, =𝑡2𝑥3−𝑡3𝑥4−180𝑡412𝑥7+84𝑡520𝑥8=𝑡2𝑥3−𝑡3𝑥4+ 𝑠𝑚𝑎𝑙𝑙 𝑡𝑒𝑟𝑚𝑠, 𝑝2: 𝑢2 = 𝐿𝑡−1𝐿𝑡−1{𝑢1𝑥𝑥 − 2𝐻1 + 𝑢1𝑥𝑥𝑥𝑥}, =𝑡4𝑥5−𝑡6𝑥7+ 𝑠𝑚𝑎𝑙𝑙 𝑡𝑒𝑟𝑚𝑠,  And so on, summing all components of u, we get  𝑢(𝑥, 𝑡) = 𝑢0 + 𝑢1 + 𝑢2 + ⋯, =1𝑥−1𝑥2𝑡 +𝑡2𝑥3−𝑡3𝑥4+𝑡4𝑥5−𝑡6𝑥7+ 𝑠𝑚𝑎𝑙𝑙 𝑡𝑒𝑟𝑚𝑠, =1𝑥+𝑡.  Fig1: behavior of u(x,t) with −40 ≤ 𝑥 ≤ 40 𝑎𝑛𝑑 − 5 ≤ 𝑡 ≤ 5. Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.6, No.9, 2016  11  Example 3.2 Consider the coupled Boussinesq equations 𝑢𝑡 + 𝑣𝑥 + 𝑢𝑢𝑥 = 0, 𝑣𝑡 + (𝑣𝑢)𝑥 + 𝑢𝑥𝑥𝑥 = 0,                                       (10) With subject to initial conditions 𝑢(𝑥, 0) =𝜆𝑘+ 2𝑘𝑡𝑎𝑛ℎ(𝑘𝑥), 𝑣(𝑥, 0) = 2𝑘2𝑠𝑒𝑐ℎ2(𝑘𝑥),                                      (11) According to the above procedure 𝑢(𝑥, 𝑡) =𝜆𝑘+ 2𝑘𝑡𝑎𝑛ℎ(𝑘𝑥) − 𝑝𝐿𝑡−1{∑ 𝑝𝑛∞𝑛=0 (𝑣𝑛𝑥) + ∑ 𝑝𝑛∞𝑛=0 (𝑢𝑛𝑢𝑛𝑥)},        (12) 𝑣(𝑥, 𝑡) = 2𝑘2𝑠𝑒𝑐ℎ2(𝑘𝑥) − 𝑝𝐿𝑡−1{∑ 𝑝𝑛(𝑣𝑢)𝑛𝑥∞𝑛=0 − ∑ 𝑝𝑛∞𝑛=0 (𝑢𝑛𝑥𝑥𝑥)},          (13) 𝑢(𝑥, 𝑡) =𝜆𝑘+ 2𝑘𝑡𝑎𝑛ℎ(𝑘𝑥) − 𝑝𝐿𝑡−1{∑ 𝑝𝑛∞𝑛=0 (𝑣𝑛𝑥) + ∑ 𝑝𝑛∞𝑛=0 (𝐻𝑛)},       (14) 𝑣(𝑥, 𝑡) = 2𝑘2𝑠𝑒𝑐ℎ2(𝑘𝑥) − 𝑝𝐿𝑡−1{∑ 𝑝𝑛(𝑀𝑛)∞𝑛=0 − ∑ 𝑝𝑛∞𝑛=0 (𝑢𝑛𝑥𝑥𝑥)},            (15) Where 𝐻𝑛 𝑎𝑛𝑑  𝑀𝑛  are nonlinear terms, equating like power components of p, we get 𝑝0: 𝑢0 =𝜆𝑘+ 2𝑘𝑡𝑎𝑛ℎ(𝑘𝑥), 𝑝0: 𝑣0 = 2𝑘2𝑠𝑒𝑐ℎ2(𝑘𝑥), 𝑝1: 𝑢1 = −𝐿𝑡−1{𝑣0𝑥 + 𝐻0}, = 4𝑘3𝑠𝑒𝑐ℎ2(𝑘𝑥) tanh(𝑘𝑥)𝑡 − 2𝜆𝑘𝑠𝑒𝑐ℎ2(𝑘𝑥)𝑡 − 4𝑘3𝑠𝑒𝑐ℎ2(𝑘𝑥) tanh(𝑘𝑥)𝑡, = −2𝜆𝑘𝑠𝑒𝑐ℎ2(𝑘𝑥)𝑡, 𝑝1: 𝑣1 = −𝐿𝑡−1{𝑀0 + 𝑢0𝑥𝑥𝑥}, = −4𝑘4𝑠𝑒𝑐ℎ4(𝑘𝑥)𝑡 + 4𝜆𝑘2𝑠𝑒𝑐ℎ2(𝑘𝑥) tanh(𝑘𝑥)𝑡 + 8𝑘4𝑠𝑒𝑐ℎ2(𝑘𝑥)𝑡− 8𝑘4𝑠𝑒𝑐ℎ4(𝑘𝑥)𝑡 − 8𝑘4𝑠𝑒𝑐ℎ2(𝑘𝑥)𝑡 + 12𝑘4𝑠𝑒𝑐ℎ4(𝑘𝑥)𝑡, =4𝜆𝑘2𝑠𝑒𝑐ℎ2(𝑘𝑥) tanh(𝑘𝑥)𝑡, And so on, summing all components of u and v, we get Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.6, No.9, 2016  12  𝑢(𝑥, 𝑡) =𝜆𝑘+ 2𝑘𝑡𝑎𝑛ℎ(𝑘𝑥) − 2𝜆𝑘𝑠𝑒𝑐ℎ2(𝑘𝑥)𝑡 − ⋯, 𝑢(𝑥, 𝑡) =𝜆𝑘+ 2𝑘𝑡𝑎𝑛ℎ(𝑘𝑥 − 𝜆𝑡). 𝑣(𝑥, 𝑡) = 2𝑘2𝑠𝑒𝑐ℎ2(𝑘𝑥) + 4𝜆𝑘2𝑠𝑒𝑐ℎ2(𝑘𝑥) tanh(𝑘𝑥)𝑡 + ⋯, 𝑣(𝑥, 𝑡) = 2𝑘2𝑠𝑒𝑐ℎ2(𝑘𝑥 − 𝜆𝑡).  Fig 2: Behavior of u(x,t) with −10 ≤ 𝑥 ≤ 10 𝑎𝑛𝑑 0 ≤ 𝑡 ≤ 1 𝑎𝑛𝑑 𝑘 = 1, 𝜆 = 0.5 .   Fig 3: Behavior of v(x, t) with −10 ≤ 𝑥 ≤ 10 𝑎𝑛𝑑 0 ≤ 𝑡 ≤ 1 𝑎𝑛𝑑 𝑘 = 1, 𝜆 = 0.5 .   Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.6, No.9, 2016  13  4. Conclusion There are two main goals that we aimed for this work. The first is to show the power of the modified decomposition method using He’s polynomials and its significant features. The second is to employ this method to obtain rational solutions of nonlinear dispersive equations. It is obvious that the method gives rapid convergent successive approximations without any restrictive assumptions or transformation that may change the physical behavior of the problem. 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Decomposition Method for Kdv Boussinesq and Coupled Kdv Boussinesq Equations

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Decomposition Method for Kdv Boussinesq and Coupled Kdv Boussinesq Equations

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