Solitary wave solutions of the Benjamin-BonaMahoney-Burgers Equation with Dual PowerLaw Nonlinearity The 5th International Conference on Mathematics & Information Sciences Zewail City Campus on Feb.11-13th 2016 , Egypt Mostafa M. A. Khater Mansoura University,Department of Mathematics, Faculty of Science, 35516 Mansoura, Egypt.
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Abstract In this paper, we employ the extended tanh function method to find the exact traveling wave solutions and solitary wave solutions involving parameters of the Benjamin-Bona-Mahoney-Burgers Equation with Dual Power-Law Nonlinearity. When these parameters are taken to be special values, the solitary wave solutions are derived from the exact traveling wave solutions. These studies reveal that the Benjamin-Bona-Mahoney-Burgers Equation with Dual Power-Law Nonlinearity has a rich variety of solutions.
Results Case 1: The exact traveling wave solutions: 1 −b23 + 2bc b3 ba1 u(ξ) = . − φ− 2 b3 b 2 b φ
(9)
√ 1 −b23 + 2bc b3 √ ba1 √ √ . u(ξ) = + −b tanh( −b ξ) + 2 b3 b2 b − −b tanh( −b ξ)
(10)
√ ba1 1 −b23 + 2bc b3 √ √ √ . + −b coth( −b ξ) + u(ξ) = 2 b3 b2 b − −b coth( −b ξ)
(11)
Description of method The solitary wave solutions: When (b < 0) , we get
Consider the following nonlinear evolution equation F (u, ut, ux, utt, uxx, ....) = 0,
(1)
where F is a polynomial in u(x, t) and its partial derivatives in which the highest order derivatives and nonlinear terms are involved. In the following,we give the main steps of this method [27-32]: Step 1. We use the wave transformation u(x, t) = u(ξ),
ξ = x − ct,
(2)
where c is a constant, to reduce Eq.(1) to the following ODE:
When (b > 0) , we get
P (u, u0, u00, u000, .....) = 0, o n 0 d . where P is a polynomial in u(ξ) and its total derivatives,while 0 = dξ Step 2. Suppose that the solution of Eq.(3) has the form: u(ξ) = a0 +
m X
or
ai φi + bi φ−i ,
(3)
(4)
where ai, bi are constants to be determined, such that am 6= 0 or bm 6= 0 and φ satisfies the Riccati equation 0 φ = b + φ2, (5)
Case 2. If b > 0, then
√ √ √ √ φ = b tan( b ξ), or φ = − b cot( b ξ).
(12)
√ 1 −b23 + 2bc b3 √ ba1 . − u(ξ) = b cot( b ξ) − √ √ 2 b3 b 2 b b cot( b ξ)
(13)
or
i=1
where b is a constant. Eq.(5) admits several types of solutions according to : Case 1. If b < 0, then √ √ √ √ φ = − −b tanh( −b ξ), or φ = − −b coth( −b ξ).
√ 1 −b23 + 2bc b3 √ ba1 . − u(ξ) = b tan( b ξ) − √ √ 2 b3 b2 b b tan( b ξ)
When (b = 0) , we get the trivial solution. and there are two an other cases in the research and each one have 4 solutions and one trivial solutions.
(6)
Conclusions (7)
Case 3. If b = 0, then
1 (8) φ=− . ξ Step 3. Determine the positive integer m in Eq.(4) by balancing the highest order derivatives and the nonlinear terms. Step 4. Substitute Eq.(4) along Eq.(5) into Eq.(3) and collecting all the terms of the same power φi, i = 0, ±1, ±2, ±3, .... and equating them to zero, we obtain a system of algebraic equations, which can be solved by Maple or Mathematica to get the values of ai and bi. Step 5. substituting these values and the solutions of Eq.(5)(1.6) into Eq.(4) we obtain the exact solutions of Eq.(1).
Main Objectives 1. Apply the extended tanh function method . 2. Finding the exact traveling wave solution of the Benjamin-Bona-Mahoney-Burgers Equation with Dual Power-Law Nonlinearity. 3. Finding the solitary wave solution of the Benjamin-Bona-Mahoney-Burgers Equation with Dual Power-Law Nonlinearity. 4. Makeing the comparision between our results obtained in the present article with the well-known results obtained by other authors using different methods.
The extended tanh function method has been applied in this paper to find the exact traveling wave solutions and then the solitary wave solutions the Benjamin-Bona-Mahoney-Burgers equation with dual power-law non linearity. Let us compare between our results obtained in the present article with the well-known results obtained by other authors using different methods as follows: Our results the Benjamin-Bona-Mahoney-Burgers equation with dual power-law non linearity are new and different from those obtained in [1, 2]. The obtained exact solutions can be used as benchmarks against the numerical simulations in theoretical physics and fluid mechanics.
References [1] Amutha. Senthilkumar. Bbm equation with non-constant coefficients. Turkish Journal of Mathematics, 37(4):652–664, 2013. [2] et al. Wang, G. W. Shock waves and other solutions to the benjaminbonamahoneyburgers equation with dual-power law nonlinearity. Acta Phys. Pol. A, 126(6):1221–1225, 2014.
Acknowledgements All the obtained results have been checked with Maple 16 by putting them back into the original equation and found correct.
