Expansion method d. R. Saparbayeva

EXACT SOLUTIONS FOR THE DOUBLE SINH-GORDON EQUATION WITH LOADED BY USING - EXPANSION METHOD
D.R.Saparbayeva .
Urgench state university, e-mail: [e-mail]

We first solve the double sinh-Gordon equation with loaded

, (1)
where is an unknown function, , , - is the given real continuous function.
We suppose that the given nonlinear partial differential equation for has the form
, (2)
where is a polynomial in its arguments. The essence of the (G^′/G)-expansion method is the following steps:
Step 1. Seek traveling wave solutions of equation (2) by taking , , and transform equation (2) to the ordinary differential equation
, (3)
where prime denotes the derivative with respect to .
Step 2. If possible, integrate equation (3) term by term one or more times. This yields constant(s) of integration. For simplicity, the integration constant(s) can be set to zero.
Step 3. Introduce the solution U (ξ) of equation (3) in the finite series form
(4)
where are real constants with to be determined, N is a positive integer to be determined. The function G(ξ) is the solution of the auxiliary linear ordinary differential equation
(5)
where λ and μ are real constants to be determined.
Step 4. Determine N. This, usually, can be accomplished by balancing the linear term(s) of highest order with the highest order nonlinear term(s) in equation (3).
Step 5. Substituting (4) together with (5) into equation (3) yields an algebraic equation involving powers of . Equating the coefficients of each power of to zero gives a system of algebraic equations for ai, λ, μ and c. Then, we solve the system with the aid of a computer algebra system, such as Maple, to determine these constants. On the other hand, depending on the sign of the discriminant , the solutions of equation (5) are well known to us. So, as a final step, we can obtain exact solutions of equation (2).


Conclusions
In this paper, the -expansion method is used to conduct an analytic study on the double sinh-Gordon with loaded equation. The exact traveling wave solutions being determined in this study are more general, and it is not difficult to arrive at some known analytic solutions for certain choices of the parameters. Comparing the proposed method with the methods used in [1–8], show that the -expansion method is not only simple and straightforward, but also avoids tedious calculations
References
[1]. H. K and A. J. Exact solutions for the double sinh-Gordon and generalized form of the double sinh-Gordon equations by using -expansion method.
[2] Z. L. Li, Constructing of new exact solutions to the GKdV-mKdV equation with any-order nonlinear terms by -expansion method, Appl. Math. Comput., 217,
[3] E. M. Zayed, The -expansion method and its applications to some nonlinear evolution equations in the mathematical physics, J. Appl. Math. Comput.,30, 89(2009)
[4] E. M. Zayed, The -expansion method combined with the Riccati equation for finding exact solutions of nonlinear PDEs, J. Appl. Math. Inform., 29, 351 (2011). 9
[5] E. M. Zayed, K. A. Alurr, Extended generalized -expansion method for solving the nonlinear quantum Zakharov Kuznetsov equation, Ricerche Mat., 65, 235 (2016).
[6] N. Shang, B. Zheng, Exact Solutions for Three Fractional Partial Differential Equations by the Method, Int. J. Appl. Math., 43, 114 (2013).
[7] A. Bekir, O. Guner, Exact solutions of nonlinear fractional differential equations by expansion method, Chin. Phys. B, 22, 110202 (2013).
[8] A. Bekir, Application of the -expansion method for nonlinear evolution equations, Phys. Lett. A, 372, 3400.