The Generalized Riccati Equation Mapping …….M. The Generalized Riccati Equation Mapping for Solving (cmZKB) and (pZK) Equations

The generalized Riccati equation mapping is extended which is powerful and straight for Word mathematical tool for solving nonlinear partial differential equations. In this paper, we construct twenty-seven traveling wave solutions for Combined (1+3) Zakharov-Kuznetsov-burgers equation (cmZKB) and potential (1+3) Zakharov-Kuznetsov Equation (Pzk)by applying this method. In this method 𝑄 ′ = 𝑙 + 𝑛𝑄 + 𝑚𝑄 2 , is used, as the auxiliary equation, called the generalized Riccati equation, where l , m and n are arbitrary constants. Further, the solutions are expressed in terms of the hyperbolic function, the trigonometric function and elliptic function.


Introduction
The study of exact traveling wave solutions for the nonlinear partial differential equations (NPDEs) is one of the attractive and remarkable research fields in all areas of science and engineering, such as plasma physics, chemical physics, optical fibres, chemistry and many others. In the recent years, many researchers implemented various methods to study different nonlinear differential equations for searching traveling wave solutions, for example, the tanh-coth method [9], the Exp-function method [5],the Inverse scattering method [2], the Inverse scattering transform method [1], the Hirota's bilinear methed [6], the painleve expansion method[12] the G'/Gexpansion method [11], the generalized Riccati equation mapping method [16] and others. In the present paper, we shall use the improved Riccati equation mapping method to find the exact solutions of (cmZKB) and (Pzk) equations.

The Extended Generalized Riccati Equation Mapping Method
Suppose the general nonlinear partial differential equation :  ( , , , ,  ,  ,  ,  ,  , , … ) = 0, (1) where = ( , , )is an unknown function, H is a polynomial in ( , , )and the subscripts indicate the partial derivatives. The most important steps of the generalized Riccati equation mapping method are as follows: : Consider the traveling wave variable: where λ and are constant, then Eq. (1) reduces to a nonlinear ordinary differential equation where the superscripts stand for the ordinary derivatives with respect to .
Step 2: We suppose that the solution of the ODE (3) can be expressed as follows: where is constant to be determined later suchas ar≠ 0 or a-r≠ 0 and = ( ) is the solution of generalized Riccati equation ′ = + + 2 (5) where , and are constants, such that ≠ 0.
Step 3: We determine the positive integer in Eq. (4) by highest order with the highest order derivative term of ( )in Eq.(3).
Step 4: Substituting Eq. (4) and along with Eq.(5) into Eq.(3) and setting all the coefficients of Q i to zero, yield a system of algebraic equations which can be solved by using the Maple to find the values of the constants , and λ.
Step 5: We have the following twenty-seven solutions, including four different types solution of Eq.(5).
Hence the formal solution of Eq.(13) takes the form: (µ) = 0 + 1 , (14) where 0 and 1 are constants to be determined. Substituting Eq. (14) in to Eq. (13), collecting the coefficients of Q and solving the resulting system using maple program,we obtain the following one solution: Univ. Aden J. Nat. and Appl. Sc. Vol. , parameters aretaken as special values, the solitary wave solutions and periodic wave solutions are obtained. We surely believe that these solutions will be of great importance for analyzing the nonlinear phenomena arising in applied physical sciences. The work shows international journal of differential equations that the improved Riccati equation method is sufficient, effective and suitable for solving other nonlinear evolution equations and it deserves further applying and studying, as well.