Exp-function method for solving nonlinear evolution equations with higher order nonlinearity

In this paper, we applied Exp-function method to some nonlinear evolution equations. The solution procedure of this method, by the help of symbolic computation of Matlab, Mathematica or so on, is of utter simplicity. The obtained results show that Exp-function method is very powerful and convenient mathematical tool for nonlinear evolution equations in science and engineering. Key WordsExp-function Method, Symbolic Computation, 2-D Bratu Type Equation, Generalized Fisher Equation 1.INTRODUCTION The investigation of the wave solutions for nonlinear evolution equations plays an important role in the study of nonlinear physical phenomena. In recent years, many effective methods have been proposed for solving the nonlinear differential equations, such that tanh-sech method [1], extended tanh method [2], sine-cosine method [3], homogenous balance method [4], Jacobi elliptic function method [5], F-expansion method [6], homotopy perturbation method [7], variational iteration method [8], Hirotas bilinear methods [9], bifurcation method [10] and so on. In 2006, a new method, called Exp-function method, was first introduced by He and Wu [11], and was successfully studied in a lot of problems [12-20] and so on. In this study, we apply Exp-function method to the two-dimensional Bratu-type equation given in [21] as   exp 0 xx yy u u su     , (1) and the generalized Fisher equation with higher order nonlinearity given in [22,23] as   1 n t xx u u u u    . (2) Two-dimensional Bratu model stimulates a thermal reaction process in a rigid material, where the process depends on a balance between chemically generated heat addition and heat transfer by conduction [24]. The nonlinear reaction-diffussion equation was first introduced by Fisher as a model for the propagation of a mutant gene. It has wide application in the fields of logistic population growth, flame propagation, neurophysiology, autocatalytic chemical reactions, and nuclear reactor theory. It is well E. Misirli and Y. Gurefe 259 known that wave phenomena of plasma media and fluid dynamics are modelled by kink-shaped and tanh-solution or bell-shaped sech-solutions [25]. 2. EXP-FUNCTION METHOD AND APPLICATION TO THE TWODIMENSIONAL BRATU TYPE EQUATION Using a complex variation  defined as kx wy    and the transformation   1 ln u s v  , we can convert Eq. (1) into ordinary different equation, which reads     2 2 2 2 2 3 0 k w vv k w v sv         , (3) where the prime denotes the derivative with respect to  . We assume that the solution of Eq. (3) can be expressed in the form       exp exp d n n c q m m p a n v b m    